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1114 lines
39 KiB
Python
1114 lines
39 KiB
Python
"""
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Histogram-related functions
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"""
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from __future__ import division, absolute_import, print_function
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import contextlib
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import functools
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import operator
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import warnings
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import numpy as np
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from numpy.compat.py3k import basestring
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from numpy.core import overrides
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__all__ = ['histogram', 'histogramdd', 'histogram_bin_edges']
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array_function_dispatch = functools.partial(
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overrides.array_function_dispatch, module='numpy')
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# range is a keyword argument to many functions, so save the builtin so they can
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# use it.
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_range = range
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def _hist_bin_sqrt(x, range):
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"""
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Square root histogram bin estimator.
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Bin width is inversely proportional to the data size. Used by many
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programs for its simplicity.
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Parameters
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----------
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x : array_like
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Input data that is to be histogrammed, trimmed to range. May not
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be empty.
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Returns
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-------
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h : An estimate of the optimal bin width for the given data.
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"""
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del range # unused
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return x.ptp() / np.sqrt(x.size)
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def _hist_bin_sturges(x, range):
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"""
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Sturges histogram bin estimator.
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A very simplistic estimator based on the assumption of normality of
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the data. This estimator has poor performance for non-normal data,
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which becomes especially obvious for large data sets. The estimate
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depends only on size of the data.
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Parameters
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----------
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x : array_like
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Input data that is to be histogrammed, trimmed to range. May not
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be empty.
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Returns
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-------
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h : An estimate of the optimal bin width for the given data.
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"""
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del range # unused
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return x.ptp() / (np.log2(x.size) + 1.0)
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def _hist_bin_rice(x, range):
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"""
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Rice histogram bin estimator.
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Another simple estimator with no normality assumption. It has better
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performance for large data than Sturges, but tends to overestimate
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the number of bins. The number of bins is proportional to the cube
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root of data size (asymptotically optimal). The estimate depends
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only on size of the data.
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Parameters
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----------
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x : array_like
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Input data that is to be histogrammed, trimmed to range. May not
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be empty.
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Returns
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-------
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h : An estimate of the optimal bin width for the given data.
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"""
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del range # unused
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return x.ptp() / (2.0 * x.size ** (1.0 / 3))
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def _hist_bin_scott(x, range):
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"""
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Scott histogram bin estimator.
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The binwidth is proportional to the standard deviation of the data
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and inversely proportional to the cube root of data size
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(asymptotically optimal).
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Parameters
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----------
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x : array_like
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Input data that is to be histogrammed, trimmed to range. May not
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be empty.
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Returns
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-------
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h : An estimate of the optimal bin width for the given data.
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"""
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del range # unused
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return (24.0 * np.pi**0.5 / x.size)**(1.0 / 3.0) * np.std(x)
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def _hist_bin_stone(x, range):
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"""
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Histogram bin estimator based on minimizing the estimated integrated squared error (ISE).
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The number of bins is chosen by minimizing the estimated ISE against the unknown true distribution.
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The ISE is estimated using cross-validation and can be regarded as a generalization of Scott's rule.
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https://en.wikipedia.org/wiki/Histogram#Scott.27s_normal_reference_rule
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This paper by Stone appears to be the origination of this rule.
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http://digitalassets.lib.berkeley.edu/sdtr/ucb/text/34.pdf
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Parameters
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----------
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x : array_like
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Input data that is to be histogrammed, trimmed to range. May not
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be empty.
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range : (float, float)
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The lower and upper range of the bins.
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Returns
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-------
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h : An estimate of the optimal bin width for the given data.
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"""
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n = x.size
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ptp_x = np.ptp(x)
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if n <= 1 or ptp_x == 0:
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return 0
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def jhat(nbins):
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hh = ptp_x / nbins
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p_k = np.histogram(x, bins=nbins, range=range)[0] / n
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return (2 - (n + 1) * p_k.dot(p_k)) / hh
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nbins_upper_bound = max(100, int(np.sqrt(n)))
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nbins = min(_range(1, nbins_upper_bound + 1), key=jhat)
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if nbins == nbins_upper_bound:
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warnings.warn("The number of bins estimated may be suboptimal.",
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RuntimeWarning, stacklevel=3)
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return ptp_x / nbins
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def _hist_bin_doane(x, range):
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"""
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Doane's histogram bin estimator.
