# Copyright 2002 Gary Strangman. All rights reserved # Copyright 2002-2016 The SciPy Developers # # The original code from Gary Strangman was heavily adapted for # use in SciPy by Travis Oliphant. The original code came with the # following disclaimer: # # This software is provided "as-is". There are no expressed or implied # warranties of any kind, including, but not limited to, the warranties # of merchantability and fitness for a given application. In no event # shall Gary Strangman be liable for any direct, indirect, incidental, # special, exemplary or consequential damages (including, but not limited # to, loss of use, data or profits, or business interruption) however # caused and on any theory of liability, whether in contract, strict # liability or tort (including negligence or otherwise) arising in any way # out of the use of this software, even if advised of the possibility of # such damage. """ A collection of basic statistical functions for Python. The function names appear below. Some scalar functions defined here are also available in the scipy.special package where they work on arbitrary sized arrays. Disclaimers: The function list is obviously incomplete and, worse, the functions are not optimized. All functions have been tested (some more so than others), but they are far from bulletproof. Thus, as with any free software, no warranty or guarantee is expressed or implied. :-) A few extra functions that don't appear in the list below can be found by interested treasure-hunters. These functions don't necessarily have both list and array versions but were deemed useful. Central Tendency ---------------- .. autosummary:: :toctree: generated/ gmean hmean mode Moments ------- .. autosummary:: :toctree: generated/ moment variation skew kurtosis normaltest Altered Versions ---------------- .. autosummary:: :toctree: generated/ tmean tvar tstd tsem describe Frequency Stats --------------- .. autosummary:: :toctree: generated/ itemfreq scoreatpercentile percentileofscore cumfreq relfreq Variability ----------- .. autosummary:: :toctree: generated/ obrientransform sem zmap zscore gstd iqr median_abs_deviation Trimming Functions ------------------ .. autosummary:: :toctree: generated/ trimboth trim1 Correlation Functions --------------------- .. autosummary:: :toctree: generated/ pearsonr fisher_exact spearmanr pointbiserialr kendalltau weightedtau linregress theilslopes multiscale_graphcorr Inferential Stats ----------------- .. autosummary:: :toctree: generated/ ttest_1samp ttest_ind ttest_ind_from_stats ttest_rel chisquare power_divergence kstest ks_1samp ks_2samp epps_singleton_2samp mannwhitneyu ranksums wilcoxon kruskal friedmanchisquare brunnermunzel combine_pvalues Statistical Distances --------------------- .. autosummary:: :toctree: generated/ wasserstein_distance energy_distance ANOVA Functions --------------- .. autosummary:: :toctree: generated/ f_oneway Support Functions ----------------- .. autosummary:: :toctree: generated/ rankdata rvs_ratio_uniforms References ---------- .. [CRCProbStat2000] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. """ import warnings import math from math import gcd from collections import namedtuple import numpy as np from numpy import array, asarray, ma from scipy.spatial.distance import cdist from scipy.ndimage import measurements from scipy._lib._util import (_lazywhere, check_random_state, MapWrapper, rng_integers, float_factorial) import scipy.special as special from scipy import linalg from . import distributions from . import mstats_basic from ._stats_mstats_common import (_find_repeats, linregress, theilslopes, siegelslopes) from ._stats import (_kendall_dis, _toint64, _weightedrankedtau, _local_correlations) from ._rvs_sampling import rvs_ratio_uniforms from ._hypotests import epps_singleton_2samp, cramervonmises __all__ = ['find_repeats', 'gmean', 'hmean', 'mode', 'tmean', 'tvar', 'tmin', 'tmax', 'tstd', 'tsem', 'moment', 'variation', 'skew', 'kurtosis', 'describe', 'skewtest', 'kurtosistest', 'normaltest', 'jarque_bera', 'itemfreq', 'scoreatpercentile', 'percentileofscore', 'cumfreq', 'relfreq', 'obrientransform', 'sem', 'zmap', 'zscore', 'iqr', 'gstd', 'median_absolute_deviation', 'median_abs_deviation', 'sigmaclip', 'trimboth', 'trim1', 'trim_mean', 'f_oneway', 'F_onewayConstantInputWarning', 'F_onewayBadInputSizesWarning', 'PearsonRConstantInputWarning', 'PearsonRNearConstantInputWarning', 'pearsonr', 'fisher_exact', 'SpearmanRConstantInputWarning', 'spearmanr', 'pointbiserialr', 'kendalltau', 'weightedtau', 'multiscale_graphcorr', 'linregress', 'siegelslopes', 'theilslopes', 'ttest_1samp', 'ttest_ind', 'ttest_ind_from_stats', 'ttest_rel', 'kstest', 'ks_1samp', 'ks_2samp', 'chisquare', 'power_divergence', 'mannwhitneyu', 'tiecorrect', 'ranksums', 'kruskal', 'friedmanchisquare', 'rankdata', 'rvs_ratio_uniforms', 'combine_pvalues', 'wasserstein_distance', 'energy_distance', 'brunnermunzel', 'epps_singleton_2samp', 'cramervonmises'] def _contains_nan(a, nan_policy='propagate'): policies = ['propagate', 'raise', 'omit'] if nan_policy not in policies: raise ValueError("nan_policy must be one of {%s}" % ', '.join("'%s'" % s for s in policies)) try: # Calling np.sum to avoid creating a huge array into memory # e.g. np.isnan(a).any() with np.errstate(invalid='ignore'): contains_nan = np.isnan(np.sum(a)) except TypeError: # This can happen when attempting to sum things which are not # numbers (e.g. as in the function `mode`). Try an alternative method: try: contains_nan = np.nan in set(a.ravel()) except TypeError: # Don't know what to do. Fall back to omitting nan values and # issue a warning. contains_nan = False nan_policy = 'omit' warnings.warn("The input array could not be properly checked for nan " "values. nan values will be ignored.", RuntimeWarning) if contains_nan and nan_policy == 'raise': raise ValueError("The input contains nan values") return contains_nan, nan_policy def _chk_asarray(a, axis): if axis is None: a = np.ravel(a) outaxis = 0 else: a = np.asarray(a) outaxis = axis if a.ndim == 0: a = np.atleast_1d(a) return a, outaxis def _chk2_asarray(a, b, axis): if axis is None: a = np.ravel(a) b = np.ravel(b) outaxis = 0 else: a = np.asarray(a) b = np.asarray(b) outaxis = axis if a.ndim == 0: a = np.atleast_1d(a) if b.ndim == 0: b = np.atleast_1d(b) return a, b, outaxis def _shape_with_dropped_axis(a, axis): """ Given an array `a` and an integer `axis`, return the shape of `a` with the `axis` dimension removed. Examples -------- >>> a = np.zeros((3, 5, 2)) >>> _shape_with_dropped_axis(a, 1) (3, 2) """ shp = list(a.shape) try: del shp[axis] except IndexError: raise np.AxisError(axis, a.ndim) from None return tuple(shp) def _broadcast_shapes(shape1, shape2): """ Given two shapes (i.e. tuples of integers), return the shape that would result from broadcasting two arrays with the given shapes. Examples -------- >>> _broadcast_shapes((2, 1), (4, 1, 3)) (4, 2, 3) """ d = len(shape1) - len(shape2) if d <= 0: shp1 = (1,)*(-d) + shape1 shp2 = shape2 elif d > 0: shp1 = shape1 shp2 = (1,)*d + shape2 shape = [] for n1, n2 in zip(shp1, shp2): if n1 == 1: n = n2 elif n2 == 1 or n1 == n2: n = n1 else: raise ValueError(f'shapes {shape1} and {shape2} could not be ' 'broadcast together') shape.append(n) return tuple(shape) def _broadcast_shapes_with_dropped_axis(a, b, axis): """ Given two arrays `a` and `b` and an integer `axis`, find the shape of the broadcast result after dropping `axis` from the shapes of `a` and `b`. Examples -------- >>> a = np.zeros((5, 2, 1)) >>> b = np.zeros((1, 9, 3)) >>> _broadcast_shapes_with_dropped_axis(a, b, 1) (5, 3) """ shp1 = _shape_with_dropped_axis(a, axis) shp2 = _shape_with_dropped_axis(b, axis) try: shp = _broadcast_shapes(shp1, shp2) except ValueError: raise ValueError(f'non-axis shapes {shp1} and {shp2} could not be ' 'broadcast together') from None return shp def gmean(a, axis=0, dtype=None): """ Compute the geometric mean along the specified axis. Return the geometric average of the array elements. That is: n-th root of (x1 * x2 * ... * xn) Parameters ---------- a : array_like Input array or object that can be converted to an array. axis : int or None, optional Axis along which the geometric mean is computed. Default is 0. If None, compute over the whole array `a`. dtype : dtype, optional Type of the returned array and of the accumulator in which the elements are summed. If dtype is not specified, it defaults to the dtype of a, unless a has an integer dtype with a precision less than that of the default platform integer. In that case, the default platform integer is used. Returns ------- gmean : ndarray See `dtype` parameter above. See Also -------- numpy.mean : Arithmetic average numpy.average : Weighted average hmean : Harmonic mean Notes ----- The geometric average is computed over a single dimension of the input array, axis=0 by default, or all values in the array if axis=None. float64 intermediate and return values are used for integer inputs. Use masked arrays to ignore any non-finite values in the input or that arise in the calculations such as Not a Number and infinity because masked arrays automatically mask any non-finite values. Examples -------- >>> from scipy.stats import gmean >>> gmean([1, 4]) 2.0 >>> gmean([1, 2, 3, 4, 5, 6, 7]) 3.3800151591412964 """ if not isinstance(a, np.ndarray): # if not an ndarray object attempt to convert it log_a = np.log(np.array(a, dtype=dtype)) elif dtype: # Must change the default dtype allowing array type if isinstance(a, np.ma.MaskedArray): log_a = np.log(np.ma.asarray(a, dtype=dtype)) else: log_a = np.log(np.asarray(a, dtype=dtype)) else: log_a = np.log(a) return np.exp(log_a.mean(axis=axis)) def hmean(a, axis=0, dtype=None): """ Calculate the harmonic mean along the specified axis. That is: n / (1/x1 + 1/x2 + ... + 1/xn) Parameters ---------- a : array_like Input array, masked array or object that can be converted to an array. axis : int or None, optional Axis along which the harmonic mean is computed. Default is 0. If None, compute over the whole array `a`. dtype : dtype, optional Type of the returned array and of the accumulator in which the elements are summed. If `dtype` is not specified, it defaults to the dtype of `a`, unless `a` has an integer `dtype` with a precision less than that of the default platform integer. In that case, the default platform integer is used. Returns ------- hmean : ndarray See `dtype` parameter above. See Also -------- numpy.mean : Arithmetic average numpy.average : Weighted average gmean : Geometric mean Notes ----- The harmonic mean is computed over a single dimension of the input array, axis=0 by default, or all values in the array if axis=None. float64 intermediate and return values are used for integer inputs. Use masked arrays to ignore any non-finite values in the input or that arise in the calculations such as Not a Number and infinity. Examples -------- >>> from scipy.stats import hmean >>> hmean([1, 4]) 1.6000000000000001 >>> hmean([1, 2, 3, 4, 5, 6, 7]) 2.6997245179063363 """ if not isinstance(a, np.ndarray): a = np.array(a, dtype=dtype) if np.all(a >= 0): # Harmonic mean only defined if greater than or equal to to zero. if isinstance(a, np.ma.MaskedArray): size = a.count(axis) else: if axis is None: a = a.ravel() size = a.shape[0] else: size = a.shape[axis] with np.errstate(divide='ignore'): return size / np.sum(1.0 / a, axis=axis, dtype=dtype) else: raise ValueError("Harmonic mean only defined if all elements greater " "than or equal to zero") ModeResult = namedtuple('ModeResult', ('mode', 'count')) def mode(a, axis=0, nan_policy='propagate'): """ Return an array of the modal (most common) value in the passed array. If there is more than one such value, only the smallest is returned. The bin-count for the modal bins is also returned. Parameters ---------- a : array_like n-dimensional array of which to find mode(s). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- mode : ndarray Array of modal values. count : ndarray Array of counts for each mode. Examples -------- >>> a = np.array([[6, 8, 3, 0], ... [3, 2, 1, 7], ... [8, 1, 8, 4], ... [5, 3, 0, 5], ... [4, 7, 5, 9]]) >>> from scipy import stats >>> stats.mode(a) ModeResult(mode=array([[3, 1, 0, 0]]), count=array([[1, 1, 1, 1]])) To get mode of whole array, specify ``axis=None``: >>> stats.mode(a, axis=None) ModeResult(mode=array([3]), count=array([3])) """ a, axis = _chk_asarray(a, axis) if a.size == 0: return ModeResult(np.array([]), np.array([])) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.mode(a, axis) if a.dtype == object and np.nan in set(a.ravel()): # Fall back to a slower method since np.unique does not work with NaN scores = set(np.ravel(a)) # get ALL unique values testshape = list(a.shape) testshape[axis] = 1 oldmostfreq = np.zeros(testshape, dtype=a.dtype) oldcounts = np.zeros(testshape, dtype=int) for score in scores: template = (a == score) counts = np.expand_dims(np.sum(template, axis), axis) mostfrequent = np.where(counts > oldcounts, score, oldmostfreq) oldcounts = np.maximum(counts, oldcounts) oldmostfreq = mostfrequent return ModeResult(mostfrequent, oldcounts) def _mode1D(a): vals, cnts = np.unique(a, return_counts=True) return vals[cnts.argmax()], cnts.max() # np.apply_along_axis will convert the _mode1D tuples to a numpy array, casting types in the process # This recreates the results without that issue # View of a, rotated so the requested axis is last in_dims = list(range(a.ndim)) a_view = np.transpose(a, in_dims[:axis] + in_dims[axis+1:] + [axis]) inds = np.ndindex(a_view.shape[:-1]) modes = np.empty(a_view.shape[:-1], dtype=a.dtype) counts = np.empty(a_view.shape[:-1], dtype=np.int_) for ind in inds: modes[ind], counts[ind] = _mode1D(a_view[ind]) newshape = list(a.shape) newshape[axis] = 1 return ModeResult(modes.reshape(newshape), counts.reshape(newshape)) def _mask_to_limits(a, limits, inclusive): """Mask an array for values outside of given limits. This is primarily a utility function. Parameters ---------- a : array limits : (float or None, float or None) A tuple consisting of the (lower limit, upper limit). Values in the input array less than the lower limit or greater than the upper limit will be masked out. None implies no limit. inclusive : (bool, bool) A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to lower or upper are allowed. Returns ------- A MaskedArray. Raises ------ A ValueError if there are no values within the given limits. """ lower_limit, upper_limit = limits lower_include, upper_include = inclusive am = ma.MaskedArray(a) if lower_limit is not None: if lower_include: am = ma.masked_less(am, lower_limit) else: am = ma.masked_less_equal(am, lower_limit) if upper_limit is not None: if upper_include: am = ma.masked_greater(am, upper_limit) else: am = ma.masked_greater_equal(am, upper_limit) if am.count() == 0: raise ValueError("No array values within given limits") return am def tmean(a, limits=None, inclusive=(True, True), axis=None): """ Compute the trimmed mean. This function finds the arithmetic mean of given values, ignoring values outside the given `limits`. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None (default), then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to compute test. Default is None. Returns ------- tmean : float Trimmed mean. See Also -------- trim_mean : Returns mean after trimming a proportion from both tails. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmean(x) 9.5 >>> stats.tmean(x, (3,17)) 10.0 """ a = asarray(a) if limits is None: return np.mean(a, None) am = _mask_to_limits(a.ravel(), limits, inclusive) return am.mean(axis=axis) def tvar(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """ Compute the trimmed variance. This function computes the sample variance of an array of values, while ignoring values which are outside of given `limits`. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tvar : float Trimmed variance. Notes ----- `tvar` computes the unbiased sample variance, i.e. it uses a correction factor ``n / (n - 1)``. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tvar(x) 35.0 >>> stats.tvar(x, (3,17)) 20.0 """ a = asarray(a) a = a.astype(float) if limits is None: return a.var(ddof=ddof, axis=axis) am = _mask_to_limits(a, limits, inclusive) amnan = am.filled(fill_value=np.nan) return np.nanvar(amnan, ddof=ddof, axis=axis) def tmin(a, lowerlimit=None, axis=0, inclusive=True, nan_policy='propagate'): """ Compute the trimmed minimum. This function finds the miminum value of an array `a` along the specified axis, but only considering values greater than a specified lower limit. Parameters ---------- a : array_like Array of values. lowerlimit : None or float, optional Values in the input array less than the given limit will be ignored. When lowerlimit is None, then all values are used. The default value is None. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. inclusive : {True, False}, optional This flag determines whether values exactly equal to the lower limit are included. The default value is True. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- tmin : float, int or ndarray Trimmed minimum. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmin(x) 0 >>> stats.tmin(x, 13) 13 >>> stats.tmin(x, 13, inclusive=False) 14 """ a, axis = _chk_asarray(a, axis) am = _mask_to_limits(a, (lowerlimit, None), (inclusive, False)) contains_nan, nan_policy = _contains_nan(am, nan_policy) if contains_nan and nan_policy == 'omit': am = ma.masked_invalid(am) res = ma.minimum.reduce(am, axis).data if res.ndim == 0: return res[()] return res def tmax(a, upperlimit=None, axis=0, inclusive=True, nan_policy='propagate'): """ Compute the trimmed maximum. This function computes the maximum value of an array along a given axis, while ignoring values larger than a specified upper limit. Parameters ---------- a : array_like Array of values. upperlimit : None or float, optional Values in the input array greater than the given limit will be ignored. When upperlimit is None, then all values are used. The default value is None. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. inclusive : {True, False}, optional This flag determines whether values exactly equal to the upper limit are included. The default value is True. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- tmax : float, int or ndarray Trimmed maximum. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tmax(x) 19 >>> stats.tmax(x, 13) 13 >>> stats.tmax(x, 13, inclusive=False) 12 """ a, axis = _chk_asarray(a, axis) am = _mask_to_limits(a, (None, upperlimit), (False, inclusive)) contains_nan, nan_policy = _contains_nan(am, nan_policy) if contains_nan and nan_policy == 'omit': am = ma.masked_invalid(am) res = ma.maximum.reduce(am, axis).data if res.ndim == 0: return res[()] return res def tstd(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """ Compute the trimmed sample standard deviation. This function finds the sample standard deviation of given values, ignoring values outside the given `limits`. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tstd : float Trimmed sample standard deviation. Notes ----- `tstd` computes the unbiased sample standard deviation, i.e. it uses a correction factor ``n / (n - 1)``. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tstd(x) 5.9160797830996161 >>> stats.tstd(x, (3,17)) 4.4721359549995796 """ return np.sqrt(tvar(a, limits, inclusive, axis, ddof)) def tsem(a, limits=None, inclusive=(True, True), axis=0, ddof=1): """ Compute the trimmed standard error of the mean. This function finds the standard error of the mean for given values, ignoring values outside the given `limits`. Parameters ---------- a : array_like Array of values. limits : None or (lower limit, upper limit), optional Values in the input array less than the lower limit or greater than the upper limit will be ignored. When limits is None, then all values are used. Either of the limit values in the tuple can also be None representing a half-open interval. The default value is None. inclusive : (bool, bool), optional A tuple consisting of the (lower flag, upper flag). These flags determine whether values exactly equal to the lower or upper limits are included. The default value is (True, True). axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees of freedom. Default is 1. Returns ------- tsem : float Trimmed standard error of the mean. Notes ----- `tsem` uses unbiased sample standard deviation, i.e. it uses a correction factor ``n / (n - 1)``. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.tsem(x) 1.3228756555322954 >>> stats.tsem(x, (3,17)) 1.1547005383792515 """ a = np.asarray(a).ravel() if limits is None: return a.std(ddof=ddof) / np.sqrt(a.size) am = _mask_to_limits(a, limits, inclusive) sd = np.sqrt(np.ma.var(am, ddof=ddof, axis=axis)) return sd / np.sqrt(am.count()) ##################################### # MOMENTS # ##################################### def moment(a, moment=1, axis=0, nan_policy='propagate'): r""" Calculate the nth moment about the mean for a sample. A moment is a specific quantitative measure of the shape of a set of points. It is often used to calculate coefficients of skewness and kurtosis due to its close relationship with them. Parameters ---------- a : array_like Input array. moment : int or array_like of ints, optional Order of central moment that is returned. Default is 1. axis : int or None, optional Axis along which the central moment is computed. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- n-th central moment : ndarray or float The appropriate moment along the given axis or over all values if axis is None. The denominator for the moment calculation is the number of observations, no degrees of freedom correction is done. See Also -------- kurtosis, skew, describe Notes ----- The k-th central moment of a data sample is: .. math:: m_k = \frac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})^k Where n is the number of samples and x-bar is the mean. This function uses exponentiation by squares [1]_ for efficiency. References ---------- .. [1] https://eli.thegreenplace.net/2009/03/21/efficient-integer-exponentiation-algorithms Examples -------- >>> from scipy.stats import moment >>> moment([1, 2, 3, 4, 5], moment=1) 0.0 >>> moment([1, 2, 3, 4, 5], moment=2) 2.0 """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.moment(a, moment, axis) if a.size == 0: # empty array, return nan(s) with shape matching `moment` if np.isscalar(moment): return np.nan else: return np.full(np.asarray(moment).shape, np.nan, dtype=np.float64) # for array_like moment input, return a value for each. if not np.isscalar(moment): mmnt = [_moment(a, i, axis) for i in moment] return np.array(mmnt) else: return _moment(a, moment, axis) def _moment(a, moment, axis): if np.abs(moment - np.round(moment)) > 0: raise ValueError("All moment parameters must be integers") if moment == 0: # When moment equals 0, the result is 1, by definition. shape = list(a.shape) del shape[axis] if shape: # return an actual array of the appropriate shape return np.ones(shape, dtype=float) else: # the input was 1D, so return a scalar instead of a rank-0 array return 1.0 elif moment == 1: # By definition the first moment about the mean is 0. shape = list(a.shape) del shape[axis] if shape: # return an actual array of the appropriate shape return np.zeros(shape, dtype=float) else: # the input was 1D, so return a scalar instead of a rank-0 array return np.float64(0.0) else: # Exponentiation by squares: form exponent sequence n_list = [moment] current_n = moment while current_n > 2: if current_n % 2: current_n = (current_n - 1) / 2 else: current_n /= 2 n_list.append(current_n) # Starting point for exponentiation by squares a_zero_mean = a - np.expand_dims(np.mean(a, axis), axis) if n_list[-1] == 1: s = a_zero_mean.copy() else: s = a_zero_mean**2 # Perform multiplications for n in n_list[-2::-1]: s = s**2 if n % 2: s *= a_zero_mean return np.mean(s, axis) def variation(a, axis=0, nan_policy='propagate'): """ Compute the coefficient of variation. The coefficient of variation is the ratio of the biased standard deviation to the mean. Parameters ---------- a : array_like Input array. axis : int or None, optional Axis along which to calculate the coefficient of variation. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- variation : ndarray The calculated variation along the requested axis. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Examples -------- >>> from scipy.stats import variation >>> variation([1, 2, 3, 4, 5]) 0.47140452079103173 """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.variation(a, axis) return a.std(axis) / a.mean(axis) def skew(a, axis=0, bias=True, nan_policy='propagate'): r""" Compute the sample skewness of a data set. For normally distributed data, the skewness should be about zero. For unimodal continuous distributions, a skewness value greater than zero means that there is more weight in the right tail of the distribution. The function `skewtest` can be used to determine if the skewness value is close enough to zero, statistically speaking. Parameters ---------- a : ndarray Input array. axis : int or None, optional Axis along which skewness is calculated. Default is 0. If None, compute over the whole array `a`. bias : bool, optional If False, then the calculations are corrected for statistical bias. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- skewness : ndarray The skewness of values along an axis, returning 0 where all values are equal. Notes ----- The sample skewness is computed as the Fisher-Pearson coefficient of skewness, i.e. .. math:: g_1=\frac{m_3}{m_2^{3/2}} where .. math:: m_i=\frac{1}{N}\sum_{n=1}^N(x[n]-\bar{x})^i is the biased sample :math:`i\texttt{th}` central moment, and :math:`\bar{x}` is the sample mean. If ``bias`` is False, the calculations are corrected for bias and the value computed is the adjusted Fisher-Pearson standardized moment coefficient, i.e. .. math:: G_1=\frac{k_3}{k_2^{3/2}}= \frac{\sqrt{N(N-1)}}{N-2}\frac{m_3}{m_2^{3/2}}. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 2.2.24.1 Examples -------- >>> from scipy.stats import skew >>> skew([1, 2, 3, 4, 5]) 0.0 >>> skew([2, 8, 0, 4, 1, 9, 9, 0]) 0.2650554122698573 """ a, axis = _chk_asarray(a, axis) n = a.shape[axis] contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.skew(a, axis, bias) m2 = moment(a, 2, axis) m3 = moment(a, 3, axis) zero = (m2 == 0) vals = _lazywhere(~zero, (m2, m3), lambda m2, m3: m3 / m2**1.5, 0.) if not bias: can_correct = (n > 2) & (m2 > 0) if can_correct.any(): m2 = np.extract(can_correct, m2) m3 = np.extract(can_correct, m3) nval = np.sqrt((n - 1.0) * n) / (n - 2.0) * m3 / m2**1.5 np.place(vals, can_correct, nval) if vals.ndim == 0: return vals.item() return vals def kurtosis(a, axis=0, fisher=True, bias=True, nan_policy='propagate'): """ Compute the kurtosis (Fisher or Pearson) of a dataset. Kurtosis is the fourth central moment divided by the square of the variance. If Fisher's definition is used, then 3.0 is subtracted from the result to give 0.0 for a normal distribution. If bias is False then the kurtosis is calculated using k statistics to eliminate bias coming from biased moment estimators Use `kurtosistest` to see if result is close enough to normal. Parameters ---------- a : array Data for which the kurtosis is calculated. axis : int or None, optional Axis along which the kurtosis is calculated. Default is 0. If None, compute over the whole array `a`. fisher : bool, optional If True, Fisher's definition is used (normal ==> 0.0). If False, Pearson's definition is used (normal ==> 3.0). bias : bool, optional If False, then the calculations are corrected for statistical bias. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. 'propagate' returns nan, 'raise' throws an error, 'omit' performs the calculations ignoring nan values. Default is 'propagate'. Returns ------- kurtosis : array The kurtosis of values along an axis. If all values are equal, return -3 for Fisher's definition and 0 for Pearson's definition. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Examples -------- In Fisher's definiton, the kurtosis of the normal distribution is zero. In the following example, the kurtosis is close to zero, because it was calculated from the dataset, not from the continuous distribution. >>> from scipy.stats import norm, kurtosis >>> data = norm.rvs(size=1000, random_state=3) >>> kurtosis(data) -0.06928694200380558 The distribution with a higher kurtosis has a heavier tail. The zero valued kurtosis of the normal distribution in Fisher's definition can serve as a reference point. >>> import matplotlib.pyplot as plt >>> import scipy.stats as stats >>> from scipy.stats import kurtosis >>> x = np.linspace(-5, 5, 100) >>> ax = plt.subplot() >>> distnames = ['laplace', 'norm', 'uniform'] >>> for distname in distnames: ... if distname == 'uniform': ... dist = getattr(stats, distname)(loc=-2, scale=4) ... else: ... dist = getattr(stats, distname) ... data = dist.rvs(size=1000) ... kur = kurtosis(data, fisher=True) ... y = dist.pdf(x) ... ax.plot(x, y, label="{}, {}".format(distname, round(kur, 3))) ... ax.legend() The Laplace distribution has a heavier tail than the normal distribution. The uniform distribution (which has negative kurtosis) has the thinnest tail. """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.kurtosis(a, axis, fisher, bias) n = a.shape[axis] m2 = moment(a, 2, axis) m4 = moment(a, 4, axis) zero = (m2 == 0) with np.errstate(all='ignore'): vals = np.where(zero, 0, m4 / m2**2.0) if not bias: can_correct = (n > 3) & (m2 > 0) if can_correct.any(): m2 = np.extract(can_correct, m2) m4 = np.extract(can_correct, m4) nval = 1.0/(n-2)/(n-3) * ((n**2-1.0)*m4/m2**2.0 - 3*(n-1)**2.0) np.place(vals, can_correct, nval + 3.0) if vals.ndim == 0: vals = vals.item() # array scalar return vals - 3 if fisher else vals DescribeResult = namedtuple('DescribeResult', ('nobs', 'minmax', 'mean', 'variance', 'skewness', 'kurtosis')) def describe(a, axis=0, ddof=1, bias=True, nan_policy='propagate'): """ Compute several descriptive statistics of the passed array. Parameters ---------- a : array_like Input data. axis : int or None, optional Axis along which statistics are calculated. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees of freedom (only for variance). Default is 1. bias : bool, optional If False, then the skewness and kurtosis calculations are corrected for statistical bias. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- nobs : int or ndarray of ints Number of observations (length of data along `axis`). When 'omit' is chosen as nan_policy, each column is counted separately. minmax: tuple of ndarrays or floats Minimum and maximum value of data array. mean : ndarray or float Arithmetic mean of data along axis. variance : ndarray or float Unbiased variance of the data along axis, denominator is number of observations minus one. skewness : ndarray or float Skewness, based on moment calculations with denominator equal to the number of observations, i.e. no degrees of freedom correction. kurtosis : ndarray or float Kurtosis (Fisher). The kurtosis is normalized so that it is zero for the normal distribution. No degrees of freedom are used. See Also -------- skew, kurtosis Examples -------- >>> from scipy import stats >>> a = np.arange(10) >>> stats.describe(a) DescribeResult(nobs=10, minmax=(0, 9), mean=4.5, variance=9.166666666666666, skewness=0.0, kurtosis=-1.2242424242424244) >>> b = [[1, 2], [3, 4]] >>> stats.describe(b) DescribeResult(nobs=2, minmax=(array([1, 2]), array([3, 4])), mean=array([2., 3.]), variance=array([2., 2.]), skewness=array([0., 0.]), kurtosis=array([-2., -2.])) """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.describe(a, axis, ddof, bias) if a.size == 0: raise ValueError("The input must not be empty.") n = a.shape[axis] mm = (np.min(a, axis=axis), np.max(a, axis=axis)) m = np.mean(a, axis=axis) v = np.var(a, axis=axis, ddof=ddof) sk = skew(a, axis, bias=bias) kurt = kurtosis(a, axis, bias=bias) return DescribeResult(n, mm, m, v, sk, kurt) ##################################### # NORMALITY TESTS # ##################################### SkewtestResult = namedtuple('SkewtestResult', ('statistic', 'pvalue')) def skewtest(a, axis=0, nan_policy='propagate'): """ Test whether the skew is different from the normal distribution. This function tests the null hypothesis that the skewness of the population that the sample was drawn from is the same as that of a corresponding normal distribution. Parameters ---------- a : array The data to be tested. axis : int or None, optional Axis along which statistics are calculated. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float The computed z-score for this test. pvalue : float Two-sided p-value for the hypothesis test. Notes ----- The sample size must be at least 8. References ---------- .. [1] R. B. D'Agostino, A. J. Belanger and R. B. D'Agostino Jr., "A suggestion for using powerful and informative tests of normality", American Statistician 44, pp. 316-321, 1990. Examples -------- >>> from scipy.stats import skewtest >>> skewtest([1, 2, 3, 4, 5, 6, 7, 8]) SkewtestResult(statistic=1.0108048609177787, pvalue=0.3121098361421897) >>> skewtest([2, 8, 0, 4, 1, 9, 9, 0]) SkewtestResult(statistic=0.44626385374196975, pvalue=0.6554066631275459) >>> skewtest([1, 2, 3, 4, 5, 6, 7, 8000]) SkewtestResult(statistic=3.571773510360407, pvalue=0.0003545719905823133) >>> skewtest([100, 100, 100, 100, 100, 100, 100, 101]) SkewtestResult(statistic=3.5717766638478072, pvalue=0.000354567720281634) """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.skewtest(a, axis) if axis is None: a = np.ravel(a) axis = 0 b2 = skew(a, axis) n = a.shape[axis] if n < 8: raise ValueError( "skewtest is not valid with less than 8 samples; %i samples" " were given." % int(n)) y = b2 * math.sqrt(((n + 1) * (n + 3)) / (6.0 * (n - 2))) beta2 = (3.0 * (n**2 + 27*n - 70) * (n+1) * (n+3) / ((n-2.0) * (n+5) * (n+7) * (n+9))) W2 = -1 + math.sqrt(2 * (beta2 - 1)) delta = 1 / math.sqrt(0.5 * math.log(W2)) alpha = math.sqrt(2.0 / (W2 - 1)) y = np.where(y == 0, 1, y) Z = delta * np.log(y / alpha + np.sqrt((y / alpha)**2 + 1)) return SkewtestResult(Z, 2 * distributions.norm.sf(np.abs(Z))) KurtosistestResult = namedtuple('KurtosistestResult', ('statistic', 'pvalue')) def kurtosistest(a, axis=0, nan_policy='propagate'): """ Test whether a dataset has normal kurtosis. This function tests the null hypothesis that the kurtosis of the population from which the sample was drawn is that of the normal distribution: ``kurtosis = 3(n-1)/(n+1)``. Parameters ---------- a : array Array of the sample data. axis : int or None, optional Axis along which to compute test. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float The computed z-score for this test. pvalue : float The two-sided p-value for the hypothesis test. Notes ----- Valid only for n>20. This function uses the method described in [1]_. References ---------- .. [1] see e.g. F. J. Anscombe, W. J. Glynn, "Distribution of the kurtosis statistic b2 for normal samples", Biometrika, vol. 70, pp. 227-234, 1983. Examples -------- >>> from scipy.stats import kurtosistest >>> kurtosistest(list(range(20))) KurtosistestResult(statistic=-1.7058104152122062, pvalue=0.08804338332528348) >>> np.random.seed(28041990) >>> s = np.random.normal(0, 1, 1000) >>> kurtosistest(s) KurtosistestResult(statistic=1.2317590987707365, pvalue=0.21803908613450895) """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.kurtosistest(a, axis) n = a.shape[axis] if n < 5: raise ValueError( "kurtosistest requires at least 5 observations; %i observations" " were given." % int(n)) if n < 20: warnings.warn("kurtosistest only valid for n>=20 ... continuing " "anyway, n=%i" % int(n)) b2 = kurtosis(a, axis, fisher=False) E = 3.0*(n-1) / (n+1) varb2 = 24.0*n*(n-2)*(n-3) / ((n+1)*(n+1.)*(n+3)*(n+5)) # [1]_ Eq. 1 x = (b2-E) / np.sqrt(varb2) # [1]_ Eq. 4 # [1]_ Eq. 2: sqrtbeta1 = 6.0*(n*n-5*n+2)/((n+7)*(n+9)) * np.sqrt((6.0*(n+3)*(n+5)) / (n*(n-2)*(n-3))) # [1]_ Eq. 3: A = 6.0 + 8.0/sqrtbeta1 * (2.0/sqrtbeta1 + np.sqrt(1+4.0/(sqrtbeta1**2))) term1 = 1 - 2/(9.0*A) denom = 1 + x*np.sqrt(2/(A-4.0)) term2 = np.sign(denom) * np.where(denom == 0.0, np.nan, np.power((1-2.0/A)/np.abs(denom), 1/3.0)) if np.any(denom == 0): msg = "Test statistic not defined in some cases due to division by " \ "zero. Return nan in that case..." warnings.warn(msg, RuntimeWarning) Z = (term1 - term2) / np.sqrt(2/(9.0*A)) # [1]_ Eq. 5 if Z.ndim == 0: Z = Z[()] # zprob uses upper tail, so Z needs to be positive return KurtosistestResult(Z, 2 * distributions.norm.sf(np.abs(Z))) NormaltestResult = namedtuple('NormaltestResult', ('statistic', 'pvalue')) def normaltest(a, axis=0, nan_policy='propagate'): """ Test whether a sample differs from a normal distribution. This function tests the null hypothesis that a sample comes from a normal distribution. It is based on D'Agostino and Pearson's [1]_, [2]_ test that combines skew and kurtosis to produce an omnibus test of normality. Parameters ---------- a : array_like The array containing the sample to be tested. axis : int or None, optional Axis along which to compute test. Default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float or array ``s^2 + k^2``, where ``s`` is the z-score returned by `skewtest` and ``k`` is the z-score returned by `kurtosistest`. pvalue : float or array A 2-sided chi squared probability for the hypothesis test. References ---------- .. [1] D'Agostino, R. B. (1971), "An omnibus test of normality for moderate and large sample size", Biometrika, 58, 341-348 .. [2] D'Agostino, R. and Pearson, E. S. (1973), "Tests for departure from normality", Biometrika, 60, 613-622 Examples -------- >>> from scipy import stats >>> pts = 1000 >>> np.random.seed(28041990) >>> a = np.random.normal(0, 1, size=pts) >>> b = np.random.normal(2, 1, size=pts) >>> x = np.concatenate((a, b)) >>> k2, p = stats.normaltest(x) >>> alpha = 1e-3 >>> print("p = {:g}".format(p)) p = 3.27207e-11 >>> if p < alpha: # null hypothesis: x comes from a normal distribution ... print("The null hypothesis can be rejected") ... else: ... print("The null hypothesis cannot be rejected") The null hypothesis can be rejected """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.normaltest(a, axis) s, _ = skewtest(a, axis) k, _ = kurtosistest(a, axis) k2 = s*s + k*k return NormaltestResult(k2, distributions.chi2.sf(k2, 2)) Jarque_beraResult = namedtuple('Jarque_beraResult', ('statistic', 'pvalue')) def jarque_bera(x): """ Perform the Jarque-Bera goodness of fit test on sample data. The Jarque-Bera test tests whether the sample data has the skewness and kurtosis matching a normal distribution. Note that this test only works for a large enough number of data samples (>2000) as the test statistic asymptotically has a Chi-squared distribution with 2 degrees of freedom. Parameters ---------- x : array_like Observations of a random variable. Returns ------- jb_value : float The test statistic. p : float The p-value for the hypothesis test. References ---------- .. [1] Jarque, C. and Bera, A. (1980) "Efficient tests for normality, homoscedasticity and serial independence of regression residuals", 6 Econometric Letters 255-259. Examples -------- >>> from scipy import stats >>> np.random.seed(987654321) >>> x = np.random.normal(0, 1, 100000) >>> jarque_bera_test = stats.jarque_bera(x) >>> jarque_bera_test Jarque_beraResult(statistic=4.716570798957913, pvalue=0.0945822550304295) >>> jarque_bera_test.statistic 4.716570798957913 >>> jarque_bera_test.pvalue 0.0945822550304295 """ x = np.asarray(x) n = x.size if n == 0: raise ValueError('At least one observation is required.') mu = x.mean() diffx = x - mu skewness = (1 / n * np.sum(diffx**3)) / (1 / n * np.sum(diffx**2))**(3 / 2.) kurtosis = (1 / n * np.sum(diffx**4)) / (1 / n * np.sum(diffx**2))**2 jb_value = n / 6 * (skewness**2 + (kurtosis - 3)**2 / 4) p = 1 - distributions.chi2.cdf(jb_value, 2) return Jarque_beraResult(jb_value, p) ##################################### # FREQUENCY FUNCTIONS # ##################################### # deindent to work around numpy/gh-16202 @np.deprecate( message="`itemfreq` is deprecated and will be removed in a " "future version. Use instead `np.unique(..., return_counts=True)`") def itemfreq(a): """ Return a 2-D array of item frequencies. Parameters ---------- a : (N,) array_like Input array. Returns ------- itemfreq : (K, 2) ndarray A 2-D frequency table. Column 1 contains sorted, unique values from `a`, column 2 contains their respective counts. Examples -------- >>> from scipy import stats >>> a = np.array([1, 1, 5, 0, 1, 2, 2, 0, 1, 4]) >>> stats.itemfreq(a) array([[ 0., 2.], [ 1., 4.], [ 2., 2.], [ 4., 1.], [ 5., 1.]]) >>> np.bincount(a) array([2, 4, 2, 0, 1, 1]) >>> stats.itemfreq(a/10.) array([[ 0. , 2. ], [ 0.1, 4. ], [ 0.2, 2. ], [ 0.4, 1. ], [ 0.5, 1. ]]) """ items, inv = np.unique(a, return_inverse=True) freq = np.bincount(inv) return np.array([items, freq]).T def scoreatpercentile(a, per, limit=(), interpolation_method='fraction', axis=None): """ Calculate the score at a given percentile of the input sequence. For example, the score at `per=50` is the median. If the desired quantile lies between two data points, we interpolate between them, according to the value of `interpolation`. If the parameter `limit` is provided, it should be a tuple (lower, upper) of two values. Parameters ---------- a : array_like A 1-D array of values from which to extract score. per : array_like Percentile(s) at which to extract score. Values should be in range [0,100]. limit : tuple, optional Tuple of two scalars, the lower and upper limits within which to compute the percentile. Values of `a` outside this (closed) interval will be ignored. interpolation_method : {'fraction', 'lower', 'higher'}, optional Specifies the interpolation method to use, when the desired quantile lies between two data points `i` and `j` The following options are available (default is 'fraction'): * 'fraction': ``i + (j - i) * fraction`` where ``fraction`` is the fractional part of the index surrounded by ``i`` and ``j`` * 'lower': ``i`` * 'higher': ``j`` axis : int, optional Axis along which the percentiles are computed. Default is None. If None, compute over the whole array `a`. Returns ------- score : float or ndarray Score at percentile(s). See Also -------- percentileofscore, numpy.percentile Notes ----- This function will become obsolete in the future. For NumPy 1.9 and higher, `numpy.percentile` provides all the functionality that `scoreatpercentile` provides. And it's significantly faster. Therefore it's recommended to use `numpy.percentile` for users that have numpy >= 1.9. Examples -------- >>> from scipy import stats >>> a = np.arange(100) >>> stats.scoreatpercentile(a, 50) 49.5 """ # adapted from NumPy's percentile function. When we require numpy >= 1.8, # the implementation of this function can be replaced by np.percentile. a = np.asarray(a) if a.size == 0: # empty array, return nan(s) with shape matching `per` if np.isscalar(per): return np.nan else: return np.full(np.asarray(per).shape, np.nan, dtype=np.float64) if limit: a = a[(limit[0] <= a) & (a <= limit[1])] sorted_ = np.sort(a, axis=axis) if axis is None: axis = 0 return _compute_qth_percentile(sorted_, per, interpolation_method, axis) # handle sequence of per's without calling sort multiple times def _compute_qth_percentile(sorted_, per, interpolation_method, axis): if not np.isscalar(per): score = [_compute_qth_percentile(sorted_, i, interpolation_method, axis) for i in per] return np.array(score) if not (0 <= per <= 100): raise ValueError("percentile must be in the range [0, 100]") indexer = [slice(None)] * sorted_.ndim idx = per / 100. * (sorted_.shape[axis] - 1) if int(idx) != idx: # round fractional indices according to interpolation method if interpolation_method == 'lower': idx = int(np.floor(idx)) elif interpolation_method == 'higher': idx = int(np.ceil(idx)) elif interpolation_method == 'fraction': pass # keep idx as fraction and interpolate else: raise ValueError("interpolation_method can only be 'fraction', " "'lower' or 'higher'") i = int(idx) if i == idx: indexer[axis] = slice(i, i + 1) weights = array(1) sumval = 1.0 else: indexer[axis] = slice(i, i + 2) j = i + 1 weights = array([(j - idx), (idx - i)], float) wshape = [1] * sorted_.ndim wshape[axis] = 2 weights.shape = wshape sumval = weights.sum() # Use np.add.reduce (== np.sum but a little faster) to coerce data type return np.add.reduce(sorted_[tuple(indexer)] * weights, axis=axis) / sumval def percentileofscore(a, score, kind='rank'): """ Compute the percentile rank of a score relative to a list of scores. A `percentileofscore` of, for example, 80% means that 80% of the scores in `a` are below the given score. In the case of gaps or ties, the exact definition depends on the optional keyword, `kind`. Parameters ---------- a : array_like Array of scores to which `score` is compared. score : int or float Score that is compared to the elements in `a`. kind : {'rank', 'weak', 'strict', 'mean'}, optional Specifies the interpretation of the resulting score. The following options are available (default is 'rank'): * 'rank': Average percentage ranking of score. In case of multiple matches, average the percentage rankings of all matching scores. * 'weak': This kind corresponds to the definition of a cumulative distribution function. A percentileofscore of 80% means that 80% of values are less than or equal to the provided score. * 'strict': Similar to "weak", except that only values that are strictly less than the given score are counted. * 'mean': The average of the "weak" and "strict" scores, often used in testing. See https://en.wikipedia.org/wiki/Percentile_rank Returns ------- pcos : float Percentile-position of score (0-100) relative to `a`. See Also -------- numpy.percentile Examples -------- Three-quarters of the given values lie below a given score: >>> from scipy import stats >>> stats.percentileofscore([1, 2, 3, 4], 3) 75.0 With multiple matches, note how the scores of the two matches, 0.6 and 0.8 respectively, are averaged: >>> stats.percentileofscore([1, 2, 3, 3, 4], 3) 70.0 Only 2/5 values are strictly less than 3: >>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='strict') 40.0 But 4/5 values are less than or equal to 3: >>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='weak') 80.0 The average between the weak and the strict scores is: >>> stats.percentileofscore([1, 2, 3, 3, 4], 3, kind='mean') 60.0 """ if np.isnan(score): return np.nan a = np.asarray(a) n = len(a) if n == 0: return 100.0 if kind == 'rank': left = np.count_nonzero(a < score) right = np.count_nonzero(a <= score) pct = (right + left + (1 if right > left else 0)) * 50.0/n return pct elif kind == 'strict': return np.count_nonzero(a < score) / n * 100 elif kind == 'weak': return np.count_nonzero(a <= score) / n * 100 elif kind == 'mean': pct = (np.count_nonzero(a < score) + np.count_nonzero(a <= score)) / n * 50 return pct else: raise ValueError("kind can only be 'rank', 'strict', 'weak' or 'mean'") HistogramResult = namedtuple('HistogramResult', ('count', 'lowerlimit', 'binsize', 'extrapoints')) def _histogram(a, numbins=10, defaultlimits=None, weights=None, printextras=False): """ Create a histogram. Separate the range into several bins and return the number of instances in each bin. Parameters ---------- a : array_like Array of scores which will be put into bins. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultlimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically ``(a.min() - s, a.max() + s)``, where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. weights : array_like, optional The weights for each value in `a`. Default is None, which gives each value a weight of 1.0 printextras : bool, optional If True, if there are extra points (i.e. the points that fall outside the bin limits) a warning is raised saying how many of those points there are. Default is False. Returns ------- count : ndarray Number of points (or sum of weights) in each bin. lowerlimit : float Lowest value of histogram, the lower limit of the first bin. binsize : float The size of the bins (all bins have the same size). extrapoints : int The number of points outside the range of the histogram. See Also -------- numpy.histogram Notes ----- This histogram is based on numpy's histogram but has a larger range by default if default limits is not set. """ a = np.ravel(a) if defaultlimits is None: if a.size == 0: # handle empty arrays. Undetermined range, so use 0-1. defaultlimits = (0, 1) else: # no range given, so use values in `a` data_min = a.min() data_max = a.max() # Have bins extend past min and max values slightly s = (data_max - data_min) / (2. * (numbins - 1.)) defaultlimits = (data_min - s, data_max + s) # use numpy's histogram method to compute bins hist, bin_edges = np.histogram(a, bins=numbins, range=defaultlimits, weights=weights) # hist are not always floats, convert to keep with old output hist = np.array(hist, dtype=float) # fixed width for bins is assumed, as numpy's histogram gives # fixed width bins for int values for 'bins' binsize = bin_edges[1] - bin_edges[0] # calculate number of extra points extrapoints = len([v for v in a if defaultlimits[0] > v or v > defaultlimits[1]]) if extrapoints > 0 and printextras: warnings.warn("Points outside given histogram range = %s" % extrapoints) return HistogramResult(hist, defaultlimits[0], binsize, extrapoints) CumfreqResult = namedtuple('CumfreqResult', ('cumcount', 'lowerlimit', 'binsize', 'extrapoints')) def cumfreq(a, numbins=10, defaultreallimits=None, weights=None): """ Return a cumulative frequency histogram, using the histogram function. A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. Parameters ---------- a : array_like Input array. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultreallimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in `a` is used. Specifically ``(a.min() - s, a.max() + s)``, where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. weights : array_like, optional The weights for each value in `a`. Default is None, which gives each value a weight of 1.0 Returns ------- cumcount : ndarray Binned values of cumulative frequency. lowerlimit : float Lower real limit binsize : float Width of each bin. extrapoints : int Extra points. Examples -------- >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> x = [1, 4, 2, 1, 3, 1] >>> res = stats.cumfreq(x, numbins=4, defaultreallimits=(1.5, 5)) >>> res.cumcount array([ 1., 2., 3., 3.]) >>> res.extrapoints 3 Create a normal distribution with 1000 random values >>> rng = np.random.RandomState(seed=12345) >>> samples = stats.norm.rvs(size=1000, random_state=rng) Calculate cumulative frequencies >>> res = stats.cumfreq(samples, numbins=25) Calculate space of values for x >>> x = res.lowerlimit + np.linspace(0, res.binsize*res.cumcount.size, ... res.cumcount.size) Plot histogram and cumulative histogram >>> fig = plt.figure(figsize=(10, 4)) >>> ax1 = fig.add_subplot(1, 2, 1) >>> ax2 = fig.add_subplot(1, 2, 2) >>> ax1.hist(samples, bins=25) >>> ax1.set_title('Histogram') >>> ax2.bar(x, res.cumcount, width=res.binsize) >>> ax2.set_title('Cumulative histogram') >>> ax2.set_xlim([x.min(), x.max()]) >>> plt.show() """ h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights) cumhist = np.cumsum(h * 1, axis=0) return CumfreqResult(cumhist, l, b, e) RelfreqResult = namedtuple('RelfreqResult', ('frequency', 'lowerlimit', 'binsize', 'extrapoints')) def relfreq(a, numbins=10, defaultreallimits=None, weights=None): """ Return a relative frequency histogram, using the histogram function. A relative frequency histogram is a mapping of the number of observations in each of the bins relative to the total of observations. Parameters ---------- a : array_like Input array. numbins : int, optional The number of bins to use for the histogram. Default is 10. defaultreallimits : tuple (lower, upper), optional The lower and upper values for the range of the histogram. If no value is given, a range slightly larger than the range of the values in a is used. Specifically ``(a.min() - s, a.max() + s)``, where ``s = (1/2)(a.max() - a.min()) / (numbins - 1)``. weights : array_like, optional The weights for each value in `a`. Default is None, which gives each value a weight of 1.0 Returns ------- frequency : ndarray Binned values of relative frequency. lowerlimit : float Lower real limit. binsize : float Width of each bin. extrapoints : int Extra points. Examples -------- >>> import matplotlib.pyplot as plt >>> from scipy import stats >>> a = np.array([2, 4, 1, 2, 3, 2]) >>> res = stats.relfreq(a, numbins=4) >>> res.frequency array([ 0.16666667, 0.5 , 0.16666667, 0.16666667]) >>> np.sum(res.frequency) # relative frequencies should add up to 1 1.0 Create a normal distribution with 1000 random values >>> rng = np.random.RandomState(seed=12345) >>> samples = stats.norm.rvs(size=1000, random_state=rng) Calculate relative frequencies >>> res = stats.relfreq(samples, numbins=25) Calculate space of values for x >>> x = res.lowerlimit + np.linspace(0, res.binsize*res.frequency.size, ... res.frequency.size) Plot relative frequency histogram >>> fig = plt.figure(figsize=(5, 4)) >>> ax = fig.add_subplot(1, 1, 1) >>> ax.bar(x, res.frequency, width=res.binsize) >>> ax.set_title('Relative frequency histogram') >>> ax.set_xlim([x.min(), x.max()]) >>> plt.show() """ a = np.asanyarray(a) h, l, b, e = _histogram(a, numbins, defaultreallimits, weights=weights) h = h / a.shape[0] return RelfreqResult(h, l, b, e) ##################################### # VARIABILITY FUNCTIONS # ##################################### def obrientransform(*args): """ Compute the O'Brien transform on input data (any number of arrays). Used to test for homogeneity of variance prior to running one-way stats. Each array in ``*args`` is one level of a factor. If `f_oneway` is run on the transformed data and found significant, the variances are unequal. From Maxwell and Delaney [1]_, p.112. Parameters ---------- args : tuple of array_like Any number of arrays. Returns ------- obrientransform : ndarray Transformed data for use in an ANOVA. The first dimension of the result corresponds to the sequence of transformed arrays. If the arrays given are all 1-D of the same length, the return value is a 2-D array; otherwise it is a 1-D array of type object, with each element being an ndarray. References ---------- .. [1] S. E. Maxwell and H. D. Delaney, "Designing Experiments and Analyzing Data: A Model Comparison Perspective", Wadsworth, 1990. Examples -------- We'll test the following data sets for differences in their variance. >>> x = [10, 11, 13, 9, 7, 12, 12, 9, 10] >>> y = [13, 21, 5, 10, 8, 14, 10, 12, 7, 15] Apply the O'Brien transform to the data. >>> from scipy.stats import obrientransform >>> tx, ty = obrientransform(x, y) Use `scipy.stats.f_oneway` to apply a one-way ANOVA test to the transformed data. >>> from scipy.stats import f_oneway >>> F, p = f_oneway(tx, ty) >>> p 0.1314139477040335 If we require that ``p < 0.05`` for significance, we cannot conclude that the variances are different. """ TINY = np.sqrt(np.finfo(float).eps) # `arrays` will hold the transformed arguments. arrays = [] sLast = None for arg in args: a = np.asarray(arg) n = len(a) mu = np.mean(a) sq = (a - mu)**2 sumsq = sq.sum() # The O'Brien transform. t = ((n - 1.5) * n * sq - 0.5 * sumsq) / ((n - 1) * (n - 2)) # Check that the mean of the transformed data is equal to the # original variance. var = sumsq / (n - 1) if abs(var - np.mean(t)) > TINY: raise ValueError('Lack of convergence in obrientransform.') arrays.append(t) sLast = a.shape if sLast: for arr in arrays[:-1]: if sLast != arr.shape: return np.array(arrays, dtype=object) return np.array(arrays) def sem(a, axis=0, ddof=1, nan_policy='propagate'): """ Compute standard error of the mean. Calculate the standard error of the mean (or standard error of measurement) of the values in the input array. Parameters ---------- a : array_like An array containing the values for which the standard error is returned. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Delta degrees-of-freedom. How many degrees of freedom to adjust for bias in limited samples relative to the population estimate of variance. Defaults to 1. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- s : ndarray or float The standard error of the mean in the sample(s), along the input axis. Notes ----- The default value for `ddof` is different to the default (0) used by other ddof containing routines, such as np.std and np.nanstd. Examples -------- Find standard error along the first axis: >>> from scipy import stats >>> a = np.arange(20).reshape(5,4) >>> stats.sem(a) array([ 2.8284, 2.8284, 2.8284, 2.8284]) Find standard error across the whole array, using n degrees of freedom: >>> stats.sem(a, axis=None, ddof=0) 1.2893796958227628 """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': a = ma.masked_invalid(a) return mstats_basic.sem(a, axis, ddof) n = a.shape[axis] s = np.std(a, axis=axis, ddof=ddof) / np.sqrt(n) return s def _isconst(x): """ Check if all values in x are the same. nans are ignored. x must be a 1d array. The return value is a 1d array with length 1, so it can be used in np.apply_along_axis. """ y = x[~np.isnan(x)] if y.size == 0: return np.array([True]) else: return (y[0] == y).all(keepdims=True) def _quiet_nanmean(x): """ Compute nanmean for the 1d array x, but quietly return nan if x is all nan. The return value is a 1d array with length 1, so it can be used in np.apply_along_axis. """ y = x[~np.isnan(x)] if y.size == 0: return np.array([np.nan]) else: return np.mean(y, keepdims=True) def _quiet_nanstd(x): """ Compute nanstd for the 1d array x, but quietly return nan if x is all nan. The return value is a 1d array with length 1, so it can be used in np.apply_along_axis. """ y = x[~np.isnan(x)] if y.size == 0: return np.array([np.nan]) else: return np.std(y, keepdims=True) def zscore(a, axis=0, ddof=0, nan_policy='propagate'): """ Compute the z score. Compute the z score of each value in the sample, relative to the sample mean and standard deviation. Parameters ---------- a : array_like An array like object containing the sample data. axis : int or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Degrees of freedom correction in the calculation of the standard deviation. Default is 0. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. 'propagate' returns nan, 'raise' throws an error, 'omit' performs the calculations ignoring nan values. Default is 'propagate'. Note that when the value is 'omit', nans in the input also propagate to the output, but they do not affect the z-scores computed for the non-nan values. Returns ------- zscore : array_like The z-scores, standardized by mean and standard deviation of input array `a`. Notes ----- This function preserves ndarray subclasses, and works also with matrices and masked arrays (it uses `asanyarray` instead of `asarray` for parameters). Examples -------- >>> a = np.array([ 0.7972, 0.0767, 0.4383, 0.7866, 0.8091, ... 0.1954, 0.6307, 0.6599, 0.1065, 0.0508]) >>> from scipy import stats >>> stats.zscore(a) array([ 1.1273, -1.247 , -0.0552, 1.0923, 1.1664, -0.8559, 0.5786, 0.6748, -1.1488, -1.3324]) Computing along a specified axis, using n-1 degrees of freedom (``ddof=1``) to calculate the standard deviation: >>> b = np.array([[ 0.3148, 0.0478, 0.6243, 0.4608], ... [ 0.7149, 0.0775, 0.6072, 0.9656], ... [ 0.6341, 0.1403, 0.9759, 0.4064], ... [ 0.5918, 0.6948, 0.904 , 0.3721], ... [ 0.0921, 0.2481, 0.1188, 0.1366]]) >>> stats.zscore(b, axis=1, ddof=1) array([[-0.19264823, -1.28415119, 1.07259584, 0.40420358], [ 0.33048416, -1.37380874, 0.04251374, 1.00081084], [ 0.26796377, -1.12598418, 1.23283094, -0.37481053], [-0.22095197, 0.24468594, 1.19042819, -1.21416216], [-0.82780366, 1.4457416 , -0.43867764, -0.1792603 ]]) An example with `nan_policy='omit'`: >>> x = np.array([[25.11, 30.10, np.nan, 32.02, 43.15], ... [14.95, 16.06, 121.25, 94.35, 29.81]]) >>> stats.zscore(x, axis=1, nan_policy='omit') array([[-1.13490897, -0.37830299, nan, -0.08718406, 1.60039602], [-0.91611681, -0.89090508, 1.4983032 , 0.88731639, -0.5785977 ]]) """ a = np.asanyarray(a) if a.size == 0: return np.empty(a.shape) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': if axis is None: mn = _quiet_nanmean(a.ravel()) std = _quiet_nanstd(a.ravel()) isconst = _isconst(a.ravel()) else: mn = np.apply_along_axis(_quiet_nanmean, axis, a) std = np.apply_along_axis(_quiet_nanstd, axis, a) isconst = np.apply_along_axis(_isconst, axis, a) else: mn = a.mean(axis=axis, keepdims=True) std = a.std(axis=axis, ddof=ddof, keepdims=True) if axis is None: isconst = (a.item(0) == a).all() else: isconst = (_first(a, axis) == a).all(axis=axis, keepdims=True) # Set std deviations that are 0 to 1 to avoid division by 0. std[isconst] = 1.0 z = (a - mn) / std # Set the outputs associated with a constant input to nan. z[np.broadcast_to(isconst, z.shape)] = np.nan return z def zmap(scores, compare, axis=0, ddof=0): """ Calculate the relative z-scores. Return an array of z-scores, i.e., scores that are standardized to zero mean and unit variance, where mean and variance are calculated from the comparison array. Parameters ---------- scores : array_like The input for which z-scores are calculated. compare : array_like The input from which the mean and standard deviation of the normalization are taken; assumed to have the same dimension as `scores`. axis : int or None, optional Axis over which mean and variance of `compare` are calculated. Default is 0. If None, compute over the whole array `scores`. ddof : int, optional Degrees of freedom correction in the calculation of the standard deviation. Default is 0. Returns ------- zscore : array_like Z-scores, in the same shape as `scores`. Notes ----- This function preserves ndarray subclasses, and works also with matrices and masked arrays (it uses `asanyarray` instead of `asarray` for parameters). Examples -------- >>> from scipy.stats import zmap >>> a = [0.5, 2.0, 2.5, 3] >>> b = [0, 1, 2, 3, 4] >>> zmap(a, b) array([-1.06066017, 0. , 0.35355339, 0.70710678]) """ scores, compare = map(np.asanyarray, [scores, compare]) mns = compare.mean(axis=axis, keepdims=True) sstd = compare.std(axis=axis, ddof=ddof, keepdims=True) return (scores - mns) / sstd def gstd(a, axis=0, ddof=1): """ Calculate the geometric standard deviation of an array. The geometric standard deviation describes the spread of a set of numbers where the geometric mean is preferred. It is a multiplicative factor, and so a dimensionless quantity. It is defined as the exponent of the standard deviation of ``log(a)``. Mathematically the population geometric standard deviation can be evaluated as:: gstd = exp(std(log(a))) .. versionadded:: 1.3.0 Parameters ---------- a : array_like An array like object containing the sample data. axis : int, tuple or None, optional Axis along which to operate. Default is 0. If None, compute over the whole array `a`. ddof : int, optional Degree of freedom correction in the calculation of the geometric standard deviation. Default is 1. Returns ------- ndarray or float An array of the geometric standard deviation. If `axis` is None or `a` is a 1d array a float is returned. Notes ----- As the calculation requires the use of logarithms the geometric standard deviation only supports strictly positive values. Any non-positive or infinite values will raise a `ValueError`. The geometric standard deviation is sometimes confused with the exponent of the standard deviation, ``exp(std(a))``. Instead the geometric standard deviation is ``exp(std(log(a)))``. The default value for `ddof` is different to the default value (0) used by other ddof containing functions, such as ``np.std`` and ``np.nanstd``. Examples -------- Find the geometric standard deviation of a log-normally distributed sample. Note that the standard deviation of the distribution is one, on a log scale this evaluates to approximately ``exp(1)``. >>> from scipy.stats import gstd >>> np.random.seed(123) >>> sample = np.random.lognormal(mean=0, sigma=1, size=1000) >>> gstd(sample) 2.7217860664589946 Compute the geometric standard deviation of a multidimensional array and of a given axis. >>> a = np.arange(1, 25).reshape(2, 3, 4) >>> gstd(a, axis=None) 2.2944076136018947 >>> gstd(a, axis=2) array([[1.82424757, 1.22436866, 1.13183117], [1.09348306, 1.07244798, 1.05914985]]) >>> gstd(a, axis=(1,2)) array([2.12939215, 1.22120169]) The geometric standard deviation further handles masked arrays. >>> a = np.arange(1, 25).reshape(2, 3, 4) >>> ma = np.ma.masked_where(a > 16, a) >>> ma masked_array( data=[[[1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12]], [[13, 14, 15, 16], [--, --, --, --], [--, --, --, --]]], mask=[[[False, False, False, False], [False, False, False, False], [False, False, False, False]], [[False, False, False, False], [ True, True, True, True], [ True, True, True, True]]], fill_value=999999) >>> gstd(ma, axis=2) masked_array( data=[[1.8242475707663655, 1.2243686572447428, 1.1318311657788478], [1.0934830582350938, --, --]], mask=[[False, False, False], [False, True, True]], fill_value=999999) """ a = np.asanyarray(a) log = ma.log if isinstance(a, ma.MaskedArray) else np.log try: with warnings.catch_warnings(): warnings.simplefilter("error", RuntimeWarning) return np.exp(np.std(log(a), axis=axis, ddof=ddof)) except RuntimeWarning as w: if np.isinf(a).any(): raise ValueError( 'Infinite value encountered. The geometric standard deviation ' 'is defined for strictly positive values only.' ) from w a_nan = np.isnan(a) a_nan_any = a_nan.any() # exclude NaN's from negativity check, but # avoid expensive masking for arrays with no NaN if ((a_nan_any and np.less_equal(np.nanmin(a), 0)) or (not a_nan_any and np.less_equal(a, 0).any())): raise ValueError( 'Non positive value encountered. The geometric standard ' 'deviation is defined for strictly positive values only.' ) from w elif 'Degrees of freedom <= 0 for slice' == str(w): raise ValueError(w) from w else: # Remaining warnings don't need to be exceptions. return np.exp(np.std(log(a, where=~a_nan), axis=axis, ddof=ddof)) except TypeError as e: raise ValueError( 'Invalid array input. The inputs could not be ' 'safely coerced to any supported types') from e # Private dictionary initialized only once at module level # See https://en.wikipedia.org/wiki/Robust_measures_of_scale _scale_conversions = {'raw': 1.0, 'normal': special.erfinv(0.5) * 2.0 * math.sqrt(2.0)} def iqr(x, axis=None, rng=(25, 75), scale=1.0, nan_policy='propagate', interpolation='linear', keepdims=False): r""" Compute the interquartile range of the data along the specified axis. The interquartile range (IQR) is the difference between the 75th and 25th percentile of the data. It is a measure of the dispersion similar to standard deviation or variance, but is much more robust against outliers [2]_. The ``rng`` parameter allows this function to compute other percentile ranges than the actual IQR. For example, setting ``rng=(0, 100)`` is equivalent to `numpy.ptp`. The IQR of an empty array is `np.nan`. .. versionadded:: 0.18.0 Parameters ---------- x : array_like Input array or object that can be converted to an array. axis : int or sequence of int, optional Axis along which the range is computed. The default is to compute the IQR for the entire array. rng : Two-element sequence containing floats in range of [0,100] optional Percentiles over which to compute the range. Each must be between 0 and 100, inclusive. The default is the true IQR: `(25, 75)`. The order of the elements is not important. scale : scalar or str, optional The numerical value of scale will be divided out of the final result. The following string values are recognized: * 'raw' : No scaling, just return the raw IQR. **Deprecated!** Use `scale=1` instead. * 'normal' : Scale by :math:`2 \sqrt{2} erf^{-1}(\frac{1}{2}) \approx 1.349`. The default is 1.0. The use of scale='raw' is deprecated. Array-like scale is also allowed, as long as it broadcasts correctly to the output such that ``out / scale`` is a valid operation. The output dimensions depend on the input array, `x`, the `axis` argument, and the `keepdims` flag. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values interpolation : {'linear', 'lower', 'higher', 'midpoint', 'nearest'}, optional Specifies the interpolation method to use when the percentile boundaries lie between two data points `i` and `j`. The following options are available (default is 'linear'): * 'linear': `i + (j - i) * fraction`, where `fraction` is the fractional part of the index surrounded by `i` and `j`. * 'lower': `i`. * 'higher': `j`. * 'nearest': `i` or `j` whichever is nearest. * 'midpoint': `(i + j) / 2`. keepdims : bool, optional If this is set to `True`, the reduced axes are left in the result as dimensions with size one. With this option, the result will broadcast correctly against the original array `x`. Returns ------- iqr : scalar or ndarray If ``axis=None``, a scalar is returned. If the input contains integers or floats of smaller precision than ``np.float64``, then the output data-type is ``np.float64``. Otherwise, the output data-type is the same as that of the input. See Also -------- numpy.std, numpy.var Notes ----- This function is heavily dependent on the version of `numpy` that is installed. Versions greater than 1.11.0b3 are highly recommended, as they include a number of enhancements and fixes to `numpy.percentile` and `numpy.nanpercentile` that affect the operation of this function. The following modifications apply: Below 1.10.0 : `nan_policy` is poorly defined. The default behavior of `numpy.percentile` is used for 'propagate'. This is a hybrid of 'omit' and 'propagate' that mostly yields a skewed version of 'omit' since NaNs are sorted to the end of the data. A warning is raised if there are NaNs in the data. Below 1.9.0: `numpy.nanpercentile` does not exist. This means that `numpy.percentile` is used regardless of `nan_policy` and a warning is issued. See previous item for a description of the behavior. Below 1.9.0: `keepdims` and `interpolation` are not supported. The keywords get ignored with a warning if supplied with non-default values. However, multiple axes are still supported. References ---------- .. [1] "Interquartile range" https://en.wikipedia.org/wiki/Interquartile_range .. [2] "Robust measures of scale" https://en.wikipedia.org/wiki/Robust_measures_of_scale .. [3] "Quantile" https://en.wikipedia.org/wiki/Quantile Examples -------- >>> from scipy.stats import iqr >>> x = np.array([[10, 7, 4], [3, 2, 1]]) >>> x array([[10, 7, 4], [ 3, 2, 1]]) >>> iqr(x) 4.0 >>> iqr(x, axis=0) array([ 3.5, 2.5, 1.5]) >>> iqr(x, axis=1) array([ 3., 1.]) >>> iqr(x, axis=1, keepdims=True) array([[ 3.], [ 1.]]) """ x = asarray(x) # This check prevents percentile from raising an error later. Also, it is # consistent with `np.var` and `np.std`. if not x.size: return np.nan # An error may be raised here, so fail-fast, before doing lengthy # computations, even though `scale` is not used until later if isinstance(scale, str): scale_key = scale.lower() if scale_key not in _scale_conversions: raise ValueError("{0} not a valid scale for `iqr`".format(scale)) if scale_key == 'raw': warnings.warn( "use of scale='raw' is deprecated, use scale=1.0 instead", np.VisibleDeprecationWarning ) scale = _scale_conversions[scale_key] # Select the percentile function to use based on nans and policy contains_nan, nan_policy = _contains_nan(x, nan_policy) if contains_nan and nan_policy == 'omit': percentile_func = np.nanpercentile else: percentile_func = np.percentile if len(rng) != 2: raise TypeError("quantile range must be two element sequence") if np.isnan(rng).any(): raise ValueError("range must not contain NaNs") rng = sorted(rng) pct = percentile_func(x, rng, axis=axis, interpolation=interpolation, keepdims=keepdims) out = np.subtract(pct[1], pct[0]) if scale != 1.0: out /= scale return out def _mad_1d(x, center, nan_policy): # Median absolute deviation for 1-d array x. # This is a helper function for `median_abs_deviation`; it assumes its # arguments have been validated already. In particular, x must be a # 1-d numpy array, center must be callable, and if nan_policy is not # 'propagate', it is assumed to be 'omit', because 'raise' is handled # in `median_abs_deviation`. # No warning is generated if x is empty or all nan. isnan = np.isnan(x) if isnan.any(): if nan_policy == 'propagate': return np.nan x = x[~isnan] if x.size == 0: # MAD of an empty array is nan. return np.nan # Edge cases have been handled, so do the basic MAD calculation. med = center(x) mad = np.median(np.abs(x - med)) return mad def median_abs_deviation(x, axis=0, center=np.median, scale=1.0, nan_policy='propagate'): r""" Compute the median absolute deviation of the data along the given axis. The median absolute deviation (MAD, [1]_) computes the median over the absolute deviations from the median. It is a measure of dispersion similar to the standard deviation but more robust to outliers [2]_. The MAD of an empty array is ``np.nan``. .. versionadded:: 1.5.0 Parameters ---------- x : array_like Input array or object that can be converted to an array. axis : int or None, optional Axis along which the range is computed. Default is 0. If None, compute the MAD over the entire array. center : callable, optional A function that will return the central value. The default is to use np.median. Any user defined function used will need to have the function signature ``func(arr, axis)``. scale : scalar or str, optional The numerical value of scale will be divided out of the final result. The default is 1.0. The string "normal" is also accepted, and results in `scale` being the inverse of the standard normal quantile function at 0.75, which is approximately 0.67449. Array-like scale is also allowed, as long as it broadcasts correctly to the output such that ``out / scale`` is a valid operation. The output dimensions depend on the input array, `x`, and the `axis` argument. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- mad : scalar or ndarray If ``axis=None``, a scalar is returned. If the input contains integers or floats of smaller precision than ``np.float64``, then the output data-type is ``np.float64``. Otherwise, the output data-type is the same as that of the input. See Also -------- numpy.std, numpy.var, numpy.median, scipy.stats.iqr, scipy.stats.tmean, scipy.stats.tstd, scipy.stats.tvar Notes ----- The `center` argument only affects the calculation of the central value around which the MAD is calculated. That is, passing in ``center=np.mean`` will calculate the MAD around the mean - it will not calculate the *mean* absolute deviation. The input array may contain `inf`, but if `center` returns `inf`, the corresponding MAD for that data will be `nan`. References ---------- .. [1] "Median absolute deviation", https://en.wikipedia.org/wiki/Median_absolute_deviation .. [2] "Robust measures of scale", https://en.wikipedia.org/wiki/Robust_measures_of_scale Examples -------- When comparing the behavior of `median_abs_deviation` with ``np.std``, the latter is affected when we change a single value of an array to have an outlier value while the MAD hardly changes: >>> from scipy import stats >>> x = stats.norm.rvs(size=100, scale=1, random_state=123456) >>> x.std() 0.9973906394005013 >>> stats.median_abs_deviation(x) 0.82832610097857 >>> x[0] = 345.6 >>> x.std() 34.42304872314415 >>> stats.median_abs_deviation(x) 0.8323442311590675 Axis handling example: >>> x = np.array([[10, 7, 4], [3, 2, 1]]) >>> x array([[10, 7, 4], [ 3, 2, 1]]) >>> stats.median_abs_deviation(x) array([3.5, 2.5, 1.5]) >>> stats.median_abs_deviation(x, axis=None) 2.0 Scale normal example: >>> x = stats.norm.rvs(size=1000000, scale=2, random_state=123456) >>> stats.median_abs_deviation(x) 1.3487398527041636 >>> stats.median_abs_deviation(x, scale='normal') 1.9996446978061115 """ if not callable(center): raise TypeError("The argument 'center' must be callable. The given " f"value {repr(center)} is not callable.") # An error may be raised here, so fail-fast, before doing lengthy # computations, even though `scale` is not used until later if isinstance(scale, str): if scale.lower() == 'normal': scale = 0.6744897501960817 # special.ndtri(0.75) else: raise ValueError(f"{scale} is not a valid scale value.") x = asarray(x) # Consistent with `np.var` and `np.std`. if not x.size: if axis is None: return np.nan nan_shape = tuple(item for i, item in enumerate(x.shape) if i != axis) if nan_shape == (): # Return nan, not array(nan) return np.nan return np.full(nan_shape, np.nan) contains_nan, nan_policy = _contains_nan(x, nan_policy) if contains_nan: if axis is None: mad = _mad_1d(x.ravel(), center, nan_policy) else: mad = np.apply_along_axis(_mad_1d, axis, x, center, nan_policy) else: if axis is None: med = center(x, axis=None) mad = np.median(np.abs(x - med)) else: # Wrap the call to center() in expand_dims() so it acts like # keepdims=True was used. med = np.expand_dims(center(x, axis=axis), axis) mad = np.median(np.abs(x - med), axis=axis) return mad / scale # Keep the top newline so that the message does not show up on the stats page _median_absolute_deviation_deprec_msg = """ To preserve the existing default behavior, use `scipy.stats.median_abs_deviation(..., scale=1/1.4826)`. The value 1.4826 is not numerically precise for scaling with a normal distribution. For a numerically precise value, use `scipy.stats.median_abs_deviation(..., scale='normal')`. """ # Due to numpy/gh-16349 we need to unindent the entire docstring @np.deprecate(old_name='median_absolute_deviation', new_name='median_abs_deviation', message=_median_absolute_deviation_deprec_msg) def median_absolute_deviation(x, axis=0, center=np.median, scale=1.4826, nan_policy='propagate'): r""" Compute the median absolute deviation of the data along the given axis. The median absolute deviation (MAD, [1]_) computes the median over the absolute deviations from the median. It is a measure of dispersion similar to the standard deviation but more robust to outliers [2]_. The MAD of an empty array is ``np.nan``. .. versionadded:: 1.3.0 Parameters ---------- x : array_like Input array or object that can be converted to an array. axis : int or None, optional Axis along which the range is computed. Default is 0. If None, compute the MAD over the entire array. center : callable, optional A function that will return the central value. The default is to use np.median. Any user defined function used will need to have the function signature ``func(arr, axis)``. scale : int, optional The scaling factor applied to the MAD. The default scale (1.4826) ensures consistency with the standard deviation for normally distributed data. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- mad : scalar or ndarray If ``axis=None``, a scalar is returned. If the input contains integers or floats of smaller precision than ``np.float64``, then the output data-type is ``np.float64``. Otherwise, the output data-type is the same as that of the input. See Also -------- numpy.std, numpy.var, numpy.median, scipy.stats.iqr, scipy.stats.tmean, scipy.stats.tstd, scipy.stats.tvar Notes ----- The `center` argument only affects the calculation of the central value around which the MAD is calculated. That is, passing in ``center=np.mean`` will calculate the MAD around the mean - it will not calculate the *mean* absolute deviation. References ---------- .. [1] "Median absolute deviation", https://en.wikipedia.org/wiki/Median_absolute_deviation .. [2] "Robust measures of scale", https://en.wikipedia.org/wiki/Robust_measures_of_scale Examples -------- When comparing the behavior of `median_absolute_deviation` with ``np.std``, the latter is affected when we change a single value of an array to have an outlier value while the MAD hardly changes: >>> from scipy import stats >>> x = stats.norm.rvs(size=100, scale=1, random_state=123456) >>> x.std() 0.9973906394005013 >>> stats.median_absolute_deviation(x) 1.2280762773108278 >>> x[0] = 345.6 >>> x.std() 34.42304872314415 >>> stats.median_absolute_deviation(x) 1.2340335571164334 Axis handling example: >>> x = np.array([[10, 7, 4], [3, 2, 1]]) >>> x array([[10, 7, 4], [ 3, 2, 1]]) >>> stats.median_absolute_deviation(x) array([5.1891, 3.7065, 2.2239]) >>> stats.median_absolute_deviation(x, axis=None) 2.9652 """ if isinstance(scale, str): if scale.lower() == 'raw': warnings.warn( "use of scale='raw' is deprecated, use scale=1.0 instead", np.VisibleDeprecationWarning ) scale = 1.0 if not isinstance(scale, str): scale = 1 / scale return median_abs_deviation(x, axis=axis, center=center, scale=scale, nan_policy=nan_policy) ##################################### # TRIMMING FUNCTIONS # ##################################### SigmaclipResult = namedtuple('SigmaclipResult', ('clipped', 'lower', 'upper')) def sigmaclip(a, low=4., high=4.): """ Perform iterative sigma-clipping of array elements. Starting from the full sample, all elements outside the critical range are removed, i.e. all elements of the input array `c` that satisfy either of the following conditions:: c < mean(c) - std(c)*low c > mean(c) + std(c)*high The iteration continues with the updated sample until no elements are outside the (updated) range. Parameters ---------- a : array_like Data array, will be raveled if not 1-D. low : float, optional Lower bound factor of sigma clipping. Default is 4. high : float, optional Upper bound factor of sigma clipping. Default is 4. Returns ------- clipped : ndarray Input array with clipped elements removed. lower : float Lower threshold value use for clipping. upper : float Upper threshold value use for clipping. Examples -------- >>> from scipy.stats import sigmaclip >>> a = np.concatenate((np.linspace(9.5, 10.5, 31), ... np.linspace(0, 20, 5))) >>> fact = 1.5 >>> c, low, upp = sigmaclip(a, fact, fact) >>> c array([ 9.96666667, 10. , 10.03333333, 10. ]) >>> c.var(), c.std() (0.00055555555555555165, 0.023570226039551501) >>> low, c.mean() - fact*c.std(), c.min() (9.9646446609406727, 9.9646446609406727, 9.9666666666666668) >>> upp, c.mean() + fact*c.std(), c.max() (10.035355339059327, 10.035355339059327, 10.033333333333333) >>> a = np.concatenate((np.linspace(9.5, 10.5, 11), ... np.linspace(-100, -50, 3))) >>> c, low, upp = sigmaclip(a, 1.8, 1.8) >>> (c == np.linspace(9.5, 10.5, 11)).all() True """ c = np.asarray(a).ravel() delta = 1 while delta: c_std = c.std() c_mean = c.mean() size = c.size critlower = c_mean - c_std * low critupper = c_mean + c_std * high c = c[(c >= critlower) & (c <= critupper)] delta = size - c.size return SigmaclipResult(c, critlower, critupper) def trimboth(a, proportiontocut, axis=0): """ Slice off a proportion of items from both ends of an array. Slice off the passed proportion of items from both ends of the passed array (i.e., with `proportiontocut` = 0.1, slices leftmost 10% **and** rightmost 10% of scores). The trimmed values are the lowest and highest ones. Slice off less if proportion results in a non-integer slice index (i.e. conservatively slices off `proportiontocut`). Parameters ---------- a : array_like Data to trim. proportiontocut : float Proportion (in range 0-1) of total data set to trim of each end. axis : int or None, optional Axis along which to trim data. Default is 0. If None, compute over the whole array `a`. Returns ------- out : ndarray Trimmed version of array `a`. The order of the trimmed content is undefined. See Also -------- trim_mean Examples -------- >>> from scipy import stats >>> a = np.arange(20) >>> b = stats.trimboth(a, 0.1) >>> b.shape (16,) """ a = np.asarray(a) if a.size == 0: return a if axis is None: a = a.ravel() axis = 0 nobs = a.shape[axis] lowercut = int(proportiontocut * nobs) uppercut = nobs - lowercut if (lowercut >= uppercut): raise ValueError("Proportion too big.") atmp = np.partition(a, (lowercut, uppercut - 1), axis) sl = [slice(None)] * atmp.ndim sl[axis] = slice(lowercut, uppercut) return atmp[tuple(sl)] def trim1(a, proportiontocut, tail='right', axis=0): """ Slice off a proportion from ONE end of the passed array distribution. If `proportiontocut` = 0.1, slices off 'leftmost' or 'rightmost' 10% of scores. The lowest or highest values are trimmed (depending on the tail). Slice off less if proportion results in a non-integer slice index (i.e. conservatively slices off `proportiontocut` ). Parameters ---------- a : array_like Input array. proportiontocut : float Fraction to cut off of 'left' or 'right' of distribution. tail : {'left', 'right'}, optional Defaults to 'right'. axis : int or None, optional Axis along which to trim data. Default is 0. If None, compute over the whole array `a`. Returns ------- trim1 : ndarray Trimmed version of array `a`. The order of the trimmed content is undefined. """ a = np.asarray(a) if axis is None: a = a.ravel() axis = 0 nobs = a.shape[axis] # avoid possible corner case if proportiontocut >= 1: return [] if tail.lower() == 'right': lowercut = 0 uppercut = nobs - int(proportiontocut * nobs) elif tail.lower() == 'left': lowercut = int(proportiontocut * nobs) uppercut = nobs atmp = np.partition(a, (lowercut, uppercut - 1), axis) return atmp[lowercut:uppercut] def trim_mean(a, proportiontocut, axis=0): """ Return mean of array after trimming distribution from both tails. If `proportiontocut` = 0.1, slices off 'leftmost' and 'rightmost' 10% of scores. The input is sorted before slicing. Slices off less if proportion results in a non-integer slice index (i.e., conservatively slices off `proportiontocut` ). Parameters ---------- a : array_like Input array. proportiontocut : float Fraction to cut off of both tails of the distribution. axis : int or None, optional Axis along which the trimmed means are computed. Default is 0. If None, compute over the whole array `a`. Returns ------- trim_mean : ndarray Mean of trimmed array. See Also -------- trimboth tmean : Compute the trimmed mean ignoring values outside given `limits`. Examples -------- >>> from scipy import stats >>> x = np.arange(20) >>> stats.trim_mean(x, 0.1) 9.5 >>> x2 = x.reshape(5, 4) >>> x2 array([[ 0, 1, 2, 3], [ 4, 5, 6, 7], [ 8, 9, 10, 11], [12, 13, 14, 15], [16, 17, 18, 19]]) >>> stats.trim_mean(x2, 0.25) array([ 8., 9., 10., 11.]) >>> stats.trim_mean(x2, 0.25, axis=1) array([ 1.5, 5.5, 9.5, 13.5, 17.5]) """ a = np.asarray(a) if a.size == 0: return np.nan if axis is None: a = a.ravel() axis = 0 nobs = a.shape[axis] lowercut = int(proportiontocut * nobs) uppercut = nobs - lowercut if (lowercut > uppercut): raise ValueError("Proportion too big.") atmp = np.partition(a, (lowercut, uppercut - 1), axis) sl = [slice(None)] * atmp.ndim sl[axis] = slice(lowercut, uppercut) return np.mean(atmp[tuple(sl)], axis=axis) F_onewayResult = namedtuple('F_onewayResult', ('statistic', 'pvalue')) class F_onewayConstantInputWarning(RuntimeWarning): """ Warning generated by `f_oneway` when an input is constant, e.g. each of the samples provided is a constant array. """ def __init__(self, msg=None): if msg is None: msg = ("Each of the input arrays is constant;" "the F statistic is not defined or infinite") self.args = (msg,) class F_onewayBadInputSizesWarning(RuntimeWarning): """ Warning generated by `f_oneway` when an input has length 0, or if all the inputs have length 1. """ pass def _create_f_oneway_nan_result(shape, axis): """ This is a helper function for f_oneway for creating the return values in certain degenerate conditions. It creates return values that are all nan with the appropriate shape for the given `shape` and `axis`. """ axis = np.core.multiarray.normalize_axis_index(axis, len(shape)) shp = shape[:axis] + shape[axis+1:] if shp == (): f = np.nan prob = np.nan else: f = np.full(shp, fill_value=np.nan) prob = f.copy() return F_onewayResult(f, prob) def _first(arr, axis): """ Return arr[..., 0:1, ...] where 0:1 is in the `axis` position. """ return np.take_along_axis(arr, np.array(0, ndmin=arr.ndim), axis) def f_oneway(*args, axis=0): """ Perform one-way ANOVA. The one-way ANOVA tests the null hypothesis that two or more groups have the same population mean. The test is applied to samples from two or more groups, possibly with differing sizes. Parameters ---------- sample1, sample2, ... : array_like The sample measurements for each group. There must be at least two arguments. If the arrays are multidimensional, then all the dimensions of the array must be the same except for `axis`. axis : int, optional Axis of the input arrays along which the test is applied. Default is 0. Returns ------- statistic : float The computed F statistic of the test. pvalue : float The associated p-value from the F distribution. Warns ----- F_onewayConstantInputWarning Raised if each of the input arrays is constant array. In this case the F statistic is either infinite or isn't defined, so ``np.inf`` or ``np.nan`` is returned. F_onewayBadInputSizesWarning Raised if the length of any input array is 0, or if all the input arrays have length 1. ``np.nan`` is returned for the F statistic and the p-value in these cases. Notes ----- The ANOVA test has important assumptions that must be satisfied in order for the associated p-value to be valid. 