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Improved version of Sturges' formula which works better for
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non-normal data. See
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stats.stackexchange.com/questions/55134/doanes-formula-for-histogram-binning
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Parameters
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----------
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x : array_like
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Input data that is to be histogrammed, trimmed to range. May not
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be empty.
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Returns
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-------
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h : An estimate of the optimal bin width for the given data.
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"""
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del range # unused
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if x.size > 2:
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sg1 = np.sqrt(6.0 * (x.size - 2) / ((x.size + 1.0) * (x.size + 3)))
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sigma = np.std(x)
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if sigma > 0.0:
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# These three operations add up to
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# g1 = np.mean(((x - np.mean(x)) / sigma)**3)
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# but use only one temp array instead of three
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temp = x - np.mean(x)
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np.true_divide(temp, sigma, temp)
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np.power(temp, 3, temp)
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g1 = np.mean(temp)
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return x.ptp() / (1.0 + np.log2(x.size) +
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np.log2(1.0 + np.absolute(g1) / sg1))
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return 0.0
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def _hist_bin_fd(x, range):
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"""
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The Freedman-Diaconis histogram bin estimator.
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The Freedman-Diaconis rule uses interquartile range (IQR) to
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estimate binwidth. It is considered a variation of the Scott rule
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with more robustness as the IQR is less affected by outliers than
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the standard deviation. However, the IQR depends on fewer points
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than the standard deviation, so it is less accurate, especially for
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long tailed distributions.
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If the IQR is 0, this function returns 1 for the number of bins.
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Binwidth is inversely proportional to the cube root of data size
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(asymptotically optimal).
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Parameters
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----------
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x : array_like
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Input data that is to be histogrammed, trimmed to range. May not
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be empty.
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Returns
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-------
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h : An estimate of the optimal bin width for the given data.
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"""
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del range # unused
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iqr = np.subtract(*np.percentile(x, [75, 25]))
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return 2.0 * iqr * x.size ** (-1.0 / 3.0)
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def _hist_bin_auto(x, range):
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"""
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Histogram bin estimator that uses the minimum width of the
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Freedman-Diaconis and Sturges estimators if the FD bandwidth is non zero
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and the Sturges estimator if the FD bandwidth is 0.
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The FD estimator is usually the most robust method, but its width
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estimate tends to be too large for small `x` and bad for data with limited
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variance. The Sturges estimator is quite good for small (<1000) datasets
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and is the default in the R language. This method gives good off the shelf
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behaviour.
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.. versionchanged:: 1.15.0
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If there is limited variance the IQR can be 0, which results in the
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FD bin width being 0 too. This is not a valid bin width, so
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``np.histogram_bin_edges`` chooses 1 bin instead, which may not be optimal.
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If the IQR is 0, it's unlikely any variance based estimators will be of
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use, so we revert to the sturges estimator, which only uses the size of the
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dataset in its calculation.
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Parameters
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----------
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x : array_like
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Input data that is to be histogrammed, trimmed to range. May not
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be empty.
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Returns
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-------
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h : An estimate of the optimal bin width for the given data.
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See Also
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--------
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_hist_bin_fd, _hist_bin_sturges
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"""
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fd_bw = _hist_bin_fd(x, range)
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sturges_bw = _hist_bin_sturges(x, range)
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del range # unused
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if fd_bw:
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return min(fd_bw, sturges_bw)
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else:
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# limited variance, so we return a len dependent bw estimator
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return sturges_bw
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# Private dict initialized at module load time
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_hist_bin_selectors = {'stone': _hist_bin_stone,
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'auto': _hist_bin_auto,
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'doane': _hist_bin_doane,
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'fd': _hist_bin_fd,
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'rice': _hist_bin_rice,
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'scott': _hist_bin_scott,
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'sqrt': _hist_bin_sqrt,
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'sturges': _hist_bin_sturges}
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def _ravel_and_check_weights(a, weights):
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""" Check a and weights have matching shapes, and ravel both """
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a = np.asarray(a)
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# Ensure that the array is a "subtractable" dtype
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if a.dtype == np.bool_:
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warnings.warn("Converting input from {} to {} for compatibility."
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.format(a.dtype, np.uint8),
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RuntimeWarning, stacklevel=3)
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a = a.astype(np.uint8)
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if weights is not None:
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weights = np.asarray(weights)
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if weights.shape != a.shape:
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raise ValueError(
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'weights should have the same shape as a.')
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weights = weights.ravel()
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a = a.ravel()
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return a, weights
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def _get_outer_edges(a, range):
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"""
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Determine the outer bin edges to use, from either the data or the range
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argument
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"""
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if range is not None:
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first_edge, last_edge = range
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if first_edge > last_edge:
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raise ValueError(
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'max must be larger than min in range parameter.')
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if not (np.isfinite(first_edge) and np.isfinite(last_edge)):
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raise ValueError(
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"supplied range of [{}, {}] is not finite".format(first_edge, last_edge))
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elif a.size == 0:
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# handle empty arrays. Can't determine range, so use 0-1.
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first_edge, last_edge = 0, 1
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else:
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first_edge, last_edge = a.min(), a.max()
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if not (np.isfinite(first_edge) and np.isfinite(last_edge)):
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raise ValueError(
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"autodetected range of [{}, {}] is not finite".format(first_edge, last_edge))
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# expand empty range to avoid divide by zero
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if first_edge == last_edge:
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first_edge = first_edge - 0.5
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last_edge = last_edge + 0.5
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return first_edge, last_edge
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def _unsigned_subtract(a, b):
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"""
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Subtract two values where a >= b, and produce an unsigned result
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This is needed when finding the difference between the upper and lower
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bound of an int16 histogram
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"""
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# coerce to a single type
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signed_to_unsigned = {
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np.byte: np.ubyte,
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np.short: np.ushort,
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np.intc: np.uintc,
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np.int_: np.uint,
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np.longlong: np.ulonglong
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}
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dt = np.result_type(a, b)
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try:
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dt = signed_to_unsigned[dt.type]
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except KeyError:
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return np.subtract(a, b, dtype=dt)
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else:
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# we know the inputs are integers, and we are deliberately casting
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# signed to unsigned
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return np.subtract(a, b, casting='unsafe', dtype=dt)
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def _get_bin_edges(a, bins, range, weights):
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"""
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Computes the bins used internally by `histogram`.
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Parameters
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==========
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a : ndarray
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Ravelled data array
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bins, range
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Forwarded arguments from `histogram`.
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weights : ndarray, optional
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Ravelled weights array, or None
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Returns
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=======
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bin_edges : ndarray
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Array of bin edges
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uniform_bins : (Number, Number, int):
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The upper bound, lowerbound, and number of bins, used in the optimized
|
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implementation of `histogram` that works on uniform bins.
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"""
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# parse the overloaded bins argument
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n_equal_bins = None
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bin_edges = None
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if isinstance(bins, basestring):
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bin_name = bins
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# if `bins` is a string for an automatic method,
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# this will replace it with the number of bins calculated
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if bin_name not in _hist_bin_selectors:
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raise ValueError(
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"{!r} is not a valid estimator for `bins`".format(bin_name))
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if weights is not None:
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raise TypeError("Automated estimation of the number of "
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"bins is not supported for weighted data")
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first_edge, last_edge = _get_outer_edges(a, range)
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|
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# truncate the range if needed
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if range is not None:
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keep = (a >= first_edge)
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keep &= (a <= last_edge)
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if not np.logical_and.reduce(keep):
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a = a[keep]
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|
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if a.size == 0:
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n_equal_bins = 1
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else:
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# Do not call selectors on empty arrays
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width = _hist_bin_selectors[bin_name](a, (first_edge, last_edge))
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if width:
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n_equal_bins = int(np.ceil(_unsigned_subtract(last_edge, first_edge) / width))
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else:
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# Width can be zero for some estimators, e.g. FD when
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# the IQR of the data is zero.