1. The samples are independent. 2. Each sample is from a normally distributed population. 3. The population standard deviations of the groups are all equal. This property is known as homoscedasticity. If these assumptions are not true for a given set of data, it may still be possible to use the Kruskal-Wallis H-test (`scipy.stats.kruskal`) although with some loss of power. The length of each group must be at least one, and there must be at least one group with length greater than one. If these conditions are not satisfied, a warning is generated and (``np.nan``, ``np.nan``) is returned. If each group contains constant values, and there exist at least two groups with different values, the function generates a warning and returns (``np.inf``, 0). If all values in all groups are the same, function generates a warning and returns (``np.nan``, ``np.nan``). The algorithm is from Heiman [2]_, pp.394-7. References ---------- .. [1] R. Lowry, "Concepts and Applications of Inferential Statistics", Chapter 14, 2014, http://vassarstats.net/textbook/ .. [2] G.W. Heiman, "Understanding research methods and statistics: An integrated introduction for psychology", Houghton, Mifflin and Company, 2001. .. [3] G.H. McDonald, "Handbook of Biological Statistics", One-way ANOVA. http://www.biostathandbook.com/onewayanova.html Examples -------- >>> from scipy.stats import f_oneway Here are some data [3]_ on a shell measurement (the length of the anterior adductor muscle scar, standardized by dividing by length) in the mussel Mytilus trossulus from five locations: Tillamook, Oregon; Newport, Oregon; Petersburg, Alaska; Magadan, Russia; and Tvarminne, Finland, taken from a much larger data set used in McDonald et al. (1991). >>> tillamook = [0.0571, 0.0813, 0.0831, 0.0976, 0.0817, 0.0859, 0.0735, ... 0.0659, 0.0923, 0.0836] >>> newport = [0.0873, 0.0662, 0.0672, 0.0819, 0.0749, 0.0649, 0.0835, ... 0.0725] >>> petersburg = [0.0974, 0.1352, 0.0817, 0.1016, 0.0968, 0.1064, 0.105] >>> magadan = [0.1033, 0.0915, 0.0781, 0.0685, 0.0677, 0.0697, 0.0764, ... 0.0689] >>> tvarminne = [0.0703, 0.1026, 0.0956, 0.0973, 0.1039, 0.1045] >>> f_oneway(tillamook, newport, petersburg, magadan, tvarminne) F_onewayResult(statistic=7.121019471642447, pvalue=0.0002812242314534544) `f_oneway` accepts multidimensional input arrays. When the inputs are multidimensional and `axis` is not given, the test is performed along the first axis of the input arrays. For the following data, the test is performed three times, once for each column. >>> a = np.array([[9.87, 9.03, 6.81], ... [7.18, 8.35, 7.00], ... [8.39, 7.58, 7.68], ... [7.45, 6.33, 9.35], ... [6.41, 7.10, 9.33], ... [8.00, 8.24, 8.44]]) >>> b = np.array([[6.35, 7.30, 7.16], ... [6.65, 6.68, 7.63], ... [5.72, 7.73, 6.72], ... [7.01, 9.19, 7.41], ... [7.75, 7.87, 8.30], ... [6.90, 7.97, 6.97]]) >>> c = np.array([[3.31, 8.77, 1.01], ... [8.25, 3.24, 3.62], ... [6.32, 8.81, 5.19], ... [7.48, 8.83, 8.91], ... [8.59, 6.01, 6.07], ... [3.07, 9.72, 7.48]]) >>> F, p = f_oneway(a, b, c) >>> F array([1.75676344, 0.03701228, 3.76439349]) >>> p array([0.20630784, 0.96375203, 0.04733157]) """ if len(args) < 2: raise TypeError(f'at least two inputs are required; got {len(args)}.') args = [np.asarray(arg, dtype=float) for arg in args] # ANOVA on N groups, each in its own array num_groups = len(args) # We haven't explicitly validated axis, but if it is bad, this call of # np.concatenate will raise np.AxisError. The call will raise ValueError # if the dimensions of all the arrays, except the axis dimension, are not # the same. alldata = np.concatenate(args, axis=axis) bign = alldata.shape[axis] # Check this after forming alldata, so shape errors are detected # and reported before checking for 0 length inputs. if any(arg.shape[axis] == 0 for arg in args): warnings.warn(F_onewayBadInputSizesWarning('at least one input ' 'has length 0')) return _create_f_oneway_nan_result(alldata.shape, axis) # Must have at least one group with length greater than 1. if all(arg.shape[axis] == 1 for arg in args): msg = ('all input arrays have length 1. f_oneway requires that at ' 'least one input has length greater than 1.') warnings.warn(F_onewayBadInputSizesWarning(msg)) return _create_f_oneway_nan_result(alldata.shape, axis) # Check if the values within each group are constant, and if the common # value in at least one group is different from that in another group. # Based on https://github.com/scipy/scipy/issues/11669 # If axis=0, say, and the groups have shape (n0, ...), (n1, ...), ..., # then is_const is a boolean array with shape (num_groups, ...). # It is True if the groups along the axis slice are each consant. # In the typical case where each input array is 1-d, is_const is a # 1-d array with length num_groups. is_const = np.concatenate([(_first(a, axis) == a).all(axis=axis, keepdims=True) for a in args], axis=axis) # all_const is a boolean array with shape (...) (see previous comment). # It is True if the values within each group along the axis slice are # the same (e.g. [[3, 3, 3], [5, 5, 5, 5], [4, 4, 4]]). all_const = is_const.all(axis=axis) if all_const.any(): warnings.warn(F_onewayConstantInputWarning()) # all_same_const is True if all the values in the groups along the axis=0 # slice are the same (e.g. [[3, 3, 3], [3, 3, 3, 3], [3, 3, 3]]). all_same_const = (_first(alldata, axis) == alldata).all(axis=axis) # Determine the mean of the data, and subtract that from all inputs to a # variance (via sum_of_sq / sq_of_sum) calculation. Variance is invariant # to a shift in location, and centering all data around zero vastly # improves numerical stability. offset = alldata.mean(axis=axis, keepdims=True) alldata -= offset normalized_ss = _square_of_sums(alldata, axis=axis) / bign sstot = _sum_of_squares(alldata, axis=axis) - normalized_ss ssbn = 0 for a in args: ssbn += _square_of_sums(a - offset, axis=axis) / a.shape[axis] # Naming: variables ending in bn/b are for "between treatments", wn/w are # for "within treatments" ssbn -= normalized_ss sswn = sstot - ssbn dfbn = num_groups - 1 dfwn = bign - num_groups msb = ssbn / dfbn msw = sswn / dfwn with np.errstate(divide='ignore', invalid='ignore'): f = msb / msw prob = special.fdtrc(dfbn, dfwn, f) # equivalent to stats.f.sf # Fix any f values that should be inf or nan because the corresponding # inputs were constant. if np.isscalar(f): if all_same_const: f = np.nan prob = np.nan elif all_const: f = np.inf prob = 0.0 else: f[all_const] = np.inf prob[all_const] = 0.0 f[all_same_const] = np.nan prob[all_same_const] = np.nan return F_onewayResult(f, prob) class PearsonRConstantInputWarning(RuntimeWarning): """Warning generated by `pearsonr` when an input is constant.""" def __init__(self, msg=None): if msg is None: msg = ("An input array is constant; the correlation coefficent " "is not defined.") self.args = (msg,) class PearsonRNearConstantInputWarning(RuntimeWarning): """Warning generated by `pearsonr` when an input is nearly constant.""" def __init__(self, msg=None): if msg is None: msg = ("An input array is nearly constant; the computed " "correlation coefficent may be inaccurate.") self.args = (msg,) def pearsonr(x, y): r""" Pearson correlation coefficient and p-value for testing non-correlation. The Pearson correlation coefficient [1]_ measures the linear relationship between two datasets. The calculation of the p-value relies on the assumption that each dataset is normally distributed. (See Kowalski [3]_ for a discussion of the effects of non-normality of the input on the distribution of the correlation coefficient.) Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact linear relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases. The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Pearson correlation at least as extreme as the one computed from these datasets. Parameters ---------- x : (N,) array_like Input array. y : (N,) array_like Input array. Returns ------- r : float Pearson's correlation coefficient. p-value : float Two-tailed p-value. Warns ----- PearsonRConstantInputWarning Raised if an input is a constant array. The correlation coefficient is not defined in this case, so ``np.nan`` is returned. PearsonRNearConstantInputWarning Raised if an input is "nearly" constant. The array ``x`` is considered nearly constant if ``norm(x - mean(x)) < 1e-13 * abs(mean(x))``. Numerical errors in the calculation ``x - mean(x)`` in this case might result in an inaccurate calculation of r. See Also -------- spearmanr : Spearman rank-order correlation coefficient. kendalltau : Kendall's tau, a correlation measure for ordinal data. Notes ----- The correlation coefficient is calculated as follows: .. math:: r = \frac{\sum (x - m_x) (y - m_y)} {\sqrt{\sum (x - m_x)^2 \sum (y - m_y)^2}} where :math:`m_x` is the mean of the vector :math:`x` and :math:`m_y` is the mean of the vector :math:`y`. Under the assumption that :math:`x` and :math:`m_y` are drawn from independent normal distributions (so the population correlation coefficient is 0), the probability density function of the sample correlation coefficient :math:`r` is ([1]_, [2]_): .. math:: f(r) = \frac{{(1-r^2)}^{n/2-2}}{\mathrm{B}(\frac{1}{2},\frac{n}{2}-1)} where n is the number of samples, and B is the beta function. This is sometimes referred to as the exact distribution of r. This is the distribution that is used in `pearsonr` to compute the p-value. The distribution is a beta distribution on the interval [-1, 1], with equal shape parameters a = b = n/2 - 1. In terms of SciPy's implementation of the beta distribution, the distribution of r is:: dist = scipy.stats.beta(n/2 - 1, n/2 - 1, loc=-1, scale=2) The p-value returned by `pearsonr` is a two-sided p-value. For a given sample with correlation coefficient r, the p-value is the probability that abs(r') of a random sample x' and y' drawn from the population with zero correlation would be greater than or equal to abs(r). In terms of the object ``dist`` shown above, the p-value for a given r and length n can be computed as:: p = 2*dist.cdf(-abs(r)) When n is 2, the above continuous distribution is not well-defined. One can interpret the limit of the beta distribution as the shape parameters a and b approach a = b = 0 as a discrete distribution with equal probability masses at r = 1 and r = -1. More directly, one can observe that, given the data x = [x1, x2] and y = [y1, y2], and assuming x1 != x2 and y1 != y2, the only possible values for r are 1 and -1. Because abs(r') for any sample x' and y' with length 2 will be 1, the two-sided p-value for a sample of length 2 is always 1. References ---------- .. [1] "Pearson correlation coefficient", Wikipedia, https://en.wikipedia.org/wiki/Pearson_correlation_coefficient .. [2] Student, "Probable error of a correlation coefficient", Biometrika, Volume 6, Issue 2-3, 1 September 1908, pp. 302-310. .. [3] C. J. Kowalski, "On the Effects of Non-Normality on the Distribution of the Sample Product-Moment Correlation Coefficient" Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 21, No. 1 (1972), pp. 1-12. Examples -------- >>> from scipy import stats >>> a = np.array([0, 0, 0, 1, 1, 1, 1]) >>> b = np.arange(7) >>> stats.pearsonr(a, b) (0.8660254037844386, 0.011724811003954649) >>> stats.pearsonr([1, 2, 3, 4, 5], [10, 9, 2.5, 6, 4]) (-0.7426106572325057, 0.1505558088534455) """ n = len(x) if n != len(y): raise ValueError('x and y must have the same length.') if n < 2: raise ValueError('x and y must have length at least 2.') x = np.asarray(x) y = np.asarray(y) # If an input is constant, the correlation coefficient is not defined. if (x == x[0]).all() or (y == y[0]).all(): warnings.warn(PearsonRConstantInputWarning()) return np.nan, np.nan # dtype is the data type for the calculations. This expression ensures # that the data type is at least 64 bit floating point. It might have # more precision if the input is, for example, np.longdouble. dtype = type(1.0 + x[0] + y[0]) if n == 2: return dtype(np.sign(x[1] - x[0])*np.sign(y[1] - y[0])), 1.0 xmean = x.mean(dtype=dtype) ymean = y.mean(dtype=dtype) # By using `astype(dtype)`, we ensure that the intermediate calculations # use at least 64 bit floating point. xm = x.astype(dtype) - xmean ym = y.astype(dtype) - ymean # Unlike np.linalg.norm or the expression sqrt((xm*xm).sum()), # scipy.linalg.norm(xm) does not overflow if xm is, for example, # [-5e210, 5e210, 3e200, -3e200] normxm = linalg.norm(xm) normym = linalg.norm(ym) threshold = 1e-13 if normxm < threshold*abs(xmean) or normym < threshold*abs(ymean): # If all the values in x (likewise y) are very close to the mean, # the loss of precision that occurs in the subtraction xm = x - xmean # might result in large errors in r. warnings.warn(PearsonRNearConstantInputWarning()) r = np.dot(xm/normxm, ym/normym) # Presumably, if abs(r) > 1, then it is only some small artifact of # floating point arithmetic. r = max(min(r, 1.0), -1.0) # As explained in the docstring, the p-value can be computed as # p = 2*dist.cdf(-abs(r)) # where dist is the beta distribution on [-1, 1] with shape parameters # a = b = n/2 - 1. `special.btdtr` is the CDF for the beta distribution # on [0, 1]. To use it, we make the transformation x = (r + 1)/2; the # shape parameters do not change. Then -abs(r) used in `cdf(-abs(r))` # becomes x = (-abs(r) + 1)/2 = 0.5*(1 - abs(r)). (r is cast to float64 # to avoid a TypeError raised by btdtr when r is higher precision.) ab = n/2 - 1 prob = 2*special.btdtr(ab, ab, 0.5*(1 - abs(np.float64(r)))) return r, prob def fisher_exact(table, alternative='two-sided'): """ Perform a Fisher exact test on a 2x2 contingency table. Parameters ---------- table : array_like of ints A 2x2 contingency table. Elements should be non-negative integers. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided * 'greater': one-sided Returns ------- oddsratio : float This is prior odds ratio and not a posterior estimate. p_value : float P-value, the probability of obtaining a distribution at least as extreme as the one that was actually observed, assuming that the null hypothesis is true. See Also -------- chi2_contingency : Chi-square test of independence of variables in a contingency table. Notes ----- The calculated odds ratio is different from the one R uses. This scipy implementation returns the (more common) "unconditional Maximum Likelihood Estimate", while R uses the "conditional Maximum Likelihood Estimate". For tables with large numbers, the (inexact) chi-square test implemented in the function `chi2_contingency` can also be used. Examples -------- Say we spend a few days counting whales and sharks in the Atlantic and Indian oceans. In the Atlantic ocean we find 8 whales and 1 shark, in the Indian ocean 2 whales and 5 sharks. Then our contingency table is:: Atlantic Indian whales 8 2 sharks 1 5 We use this table to find the p-value: >>> import scipy.stats as stats >>> oddsratio, pvalue = stats.fisher_exact([[8, 2], [1, 5]]) >>> pvalue 0.0349... The probability that we would observe this or an even more imbalanced ratio by chance is about 3.5%. A commonly used significance level is 5%--if we adopt that, we can therefore conclude that our observed imbalance is statistically significant; whales prefer the Atlantic while sharks prefer the Indian ocean. """ hypergeom = distributions.hypergeom c = np.asarray(table, dtype=np.int64) # int32 is not enough for the algorithm if not c.shape == (2, 2): raise ValueError("The input `table` must be of shape (2, 2).") if np.any(c < 0): raise ValueError("All values in `table` must be nonnegative.") if 0 in c.sum(axis=0) or 0 in c.sum(axis=1): # If both values in a row or column are zero, the p-value is 1 and # the odds ratio is NaN. return np.nan, 1.0 if c[1, 0] > 0 and c[0, 1] > 0: oddsratio = c[0, 0] * c[1, 1] / (c[1, 0] * c[0, 1]) else: oddsratio = np.inf n1 = c[0, 0] + c[0, 1] n2 = c[1, 0] + c[1, 1] n = c[0, 0] + c[1, 0] def binary_search(n, n1, n2, side): """Binary search for where to begin halves in two-sided test.""" if side == "upper": minval = mode maxval = n else: minval = 0 maxval = mode guess = -1 while maxval - minval > 1: if maxval == minval + 1 and guess == minval: guess = maxval else: guess = (maxval + minval) // 2 pguess = hypergeom.pmf(guess, n1 + n2, n1, n) if side == "upper": ng = guess - 1 else: ng = guess + 1 if pguess <= pexact < hypergeom.pmf(ng, n1 + n2, n1, n): break elif pguess < pexact: maxval = guess else: minval = guess if guess == -1: guess = minval if side == "upper": while guess > 0 and hypergeom.pmf(guess, n1 + n2, n1, n) < pexact * epsilon: guess -= 1 while hypergeom.pmf(guess, n1 + n2, n1, n) > pexact / epsilon: guess += 1 else: while hypergeom.pmf(guess, n1 + n2, n1, n) < pexact * epsilon: guess += 1 while guess > 0 and hypergeom.pmf(guess, n1 + n2, n1, n) > pexact / epsilon: guess -= 1 return guess if alternative == 'less': pvalue = hypergeom.cdf(c[0, 0], n1 + n2, n1, n) elif alternative == 'greater': # Same formula as the 'less' case, but with the second column. pvalue = hypergeom.cdf(c[0, 1], n1 + n2, n1, c[0, 1] + c[1, 1]) elif alternative == 'two-sided': mode = int((n + 1) * (n1 + 1) / (n1 + n2 + 2)) pexact = hypergeom.pmf(c[0, 0], n1 + n2, n1, n) pmode = hypergeom.pmf(mode, n1 + n2, n1, n) epsilon = 1 - 1e-4 if np.abs(pexact - pmode) / np.maximum(pexact, pmode) <= 1 - epsilon: return oddsratio, 1. elif c[0, 0] < mode: plower = hypergeom.cdf(c[0, 0], n1 + n2, n1, n) if hypergeom.pmf(n, n1 + n2, n1, n) > pexact / epsilon: return oddsratio, plower guess = binary_search(n, n1, n2, "upper") pvalue = plower + hypergeom.sf(guess - 1, n1 + n2, n1, n) else: pupper = hypergeom.sf(c[0, 0] - 1, n1 + n2, n1, n) if hypergeom.pmf(0, n1 + n2, n1, n) > pexact / epsilon: return oddsratio, pupper guess = binary_search(n, n1, n2, "lower") pvalue = pupper + hypergeom.cdf(guess, n1 + n2, n1, n) else: msg = "`alternative` should be one of {'two-sided', 'less', 'greater'}" raise ValueError(msg) pvalue = min(pvalue, 1.0) return oddsratio, pvalue class SpearmanRConstantInputWarning(RuntimeWarning): """Warning generated by `spearmanr` when an input is constant.""" def __init__(self, msg=None): if msg is None: msg = ("An input array is constant; the correlation coefficent " "is not defined.") self.args = (msg,) SpearmanrResult = namedtuple('SpearmanrResult', ('correlation', 'pvalue')) def spearmanr(a, b=None, axis=0, nan_policy='propagate'): """ Calculate a Spearman correlation coefficient with associated p-value. The Spearman rank-order correlation coefficient is a nonparametric measure of the monotonicity of the relationship between two datasets. Unlike the Pearson correlation, the Spearman correlation does not assume that both datasets are normally distributed. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact monotonic relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases. The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Spearman correlation at least as extreme as the one computed from these datasets. The p-values are not entirely reliable but are probably reasonable for datasets larger than 500 or so. Parameters ---------- a, b : 1D or 2D array_like, b is optional One or two 1-D or 2-D arrays containing multiple variables and observations. When these are 1-D, each represents a vector of observations of a single variable. For the behavior in the 2-D case, see under ``axis``, below. Both arrays need to have the same length in the ``axis`` dimension. axis : int or None, optional If axis=0 (default), then each column represents a variable, with observations in the rows. If axis=1, the relationship is transposed: each row represents a variable, while the columns contain observations. If axis=None, then both arrays will be raveled. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- correlation : float or ndarray (2-D square) Spearman correlation matrix or correlation coefficient (if only 2 variables are given as parameters. Correlation matrix is square with length equal to total number of variables (columns or rows) in ``a`` and ``b`` combined. pvalue : float The two-sided p-value for a hypothesis test whose null hypothesis is that two sets of data are uncorrelated, has same dimension as rho. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 14.7 Examples -------- >>> from scipy import stats >>> stats.spearmanr([1,2,3,4,5], [5,6,7,8,7]) (0.82078268166812329, 0.088587005313543798) >>> np.random.seed(1234321) >>> x2n = np.random.randn(100, 2) >>> y2n = np.random.randn(100, 2) >>> stats.spearmanr(x2n) (0.059969996999699973, 0.55338590803773591) >>> stats.spearmanr(x2n[:,0], x2n[:,1]) (0.059969996999699973, 0.55338590803773591) >>> rho, pval = stats.spearmanr(x2n, y2n) >>> rho array([[ 1. , 0.05997 , 0.18569457, 0.06258626], [ 0.05997 , 1. , 0.110003 , 0.02534653], [ 0.18569457, 0.110003 , 1. , 0.03488749], [ 0.06258626, 0.02534653, 0.03488749, 1. ]]) >>> pval array([[ 0. , 0.55338591, 0.06435364, 0.53617935], [ 0.55338591, 0. , 0.27592895, 0.80234077], [ 0.06435364, 0.27592895, 0. , 0.73039992], [ 0.53617935, 0.80234077, 0.73039992, 0. ]]) >>> rho, pval = stats.spearmanr(x2n.T, y2n.T, axis=1) >>> rho array([[ 1. , 0.05997 , 0.18569457, 0.06258626], [ 0.05997 , 1. , 0.110003 , 0.02534653], [ 0.18569457, 0.110003 , 1. , 0.03488749], [ 0.06258626, 0.02534653, 0.03488749, 1. ]]) >>> stats.spearmanr(x2n, y2n, axis=None) (0.10816770419260482, 0.1273562188027364) >>> stats.spearmanr(x2n.ravel(), y2n.ravel()) (0.10816770419260482, 0.1273562188027364) >>> xint = np.random.randint(10, size=(100, 2)) >>> stats.spearmanr(xint) (0.052760927029710199, 0.60213045837062351) """ if axis is not None and axis > 1: raise ValueError("spearmanr only handles 1-D or 2-D arrays, supplied axis argument {}, please use only values 0, 1 or None for axis".format(axis)) a, axisout = _chk_asarray(a, axis) if a.ndim > 2: raise ValueError("spearmanr only handles 1-D or 2-D arrays") if b is None: if a.ndim < 2: raise ValueError("`spearmanr` needs at least 2 variables to compare") else: # Concatenate a and b, so that we now only have to handle the case # of a 2-D `a`. b, _ = _chk_asarray(b, axis) if axisout == 0: a = np.column_stack((a, b)) else: a = np.row_stack((a, b)) n_vars = a.shape[1 - axisout] n_obs = a.shape[axisout] if n_obs <= 1: # Handle empty arrays or single observations. return SpearmanrResult(np.nan, np.nan) if axisout == 0: if (a[:, 0][0] == a[:, 0]).all() or (a[:, 1][0] == a[:, 1]).all(): # If an input is constant, the correlation coefficient is not defined. warnings.warn(SpearmanRConstantInputWarning()) return SpearmanrResult(np.nan, np.nan) else: # case when axisout == 1 b/c a is 2 dim only if (a[0, :][0] == a[0, :]).all() or (a[1, :][0] == a[1, :]).all(): # If an input is constant, the correlation coefficient is not defined. warnings.warn(SpearmanRConstantInputWarning()) return SpearmanrResult(np.nan, np.nan) a_contains_nan, nan_policy = _contains_nan(a, nan_policy) variable_has_nan = np.zeros(n_vars, dtype=bool) if a_contains_nan: if nan_policy == 'omit': return mstats_basic.spearmanr(a, axis=axis, nan_policy=nan_policy) elif nan_policy == 'propagate': if a.ndim == 1 or n_vars <= 2: return SpearmanrResult(np.nan, np.nan) else: # Keep track of variables with NaNs, set the outputs to NaN # only for those variables variable_has_nan = np.isnan(a).any(axis=axisout) a_ranked = np.apply_along_axis(rankdata, axisout, a) rs = np.corrcoef(a_ranked, rowvar=axisout) dof = n_obs - 2 # degrees of freedom # rs can have elements equal to 1, so avoid zero division warnings with np.errstate(divide='ignore'): # clip the small negative values possibly caused by rounding # errors before taking the square root t = rs * np.sqrt((dof/((rs+1.0)*(1.0-rs))).clip(0)) prob = 2 * distributions.t.sf(np.abs(t), dof) # For backwards compatibility, return scalars when comparing 2 columns if rs.shape == (2, 2): return SpearmanrResult(rs[1, 0], prob[1, 0]) else: rs[variable_has_nan, :] = np.nan rs[:, variable_has_nan] = np.nan return SpearmanrResult(rs, prob) PointbiserialrResult = namedtuple('PointbiserialrResult', ('correlation', 'pvalue')) def pointbiserialr(x, y): r""" Calculate a point biserial correlation coefficient and its p-value. The point biserial correlation is used to measure the relationship between a binary variable, x, and a continuous variable, y. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply a determinative relationship. This function uses a shortcut formula but produces the same result as `pearsonr`. Parameters ---------- x : array_like of bools Input array. y : array_like Input array. Returns ------- correlation : float R value. pvalue : float Two-sided p-value. Notes ----- `pointbiserialr` uses a t-test with ``n-1`` degrees of freedom. It is equivalent to `pearsonr`. The value of the point-biserial correlation can be calculated from: .. math:: r_{pb} = \frac{\overline{Y_{1}} - \overline{Y_{0}}}{s_{y}}\sqrt{\frac{N_{1} N_{2}}{N (N - 1))}} Where :math:`Y_{0}` and :math:`Y_{1}` are means of the metric observations coded 0 and 1 respectively; :math:`N_{0}` and :math:`N_{1}` are number of observations coded 0 and 1 respectively; :math:`N` is the total number of observations and :math:`s_{y}` is the standard deviation of all the metric observations. A value of :math:`r_{pb}` that is significantly different from zero is completely equivalent to a significant difference in means between the two groups. Thus, an independent groups t Test with :math:`N-2` degrees of freedom may be used to test whether :math:`r_{pb}` is nonzero. The relation between the t-statistic for comparing two independent groups and :math:`r_{pb}` is given by: .. math:: t = \sqrt{N - 2}\frac{r_{pb}}{\sqrt{1 - r^{2}_{pb}}} References ---------- .. [1] J. Lev, "The Point Biserial Coefficient of Correlation", Ann. Math. Statist., Vol. 20, no.1, pp. 125-126, 1949. .. [2] R.F. Tate, "Correlation Between a Discrete and a Continuous Variable. Point-Biserial Correlation.", Ann. Math. Statist., Vol. 25, np. 3, pp. 603-607, 1954. .. [3] D. Kornbrot "Point Biserial Correlation", In Wiley StatsRef: Statistics Reference Online (eds N. Balakrishnan, et al.), 2014. :doi:`10.1002/9781118445112.stat06227` Examples -------- >>> from scipy import stats >>> a = np.array([0, 0, 0, 1, 1, 1, 1]) >>> b = np.arange(7) >>> stats.pointbiserialr(a, b) (0.8660254037844386, 0.011724811003954652) >>> stats.pearsonr(a, b) (0.86602540378443871, 0.011724811003954626) >>> np.corrcoef(a, b) array([[ 1. , 0.8660254], [ 0.8660254, 1. ]]) """ rpb, prob = pearsonr(x, y) return PointbiserialrResult(rpb, prob) KendalltauResult = namedtuple('KendalltauResult', ('correlation', 'pvalue')) def kendalltau(x, y, initial_lexsort=None, nan_policy='propagate', method='auto', variant='b'): """ Calculate Kendall's tau, a correlation measure for ordinal data. Kendall's tau is a measure of the correspondence between two rankings. Values close to 1 indicate strong agreement, and values close to -1 indicate strong disagreement. This implements two variants of Kendall's tau: tau-b (the default) and tau-c (also known as Stuart's tau-c). These differ only in how they are normalized to lie within the range -1 to 1; the hypothesis tests (their p-values) are identical. Kendall's original tau-a is not implemented separately because both tau-b and tau-c reduce to tau-a in the absence of ties. Parameters ---------- x, y : array_like Arrays of rankings, of the same shape. If arrays are not 1-D, they will be flattened to 1-D. initial_lexsort : bool, optional Unused (deprecated). nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values method : {'auto', 'asymptotic', 'exact'}, optional Defines which method is used to calculate the p-value [5]_. The following options are available (default is 'auto'): * 'auto': selects the appropriate method based on a trade-off between speed and accuracy * 'asymptotic': uses a normal approximation valid for large samples * 'exact': computes the exact p-value, but can only be used if no ties are present. As the sample size increases, the 'exact' computation time may grow and the result may lose some precision. variant: {'b', 'c'}, optional Defines which variant of Kendall's tau is returned. Default is 'b'. Returns ------- correlation : float The tau statistic. pvalue : float The two-sided p-value for a hypothesis test whose null hypothesis is an absence of association, tau = 0. See Also -------- spearmanr : Calculates a Spearman rank-order correlation coefficient. theilslopes : Computes the Theil-Sen estimator for a set of points (x, y). weightedtau : Computes a weighted version of Kendall's tau. Notes ----- The definition of Kendall's tau that is used is [2]_:: tau_b = (P - Q) / sqrt((P + Q + T) * (P + Q + U)) tau_c = 2 (P - Q) / (n**2 * (m - 1) / m) where P is the number of concordant pairs, Q the number of discordant pairs, T the number of ties only in `x`, and U the number of ties only in `y`. If a tie occurs for the same pair in both `x` and `y`, it is not added to either T or U. n is the total number of samples, and m is the number of unique values in either `x` or `y`, whichever is smaller. References ---------- .. [1] Maurice G. Kendall, "A New Measure of Rank Correlation", Biometrika Vol. 30, No. 1/2, pp. 81-93, 1938. .. [2] Maurice G. Kendall, "The treatment of ties in ranking problems", Biometrika Vol. 33, No. 3, pp. 239-251. 1945. .. [3] Gottfried E. Noether, "Elements of Nonparametric Statistics", John Wiley & Sons, 1967. .. [4] Peter M. Fenwick, "A new data structure for cumulative frequency tables", Software: Practice and Experience, Vol. 24, No. 3, pp. 327-336, 1994. .. [5] Maurice G. Kendall, "Rank Correlation Methods" (4th Edition), Charles Griffin & Co., 1970. Examples -------- >>> from scipy import stats >>> x1 = [12, 2, 1, 12, 2] >>> x2 = [1, 4, 7, 1, 0] >>> tau, p_value = stats.kendalltau(x1, x2) >>> tau -0.47140452079103173 >>> p_value 0.2827454599327748 """ x = np.asarray(x).ravel() y = np.asarray(y).ravel() if x.size != y.size: raise ValueError("All inputs to `kendalltau` must be of the same " f"size, found x-size {x.size} and y-size {y.size}") elif not x.size or not y.size: # Return NaN if arrays are empty return KendalltauResult(np.nan, np.nan) # check both x and y cnx, npx = _contains_nan(x, nan_policy) cny, npy = _contains_nan(y, nan_policy) contains_nan = cnx or cny if npx == 'omit' or npy == 'omit': nan_policy = 'omit' if contains_nan and nan_policy == 'propagate': return KendalltauResult(np.nan, np.nan) elif contains_nan and nan_policy == 'omit': x = ma.masked_invalid(x) y = ma.masked_invalid(y) if variant == 'b': return mstats_basic.kendalltau(x, y, method=method, use_ties=True) else: raise ValueError("Only variant 'b' is supported for masked arrays") if initial_lexsort is not None: # deprecate to drop! warnings.warn('"initial_lexsort" is gone!') def count_rank_tie(ranks): cnt = np.bincount(ranks).astype('int64', copy=False) cnt = cnt[cnt > 1] return ((cnt * (cnt - 1) // 2).sum(), (cnt * (cnt - 1.) * (cnt - 2)).sum(), (cnt * (cnt - 1.) * (2*cnt + 5)).sum()) size = x.size perm = np.argsort(y) # sort on y and convert y to dense ranks x, y = x[perm], y[perm] y = np.r_[True, y[1:] != y[:-1]].cumsum(dtype=np.intp) # stable sort on x and convert x to dense ranks perm = np.argsort(x, kind='mergesort') x, y = x[perm], y[perm] x = np.r_[True, x[1:] != x[:-1]].cumsum(dtype=np.intp) dis = _kendall_dis(x, y) # discordant pairs obs = np.r_[True, (x[1:] != x[:-1]) | (y[1:] != y[:-1]), True] cnt = np.diff(np.nonzero(obs)[0]).astype('int64', copy=False) ntie = (cnt * (cnt - 1) // 2).sum() # joint ties xtie, x0, x1 = count_rank_tie(x) # ties in x, stats ytie, y0, y1 = count_rank_tie(y) # ties in y, stats tot = (size * (size - 1)) // 2 if xtie == tot or ytie == tot: return KendalltauResult(np.nan, np.nan) # Note that tot = con + dis + (xtie - ntie) + (ytie - ntie) + ntie # = con + dis + xtie + ytie - ntie con_minus_dis = tot - xtie - ytie + ntie - 2 * dis if variant == 'b': tau = con_minus_dis / np.sqrt(tot - xtie) / np.sqrt(tot - ytie) elif variant == 'c': minclasses = min(len(set(x)), len(set(y))) tau = 2*con_minus_dis / (size**2 * (minclasses-1)/minclasses) else: raise ValueError(f"Unknown variant of the method chosen: {variant}. " "variant must be 'b' or 'c'.") # Limit range to fix computational errors tau = min(1., max(-1., tau)) # The p-value calculation is the same for all variants since the p-value # depends only on con_minus_dis. if method == 'exact' and (xtie != 0 or ytie != 0): raise ValueError("Ties found, exact method cannot be used.") if method == 'auto': if (xtie == 0 and ytie == 0) and (size <= 33 or min(dis, tot-dis) <= 1): method = 'exact' else: method = 'asymptotic' if xtie == 0 and ytie == 0 and method == 'exact': pvalue = mstats_basic._kendall_p_exact(size, min(dis, tot-dis)) elif method == 'asymptotic': # con_minus_dis is approx normally distributed with this variance [3]_ m = size * (size - 1.) var = ((m * (2*size + 5) - x1 - y1) / 18 + (2 * xtie * ytie) / m + x0 * y0 / (9 * m * (size - 2))) pvalue = (special.erfc(np.abs(con_minus_dis) / np.sqrt(var) / np.sqrt(2))) else: raise ValueError(f"Unknown method {method} specified. Use 'auto', " "'exact' or 'asymptotic'.") return KendalltauResult(tau, pvalue) WeightedTauResult = namedtuple('WeightedTauResult', ('correlation', 'pvalue')) def weightedtau(x, y, rank=True, weigher=None, additive=True): r""" Compute a weighted version of Kendall's :math:`\tau`. The weighted :math:`\tau` is a weighted version of Kendall's :math:`\tau` in which exchanges of high weight are more influential than exchanges of low weight. The default parameters compute the additive hyperbolic version of the index, :math:`\tau_\mathrm h`, which has been shown to provide the best balance between important and unimportant elements [1]_. The weighting is defined by means of a rank array, which assigns a nonnegative rank to each element, and a weigher function, which assigns a weight based from the rank to each element. The weight of an exchange is then the sum or the product of the weights of the ranks of the exchanged elements. The default parameters compute :math:`\tau_\mathrm h`: an exchange between elements with rank :math:`r` and :math:`s` (starting from zero) has weight :math:`1/(r+1) + 1/(s+1)`. Specifying a rank array is meaningful only if you have in mind an external criterion of importance. If, as it usually happens, you do not have in mind a specific rank, the weighted :math:`\tau` is defined by averaging the values obtained using the decreasing lexicographical rank by (`x`, `y`) and by (`y`, `x`). This is the behavior with default parameters. Note that if you are computing the weighted :math:`\tau` on arrays of ranks, rather than of scores (i.e., a larger value implies a lower rank) you must negate the ranks, so that elements of higher rank are associated with a larger value. Parameters ---------- x, y : array_like Arrays of scores, of the same shape. If arrays are not 1-D, they will be flattened to 1-D. rank : array_like of ints or bool, optional A nonnegative rank assigned to each element. If it is None, the decreasing lexicographical rank by (`x`, `y`) will be used: elements of higher rank will be those with larger `x`-values, using `y`-values to break ties (in particular, swapping `x` and `y` will give a different result). If it is False, the element indices will be used directly as ranks. The default is True, in which case this function returns the average of the values obtained using the decreasing lexicographical rank by (`x`, `y`) and by (`y`, `x`). weigher : callable, optional The weigher function. Must map nonnegative integers (zero representing the most important element) to a nonnegative weight. The default, None, provides hyperbolic weighing, that is, rank :math:`r` is mapped to weight :math:`1/(r+1)`. additive : bool, optional If True, the weight of an exchange is computed by adding the weights of the ranks of the exchanged elements; otherwise, the weights are multiplied. The default is True. Returns ------- correlation : float The weighted :math:`\tau` correlation index. pvalue : float Presently ``np.nan``, as the null statistics is unknown (even in the additive hyperbolic case). See Also -------- kendalltau : Calculates Kendall's tau. spearmanr : Calculates a Spearman rank-order correlation coefficient. theilslopes : Computes the Theil-Sen estimator for a set of points (x, y). Notes ----- This function uses an :math:`O(n \log n)`, mergesort-based algorithm [1]_ that is a weighted extension of Knight's algorithm for Kendall's :math:`\tau` [2]_. It can compute Shieh's weighted :math:`\tau` [3]_ between rankings without ties (i.e., permutations) by setting `additive` and `rank` to False, as the definition given in [1]_ is a generalization of Shieh's. NaNs are considered the smallest possible score. .. versionadded:: 0.19.0 References ---------- .. [1] Sebastiano Vigna, "A weighted correlation index for rankings with ties", Proceedings of the 24th international conference on World Wide Web, pp. 1166-1176, ACM, 2015. .. [2] W.R. Knight, "A Computer Method for Calculating Kendall's Tau with Ungrouped Data", Journal of the American Statistical Association, Vol. 61, No. 314, Part 1, pp. 436-439, 1966. .. [3] Grace S. Shieh. "A weighted Kendall's tau statistic", Statistics & Probability Letters, Vol. 39, No. 1, pp. 17-24, 1998. Examples -------- >>> from scipy import stats >>> x = [12, 2, 1, 12, 2] >>> y = [1, 4, 7, 1, 0] >>> tau, p_value = stats.weightedtau(x, y) >>> tau -0.56694968153682723 >>> p_value nan >>> tau, p_value = stats.weightedtau(x, y, additive=False) >>> tau -0.62205716951801038 NaNs are considered the smallest possible score: >>> x = [12, 2, 1, 12, 2] >>> y = [1, 4, 7, 1, np.nan] >>> tau, _ = stats.weightedtau(x, y) >>> tau -0.56694968153682723 This is exactly Kendall's tau: >>> x = [12, 2, 1, 12, 2] >>> y = [1, 4, 7, 1, 0] >>> tau, _ = stats.weightedtau(x, y, weigher=lambda x: 1) >>> tau -0.47140452079103173 >>> x = [12, 2, 1, 12, 2] >>> y = [1, 4, 7, 1, 0] >>> stats.weightedtau(x, y, rank=None) WeightedTauResult(correlation=-0.4157652301037516, pvalue=nan) >>> stats.weightedtau(y, x, rank=None) WeightedTauResult(correlation=-0.7181341329699028, pvalue=nan) """ x = np.asarray(x).ravel() y = np.asarray(y).ravel() if x.size != y.size: raise ValueError("All inputs to `weightedtau` must be of the same size, " "found x-size %s and y-size %s" % (x.size, y.size)) if not x.size: return WeightedTauResult(np.nan, np.nan) # Return NaN if arrays are empty # If there are NaNs we apply _toint64() if np.isnan(np.sum(x)): x = _toint64(x) if np.isnan(np.sum(y)): y = _toint64(y) # Reduce to ranks unsupported types if x.dtype != y.dtype: if x.dtype != np.int64: x = _toint64(x) if y.dtype != np.int64: y = _toint64(y) else: if x.dtype not in (np.int32, np.int64, np.float32, np.float64): x = _toint64(x) y = _toint64(y) if rank is True: return WeightedTauResult(( _weightedrankedtau(x, y, None, weigher, additive) + _weightedrankedtau(y, x, None, weigher, additive) ) / 2, np.nan) if rank is False: rank = np.arange(x.size, dtype=np.intp) elif rank is not None: rank = np.asarray(rank).ravel() if rank.size != x.size: raise ValueError("All inputs to `weightedtau` must be of the same size, " "found x-size %s and rank-size %s" % (x.size, rank.size)) return WeightedTauResult(_weightedrankedtau(x, y, rank, weigher, additive), np.nan) # FROM MGCPY: https://github.com/neurodata/mgcpy class _ParallelP(object): """ Helper function to calculate parallel p-value. """ def __init__(self, x, y, random_states): self.x = x self.y = y self.random_states = random_states def __call__(self, index): order = self.random_states[index].permutation(self.y.shape[0]) permy = self.y[order][:, order] # calculate permuted stats, store in null distribution perm_stat = _mgc_stat(self.x, permy)[0] return perm_stat def _perm_test(x, y, stat, reps=1000, workers=-1, random_state=None): r""" Helper function that calculates the p-value. See below for uses. Parameters ---------- x, y : ndarray `x` and `y` have shapes `(n, p)` and `(n, q)`. stat : float The sample test statistic. reps : int, optional The number of replications used to estimate the null when using the permutation test. The default is 1000 replications. workers : int or map-like callable, optional If `workers` is an int the population is subdivided into `workers` sections and evaluated in parallel (uses `multiprocessing.Pool `). Supply `-1` to use all cores available to the Process. Alternatively supply a map-like callable, such as `multiprocessing.Pool.map` for evaluating the population in parallel. This evaluation is carried out as `workers(func, iterable)`. Requires that `func` be pickleable. random_state : int or np.random.RandomState instance, optional If already a RandomState instance, use it. If seed is an int, return a new RandomState instance seeded with seed. If None, use np.random.RandomState. Default is None. Returns ------- pvalue : float The sample test p-value. null_dist : list The approximated null distribution. """ # generate seeds for each rep (change to new parallel random number # capabilities in numpy >= 1.17+) random_state = check_random_state(random_state) random_states = [np.random.RandomState(rng_integers(random_state, 1 << 32, size=4, dtype=np.uint32)) for _ in range(reps)] # parallelizes with specified workers over number of reps and set seeds parallelp = _ParallelP(x=x, y=y, random_states=random_states) with MapWrapper(workers) as mapwrapper: null_dist = np.array(list(mapwrapper(parallelp, range(reps)))) # calculate p-value and significant permutation map through list pvalue = (null_dist >= stat).sum() / reps # correct for a p-value of 0. This is because, with bootstrapping # permutations, a p-value of 0 is incorrect if pvalue == 0: pvalue = 1 / reps return pvalue, null_dist def _euclidean_dist(x): return cdist(x, x) MGCResult = namedtuple('MGCResult', ('stat', 'pvalue', 'mgc_dict')) def multiscale_graphcorr(x, y, compute_distance=_euclidean_dist, reps=1000, workers=1, is_twosamp=False, random_state=None): r""" Computes the Multiscale Graph Correlation (MGC) test statistic. Specifically, for each point, MGC finds the :math:`k`-nearest neighbors for one property (e.g. cloud density), and the :math:`l`-nearest neighbors for the other property (e.g. grass wetness) [1]_. This pair :math:`(k, l)` is called the "scale". A priori, however, it is not know which scales will be most informative. So, MGC computes all distance pairs, and then efficiently computes the distance correlations for all scales. The local correlations illustrate which scales are relatively informative about the relationship. The key, therefore, to successfully discover and decipher relationships between disparate data modalities is to adaptively determine which scales are the most informative, and the geometric implication for the most informative scales. Doing so not only provides an estimate of whether the modalities are related, but also provides insight into how the determination was made. This is especially important in high-dimensional data, where simple visualizations do not reveal relationships to the unaided human eye. Characterizations of this implementation in particular have been derived from and benchmarked within in [2]_. Parameters ---------- x, y : ndarray If ``x`` and ``y`` have shapes ``(n, p)`` and ``(n, q)`` where `n` is the number of samples and `p` and `q` are the number of dimensions, then the MGC independence test will be run. Alternatively, ``x`` and ``y`` can have shapes ``(n, n)`` if they are distance or similarity matrices, and ``compute_distance`` must be sent to ``None``. If ``x`` and ``y`` have shapes ``(n, p)`` and ``(m, p)``, an unpaired two-sample MGC test will be run. compute_distance : callable, optional A function that computes the distance or similarity among the samples within each data matrix. Set to ``None`` if ``x`` and ``y`` are already distance matrices. The default uses the euclidean norm metric. If you are calling a custom function, either create the distance matrix before-hand or create a function of the form ``compute_distance(x)`` where `x` is the data matrix for which pairwise distances are calculated. reps : int, optional The number of replications used to estimate the null when using the permutation test. The default is ``1000``. workers : int or map-like callable, optional If ``workers`` is an int the population is subdivided into ``workers`` sections and evaluated in parallel (uses ``multiprocessing.Pool ``). Supply ``-1`` to use all cores available to the Process. Alternatively supply a map-like callable, such as ``multiprocessing.Pool.map`` for evaluating the p-value in parallel. This evaluation is carried out as ``workers(func, iterable)``. Requires that `func` be pickleable. The default is ``1``. is_twosamp : bool, optional If `True`, a two sample test will be run. If ``x`` and ``y`` have shapes ``(n, p)`` and ``(m, p)``, this optional will be overriden and set to ``True``. Set to ``True`` if ``x`` and ``y`` both have shapes ``(n, p)`` and a two sample test is desired. The default is ``False``. Note that this will not run if inputs are distance matrices. random_state : int or np.random.RandomState instance, optional If already a RandomState instance, use it. If seed is an int, return a new RandomState instance seeded with seed. If None, use np.random.RandomState. Default is None. Returns ------- stat : float The sample MGC test statistic within `[-1, 1]`. pvalue : float The p-value obtained via permutation. mgc_dict : dict Contains additional useful additional returns containing the following keys: - mgc_map : ndarray A 2D representation of the latent geometry of the relationship. of the relationship. - opt_scale : (int, int) The estimated optimal scale as a `(x, y)` pair. - null_dist : list The null distribution derived from the permuted matrices See Also -------- pearsonr : Pearson correlation coefficient and p-value for testing non-correlation. kendalltau : Calculates Kendall's tau. spearmanr : Calculates a Spearman rank-order correlation coefficient. Notes ----- A description of the process of MGC and applications on neuroscience data can be found in [1]_. It is performed using the following steps: #. Two distance matrices :math:`D^X` and :math:`D^Y` are computed and modified to be mean zero columnwise. This results in two :math:`n \times n` distance matrices :math:`A` and :math:`B` (the centering and unbiased modification) [3]_. #. For all values :math:`k` and :math:`l` from :math:`1, ..., n`, * The :math:`k`-nearest neighbor and :math:`l`-nearest neighbor graphs are calculated for each property. Here, :math:`G_k (i, j)` indicates the :math:`k`-smallest values of the :math:`i`-th row of :math:`A` and :math:`H_l (i, j)` indicates the :math:`l` smallested values of the :math:`i`-th row of :math:`B` * Let :math:`\circ` denotes the entry-wise matrix product, then local correlations are summed and normalized using the following statistic: .. math:: c^{kl} = \frac{\sum_{ij} A G_k B H_l} {\sqrt{\sum_{ij} A^2 G_k \times \sum_{ij} B^2 H_l}} #. The MGC test statistic is the smoothed optimal local correlation of :math:`\{ c^{kl} \}`. Denote the smoothing operation as :math:`R(\cdot)` (which essentially set all isolated large correlations) as 0 and connected large correlations the same as before, see [3]_.) MGC is, .. math:: MGC_n (x, y) = \max_{(k, l)} R \left(c^{kl} \left( x_n, y_n \right) \right) The test statistic returns a value between :math:`(-1, 1)` since it is normalized. The p-value returned is calculated using a permutation test. This process is completed by first randomly permuting :math:`y` to estimate the null distribution and then calculating the probability of observing a test statistic, under the null, at least as extreme as the observed test statistic. MGC requires at least 5 samples to run with reliable results. It can also handle high-dimensional data sets. In addition, by manipulating the input data matrices, the two-sample testing problem can be reduced to the independence testing problem [4]_. Given sample data :math:`U` and :math:`V` of sizes :math:`p \times n` :math:`p \times m`, data matrix :math:`X` and :math:`Y` can be created as follows: .. math:: X = [U | V] \in \mathcal{R}^{p \times (n + m)} Y = [0_{1 \times n} | 1_{1 \times m}] \in \mathcal{R}^{(n + m)} Then, the MGC statistic can be calculated as normal. This methodology can be extended to similar tests such as distance correlation [4]_. .. versionadded:: 1.4.0 References ---------- .. [1] Vogelstein, J. T., Bridgeford, E. W., Wang, Q., Priebe, C. E., Maggioni, M., & Shen, C. (2019). Discovering and deciphering relationships across disparate data modalities. ELife. .. [2] Panda, S., Palaniappan, S., Xiong, J., Swaminathan, A., Ramachandran, S., Bridgeford, E. W., ... Vogelstein, J. T. (2019). mgcpy: A Comprehensive High Dimensional Independence Testing Python Package. :arXiv:`1907.02088` .. [3] Shen, C., Priebe, C.E., & Vogelstein, J. T. (2019). From distance correlation to multiscale graph correlation. Journal of the American Statistical Association. .. [4] Shen, C. & Vogelstein, J. T. (2018). The Exact Equivalence of Distance and Kernel Methods for Hypothesis Testing. :arXiv:`1806.05514` Examples -------- >>> from scipy.stats import multiscale_graphcorr >>> x = np.arange(100) >>> y = x >>> stat, pvalue, _ = multiscale_graphcorr(x, y, workers=-1) >>> '%.1f, %.3f' % (stat, pvalue) '1.0, 0.001' Alternatively, >>> x = np.arange(100) >>> y = x >>> mgc = multiscale_graphcorr(x, y) >>> '%.1f, %.3f' % (mgc.stat, mgc.pvalue) '1.0, 0.001' To run an unpaired two-sample test, >>> x = np.arange(100) >>> y = np.arange(79) >>> mgc = multiscale_graphcorr(x, y, random_state=1) >>> '%.3f, %.2f' % (mgc.stat, mgc.pvalue) '0.033, 0.02' or, if shape of the inputs are the same, >>> x = np.arange(100) >>> y = x >>> mgc = multiscale_graphcorr(x, y, is_twosamp=True) >>> '%.3f, %.1f' % (mgc.stat, mgc.pvalue) '-0.008, 1.0' """ if not isinstance(x, np.ndarray) or not isinstance(y, np.ndarray): raise ValueError("x and y must be ndarrays") # convert arrays of type (n,) to (n, 1) if x.ndim == 1: x = x[:, np.newaxis] elif x.ndim != 2: raise ValueError("Expected a 2-D array `x`, found shape " "{}".format(x.shape)) if y.ndim == 1: y = y[:, np.newaxis] elif y.ndim != 2: raise ValueError("Expected a 2-D array `y`, found shape " "{}".format(y.shape)) nx, px = x.shape ny, py = y.shape # check for NaNs _contains_nan(x, nan_policy='raise') _contains_nan(y, nan_policy='raise') # check for positive or negative infinity and raise error if np.sum(np.isinf(x)) > 0 or np.sum(np.isinf(y)) > 0: raise ValueError("Inputs contain infinities") if nx != ny: if px == py: # reshape x and y for two sample testing is_twosamp = True else: raise ValueError("Shape mismatch, x and y must have shape [n, p] " "and [n, q] or have shape [n, p] and [m, p].") if nx < 5 or ny < 5: raise ValueError("MGC requires at least 5 samples to give reasonable " "results.") # convert x and y to float x = x.astype(np.float64) y = y.astype(np.float64) # check if compute_distance_matrix if a callable() if not callable(compute_distance) and compute_distance is not None: raise ValueError("Compute_distance must be a function.") # check if number of reps exists, integer, or > 0 (if under 1000 raises # warning) if not isinstance(reps, int) or reps < 0: raise ValueError("Number of reps must be an integer greater than 0.") elif reps < 1000: msg = ("The number of replications is low (under 1000), and p-value " "calculations may be unreliable. Use the p-value result, with " "caution!") warnings.warn(msg, RuntimeWarning) if is_twosamp: if compute_distance is None: raise ValueError("Cannot run if inputs are distance matrices") x, y = _two_sample_transform(x, y) if compute_distance is not None: # compute distance matrices for x and y x = compute_distance(x) y = compute_distance(y) # calculate MGC stat stat, stat_dict = _mgc_stat(x, y) stat_mgc_map = stat_dict["stat_mgc_map"] opt_scale = stat_dict["opt_scale"] # calculate permutation MGC p-value pvalue, null_dist = _perm_test(x, y, stat, reps=reps, workers=workers, random_state=random_state) # save all stats (other than stat/p-value) in dictionary mgc_dict = {"mgc_map": stat_mgc_map, "opt_scale": opt_scale, "null_dist": null_dist} return MGCResult(stat, pvalue, mgc_dict) def _mgc_stat(distx, disty): r""" Helper function that calculates the MGC stat. See above for use. Parameters ---------- x, y : ndarray `x` and `y` have shapes `(n, p)` and `(n, q)` or `(n, n)` and `(n, n)` if distance matrices. Returns ------- stat : float The sample MGC test statistic within `[-1, 1]`. stat_dict : dict Contains additional useful additional returns containing the following keys: - stat_mgc_map : ndarray MGC-map of the statistics. - opt_scale : (float, float) The estimated optimal scale as a `(x, y)` pair. """ # calculate MGC map and optimal scale stat_mgc_map = _local_correlations(distx, disty, global_corr='mgc') n, m = stat_mgc_map.shape if m == 1 or n == 1: # the global scale at is the statistic calculated at maximial nearest # neighbors. There is not enough local scale to search over, so # default to global scale stat = stat_mgc_map[m - 1][n - 1] opt_scale = m * n else: samp_size = len(distx) - 1 # threshold to find connected region of significant local correlations sig_connect = _threshold_mgc_map(stat_mgc_map, samp_size) # maximum within the significant region stat, opt_scale = _smooth_mgc_map(sig_connect, stat_mgc_map) stat_dict = {"stat_mgc_map": stat_mgc_map, "opt_scale": opt_scale} return stat, stat_dict def _threshold_mgc_map(stat_mgc_map, samp_size): r""" Finds a connected region of significance in the MGC-map by thresholding. Parameters ---------- stat_mgc_map : ndarray All local correlations within `[-1,1]`. samp_size : int The sample size of original data. Returns ------- sig_connect : ndarray A binary matrix with 1's indicating the significant region. """ m, n = stat_mgc_map.shape # 0.02 is simply an empirical threshold, this can be set to 0.01 or 0.05 # with varying levels of performance. Threshold is based on a beta # approximation. per_sig = 1 - (0.02 / samp_size) # Percentile to consider as significant threshold = samp_size * (samp_size - 3)/4 - 1/2 # Beta approximation threshold = distributions.beta.ppf(per_sig, threshold, threshold) * 2 - 1 # the global scale at is the statistic calculated at maximial nearest # neighbors. Threshold is the maximium on the global and local scales threshold = max(threshold, stat_mgc_map[m - 1][n - 1]) # find the largest connected component of significant correlations sig_connect = stat_mgc_map > threshold if np.sum(sig_connect) > 0: sig_connect, _ = measurements.label(sig_connect) _, label_counts = np.unique(sig_connect, return_counts=True) # skip the first element in label_counts, as it is count(zeros) max_label = np.argmax(label_counts[1:]) + 1 sig_connect = sig_connect == max_label else: sig_connect = np.array([[False]]) return sig_connect def _smooth_mgc_map(sig_connect, stat_mgc_map): """ Finds the smoothed maximal within the significant region R. If area of R is too small it returns the last local correlation. Otherwise, returns the maximum within significant_connected_region. Parameters ---------- sig_connect: ndarray A binary matrix with 1's indicating the significant region. stat_mgc_map: ndarray All local correlations within `[-1, 1]`. Returns ------- stat : float The sample MGC statistic within `[-1, 1]`. opt_scale: (float, float) The estimated optimal scale as an `(x, y)` pair. """ m, n = stat_mgc_map.shape # the global scale at is the statistic calculated at maximial nearest # neighbors. By default, statistic and optimal scale are global. stat = stat_mgc_map[m - 1][n - 1] opt_scale = [m, n] if np.linalg.norm(sig_connect) != 0: # proceed only when the connected region's area is sufficiently large # 0.02 is simply an empirical threshold, this can be set to 0.01 or 0.05 # with varying levels of performance if np.sum(sig_connect) >= np.ceil(0.02 * max(m, n)) * min(m, n): max_corr = max(stat_mgc_map[sig_connect]) # find all scales within significant_connected_region that maximize # the local correlation max_corr_index = np.where((stat_mgc_map >= max_corr) & sig_connect) if max_corr >= stat: stat = max_corr k, l = max_corr_index one_d_indices = k * n + l # 2D to 1D indexing k = np.max(one_d_indices) // n l = np.max(one_d_indices) % n opt_scale = [k+1, l+1] # adding 1s to match R indexing return stat, opt_scale def _two_sample_transform(u, v): """ Helper function that concatenates x and y for two sample MGC stat. See above for use. Parameters ---------- u, v : ndarray `u` and `v` have shapes `(n, p)` and `(m, p)`. Returns ------- x : ndarray Concatenate `u` and `v` along the `axis = 0`. `x` thus has shape `(2n, p)`. y : ndarray Label matrix for `x` where 0 refers to samples that comes from `u` and 1 refers to samples that come from `v`. `y` thus has shape `(2n, 1)`. """ nx = u.shape[0] ny = v.shape[0] x = np.concatenate([u, v], axis=0) y = np.concatenate([np.zeros(nx), np.ones(ny)], axis=0).reshape(-1, 1) return x, y ##################################### # INFERENTIAL STATISTICS # ##################################### Ttest_1sampResult = namedtuple('Ttest_1sampResult', ('statistic', 'pvalue')) def ttest_1samp(a, popmean, axis=0, nan_policy='propagate', alternative="two-sided"): """ Calculate the T-test for the mean of ONE group of scores. This is a two-sided test for the null hypothesis that the expected value (mean) of a sample of independent observations `a` is equal to the given population mean, `popmean`. Parameters ---------- a : array_like Sample observation. popmean : float or array_like Expected value in null hypothesis. If array_like, then it must have the same shape as `a` excluding the axis dimension. axis : int or None, optional Axis along which to compute test; default is 0. If None, compute over the whole array `a`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided * 'greater': one-sided .. versionadded:: 1.6.0 Returns ------- statistic : float or array t-statistic. pvalue : float or array Two-sided p-value. Examples -------- >>> from scipy import stats >>> np.random.seed(7654567) # fix seed to get the same result >>> rvs = stats.norm.rvs(loc=5, scale=10, size=(50,2)) Test if mean of random sample is equal to true mean, and different mean. We reject the null hypothesis in the second case and don't reject it in the first case. >>> stats.ttest_1samp(rvs,5.0) (array([-0.68014479, -0.04323899]), array([ 0.49961383, 0.96568674])) >>> stats.ttest_1samp(rvs,0.0) (array([ 2.77025808, 4.11038784]), array([ 0.00789095, 0.00014999])) Examples using axis and non-scalar dimension for population mean. >>> result = stats.ttest_1samp(rvs, [5.0, 0.0]) >>> result.statistic array([-0.68014479, 4.11038784]), >>> result.pvalue array([4.99613833e-01, 1.49986458e-04]) >>> result = stats.ttest_1samp(rvs.T, [5.0, 0.0], axis=1) >>> result.statistic array([-0.68014479, 4.11038784]) >>> result.pvalue array([4.99613833e-01, 1.49986458e-04]) >>> result = stats.ttest_1samp(rvs, [[5.0], [0.0]]) >>> result.statistic array([[-0.68014479, -0.04323899], [ 2.77025808, 4.11038784]]) >>> result.pvalue array([[4.99613833e-01, 9.65686743e-01], [7.89094663e-03, 1.49986458e-04]]) """ a, axis = _chk_asarray(a, axis) contains_nan, nan_policy = _contains_nan(a, nan_policy) if contains_nan and nan_policy == 'omit': if alternative != 'two-sided': raise ValueError("nan-containing/masked inputs with " "nan_policy='omit' are currently not " "supported by one-sided alternatives.") a = ma.masked_invalid(a) return mstats_basic.ttest_1samp(a, popmean, axis) n = a.shape[axis] df = n - 1 d = np.mean(a, axis) - popmean v = np.var(a, axis, ddof=1) denom = np.sqrt(v / n) with np.errstate(divide='ignore', invalid='ignore'): t = np.divide(d, denom) t, prob = _ttest_finish(df, t, alternative) return Ttest_1sampResult(t, prob) def _ttest_finish(df, t, alternative): """Common code between all 3 t-test functions.""" if alternative == 'less': prob = distributions.t.cdf(t, df) elif alternative == 'greater': prob = distributions.t.sf(t, df) elif alternative == 'two-sided': prob = 2 * distributions.t.sf(np.abs(t), df) else: raise ValueError("alternative must be " "'less', 'greater' or 'two-sided'") if t.ndim == 0: t = t[()] return t, prob def _ttest_ind_from_stats(mean1, mean2, denom, df, alternative): d = mean1 - mean2 with np.errstate(divide='ignore', invalid='ignore'): t = np.divide(d, denom) t, prob = _ttest_finish(df, t, alternative) return (t, prob) def _unequal_var_ttest_denom(v1, n1, v2, n2): vn1 = v1 / n1 vn2 = v2 / n2 with np.errstate(divide='ignore', invalid='ignore'): df = (vn1 + vn2)**2 / (vn1**2 / (n1 - 1) + vn2**2 / (n2 - 1)) # If df is undefined, variances are zero (assumes n1 > 0 & n2 > 0). # Hence it doesn't matter what df is as long as it's not NaN. df = np.where(np.isnan(df), 1, df) denom = np.sqrt(vn1 + vn2) return df, denom def _equal_var_ttest_denom(v1, n1, v2, n2): df = n1 + n2 - 2.0 svar = ((n1 - 1) * v1 + (n2 - 1) * v2) / df denom = np.sqrt(svar * (1.0 / n1 + 1.0 / n2)) return df, denom Ttest_indResult = namedtuple('Ttest_indResult', ('statistic', 'pvalue')) def ttest_ind_from_stats(mean1, std1, nobs1, mean2, std2, nobs2, equal_var=True, alternative="two-sided"): r""" T-test for means of two independent samples from descriptive statistics. This is a two-sided test for the null hypothesis that two independent samples have identical average (expected) values. Parameters ---------- mean1 : array_like The mean(s) of sample 1. std1 : array_like The standard deviation(s) of sample 1. nobs1 : array_like The number(s) of observations of sample 1. mean2 : array_like The mean(s) of sample 2. std2 : array_like The standard deviations(s) of sample 2. nobs2 : array_like The number(s) of observations of sample 2. equal_var : bool, optional If True (default), perform a standard independent 2 sample test that assumes equal population variances [1]_. If False, perform Welch's t-test, which does not assume equal population variance [2]_. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided * 'greater': one-sided .. versionadded:: 1.6.0 Returns ------- statistic : float or array The calculated t-statistics. pvalue : float or array The two-tailed p-value. See Also -------- scipy.stats.ttest_ind Notes ----- .. versionadded:: 0.16.0 References ---------- .. [1] https://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test .. [2] https://en.wikipedia.org/wiki/Welch%27s_t-test Examples -------- Suppose we have the summary data for two samples, as follows:: Sample Sample Size Mean Variance Sample 1 13 15.0 87.5 Sample 2 11 12.0 39.0 Apply the t-test to this data (with the assumption that the population variances are equal): >>> from scipy.stats import ttest_ind_from_stats >>> ttest_ind_from_stats(mean1=15.0, std1=np.sqrt(87.5), nobs1=13, ... mean2=12.0, std2=np.sqrt(39.0), nobs2=11) Ttest_indResult(statistic=0.9051358093310269, pvalue=0.3751996797581487) For comparison, here is the data from which those summary statistics were taken. With this data, we can compute the same result using `scipy.stats.ttest_ind`: >>> a = np.array([1, 3, 4, 6, 11, 13, 15, 19, 22, 24, 25, 26, 26]) >>> b = np.array([2, 4, 6, 9, 11, 13, 14, 15, 18, 19, 21]) >>> from scipy.stats import ttest_ind >>> ttest_ind(a, b) Ttest_indResult(statistic=0.905135809331027, pvalue=0.3751996797581486) Suppose we instead have binary data and would like to apply a t-test to compare the proportion of 1s in two independent groups:: Number of Sample Sample Size ones Mean Variance Sample 1 150 30 0.2 0.16 Sample 2 200 45 0.225 0.174375 The sample mean :math:`\hat{p}` is the proportion of ones in the sample and the variance for a binary observation is estimated by :math:`\hat{p}(1-\hat{p})`. >>> ttest_ind_from_stats(mean1=0.2, std1=np.sqrt(0.16), nobs1=150, ... mean2=0.225, std2=np.sqrt(0.17437), nobs2=200) Ttest_indResult(statistic=-0.564327545549774, pvalue=0.5728947691244874) For comparison, we could compute the t statistic and p-value using arrays of 0s and 1s and `scipy.stat.ttest_ind`, as above. >>> group1 = np.array([1]*30 + [0]*(150-30)) >>> group2 = np.array([1]*45 + [0]*(200-45)) >>> ttest_ind(group1, group2) Ttest_indResult(statistic=-0.5627179589855622, pvalue=0.573989277115258) """ if equal_var: df, denom = _equal_var_ttest_denom(std1**2, nobs1, std2**2, nobs2) else: df, denom = _unequal_var_ttest_denom(std1**2, nobs1, std2**2, nobs2) res = _ttest_ind_from_stats(mean1, mean2, denom, df, alternative) return Ttest_indResult(*res) def _ttest_nans(a, b, axis, namedtuple_type): """ Generate an array of `nan`, with shape determined by `a`, `b` and `axis`. This function is used by ttest_ind and ttest_rel to create the return value when one of the inputs has size 0. The shapes of the arrays are determined by dropping `axis` from the shapes of `a` and `b` and broadcasting what is left. The return value is a named tuple of the type given in `namedtuple_type`. Examples -------- >>> a = np.zeros((9, 2)) >>> b = np.zeros((5, 1)) >>> _ttest_nans(a, b, 0, Ttest_indResult) Ttest_indResult(statistic=array([nan, nan]), pvalue=array([nan, nan])) >>> a = np.zeros((3, 0, 9)) >>> b = np.zeros((1, 10)) >>> stat, p = _ttest_nans(a, b, -1, Ttest_indResult) >>> stat array([], shape=(3, 0), dtype=float64) >>> p array([], shape=(3, 0), dtype=float64) >>> a = np.zeros(10) >>> b = np.zeros(7) >>> _ttest_nans(a, b, 0, Ttest_indResult) Ttest_indResult(statistic=nan, pvalue=nan) """ shp = _broadcast_shapes_with_dropped_axis(a, b, axis) if len(shp) == 0: t = np.nan p = np.nan else: t = np.full(shp, fill_value=np.nan) p = t.copy() return namedtuple_type(t, p) def ttest_ind(a, b, axis=0, equal_var=True, nan_policy='propagate', alternative="two-sided"): """ Calculate the T-test for the means of *two independent* samples of scores. This is a two-sided test for the null hypothesis that 2 independent samples have identical average (expected) values. This test assumes that the populations have identical variances by default. Parameters ---------- a, b : array_like The arrays must have the same shape, except in the dimension corresponding to `axis` (the first, by default). axis : int or None, optional Axis along which to compute test. If None, compute over the whole arrays, `a`, and `b`. equal_var : bool, optional If True (default), perform a standard independent 2 sample test that assumes equal population variances [1]_. If False, perform Welch's t-test, which does not assume equal population variance [2]_. .. versionadded:: 0.11.0 nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided * 'greater': one-sided .. versionadded:: 1.6.0 Returns ------- statistic : float or array The calculated t-statistic. pvalue : float or array The two-tailed p-value. Notes ----- We can use this test, if we observe two independent samples from the same or different population, e.g. exam scores of boys and girls or of two ethnic groups. The test measures whether the average (expected) value differs significantly across samples. If we observe a large p-value, for example larger than 0.05 or 0.1, then we cannot reject the null hypothesis of identical average scores. If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%, then we reject the null hypothesis of equal averages. References ---------- .. [1] https://en.wikipedia.org/wiki/T-test#Independent_two-sample_t-test .. [2] https://en.wikipedia.org/wiki/Welch%27s_t-test Examples -------- >>> from scipy import stats >>> np.random.seed(12345678) Test with sample with identical means: >>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500) >>> rvs2 = stats.norm.rvs(loc=5,scale=10,size=500) >>> stats.ttest_ind(rvs1,rvs2) (0.26833823296239279, 0.78849443369564776) >>> stats.ttest_ind(rvs1,rvs2, equal_var = False) (0.26833823296239279, 0.78849452749500748) `ttest_ind` underestimates p for unequal variances: >>> rvs3 = stats.norm.rvs(loc=5, scale=20, size=500) >>> stats.ttest_ind(rvs1, rvs3) (-0.46580283298287162, 0.64145827413436174) >>> stats.ttest_ind(rvs1, rvs3, equal_var = False) (-0.46580283298287162, 0.64149646246569292) When n1 != n2, the equal variance t-statistic is no longer equal to the unequal variance t-statistic: >>> rvs4 = stats.norm.rvs(loc=5, scale=20, size=100) >>> stats.ttest_ind(rvs1, rvs4) (-0.99882539442782481, 0.3182832709103896) >>> stats.ttest_ind(rvs1, rvs4, equal_var = False) (-0.69712570584654099, 0.48716927725402048) T-test with different means, variance, and n: >>> rvs5 = stats.norm.rvs(loc=8, scale=20, size=100) >>> stats.ttest_ind(rvs1, rvs5) (-1.4679669854490653, 0.14263895620529152) >>> stats.ttest_ind(rvs1, rvs5, equal_var = False) (-0.94365973617132992, 0.34744170334794122) """ a, b, axis = _chk2_asarray(a, b, axis) # check both a and b cna, npa = _contains_nan(a, nan_policy) cnb, npb = _contains_nan(b, nan_policy) contains_nan = cna or cnb if npa == 'omit' or npb == 'omit': nan_policy = 'omit' if contains_nan and nan_policy == 'omit': if alternative != 'two-sided': raise ValueError("nan-containing/masked inputs with " "nan_policy='omit' are currently not " "supported by one-sided alternatives.") a = ma.masked_invalid(a) b = ma.masked_invalid(b) return mstats_basic.ttest_ind(a, b, axis, equal_var) if a.size == 0 or b.size == 0: return _ttest_nans(a, b, axis, Ttest_indResult) v1 = np.var(a, axis, ddof=1) v2 = np.var(b, axis, ddof=1) n1 = a.shape[axis] n2 = b.shape[axis] if equal_var: df, denom = _equal_var_ttest_denom(v1, n1, v2, n2) else: df, denom = _unequal_var_ttest_denom(v1, n1, v2, n2) res = _ttest_ind_from_stats(np.mean(a, axis), np.mean(b, axis), denom, df, alternative) return Ttest_indResult(*res) def _get_len(a, axis, msg): try: n = a.shape[axis] except IndexError: raise np.AxisError(axis, a.ndim, msg) from None return n Ttest_relResult = namedtuple('Ttest_relResult', ('statistic', 'pvalue')) def ttest_rel(a, b, axis=0, nan_policy='propagate', alternative="two-sided"): """ Calculate the t-test on TWO RELATED samples of scores, a and b. This is a two-sided test for the null hypothesis that 2 related or repeated samples have identical average (expected) values. Parameters ---------- a, b : array_like The arrays must have the same shape. axis : int or None, optional Axis along which to compute test. If None, compute over the whole arrays, `a`, and `b`. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided * 'greater': one-sided .. versionadded:: 1.6.0 Returns ------- statistic : float or array t-statistic. pvalue : float or array Two-sided p-value. Notes ----- Examples for use are scores of the same set of student in different exams, or repeated sampling from the same units. The test measures whether the average score differs significantly across samples (e.g. exams). If we observe a large p-value, for example greater than 0.05 or 0.1 then we cannot reject the null hypothesis of identical average scores. If the p-value is smaller than the threshold, e.g. 1%, 5% or 10%, then we reject the null hypothesis of equal averages. Small p-values are associated with large t-statistics. References ---------- https://en.wikipedia.org/wiki/T-test#Dependent_t-test_for_paired_samples Examples -------- >>> from scipy import stats >>> np.random.seed(12345678) # fix random seed to get same numbers >>> rvs1 = stats.norm.rvs(loc=5,scale=10,size=500) >>> rvs2 = (stats.norm.rvs(loc=5,scale=10,size=500) + ... stats.norm.rvs(scale=0.2,size=500)) >>> stats.ttest_rel(rvs1,rvs2) (0.24101764965300962, 0.80964043445811562) >>> rvs3 = (stats.norm.rvs(loc=8,scale=10,size=500) + ... stats.norm.rvs(scale=0.2,size=500)) >>> stats.ttest_rel(rvs1,rvs3) (-3.9995108708727933, 7.3082402191726459e-005) """ a, b, axis = _chk2_asarray(a, b, axis) cna, npa = _contains_nan(a, nan_policy) cnb, npb = _contains_nan(b, nan_policy) contains_nan = cna or cnb if npa == 'omit' or npb == 'omit': nan_policy = 'omit' if contains_nan and nan_policy == 'omit': if alternative != 'two-sided': raise ValueError("nan-containing/masked inputs with " "nan_policy='omit' are currently not " "supported by one-sided alternatives.") a = ma.masked_invalid(a) b = ma.masked_invalid(b) m = ma.mask_or(ma.getmask(a), ma.getmask(b)) aa = ma.array(a, mask=m, copy=True) bb = ma.array(b, mask=m, copy=True) return mstats_basic.ttest_rel(aa, bb, axis) na = _get_len(a, axis, "first argument") nb = _get_len(b, axis, "second argument") if na != nb: raise ValueError('unequal length arrays') if na == 0: return _ttest_nans(a, b, axis, Ttest_relResult) n = a.shape[axis] df = n - 1 d = (a - b).astype(np.float64) v = np.var(d, axis, ddof=1) dm = np.mean(d, axis) denom = np.sqrt(v / n) with np.errstate(divide='ignore', invalid='ignore'): t = np.divide(dm, denom) t, prob = _ttest_finish(df, t, alternative) return Ttest_relResult(t, prob) # Map from names to lambda_ values used in power_divergence(). _power_div_lambda_names = { "pearson": 1, "log-likelihood": 0, "freeman-tukey": -0.5, "mod-log-likelihood": -1, "neyman": -2, "cressie-read": 2/3, } def _count(a, axis=None): """ Count the number of non-masked elements of an array. This function behaves like np.ma.count(), but is much faster for ndarrays. """ if hasattr(a, 'count'): num = a.count(axis=axis) if isinstance(num, np.ndarray) and num.ndim == 0: # In some cases, the `count` method returns a scalar array (e.g. # np.array(3)), but we want a plain integer. num = int(num) else: if axis is None: num = a.size else: num = a.shape[axis] return num Power_divergenceResult = namedtuple('Power_divergenceResult', ('statistic', 'pvalue')) def power_divergence(f_obs, f_exp=None, ddof=0, axis=0, lambda_=None): """ Cressie-Read power divergence statistic and goodness of fit test. This function tests the null hypothesis that the categorical data has the given frequencies, using the Cressie-Read power divergence statistic. Parameters ---------- f_obs : array_like Observed frequencies in each category. f_exp : array_like, optional Expected frequencies in each category. By default the categories are assumed to be equally likely. ddof : int, optional "Delta degrees of freedom": adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with ``k - 1 - ddof`` degrees of freedom, where `k` is the number of observed frequencies. The default value of `ddof` is 0. axis : int or None, optional The axis of the broadcast result of `f_obs` and `f_exp` along which to apply the test. If axis is None, all values in `f_obs` are treated as a single data set. Default is 0. lambda_ : float or str, optional The power in the Cressie-Read power divergence statistic. The default is 1. For convenience, `lambda_` may be assigned one of the following strings, in which case the corresponding numerical value is used:: String Value Description "pearson" 1 Pearson's chi-squared statistic. In this case, the function is equivalent to `stats.chisquare`. "log-likelihood" 0 Log-likelihood ratio. Also known as the G-test [3]_. "freeman-tukey" -1/2 Freeman-Tukey statistic. "mod-log-likelihood" -1 Modified log-likelihood ratio. "neyman" -2 Neyman's statistic. "cressie-read" 2/3 The power recommended in [5]_. Returns ------- statistic : float or ndarray The Cressie-Read power divergence test statistic. The value is a float if `axis` is None or if` `f_obs` and `f_exp` are 1-D. pvalue : float or ndarray The p-value of the test. The value is a float if `ddof` and the return value `stat` are scalars. See Also -------- chisquare Notes ----- This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5. When `lambda_` is less than zero, the formula for the statistic involves dividing by `f_obs`, so a warning or error may be generated if any value in `f_obs` is 0. Similarly, a warning or error may be generated if any value in `f_exp` is zero when `lambda_` >= 0. The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not a chisquare, in which case this test is not appropriate. This function handles masked arrays. If an element of `f_obs` or `f_exp` is masked, then data at that position is ignored, and does not count towards the size of the data set. .. versionadded:: 0.13.0 References ---------- .. [1] Lowry, Richard. "Concepts and Applications of Inferential Statistics". Chapter 8. https://web.archive.org/web/20171015035606/http://faculty.vassar.edu/lowry/ch8pt1.html .. [2] "Chi-squared test", https://en.wikipedia.org/wiki/Chi-squared_test .. [3] "G-test", https://en.wikipedia.org/wiki/G-test .. [4] Sokal, R. R. and Rohlf, F. J. "Biometry: the principles and practice of statistics in biological research", New York: Freeman (1981) .. [5] Cressie, N. and Read, T. R. C., "Multinomial Goodness-of-Fit Tests", J. Royal Stat. Soc. Series B, Vol. 46, No. 3 (1984), pp. 440-464. Examples -------- (See `chisquare` for more examples.) When just `f_obs` is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies. Here we perform a G-test (i.e. use the log-likelihood ratio statistic): >>> from scipy.stats import power_divergence >>> power_divergence([16, 18, 16, 14, 12, 12], lambda_='log-likelihood') (2.006573162632538, 0.84823476779463769) The expected frequencies can be given with the `f_exp` argument: >>> power_divergence([16, 18, 16, 14, 12, 12], ... f_exp=[16, 16, 16, 16, 16, 8], ... lambda_='log-likelihood') (3.3281031458963746, 0.6495419288047497) When `f_obs` is 2-D, by default the test is applied to each column. >>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T >>> obs.shape (6, 2) >>> power_divergence(obs, lambda_="log-likelihood") (array([ 2.00657316, 6.77634498]), array([ 0.84823477, 0.23781225])) By setting ``axis=None``, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array. >>> power_divergence(obs, axis=None) (23.31034482758621, 0.015975692534127565) >>> power_divergence(obs.ravel()) (23.31034482758621, 0.015975692534127565) `ddof` is the change to make to the default degrees of freedom. >>> power_divergence([16, 18, 16, 14, 12, 12], ddof=1) (2.0, 0.73575888234288467) The calculation of the p-values is done by broadcasting the test statistic with `ddof`. >>> power_divergence([16, 18, 16, 14, 12, 12], ddof=[0,1,2]) (2.0, array([ 0.84914504, 0.73575888, 0.5724067 ])) `f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting `f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared statistics, we must use ``axis=1``: >>> power_divergence([16, 18, 16, 14, 12, 12], ... f_exp=[[16, 16, 16, 16, 16, 8], ... [8, 20, 20, 16, 12, 12]], ... axis=1) (array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846])) """ # Convert the input argument `lambda_` to a numerical value. if isinstance(lambda_, str): if lambda_ not in _power_div_lambda_names: names = repr(list(_power_div_lambda_names.keys()))[1:-1] raise ValueError("invalid string for lambda_: {0!r}. Valid strings " "are {1}".format(lambda_, names)) lambda_ = _power_div_lambda_names[lambda_] elif lambda_ is None: lambda_ = 1 f_obs = np.asanyarray(f_obs) if f_exp is not None: f_exp = np.asanyarray(f_exp) else: # Ignore 'invalid' errors so the edge case of a data set with length 0 # is handled without spurious warnings. with np.errstate(invalid='ignore'): f_exp = f_obs.mean(axis=axis, keepdims=True) # `terms` is the array of terms that are summed along `axis` to create # the test statistic. We use some specialized code for a few special # cases of lambda_. if lambda_ == 1: # Pearson's chi-squared statistic terms = (f_obs.astype(np.float64) - f_exp)**2 / f_exp elif lambda_ == 0: # Log-likelihood ratio (i.e. G-test) terms = 2.0 * special.xlogy(f_obs, f_obs / f_exp) elif lambda_ == -1: # Modified log-likelihood ratio terms = 2.0 * special.xlogy(f_exp, f_exp / f_obs) else: # General Cressie-Read power divergence. terms = f_obs * ((f_obs / f_exp)**lambda_ - 1) terms /= 0.5 * lambda_ * (lambda_ + 1) stat = terms.sum(axis=axis) num_obs = _count(terms, axis=axis) ddof = asarray(ddof) p = distributions.chi2.sf(stat, num_obs - 1 - ddof) return Power_divergenceResult(stat, p) def chisquare(f_obs, f_exp=None, ddof=0, axis=0): """ Calculate a one-way chi-square test. The chi-square test tests the null hypothesis that the categorical data has the given frequencies. Parameters ---------- f_obs : array_like Observed frequencies in each category. f_exp : array_like, optional Expected frequencies in each category. By default the categories are assumed to be equally likely. ddof : int, optional "Delta degrees of freedom": adjustment to the degrees of freedom for the p-value. The p-value is computed using a chi-squared distribution with ``k - 1 - ddof`` degrees of freedom, where `k` is the number of observed frequencies. The default value of `ddof` is 0. axis : int or None, optional The axis of the broadcast result of `f_obs` and `f_exp` along which to apply the test. If axis is None, all values in `f_obs` are treated as a single data set. Default is 0. Returns ------- chisq : float or ndarray The chi-squared test statistic. The value is a float if `axis` is None or `f_obs` and `f_exp` are 1-D. p : float or ndarray The p-value of the test. The value is a float if `ddof` and the return value `chisq` are scalars. See Also -------- scipy.stats.power_divergence Notes ----- This test is invalid when the observed or expected frequencies in each category are too small. A typical rule is that all of the observed and expected frequencies should be at least 5. The default degrees of freedom, k-1, are for the case when no parameters of the distribution are estimated. If p parameters are estimated by efficient maximum likelihood then the correct degrees of freedom are k-1-p. If the parameters are estimated in a different way, then the dof can be between k-1-p and k-1. However, it is also possible that the asymptotic distribution is not chi-square, in which case this test is not appropriate. References ---------- .. [1] Lowry, Richard. "Concepts and Applications of Inferential Statistics". Chapter 8. https://web.archive.org/web/20171022032306/http://vassarstats.net:80/textbook/ch8pt1.html .. [2] "Chi-squared test", https://en.wikipedia.org/wiki/Chi-squared_test Examples -------- When just `f_obs` is given, it is assumed that the expected frequencies are uniform and given by the mean of the observed frequencies. >>> from scipy.stats import chisquare >>> chisquare([16, 18, 16, 14, 12, 12]) (2.0, 0.84914503608460956) With `f_exp` the expected frequencies can be given. >>> chisquare([16, 18, 16, 14, 12, 12], f_exp=[16, 16, 16, 16, 16, 8]) (3.5, 0.62338762774958223) When `f_obs` is 2-D, by default the test is applied to each column. >>> obs = np.array([[16, 18, 16, 14, 12, 12], [32, 24, 16, 28, 20, 24]]).T >>> obs.shape (6, 2) >>> chisquare(obs) (array([ 2. , 6.66666667]), array([ 0.84914504, 0.24663415])) By setting ``axis=None``, the test is applied to all data in the array, which is equivalent to applying the test to the flattened array. >>> chisquare(obs, axis=None) (23.31034482758621, 0.015975692534127565) >>> chisquare(obs.ravel()) (23.31034482758621, 0.015975692534127565) `ddof` is the change to make to the default degrees of freedom. >>> chisquare([16, 18, 16, 14, 12, 12], ddof=1) (2.0, 0.73575888234288467) The calculation of the p-values is done by broadcasting the chi-squared statistic with `ddof`. >>> chisquare([16, 18, 16, 14, 12, 12], ddof=[0,1,2]) (2.0, array([ 0.84914504, 0.73575888, 0.5724067 ])) `f_obs` and `f_exp` are also broadcast. In the following, `f_obs` has shape (6,) and `f_exp` has shape (2, 6), so the result of broadcasting `f_obs` and `f_exp` has shape (2, 6). To compute the desired chi-squared statistics, we use ``axis=1``: >>> chisquare([16, 18, 16, 14, 12, 12], ... f_exp=[[16, 16, 16, 16, 16, 8], [8, 20, 20, 16, 12, 12]], ... axis=1) (array([ 3.5 , 9.25]), array([ 0.62338763, 0.09949846])) """ return power_divergence(f_obs, f_exp=f_exp, ddof=ddof, axis=axis, lambda_="pearson") KstestResult = namedtuple('KstestResult', ('statistic', 'pvalue')) def _compute_dplus(cdfvals): """Computes D+ as used in the Kolmogorov-Smirnov test. Parameters ---------- cdfvals: array_like Sorted array of CDF values between 0 and 1 Returns ------- Maximum distance of the CDF values below Uniform(0, 1) """ n = len(cdfvals) return (np.arange(1.0, n + 1) / n - cdfvals).max() def _compute_dminus(cdfvals): """Computes D- as used in the Kolmogorov-Smirnov test. Parameters ---------- cdfvals: array_like Sorted array of CDF values between 0 and 1 Returns ------- Maximum distance of the CDF values above Uniform(0, 1) """ n = len(cdfvals) return (cdfvals - np.arange(0.0, n)/n).max() def ks_1samp(x, cdf, args=(), alternative='two-sided', mode='auto'): """ Performs the Kolmogorov-Smirnov test for goodness of fit. This performs a test of the distribution F(x) of an observed random variable against a given distribution G(x). Under the null hypothesis, the two distributions are identical, F(x)=G(x). The alternative hypothesis can be either 'two-sided' (default), 'less' or 'greater'. The KS test is only valid for continuous distributions. Parameters ---------- x : array_like a 1-D array of observations of iid random variables. cdf : callable callable used to calculate the cdf. args : tuple, sequence, optional Distribution parameters, used with `cdf`. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided, see explanation in Notes * 'greater': one-sided, see explanation in Notes mode : {'auto', 'exact', 'approx', 'asymp'}, optional Defines the distribution used for calculating the p-value. The following options are available (default is 'auto'): * 'auto' : selects one of the other options. * 'exact' : uses the exact distribution of test statistic. * 'approx' : approximates the two-sided probability with twice the one-sided probability * 'asymp': uses asymptotic distribution of test statistic Returns ------- statistic : float KS test statistic, either D, D+ or D- (depending on the value of 'alternative') pvalue : float One-tailed or two-tailed p-value. See Also -------- ks_2samp, kstest Notes ----- In the one-sided test, the alternative is that the empirical cumulative distribution function of the random variable is "less" or "greater" than the cumulative distribution function G(x) of the hypothesis, ``F(x)<=G(x)``, resp. ``F(x)>=G(x)``. Examples -------- >>> from scipy import stats >>> x = np.linspace(-15, 15, 9) >>> stats.ks_1samp(x, stats.norm.cdf) (0.44435602715924361, 0.038850142705171065) >>> np.random.seed(987654321) # set random seed to get the same result >>> stats.ks_1samp(stats.norm.rvs(size=100), stats.norm.cdf) (0.058352892479417884, 0.8653960860778898) *Test against one-sided alternative hypothesis* Shift distribution to larger values, so that `` CDF(x) < norm.cdf(x)``: >>> np.random.seed(987654321) >>> x = stats.norm.rvs(loc=0.2, size=100) >>> stats.ks_1samp(x, stats.norm.cdf, alternative='less') (0.12464329735846891, 0.040989164077641749) Reject equal distribution against alternative hypothesis: less >>> stats.ks_1samp(x, stats.norm.cdf, alternative='greater') (0.0072115233216311081, 0.98531158590396395) Don't reject equal distribution against alternative hypothesis: greater >>> stats.ks_1samp(x, stats.norm.cdf) (0.12464329735846891, 0.08197335233541582) Don't reject equal distribution against alternative hypothesis: two-sided *Testing t distributed random variables against normal distribution* With 100 degrees of freedom the t distribution looks close to the normal distribution, and the K-S test does not reject the hypothesis that the sample came from the normal distribution: >>> np.random.seed(987654321) >>> stats.ks_1samp(stats.t.rvs(100,size=100), stats.norm.cdf) (0.072018929165471257, 0.6505883498379312) With 3 degrees of freedom the t distribution looks sufficiently different from the normal distribution, that we can reject the hypothesis that the sample came from the normal distribution at the 10% level: >>> np.random.seed(987654321) >>> stats.ks_1samp(stats.t.rvs(3,size=100), stats.norm.cdf) (0.131016895759829, 0.058826222555312224) """ alternative = {'t': 'two-sided', 'g': 'greater', 'l': 'less'}.get( alternative.lower()[0], alternative) if alternative not in ['two-sided', 'greater', 'less']: raise ValueError("Unexpected alternative %s" % alternative) if np.ma.is_masked(x): x = x.compressed() N = len(x) x = np.sort(x) cdfvals = cdf(x, *args) if alternative == 'greater': Dplus = _compute_dplus(cdfvals) return KstestResult(Dplus, distributions.ksone.sf(Dplus, N)) if alternative == 'less': Dminus = _compute_dminus(cdfvals) return KstestResult(Dminus, distributions.ksone.sf(Dminus, N)) # alternative == 'two-sided': Dplus = _compute_dplus(cdfvals) Dminus = _compute_dminus(cdfvals) D = np.max([Dplus, Dminus]) if mode == 'auto': # Always select exact mode = 'exact' if mode == 'exact': prob = distributions.kstwo.sf(D, N) elif mode == 'asymp': prob = distributions.kstwobign.sf(D * np.sqrt(N)) else: # mode == 'approx' prob = 2 * distributions.ksone.sf(D, N) prob = np.clip(prob, 0, 1) return KstestResult(D, prob) Ks_2sampResult = KstestResult def _compute_prob_inside_method(m, n, g, h): """ Count the proportion of paths that stay strictly inside two diagonal lines. Parameters ---------- m : integer m > 0 n : integer n > 0 g : integer g is greatest common divisor of m and n h : integer 0 <= h <= lcm(m,n) Returns ------- p : float The proportion of paths that stay inside the two lines. Count the integer lattice paths from (0, 0) to (m, n) which satisfy |x/m - y/n| < h / lcm(m, n). The paths make steps of size +1 in either positive x or positive y directions. We generally follow Hodges' treatment of Drion/Gnedenko/Korolyuk. Hodges, J.L. Jr., "The Significance Probability of the Smirnov Two-Sample Test," Arkiv fiur Matematik, 3, No. 43 (1958), 469-86. """ # Probability is symmetrical in m, n. Computation below uses m >= n. if m < n: m, n = n, m mg = m // g ng = n // g # Count the integer lattice paths from (0, 0) to (m, n) which satisfy # |nx/g - my/g| < h. # Compute matrix A such that: # A(x, 0) = A(0, y) = 1 # A(x, y) = A(x, y-1) + A(x-1, y), for x,y>=1, except that # A(x, y) = 0 if |x/m - y/n|>= h # Probability is A(m, n)/binom(m+n, n) # Optimizations exist for m==n, m==n*p. # Only need to preserve a single column of A, and only a sliding window of it. # minj keeps track of the slide. minj, maxj = 0, min(int(np.ceil(h / mg)), n + 1) curlen = maxj - minj # Make a vector long enough to hold maximum window needed. lenA = min(2 * maxj + 2, n + 1) # This is an integer calculation, but the entries are essentially # binomial coefficients, hence grow quickly. # Scaling after each column is computed avoids dividing by a # large binomial coefficent at the end, but is not sufficient to avoid # the large dyanamic range which appears during the calculation. # Instead we rescale based on the magnitude of the right most term in # the column and keep track of an exponent separately and apply # it at the end of the calculation. Similarly when multiplying by # the binomial coefficint dtype = np.float64 A = np.zeros(lenA, dtype=dtype) # Initialize the first column A[minj:maxj] = 1 expnt = 0 for i in range(1, m + 1): # Generate the next column. # First calculate the sliding window lastminj, lastlen = minj, curlen minj = max(int(np.floor((ng * i - h) / mg)) + 1, 0) minj = min(minj, n) maxj = min(int(np.ceil((ng * i + h) / mg)), n + 1) if maxj <= minj: return 0 # Now fill in the values A[0:maxj - minj] = np.cumsum(A[minj - lastminj:maxj - lastminj]) curlen = maxj - minj if lastlen > curlen: # Set some carried-over elements to 0 A[maxj - minj:maxj - minj + (lastlen - curlen)] = 0 # Rescale if the right most value is over 2**900 val = A[maxj - minj - 1] _, valexpt = math.frexp(val) if valexpt > 900: # Scaling to bring down to about 2**800 appears # sufficient for sizes under 10000. valexpt -= 800 A = np.ldexp(A, -valexpt) expnt += valexpt val = A[maxj - minj - 1] # Now divide by the binomial (m+n)!/m!/n! for i in range(1, n + 1): val = (val * i) / (m + i) _, valexpt = math.frexp(val) if valexpt < -128: val = np.ldexp(val, -valexpt) expnt += valexpt # Finally scale if needed. return np.ldexp(val, expnt) def _compute_prob_outside_square(n, h): """ Compute the proportion of paths that pass outside the two diagonal lines. Parameters ---------- n : integer n > 0 h : integer 0 <= h <= n Returns ------- p : float The proportion of paths that pass outside the lines x-y = +/-h. """ # Compute Pr(D_{n,n} >= h/n) # Prob = 2 * ( binom(2n, n-h) - binom(2n, n-2a) + binom(2n, n-3a) - ... ) / binom(2n, n) # This formulation exhibits subtractive cancellation. # Instead divide each term by binom(2n, n), then factor common terms # and use a Horner-like algorithm # P = 2 * A0 * (1 - A1*(1 - A2*(1 - A3*(1 - A4*(...))))) P = 0.0 k = int(np.floor(n / h)) while k >= 0: p1 = 1.0 # Each of the Ai terms has numerator and denominator with h simple terms. for j in range(h): p1 = (n - k * h - j) * p1 / (n + k * h + j + 1) P = p1 * (1.0 - P) k -= 1 return 2 * P def _count_paths_outside_method(m, n, g, h): """ Count the number of paths that pass outside the specified diagonal. Parameters ---------- m : integer m > 0 n : integer n > 0 g : integer g is greatest common divisor of m and n h : integer 0 <= h <= lcm(m,n) Returns ------- p : float The number of paths that go low. The calculation may overflow - check for a finite answer. Raises ------ FloatingPointError: Raised if the intermediate computation goes outside the range of a float. Notes ----- Count the integer lattice paths from (0, 0) to (m, n), which at some point (x, y) along the path, satisfy: m*y <= n*x - h*g The paths make steps of size +1 in either positive x or positive y directions. We generally follow Hodges' treatment of Drion/Gnedenko/Korolyuk. Hodges, J.L. Jr., "The Significance Probability of the Smirnov Two-Sample Test," Arkiv fiur Matematik, 3, No. 43 (1958), 469-86. """ # Compute #paths which stay lower than x/m-y/n = h/lcm(m,n) # B(x, y) = #{paths from (0,0) to (x,y) without previously crossing the boundary} # = binom(x, y) - #{paths which already reached the boundary} # Multiply by the number of path extensions going from (x, y) to (m, n) # Sum. # Probability is symmetrical in m, n. Computation below assumes m >= n. if m < n: m, n = n, m mg = m // g ng = n // g # Not every x needs to be considered. # xj holds the list of x values to be checked. # Wherever n*x/m + ng*h crosses an integer lxj = n + (mg-h)//mg xj = [(h + mg * j + ng-1)//ng for j in range(lxj)] # B is an array just holding a few values of B(x,y), the ones needed. # B[j] == B(x_j, j) if lxj == 0: return np.round(special.binom(m + n, n)) B = np.zeros(lxj) B[0] = 1 # Compute the B(x, y) terms # The binomial coefficient is an integer, but special.binom() may return a float. # Round it to the nearest integer. for j in range(1, lxj): Bj = np.round(special.binom(xj[j] + j, j)) if not np.isfinite(Bj): raise FloatingPointError() for i in range(j): bin = np.round(special.binom(xj[j] - xj[i] + j - i, j-i)) Bj -= bin * B[i] B[j] = Bj if not np.isfinite(Bj): raise FloatingPointError() # Compute the number of path extensions... num_paths = 0 for j in range(lxj): bin = np.round(special.binom((m-xj[j]) + (n - j), n-j)) term = B[j] * bin if not np.isfinite(term): raise FloatingPointError() num_paths += term return np.round(num_paths) def _attempt_exact_2kssamp(n1, n2, g, d, alternative): """Attempts to compute the exact 2sample probability. n1, n2 are the sample sizes g is the gcd(n1, n2) d is the computed max difference in ECDFs Returns (success, d, probability) """ lcm = (n1 // g) * n2 h = int(np.round(d * lcm)) d = h * 1.0 / lcm if h == 0: return True, d, 1.0 saw_fp_error, prob = False, np.nan try: if alternative == 'two-sided': if n1 == n2: prob = _compute_prob_outside_square(n1, h) else: prob = 1 - _compute_prob_inside_method(n1, n2, g, h) else: if n1 == n2: # prob = binom(2n, n-h) / binom(2n, n) # Evaluating in that form incurs roundoff errors # from special.binom. Instead calculate directly jrange = np.arange(h) prob = np.prod((n1 - jrange) / (n1 + jrange + 1.0)) else: num_paths = _count_paths_outside_method(n1, n2, g, h) bin = special.binom(n1 + n2, n1) if not np.isfinite(bin) or not np.isfinite(num_paths) or num_paths > bin: saw_fp_error = True else: prob = num_paths / bin except FloatingPointError: saw_fp_error = True if saw_fp_error: return False, d, np.nan if not (0 <= prob <= 1): return False, d, prob return True, d, prob def ks_2samp(data1, data2, alternative='two-sided', mode='auto'): """ Compute the Kolmogorov-Smirnov statistic on 2 samples. This is a two-sided test for the null hypothesis that 2 independent samples are drawn from the same continuous distribution. The alternative hypothesis can be either 'two-sided' (default), 'less' or 'greater'. Parameters ---------- data1, data2 : array_like, 1-Dimensional Two arrays of sample observations assumed to be drawn from a continuous distribution, sample sizes can be different. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided, see explanation in Notes * 'greater': one-sided, see explanation in Notes mode : {'auto', 'exact', 'asymp'}, optional Defines the method used for calculating the p-value. The following options are available (default is 'auto'): * 'auto' : use 'exact' for small size arrays, 'asymp' for large * 'exact' : use exact distribution of test statistic * 'asymp' : use asymptotic distribution of test statistic Returns ------- statistic : float KS statistic. pvalue : float Two-tailed p-value. See Also -------- kstest, ks_1samp, epps_singleton_2samp, anderson_ksamp Notes ----- This tests whether 2 samples are drawn from the same distribution. Note that, like in the case of the one-sample KS test, the distribution is assumed to be continuous. In the one-sided test, the alternative is that the empirical cumulative distribution function F(x) of the data1 variable is "less" or "greater" than the empirical cumulative distribution function G(x) of the data2 variable, ``F(x)<=G(x)``, resp. ``F(x)>=G(x)``. If the KS statistic is small or the p-value is high, then we cannot reject the hypothesis that the distributions of the two samples are the same. If the mode is 'auto', the computation is exact if the sample sizes are less than 10000. For larger sizes, the computation uses the Kolmogorov-Smirnov distributions to compute an approximate value. The 'two-sided' 'exact' computation computes the complementary probability and then subtracts from 1. As such, the minimum probability it can return is about 1e-16. While the algorithm itself is exact, numerical errors may accumulate for large sample sizes. It is most suited to situations in which one of the sample sizes is only a few thousand. We generally follow Hodges' treatment of Drion/Gnedenko/Korolyuk [1]_. References ---------- .. [1] Hodges, J.L. Jr., "The Significance Probability of the Smirnov Two-Sample Test," Arkiv fiur Matematik, 3, No. 43 (1958), 469-86. Examples -------- >>> from scipy import stats >>> np.random.seed(12345678) #fix random seed to get the same result >>> n1 = 200 # size of first sample >>> n2 = 300 # size of second sample For a different distribution, we can reject the null hypothesis since the pvalue is below 1%: >>> rvs1 = stats.norm.rvs(size=n1, loc=0., scale=1) >>> rvs2 = stats.norm.rvs(size=n2, loc=0.5, scale=1.5) >>> stats.ks_2samp(rvs1, rvs2) (0.20833333333333334, 5.129279597781977e-05) For a slightly different distribution, we cannot reject the null hypothesis at a 10% or lower alpha since the p-value at 0.144 is higher than 10% >>> rvs3 = stats.norm.rvs(size=n2, loc=0.01, scale=1.0) >>> stats.ks_2samp(rvs1, rvs3) (0.10333333333333333, 0.14691437867433876) For an identical distribution, we cannot reject the null hypothesis since the p-value is high, 41%: >>> rvs4 = stats.norm.rvs(size=n2, loc=0.0, scale=1.0) >>> stats.ks_2samp(rvs1, rvs4) (0.07999999999999996, 0.41126949729859719) """ if mode not in ['auto', 'exact', 'asymp']: raise ValueError(f'Invalid value for mode: {mode}') alternative = {'t': 'two-sided', 'g': 'greater', 'l': 'less'}.get( alternative.lower()[0], alternative) if alternative not in ['two-sided', 'less', 'greater']: raise ValueError(f'Invalid value for alternative: {alternative}') MAX_AUTO_N = 10000 # 'auto' will attempt to be exact if n1,n2 <= MAX_AUTO_N if np.ma.is_masked(data1): data1 = data1.compressed() if np.ma.is_masked(data2): data2 = data2.compressed() data1 = np.sort(data1) data2 = np.sort(data2) n1 = data1.shape[0] n2 = data2.shape[0] if min(n1, n2) == 0: raise ValueError('Data passed to ks_2samp must not be empty') data_all = np.concatenate([data1, data2]) # using searchsorted solves equal data problem cdf1 = np.searchsorted(data1, data_all, side='right') / n1 cdf2 = np.searchsorted(data2, data_all, side='right') / n2 cddiffs = cdf1 - cdf2 minS = np.clip(-np.min(cddiffs), 0, 1) # Ensure sign of minS is not negative. maxS = np.max(cddiffs) alt2Dvalue = {'less': minS, 'greater': maxS, 'two-sided': max(minS, maxS)} d = alt2Dvalue[alternative] g = gcd(n1, n2) n1g = n1 // g n2g = n2 // g prob = -np.inf original_mode = mode if mode == 'auto': mode = 'exact' if max(n1, n2) <= MAX_AUTO_N else 'asymp' elif mode == 'exact': # If lcm(n1, n2) is too big, switch from exact to asymp if n1g >= np.iinfo(np.int_).max / n2g: mode = 'asymp' warnings.warn( f"Exact ks_2samp calculation not possible with samples sizes " f"{n1} and {n2}. Switching to 'asymp'.", RuntimeWarning) if mode == 'exact': success, d, prob = _attempt_exact_2kssamp(n1, n2, g, d, alternative) if not success: mode = 'asymp' if original_mode == 'exact': warnings.warn(f"ks_2samp: Exact calculation unsuccessful. " f"Switching to mode={mode}.", RuntimeWarning) if mode == 'asymp': # The product n1*n2 is large. Use Smirnov's asymptoptic formula. # Ensure float to avoid overflow in multiplication # sorted because the one-sided formula is not symmetric in n1, n2 m, n = sorted([float(n1), float(n2)], reverse=True) en = m * n / (m + n) if alternative == 'two-sided': prob = distributions.kstwo.sf(d, np.round(en)) else: z = np.sqrt(en) * d # Use Hodges' suggested approximation Eqn 5.3 # Requires m to be the larger of (n1, n2) expt = -2 * z**2 - 2 * z * (m + 2*n)/np.sqrt(m*n*(m+n))/3.0 prob = np.exp(expt) prob = np.clip(prob, 0, 1) return KstestResult(d, prob) def _parse_kstest_args(data1, data2, args, N): # kstest allows many different variations of arguments. # Pull out the parsing into a separate function # (xvals, yvals, ) # 2sample # (xvals, cdf function,..) # (xvals, name of distribution, ...) # (name of distribution, name of distribution, ...) # Returns xvals, yvals, cdf # where cdf is a cdf function, or None # and yvals is either an array_like of values, or None # and xvals is array_like. rvsfunc, cdf = None, None if isinstance(data1, str): rvsfunc = getattr(distributions, data1).rvs elif callable(data1): rvsfunc = data1 if isinstance(data2, str): cdf = getattr(distributions, data2).cdf data2 = None elif callable(data2): cdf = data2 data2 = None data1 = np.sort(rvsfunc(*args, size=N) if rvsfunc else data1) return data1, data2, cdf def kstest(rvs, cdf, args=(), N=20, alternative='two-sided', mode='auto'): """ Performs the (one sample or two samples) Kolmogorov-Smirnov test for goodness of fit. The one-sample test performs a test of the distribution F(x) of an observed random variable against a given distribution G(x). Under the null hypothesis, the two distributions are identical, F(x)=G(x). The alternative hypothesis can be either 'two-sided' (default), 'less' or 'greater'. The KS test is only valid for continuous distributions. The two-sample test tests whether the two independent samples are drawn from the same continuous distribution. Parameters ---------- rvs : str, array_like, or callable If an array, it should be a 1-D array of observations of random variables. If a callable, it should be a function to generate random variables; it is required to have a keyword argument `size`. If a string, it should be the name of a distribution in `scipy.stats`, which will be used to generate random variables. cdf : str, array_like or callable If array_like, it should be a 1-D array of observations of random variables, and the two-sample test is performed (and rvs must be array_like) If a callable, that callable is used to calculate the cdf. If a string, it should be the name of a distribution in `scipy.stats`, which will be used as the cdf function. args : tuple, sequence, optional Distribution parameters, used if `rvs` or `cdf` are strings or callables. N : int, optional Sample size if `rvs` is string or callable. Default is 20. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided, see explanation in Notes * 'greater': one-sided, see explanation in Notes mode : {'auto', 'exact', 'approx', 'asymp'}, optional Defines the distribution used for calculating the p-value. The following options are available (default is 'auto'): * 'auto' : selects one of the other options. * 'exact' : uses the exact distribution of test statistic. * 'approx' : approximates the two-sided probability with twice the one-sided probability * 'asymp': uses asymptotic distribution of test statistic Returns ------- statistic : float KS test statistic, either D, D+ or D-. pvalue : float One-tailed or two-tailed p-value. See Also -------- ks_2samp Notes ----- In the one-sided test, the alternative is that the empirical cumulative distribution function of the random variable is "less" or "greater" than the cumulative distribution function G(x) of the hypothesis, ``F(x)<=G(x)``, resp. ``F(x)>=G(x)``. Examples -------- >>> from scipy import stats >>> x = np.linspace(-15, 15, 9) >>> stats.kstest(x, 'norm') (0.44435602715924361, 0.038850142705171065) >>> np.random.seed(987654321) # set random seed to get the same result >>> stats.kstest(stats.norm.rvs(size=100), stats.norm.cdf) (0.058352892479417884, 0.8653960860778898) The above lines are equivalent to: >>> np.random.seed(987654321) >>> stats.kstest(stats.norm.rvs, 'norm', N=100) (0.058352892479417884, 0.8653960860778898) *Test against one-sided alternative hypothesis* Shift distribution to larger values, so that ``CDF(x) < norm.cdf(x)``: >>> np.random.seed(987654321) >>> x = stats.norm.rvs(loc=0.2, size=100) >>> stats.kstest(x, 'norm', alternative='less') (0.12464329735846891, 0.040989164077641749) Reject equal distribution against alternative hypothesis: less >>> stats.kstest(x, 'norm', alternative='greater') (0.0072115233216311081, 0.98531158590396395) Don't reject equal distribution against alternative hypothesis: greater >>> stats.kstest(x, 'norm') (0.12464329735846891, 0.08197335233541582) *Testing t distributed random variables against normal distribution* With 100 degrees of freedom the t distribution looks close to the normal distribution, and the K-S test does not reject the hypothesis that the sample came from the normal distribution: >>> np.random.seed(987654321) >>> stats.kstest(stats.t.rvs(100, size=100), 'norm') (0.072018929165471257, 0.6505883498379312) With 3 degrees of freedom the t distribution looks sufficiently different from the normal distribution, that we can reject the hypothesis that the sample came from the normal distribution at the 10% level: >>> np.random.seed(987654321) >>> stats.kstest(stats.t.rvs(3, size=100), 'norm') (0.131016895759829, 0.058826222555312224) """ # to not break compatibility with existing code if alternative == 'two_sided': alternative = 'two-sided' if alternative not in ['two-sided', 'greater', 'less']: raise ValueError("Unexpected alternative %s" % alternative) xvals, yvals, cdf = _parse_kstest_args(rvs, cdf, args, N) if cdf: return ks_1samp(xvals, cdf, args=args, alternative=alternative, mode=mode) return ks_2samp(xvals, yvals, alternative=alternative, mode=mode) def tiecorrect(rankvals): """ Tie correction factor for Mann-Whitney U and Kruskal-Wallis H tests. Parameters ---------- rankvals : array_like A 1-D sequence of ranks. Typically this will be the array returned by `~scipy.stats.rankdata`. Returns ------- factor : float Correction factor for U or H. See Also -------- rankdata : Assign ranks to the data mannwhitneyu : Mann-Whitney rank test kruskal : Kruskal-Wallis H test References ---------- .. [1] Siegel, S. (1956) Nonparametric Statistics for the Behavioral Sciences. New York: McGraw-Hill. Examples -------- >>> from scipy.stats import tiecorrect, rankdata >>> tiecorrect([1, 2.5, 2.5, 4]) 0.9 >>> ranks = rankdata([1, 3, 2, 4, 5, 7, 2, 8, 4]) >>> ranks array([ 1. , 4. , 2.5, 5.5, 7. , 8. , 2.5, 9. , 5.5]) >>> tiecorrect(ranks) 0.9833333333333333 """ arr = np.sort(rankvals) idx = np.nonzero(np.r_[True, arr[1:] != arr[:-1], True])[0] cnt = np.diff(idx).astype(np.float64) size = np.float64(arr.size) return 1.0 if size < 2 else 1.0 - (cnt**3 - cnt).sum() / (size**3 - size) MannwhitneyuResult = namedtuple('MannwhitneyuResult', ('statistic', 'pvalue')) def mannwhitneyu(x, y, use_continuity=True, alternative=None): """ Compute the Mann-Whitney rank test on samples x and y. Parameters ---------- x, y : array_like Array of samples, should be one-dimensional. use_continuity : bool, optional Whether a continuity correction (1/2.) should be taken into account. Default is True. alternative : {None, 'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is None): * None: computes p-value half the size of the 'two-sided' p-value and a different U statistic. The default behavior is not the same as using 'less' or 'greater'; it only exists for backward compatibility and is deprecated. * 'two-sided' * 'less': one-sided * 'greater': one-sided Use of the None option is deprecated. Returns ------- statistic : float The Mann-Whitney U statistic, equal to min(U for x, U for y) if `alternative` is equal to None (deprecated; exists for backward compatibility), and U for y otherwise. pvalue : float p-value assuming an asymptotic normal distribution. One-sided or two-sided, depending on the choice of `alternative`. Notes ----- Use only when the number of observation in each sample is > 20 and you have 2 independent samples of ranks. Mann-Whitney U is significant if the u-obtained is LESS THAN or equal to the critical value of U. This test corrects for ties and by default uses a continuity correction. References ---------- .. [1] https://en.wikipedia.org/wiki/Mann-Whitney_U_test .. [2] H.B. Mann and D.R. Whitney, "On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other," The Annals of Mathematical Statistics, vol. 18, no. 1, pp. 50-60, 1947. """ if alternative is None: warnings.warn("Calling `mannwhitneyu` without specifying " "`alternative` is deprecated.", DeprecationWarning) x = np.asarray(x) y = np.asarray(y) n1 = len(x) n2 = len(y) ranked = rankdata(np.concatenate((x, y))) rankx = ranked[0:n1] # get the x-ranks u1 = n1*n2 + (n1*(n1+1))/2.0 - np.sum(rankx, axis=0) # calc U for x u2 = n1*n2 - u1 # remainder is U for y T = tiecorrect(ranked) if T == 0: raise ValueError('All numbers are identical in mannwhitneyu') sd = np.sqrt(T * n1 * n2 * (n1+n2+1) / 12.0) meanrank = n1*n2/2.0 + 0.5 * use_continuity if alternative is None or alternative == 'two-sided': bigu = max(u1, u2) elif alternative == 'less': bigu = u1 elif alternative == 'greater': bigu = u2 else: raise ValueError("alternative should be None, 'less', 'greater' " "or 'two-sided'") z = (bigu - meanrank) / sd if alternative is None: # This behavior, equal to half the size of the two-sided # p-value, is deprecated. p = distributions.norm.sf(abs(z)) elif alternative == 'two-sided': p = 2 * distributions.norm.sf(abs(z)) else: p = distributions.norm.sf(z) u = u2 # This behavior is deprecated. if alternative is None: u = min(u1, u2) return MannwhitneyuResult(u, p) RanksumsResult = namedtuple('RanksumsResult', ('statistic', 'pvalue')) def ranksums(x, y): """ Compute the Wilcoxon rank-sum statistic for two samples. The Wilcoxon rank-sum test tests the null hypothesis that two sets of measurements are drawn from the same distribution. The alternative hypothesis is that values in one sample are more likely to be larger than the values in the other sample. This test should be used to compare two samples from continuous distributions. It does not handle ties between measurements in x and y. For tie-handling and an optional continuity correction see `scipy.stats.mannwhitneyu`. Parameters ---------- x,y : array_like The data from the two samples. Returns ------- statistic : float The test statistic under the large-sample approximation that the rank sum statistic is normally distributed. pvalue : float The two-sided p-value of the test. References ---------- .. [1] https://en.wikipedia.org/wiki/Wilcoxon_rank-sum_test Examples -------- We can test the hypothesis that two independent unequal-sized samples are drawn from the same distribution with computing the Wilcoxon rank-sum statistic. >>> from scipy.stats import ranksums >>> sample1 = np.random.uniform(-1, 1, 200) >>> sample2 = np.random.uniform(-0.5, 1.5, 300) # a shifted distribution >>> ranksums(sample1, sample2) RanksumsResult(statistic=-7.887059, pvalue=3.09390448e-15) # may vary The p-value of less than ``0.05`` indicates that this test rejects the hypothesis at the 5% significance level. """ x, y = map(np.asarray, (x, y)) n1 = len(x) n2 = len(y) alldata = np.concatenate((x, y)) ranked = rankdata(alldata) x = ranked[:n1] s = np.sum(x, axis=0) expected = n1 * (n1+n2+1) / 2.0 z = (s - expected) / np.sqrt(n1*n2*(n1+n2+1)/12.0) prob = 2 * distributions.norm.sf(abs(z)) return RanksumsResult(z, prob) KruskalResult = namedtuple('KruskalResult', ('statistic', 'pvalue')) def kruskal(*args, nan_policy='propagate'): """ Compute the Kruskal-Wallis H-test for independent samples. The Kruskal-Wallis H-test tests the null hypothesis that the population median of all of the groups are equal. It is a non-parametric version of ANOVA. The test works on 2 or more independent samples, which may have different sizes. Note that rejecting the null hypothesis does not indicate which of the groups differs. Post hoc comparisons between groups are required to determine which groups are different. Parameters ---------- sample1, sample2, ... : array_like Two or more arrays with the sample measurements can be given as arguments. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float The Kruskal-Wallis H statistic, corrected for ties. pvalue : float The p-value for the test using the assumption that H has a chi square distribution. The p-value returned is the survival function of the chi square distribution evaluated at H. See Also -------- f_oneway : 1-way ANOVA. mannwhitneyu : Mann-Whitney rank test on two samples. friedmanchisquare : Friedman test for repeated measurements. Notes ----- Due to the assumption that H has a chi square distribution, the number of samples in each group must not be too small. A typical rule is that each sample must have at least 5 measurements. References ---------- .. [1] W. H. Kruskal & W. W. Wallis, "Use of Ranks in One-Criterion Variance Analysis", Journal of the American Statistical Association, Vol. 47, Issue 260, pp. 583-621, 1952. .. [2] https://en.wikipedia.org/wiki/Kruskal-Wallis_one-way_analysis_of_variance Examples -------- >>> from scipy import stats >>> x = [1, 3, 5, 7, 9] >>> y = [2, 4, 6, 8, 10] >>> stats.kruskal(x, y) KruskalResult(statistic=0.2727272727272734, pvalue=0.6015081344405895) >>> x = [1, 1, 1] >>> y = [2, 2, 2] >>> z = [2, 2] >>> stats.kruskal(x, y, z) KruskalResult(statistic=7.0, pvalue=0.0301973834223185) """ args = list(map(np.asarray, args)) num_groups = len(args) if num_groups < 2: raise ValueError("Need at least two groups in stats.kruskal()") for arg in args: if arg.size == 0: return KruskalResult(np.nan, np.nan) n = np.asarray(list(map(len, args))) if nan_policy not in ('propagate', 'raise', 'omit'): raise ValueError("nan_policy must be 'propagate', 'raise' or 'omit'") contains_nan = False for arg in args: cn = _contains_nan(arg, nan_policy) if cn[0]: contains_nan = True break if contains_nan and nan_policy == 'omit': for a in args: a = ma.masked_invalid(a) return mstats_basic.kruskal(*args) if contains_nan and nan_policy == 'propagate': return KruskalResult(np.nan, np.nan) alldata = np.concatenate(args) ranked = rankdata(alldata) ties = tiecorrect(ranked) if ties == 0: raise ValueError('All numbers are identical in kruskal') # Compute sum^2/n for each group and sum j = np.insert(np.cumsum(n), 0, 0) ssbn = 0 for i in range(num_groups): ssbn += _square_of_sums(ranked[j[i]:j[i+1]]) / n[i] totaln = np.sum(n, dtype=float) h = 12.0 / (totaln * (totaln + 1)) * ssbn - 3 * (totaln + 1) df = num_groups - 1 h /= ties return KruskalResult(h, distributions.chi2.sf(h, df)) FriedmanchisquareResult = namedtuple('FriedmanchisquareResult', ('statistic', 'pvalue')) def friedmanchisquare(*args): """ Compute the Friedman test for repeated measurements. The Friedman test tests the null hypothesis that repeated measurements of the same individuals have the same distribution. It is often used to test for consistency among measurements obtained in different ways. For example, if two measurement techniques are used on the same set of individuals, the Friedman test can be used to determine if the two measurement techniques are consistent. Parameters ---------- measurements1, measurements2, measurements3... : array_like Arrays of measurements. All of the arrays must have the same number of elements. At least 3 sets of measurements must be given. Returns ------- statistic : float The test statistic, correcting for ties. pvalue : float The associated p-value assuming that the test statistic has a chi squared distribution. Notes ----- Due to the assumption that the test statistic has a chi squared distribution, the p-value is only reliable for n > 10 and more than 6 repeated measurements. References ---------- .. [1] https://en.wikipedia.org/wiki/Friedman_test """ k = len(args) if k < 3: raise ValueError('At least 3 sets of measurements must be given for Friedman test, got {}.'.format(k)) n = len(args[0]) for i in range(1, k): if len(args[i]) != n: raise ValueError('Unequal N in friedmanchisquare. Aborting.') # Rank data data = np.vstack(args).T data = data.astype(float) for i in range(len(data)): data[i] = rankdata(data[i]) # Handle ties ties = 0 for i in range(len(data)): replist, repnum = find_repeats(array(data[i])) for t in repnum: ties += t * (t*t - 1) c = 1 - ties / (k*(k*k - 1)*n) ssbn = np.sum(data.sum(axis=0)**2) chisq = (12.0 / (k*n*(k+1)) * ssbn - 3*n*(k+1)) / c return FriedmanchisquareResult(chisq, distributions.chi2.sf(chisq, k - 1)) BrunnerMunzelResult = namedtuple('BrunnerMunzelResult', ('statistic', 'pvalue')) def brunnermunzel(x, y, alternative="two-sided", distribution="t", nan_policy='propagate'): """ Compute the Brunner-Munzel test on samples x and y. The Brunner-Munzel test is a nonparametric test of the null hypothesis that when values are taken one by one from each group, the probabilities of getting large values in both groups are equal. Unlike the Wilcoxon-Mann-Whitney's U test, this does not require the assumption of equivariance of two groups. Note that this does not assume the distributions are same. This test works on two independent samples, which may have different sizes. Parameters ---------- x, y : array_like Array of samples, should be one-dimensional. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. The following options are available (default is 'two-sided'): * 'two-sided' * 'less': one-sided * 'greater': one-sided distribution : {'t', 'normal'}, optional Defines how to get the p-value. The following options are available (default is 't'): * 't': get the p-value by t-distribution * 'normal': get the p-value by standard normal distribution. nan_policy : {'propagate', 'raise', 'omit'}, optional Defines how to handle when input contains nan. The following options are available (default is 'propagate'): * 'propagate': returns nan * 'raise': throws an error * 'omit': performs the calculations ignoring nan values Returns ------- statistic : float The Brunner-Munzer W statistic. pvalue : float p-value assuming an t distribution. One-sided or two-sided, depending on the choice of `alternative` and `distribution`. See Also -------- mannwhitneyu : Mann-Whitney rank test on two samples. Notes ----- Brunner and Munzel recommended to estimate the p-value by t-distribution when the size of data is 50 or less. If the size is lower than 10, it would be better to use permuted Brunner Munzel test (see [2]_). References ---------- .. [1] Brunner, E. and Munzel, U. "The nonparametric Benhrens-Fisher problem: Asymptotic theory and a small-sample approximation". Biometrical Journal. Vol. 42(2000): 17-25. .. [2] Neubert, K. and Brunner, E. "A studentized permutation test for the non-parametric Behrens-Fisher problem". Computational Statistics and Data Analysis. Vol. 51(2007): 5192-5204. Examples -------- >>> from scipy import stats >>> x1 = [1,2,1,1,1,1,1,1,1,1,2,4,1,1] >>> x2 = [3,3,4,3,1,2,3,1,1,5,4] >>> w, p_value = stats.brunnermunzel(x1, x2) >>> w 3.1374674823029505 >>> p_value 0.0057862086661515377 """ x = np.asarray(x) y = np.asarray(y) # check both x and y cnx, npx = _contains_nan(x, nan_policy) cny, npy = _contains_nan(y, nan_policy) contains_nan = cnx or cny if npx == "omit" or npy == "omit": nan_policy = "omit" if contains_nan and nan_policy == "propagate": return BrunnerMunzelResult(np.nan, np.nan) elif contains_nan and nan_policy == "omit": x = ma.masked_invalid(x) y = ma.masked_invalid(y) return mstats_basic.brunnermunzel(x, y, alternative, distribution) nx = len(x) ny = len(y) if nx == 0 or ny == 0: return BrunnerMunzelResult(np.nan, np.nan) rankc = rankdata(np.concatenate((x, y))) rankcx = rankc[0:nx] rankcy = rankc[nx:nx+ny] rankcx_mean = np.mean(rankcx) rankcy_mean = np.mean(rankcy) rankx = rankdata(x) ranky = rankdata(y) rankx_mean = np.mean(rankx) ranky_mean = np.mean(ranky) Sx = np.sum(np.power(rankcx - rankx - rankcx_mean + rankx_mean, 2.0)) Sx /= nx - 1 Sy = np.sum(np.power(rankcy - ranky - rankcy_mean + ranky_mean, 2.0)) Sy /= ny - 1 wbfn = nx * ny * (rankcy_mean - rankcx_mean) wbfn /= (nx + ny) * np.sqrt(nx * Sx + ny * Sy) if distribution == "t": df_numer = np.power(nx * Sx + ny * Sy, 2.0) df_denom = np.power(nx * Sx, 2.0) / (nx - 1) df_denom += np.power(ny * Sy, 2.0) / (ny - 1) df = df_numer / df_denom p = distributions.t.cdf(wbfn, df) elif distribution == "normal": p = distributions.norm.cdf(wbfn) else: raise ValueError( "distribution should be 't' or 'normal'") if alternative == "greater": pass elif alternative == "less": p = 1 - p elif alternative == "two-sided": p = 2 * np.min([p, 1-p]) else: raise ValueError( "alternative should be 'less', 'greater' or 'two-sided'") return BrunnerMunzelResult(wbfn, p) def combine_pvalues(pvalues, method='fisher', weights=None): """ Combine p-values from independent tests bearing upon the same hypothesis. Parameters ---------- pvalues : array_like, 1-D Array of p-values assumed to come from independent tests. method : {'fisher', 'pearson', 'tippett', 'stouffer', 'mudholkar_george'}, optional Name of method to use to combine p-values. The following methods are available (default is 'fisher'): * 'fisher': Fisher's method (Fisher's combined probability test), the sum of the logarithm of the p-values * 'pearson': Pearson's method (similar to Fisher's but uses sum of the complement of the p-values inside the logarithms) * 'tippett': Tippett's method (minimum of p-values) * 'stouffer': Stouffer's Z-score method * 'mudholkar_george': the difference of Fisher's and Pearson's methods divided by 2 weights : array_like, 1-D, optional Optional array of weights used only for Stouffer's Z-score method. Returns ------- statistic: float The statistic calculated by the specified method. pval: float The combined p-value. Notes ----- Fisher's method (also known as Fisher's combined probability test) [1]_ uses a chi-squared statistic to compute a combined p-value. The closely related Stouffer's Z-score method [2]_ uses Z-scores rather than p-values. The advantage of Stouffer's method is that it is straightforward to introduce weights, which can make Stouffer's method more powerful than Fisher's method when the p-values are from studies of different size [6]_ [7]_. The Pearson's method uses :math:`log(1-p_i)` inside the sum whereas Fisher's method uses :math:`log(p_i)` [4]_. For Fisher's and Pearson's method, the sum of the logarithms is multiplied by -2 in the implementation. This quantity has a chi-square distribution that determines the p-value. The `mudholkar_george` method is the difference of the Fisher's and Pearson's test statistics, each of which include the -2 factor [4]_. However, the `mudholkar_george` method does not include these -2 factors. The test statistic of `mudholkar_george` is the sum of logisitic random variables and equation 3.6 in [3]_ is used to approximate the p-value based on Student's t-distribution. Fisher's method may be extended to combine p-values from dependent tests [5]_. Extensions such as Brown's method and Kost's method are not currently implemented. .. versionadded:: 0.15.0 References ---------- .. [1] https://en.wikipedia.org/wiki/Fisher%27s_method .. [2] https://en.wikipedia.org/wiki/Fisher%27s_method#Relation_to_Stouffer.27s_Z-score_method .. [3] George, E. O., and G. S. Mudholkar. "On the convolution of logistic random variables." Metrika 30.1 (1983): 1-13. .. [4] Heard, N. and Rubin-Delanchey, P. "Choosing between methods of combining p-values." Biometrika 105.1 (2018): 239-246. .. [5] Whitlock, M. C. "Combining probability from independent tests: the weighted Z-method is superior to Fisher's approach." Journal of Evolutionary Biology 18, no. 5 (2005): 1368-1373. .. [6] Zaykin, Dmitri V. "Optimally weighted Z-test is a powerful method for combining probabilities in meta-analysis." Journal of Evolutionary Biology 24, no. 8 (2011): 1836-1841. .. [7] https://en.wikipedia.org/wiki/Extensions_of_Fisher%27s_method """ pvalues = np.asarray(pvalues) if pvalues.ndim != 1: raise ValueError("pvalues is not 1-D") if method == 'fisher': statistic = -2 * np.sum(np.log(pvalues)) pval = distributions.chi2.sf(statistic, 2 * len(pvalues)) elif method == 'pearson': statistic = -2 * np.sum(np.log1p(-pvalues)) pval = distributions.chi2.sf(statistic, 2 * len(pvalues)) elif method == 'mudholkar_george': normalizing_factor = np.sqrt(3/len(pvalues))/np.pi statistic = -np.sum(np.log(pvalues)) + np.sum(np.log1p(-pvalues)) nu = 5 * len(pvalues) + 4 approx_factor = np.sqrt(nu / (nu - 2)) pval = distributions.t.sf(statistic * normalizing_factor * approx_factor, nu) elif method == 'tippett': statistic = np.min(pvalues) pval = distributions.beta.sf(statistic, 1, len(pvalues)) elif method == 'stouffer': if weights is None: weights = np.ones_like(pvalues) elif len(weights) != len(pvalues): raise ValueError("pvalues and weights must be of the same size.") weights = np.asarray(weights) if weights.ndim != 1: raise ValueError("weights is not 1-D") Zi = distributions.norm.isf(pvalues) statistic = np.dot(weights, Zi) / np.linalg.norm(weights) pval = distributions.norm.sf(statistic) else: raise ValueError( "Invalid method '%s'. Options are 'fisher', 'pearson', \ 'mudholkar_george', 'tippett', 'or 'stouffer'", method) return (statistic, pval) ##################################### # STATISTICAL DISTANCES # ##################################### def wasserstein_distance(u_values, v_values, u_weights=None, v_weights=None): r""" Compute the first Wasserstein distance between two 1D distributions. This distance is also known as the earth mover's distance, since it can be seen as the minimum amount of "work" required to transform :math:`u` into :math:`v`, where "work" is measured as the amount of distribution weight that must be moved, multiplied by the distance it has to be moved. .. versionadded:: 1.0.0 Parameters ---------- u_values, v_values : array_like Values observed in the (empirical) distribution. u_weights, v_weights : array_like, optional Weight for each value. If unspecified, each value is assigned the same weight. `u_weights` (resp. `v_weights`) must have the same length as `u_values` (resp. `v_values`). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1. Returns ------- distance : float The computed distance between the distributions. Notes ----- The first Wasserstein distance between the distributions :math:`u` and :math:`v` is: .. math:: l_1 (u, v) = \inf_{\pi \in \Gamma (u, v)} \int_{\mathbb{R} \times \mathbb{R}} |x-y| \mathrm{d} \pi (x, y) where :math:`\Gamma (u, v)` is the set of (probability) distributions on :math:`\mathbb{R} \times \mathbb{R}` whose marginals are :math:`u` and :math:`v` on the first and second factors respectively. If :math:`U` and :math:`V` are the respective CDFs of :math:`u` and :math:`v`, this distance also equals to: .. math:: l_1(u, v) = \int_{-\infty}^{+\infty} |U-V| See [2]_ for a proof of the equivalence of both definitions. The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values. References ---------- .. [1] "Wasserstein metric", https://en.wikipedia.org/wiki/Wasserstein_metric .. [2] Ramdas, Garcia, Cuturi "On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests" (2015). :arXiv:`1509.02237`. Examples -------- >>> from scipy.stats import wasserstein_distance >>> wasserstein_distance([0, 1, 3], [5, 6, 8]) 5.0 >>> wasserstein_distance([0, 1], [0, 1], [3, 1], [2, 2]) 0.25 >>> wasserstein_distance([3.4, 3.9, 7.5, 7.8], [4.5, 1.4], ... [1.4, 0.9, 3.1, 7.2], [3.2, 3.5]) 4.0781331438047861 """ return _cdf_distance(1, u_values, v_values, u_weights, v_weights) def energy_distance(u_values, v_values, u_weights=None, v_weights=None): r""" Compute the energy distance between two 1D distributions. .. versionadded:: 1.0.0 Parameters ---------- u_values, v_values : array_like Values observed in the (empirical) distribution. u_weights, v_weights : array_like, optional Weight for each value. If unspecified, each value is assigned the same weight. `u_weights` (resp. `v_weights`) must have the same length as `u_values` (resp. `v_values`). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1. Returns ------- distance : float The computed distance between the distributions. Notes ----- The energy distance between two distributions :math:`u` and :math:`v`, whose respective CDFs are :math:`U` and :math:`V`, equals to: .. math:: D(u, v) = \left( 2\mathbb E|X - Y| - \mathbb E|X - X'| - \mathbb E|Y - Y'| \right)^{1/2} where :math:`X` and :math:`X'` (resp. :math:`Y` and :math:`Y'`) are independent random variables whose probability distribution is :math:`u` (resp. :math:`v`). As shown in [2]_, for one-dimensional real-valued variables, the energy distance is linked to the non-distribution-free version of the Cramér-von Mises distance: .. math:: D(u, v) = \sqrt{2} l_2(u, v) = \left( 2 \int_{-\infty}^{+\infty} (U-V)^2 \right)^{1/2} Note that the common Cramér-von Mises criterion uses the distribution-free version of the distance. See [2]_ (section 2), for more details about both versions of the distance. The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values. References ---------- .. [1] "Energy distance", https://en.wikipedia.org/wiki/Energy_distance .. [2] Szekely "E-statistics: The energy of statistical samples." Bowling Green State University, Department of Mathematics and Statistics, Technical Report 02-16 (2002). .. [3] Rizzo, Szekely "Energy distance." Wiley Interdisciplinary Reviews: Computational Statistics, 8(1):27-38 (2015). .. [4] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer, Munos "The Cramer Distance as a Solution to Biased Wasserstein Gradients" (2017). :arXiv:`1705.10743`. Examples -------- >>> from scipy.stats import energy_distance >>> energy_distance([0], [2]) 2.0000000000000004 >>> energy_distance([0, 8], [0, 8], [3, 1], [2, 2]) 1.0000000000000002 >>> energy_distance([0.7, 7.4, 2.4, 6.8], [1.4, 8. ], ... [2.1, 4.2, 7.4, 8. ], [7.6, 8.8]) 0.88003340976158217 """ return np.sqrt(2) * _cdf_distance(2, u_values, v_values, u_weights, v_weights) def _cdf_distance(p, u_values, v_values, u_weights=None, v_weights=None): r""" Compute, between two one-dimensional distributions :math:`u` and :math:`v`, whose respective CDFs are :math:`U` and :math:`V`, the statistical distance that is defined as: .. math:: l_p(u, v) = \left( \int_{-\infty}^{+\infty} |U-V|^p \right)^{1/p} p is a positive parameter; p = 1 gives the Wasserstein distance, p = 2 gives the energy distance. Parameters ---------- u_values, v_values : array_like Values observed in the (empirical) distribution. u_weights, v_weights : array_like, optional Weight for each value. If unspecified, each value is assigned the same weight. `u_weights` (resp. `v_weights`) must have the same length as `u_values` (resp. `v_values`). If the weight sum differs from 1, it must still be positive and finite so that the weights can be normalized to sum to 1. Returns ------- distance : float The computed distance between the distributions. Notes ----- The input distributions can be empirical, therefore coming from samples whose values are effectively inputs of the function, or they can be seen as generalized functions, in which case they are weighted sums of Dirac delta functions located at the specified values. References ---------- .. [1] Bellemare, Danihelka, Dabney, Mohamed, Lakshminarayanan, Hoyer, Munos "The Cramer Distance as a Solution to Biased Wasserstein Gradients" (2017). :arXiv:`1705.10743`. """ u_values, u_weights = _validate_distribution(u_values, u_weights) v_values, v_weights = _validate_distribution(v_values, v_weights) u_sorter = np.argsort(u_values) v_sorter = np.argsort(v_values) all_values = np.concatenate((u_values, v_values)) all_values.sort(kind='mergesort') # Compute the differences between pairs of successive values of u and v. deltas = np.diff(all_values) # Get the respective positions of the values of u and v among the values of # both distributions. u_cdf_indices = u_values[u_sorter].searchsorted(all_values[:-1], 'right') v_cdf_indices = v_values[v_sorter].searchsorted(all_values[:-1], 'right') # Calculate the CDFs of u and v using their weights, if specified. if u_weights is None: u_cdf = u_cdf_indices / u_values.size else: u_sorted_cumweights = np.concatenate(([0], np.cumsum(u_weights[u_sorter]))) u_cdf = u_sorted_cumweights[u_cdf_indices] / u_sorted_cumweights[-1] if v_weights is None: v_cdf = v_cdf_indices / v_values.size else: v_sorted_cumweights = np.concatenate(([0], np.cumsum(v_weights[v_sorter]))) v_cdf = v_sorted_cumweights[v_cdf_indices] / v_sorted_cumweights[-1] # Compute the value of the integral based on the CDFs. # If p = 1 or p = 2, we avoid using np.power, which introduces an overhead # of about 15%. if p == 1: return np.sum(np.multiply(np.abs(u_cdf - v_cdf), deltas)) if p == 2: return np.sqrt(np.sum(np.multiply(np.square(u_cdf - v_cdf), deltas))) return np.power(np.sum(np.multiply(np.power(np.abs(u_cdf - v_cdf), p), deltas)), 1/p) def _validate_distribution(values, weights): """ Validate the values and weights from a distribution input of `cdf_distance` and return them as ndarray objects. Parameters ---------- values : array_like Values observed in the (empirical) distribution. weights : array_like Weight for each value. Returns ------- values : ndarray Values as ndarray. weights : ndarray Weights as ndarray. """ # Validate the value array. values = np.asarray(values, dtype=float) if len(values) == 0: raise ValueError("Distribution can't be empty.") # Validate the weight array, if specified. if weights is not None: weights = np.asarray(weights, dtype=float) if len(weights) != len(values): raise ValueError('Value and weight array-likes for the same ' 'empirical distribution must be of the same size.') if np.any(weights < 0): raise ValueError('All weights must be non-negative.') if not 0 < np.sum(weights) < np.inf: raise ValueError('Weight array-like sum must be positive and ' 'finite. Set as None for an equal distribution of ' 'weight.') return values, weights return values, None ##################################### # SUPPORT FUNCTIONS # ##################################### RepeatedResults = namedtuple('RepeatedResults', ('values', 'counts')) def find_repeats(arr): """ Find repeats and repeat counts. Parameters ---------- arr : array_like Input array. This is cast to float64. Returns ------- values : ndarray The unique values from the (flattened) input that are repeated. counts : ndarray Number of times the corresponding 'value' is repeated. Notes ----- In numpy >= 1.9 `numpy.unique` provides similar functionality. The main difference is that `find_repeats` only returns repeated values. Examples -------- >>> from scipy import stats >>> stats.find_repeats([2, 1, 2, 3, 2, 2, 5]) RepeatedResults(values=array([2.]), counts=array([4])) >>> stats.find_repeats([[10, 20, 1, 2], [5, 5, 4, 4]]) RepeatedResults(values=array([4., 5.]), counts=array([2, 2])) """ # Note: always copies. return RepeatedResults(*_find_repeats(np.array(arr, dtype=np.float64))) def _sum_of_squares(a, axis=0): """ Square each element of the input array, and return the sum(s) of that. Parameters ---------- a : array_like Input array. axis : int or None, optional Axis along which to calculate. Default is 0. If None, compute over the whole array `a`. Returns ------- sum_of_squares : ndarray The sum along the given axis for (a**2). See Also -------- _square_of_sums : The square(s) of the sum(s) (the opposite of `_sum_of_squares`). """ a, axis = _chk_asarray(a, axis) return np.sum(a*a, axis) def _square_of_sums(a, axis=0): """ Sum elements of the input array, and return the square(s) of that sum. Parameters ---------- a : array_like Input array. axis : int or None, optional Axis along which to calculate. Default is 0. If None, compute over the whole array `a`. Returns ------- square_of_sums : float or ndarray The square of the sum over `axis`. See Also -------- _sum_of_squares : The sum of squares (the opposite of `square_of_sums`). """ a, axis = _chk_asarray(a, axis) s = np.sum(a, axis) if not np.isscalar(s): return s.astype(float) * s else: return float(s) * s def rankdata(a, method='average', *, axis=None): """ Assign ranks to data, dealing with ties appropriately. By default (``axis=None``), the data array is first flattened, and a flat array of ranks is returned. Separately reshape the rank array to the shape of the data array if desired (see Examples). Ranks begin at 1. The `method` argument controls how ranks are assigned to equal values. See [1]_ for further discussion of ranking methods. Parameters ---------- a : array_like The array of values to be ranked. method : {'average', 'min', 'max', 'dense', 'ordinal'}, optional The method used to assign ranks to tied elements. The following methods are available (default is 'average'): * 'average': The average of the ranks that would have been assigned to all the tied values is assigned to each value. * 'min': The minimum of the ranks that would have been assigned to all the tied values is assigned to each value. (This is also referred to as "competition" ranking.) * 'max': The maximum of the ranks that would have been assigned to all the tied values is assigned to each value. * 'dense': Like 'min', but the rank of the next highest element is assigned the rank immediately after those assigned to the tied elements. * 'ordinal': All values are given a distinct rank, corresponding to the order that the values occur in `a`. axis : {None, int}, optional Axis along which to perform the ranking. If ``None``, the data array is first flattened. Returns ------- ranks : ndarray An array of size equal to the size of `a`, containing rank scores. References ---------- .. [1] "Ranking", https://en.wikipedia.org/wiki/Ranking Examples -------- >>> from scipy.stats import rankdata >>> rankdata([0, 2, 3, 2]) array([ 1. , 2.5, 4. , 2.5]) >>> rankdata([0, 2, 3, 2], method='min') array([ 1, 2, 4, 2]) >>> rankdata([0, 2, 3, 2], method='max') array([ 1, 3, 4, 3]) >>> rankdata([0, 2, 3, 2], method='dense') array([ 1, 2, 3, 2]) >>> rankdata([0, 2, 3, 2], method='ordinal') array([ 1, 2, 4, 3]) >>> rankdata([[0, 2], [3, 2]]).reshape(2,2) array([[1. , 2.5], [4. , 2.5]]) >>> rankdata([[0, 2, 2], [3, 2, 5]], axis=1) array([[1. , 2.5, 2.5], [2. , 1. , 3. ]]) """ if method not in ('average', 'min', 'max', 'dense', 'ordinal'): raise ValueError('unknown method "{0}"'.format(method)) if axis is not None: a = np.asarray(a) if a.size == 0: # The return values of `normalize_axis_index` are ignored. The # call validates `axis`, even though we won't use it. # use scipy._lib._util._normalize_axis_index when available np.core.multiarray.normalize_axis_index(axis, a.ndim) dt = np.float64 if method == 'average' else np.int_ return np.empty(a.shape, dtype=dt) return np.apply_along_axis(rankdata, axis, a, method) arr = np.ravel(np.asarray(a)) algo = 'mergesort' if method == 'ordinal' else 'quicksort' sorter = np.argsort(arr, kind=algo) inv = np.empty(sorter.size, dtype=np.intp) inv[sorter] = np.arange(sorter.size, dtype=np.intp) if method == 'ordinal': return inv + 1 arr = arr[sorter] obs = np.r_[True, arr[1:] != arr[:-1]] dense = obs.cumsum()[inv] if method == 'dense': return dense # cumulative counts of each unique value count = np.r_[np.nonzero(obs)[0], len(obs)] if method == 'max': return count[dense] if method == 'min': return count[dense - 1] + 1 # average method return .5 * (count[dense] + count[dense - 1] + 1)