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n_equal_bins = 1
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elif np.ndim(bins) == 0:
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try:
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n_equal_bins = operator.index(bins)
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except TypeError:
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raise TypeError(
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'`bins` must be an integer, a string, or an array')
|
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if n_equal_bins < 1:
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raise ValueError('`bins` must be positive, when an integer')
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|
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first_edge, last_edge = _get_outer_edges(a, range)
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|
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elif np.ndim(bins) == 1:
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bin_edges = np.asarray(bins)
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if np.any(bin_edges[:-1] > bin_edges[1:]):
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raise ValueError(
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'`bins` must increase monotonically, when an array')
|
|
|
|
else:
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raise ValueError('`bins` must be 1d, when an array')
|
|
|
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if n_equal_bins is not None:
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# gh-10322 means that type resolution rules are dependent on array
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|
# shapes. To avoid this causing problems, we pick a type now and stick
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# with it throughout.
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bin_type = np.result_type(first_edge, last_edge, a)
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if np.issubdtype(bin_type, np.integer):
|
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bin_type = np.result_type(bin_type, float)
|
|
|
|
# bin edges must be computed
|
|
bin_edges = np.linspace(
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first_edge, last_edge, n_equal_bins + 1,
|
|
endpoint=True, dtype=bin_type)
|
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return bin_edges, (first_edge, last_edge, n_equal_bins)
|
|
else:
|
|
return bin_edges, None
|
|
|
|
|
|
def _search_sorted_inclusive(a, v):
|
|
"""
|
|
Like `searchsorted`, but where the last item in `v` is placed on the right.
|
|
|
|
In the context of a histogram, this makes the last bin edge inclusive
|
|
"""
|
|
return np.concatenate((
|
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a.searchsorted(v[:-1], 'left'),
|
|
a.searchsorted(v[-1:], 'right')
|
|
))
|
|
|
|
|
|
def _histogram_bin_edges_dispatcher(a, bins=None, range=None, weights=None):
|
|
return (a, bins, weights)
|
|
|
|
|
|
@array_function_dispatch(_histogram_bin_edges_dispatcher)
|
|
def histogram_bin_edges(a, bins=10, range=None, weights=None):
|
|
r"""
|
|
Function to calculate only the edges of the bins used by the `histogram`
|
|
function.
|
|
|
|
Parameters
|
|
----------
|
|
a : array_like
|
|
Input data. The histogram is computed over the flattened array.
|
|
bins : int or sequence of scalars or str, optional
|
|
If `bins` is an int, it defines the number of equal-width
|
|
bins in the given range (10, by default). If `bins` is a
|
|
sequence, it defines the bin edges, including the rightmost
|
|
edge, allowing for non-uniform bin widths.
|
|
|
|
If `bins` is a string from the list below, `histogram_bin_edges` will use
|
|
the method chosen to calculate the optimal bin width and
|
|
consequently the number of bins (see `Notes` for more detail on
|
|
the estimators) from the data that falls within the requested
|
|
range. While the bin width will be optimal for the actual data
|
|
in the range, the number of bins will be computed to fill the
|
|
entire range, including the empty portions. For visualisation,
|
|
using the 'auto' option is suggested. Weighted data is not
|
|
supported for automated bin size selection.
|
|
|
|
'auto'
|
|
Maximum of the 'sturges' and 'fd' estimators. Provides good
|
|
all around performance.
|
|
|
|
'fd' (Freedman Diaconis Estimator)
|
|
Robust (resilient to outliers) estimator that takes into
|
|
account data variability and data size.
|
|
|
|
'doane'
|
|
An improved version of Sturges' estimator that works better
|
|
with non-normal datasets.
|
|
|
|
'scott'
|
|
Less robust estimator that that takes into account data
|
|
variability and data size.
|
|
|
|
'stone'
|
|
Estimator based on leave-one-out cross-validation estimate of
|
|
the integrated squared error. Can be regarded as a generalization
|
|
of Scott's rule.
|
|
|
|
'rice'
|
|
Estimator does not take variability into account, only data
|
|
size. Commonly overestimates number of bins required.
|
|
|
|
'sturges'
|
|
R's default method, only accounts for data size. Only
|
|
optimal for gaussian data and underestimates number of bins
|
|
for large non-gaussian datasets.
|
|
|
|
'sqrt'
|
|
Square root (of data size) estimator, used by Excel and
|
|
other programs for its speed and simplicity.
|
|
|
|
range : (float, float), optional
|
|
The lower and upper range of the bins. If not provided, range
|
|
is simply ``(a.min(), a.max())``. Values outside the range are
|
|
ignored. The first element of the range must be less than or
|
|
equal to the second. `range` affects the automatic bin
|
|
computation as well. While bin width is computed to be optimal
|
|
based on the actual data within `range`, the bin count will fill
|
|
the entire range including portions containing no data.
|
|
|
|
weights : array_like, optional
|
|
An array of weights, of the same shape as `a`. Each value in
|
|
`a` only contributes its associated weight towards the bin count
|
|
(instead of 1). This is currently not used by any of the bin estimators,
|
|
but may be in the future.
|
|
|
|
Returns
|
|
-------
|
|
bin_edges : array of dtype float
|
|
The edges to pass into `histogram`
|
|
|
|
See Also
|
|
--------
|
|
histogram
|
|
|
|
Notes
|
|
-----
|
|
The methods to estimate the optimal number of bins are well founded
|
|
in literature, and are inspired by the choices R provides for
|
|
histogram visualisation. Note that having the number of bins
|
|
proportional to :math:`n^{1/3}` is asymptotically optimal, which is
|
|
why it appears in most estimators. These are simply plug-in methods
|
|
that give good starting points for number of bins. In the equations
|
|
below, :math:`h` is the binwidth and :math:`n_h` is the number of
|
|
bins. All estimators that compute bin counts are recast to bin width
|
|
using the `ptp` of the data. The final bin count is obtained from
|
|
``np.round(np.ceil(range / h))``.
|
|
|
|
'auto' (maximum of the 'sturges' and 'fd' estimators)
|
|
A compromise to get a good value. For small datasets the Sturges
|
|
value will usually be chosen, while larger datasets will usually
|
|
default to FD. Avoids the overly conservative behaviour of FD
|
|
and Sturges for small and large datasets respectively.
|
|
Switchover point is usually :math:`a.size \approx 1000`.
|
|
|
|
'fd' (Freedman Diaconis Estimator)
|
|
.. math:: h = 2 \frac{IQR}{n^{1/3}}
|
|
|
|
The binwidth is proportional to the interquartile range (IQR)
|
|
and inversely proportional to cube root of a.size. Can be too
|
|
conservative for small datasets, but is quite good for large
|
|
datasets. The IQR is very robust to outliers.
|
|
|
|
'scott'
|
|
.. math:: h = \sigma \sqrt[3]{\frac{24 * \sqrt{\pi}}{n}}
|
|
|
|
The binwidth is proportional to the standard deviation of the
|
|
data and inversely proportional to cube root of ``x.size``. Can
|
|
be too conservative for small datasets, but is quite good for
|
|
large datasets. The standard deviation is not very robust to
|
|
outliers. Values are very similar to the Freedman-Diaconis
|
|
estimator in the absence of outliers.
|
|
|
|
'rice'
|
|
.. math:: n_h = 2n^{1/3}
|
|
|
|
The number of bins is only proportional to cube root of
|
|
``a.size``. It tends to overestimate the number of bins and it
|
|
does not take into account data variability.
|
|
|
|
'sturges'
|
|
.. math:: n_h = \log _{2}n+1
|
|
|
|
The number of bins is the base 2 log of ``a.size``. This
|
|
estimator assumes normality of data and is too conservative for
|
|
larger, non-normal datasets. This is the default method in R's
|
|
``hist`` method.
|
|
|
|
'doane'
|
|
.. math:: n_h = 1 + \log_{2}(n) +
|
|
\log_{2}(1 + \frac{|g_1|}{\sigma_{g_1}})
|
|
|
|
g_1 = mean[(\frac{x - \mu}{\sigma})^3]
|
|
|
|
\sigma_{g_1} = \sqrt{\frac{6(n - 2)}{(n + 1)(n + 3)}}
|
|
|
|
An improved version of Sturges' formula that produces better
|
|
estimates for non-normal datasets. This estimator attempts to
|
|
account for the skew of the data.
|
|
|
|
'sqrt'
|
|
.. math:: n_h = \sqrt n
|
|
|
|
The simplest and fastest estimator. Only takes into account the
|
|
data size.
|
|
|
|
Examples
|
|
--------
|
|
>>> arr = np.array([0, 0, 0, 1, 2, 3, 3, 4, 5])
|
|
>>> np.histogram_bin_edges(arr, bins='auto', range=(0, 1))
|
|
array([0. , 0.25, 0.5 , 0.75, 1. ])
|
|
>>> np.histogram_bin_edges(arr, bins=2)
|
|
array([0. , 2.5, 5. ])
|
|
|
|
For consistency with histogram, an array of pre-computed bins is
|
|
passed through unmodified:
|
|
|
|
>>> np.histogram_bin_edges(arr, [1, 2])
|
|
array([1, 2])
|
|
|
|
This function allows one set of bins to be computed, and reused across
|
|
multiple histograms:
|
|
|
|
>>> shared_bins = np.histogram_bin_edges(arr, bins='auto')
|
|
>>> shared_bins
|
|
array([0., 1., 2., 3., 4., 5.])
|
|
|
|
>>> group_id = np.array([0, 1, 1, 0, 1, 1, 0, 1, 1])
|
|
>>> hist_0, _ = np.histogram(arr[group_id == 0], bins=shared_bins)
|
|
>>> hist_1, _ = np.histogram(arr[group_id == 1], bins=shared_bins)
|
|
|
|
>>> hist_0; hist_1
|
|
array([1, 1, 0, 1, 0])
|
|
array([2, 0, 1, 1, 2])
|
|
|
|
Which gives more easily comparable results than using separate bins for
|
|
each histogram:
|
|
|
|
>>> hist_0, bins_0 = np.histogram(arr[group_id == 0], bins='auto')
|
|
>>> hist_1, bins_1 = np.histogram(arr[group_id == 1], bins='auto')
|
|
>>> hist_0; hist_1
|
|
array([1, 1, 1])
|
|
array([2, 1, 1, 2])
|
|
>>> bins_0; bins_1
|
|
array([0., 1., 2., 3.])
|
|
array([0. , 1.25, 2.5 , 3.75, 5. ])
|
|
|
|
"""
|
|
a, weights = _ravel_and_check_weights(a, weights)
|
|
bin_edges, _ = _get_bin_edges(a, bins, range, weights)
|
|
return bin_edges
|
|
|
|
|
|
def _histogram_dispatcher(
|
|
a, bins=None, range=None, normed=None, weights=None, density=None):
|
|
return (a, bins, weights)
|
|
|
|
|
|
@array_function_dispatch(_histogram_dispatcher)
|
|
def histogram(a, bins=10, range=None, normed=None, weights=None,
|
|
density=None):
|
|
r"""
|
|
Compute the histogram of a set of data.
|
|
|
|
Parameters
|
|
----------
|
|
a : array_like
|
|
Input data. The histogram is computed over the flattened array.
|
|
bins : int or sequence of scalars or str, optional
|
|
If `bins` is an int, it defines the number of equal-width
|
|
bins in the given range (10, by default). If `bins` is a
|
|
sequence, it defines a monotonically increasing array of bin edges,
|
|
including the rightmost edge, allowing for non-uniform bin widths.
|
|
|
|
.. versionadded:: 1.11.0
|
|
|
|
If `bins` is a string, it defines the method used to calculate the
|
|
optimal bin width, as defined by `histogram_bin_edges`.
|
|
|
|
range : (float, float), optional
|
|
The lower and upper range of the bins. If not provided, range
|
|
is simply ``(a.min(), a.max())``. Values outside the range are
|
|
ignored. The first element of the range must be less than or
|
|
equal to the second. `range` affects the automatic bin
|
|
computation as well. While bin width is computed to be optimal
|
|
based on the actual data within `range`, the bin count will fill
|
|
the entire range including portions containing no data.
|
|
normed : bool, optional
|
|
|
|
.. deprecated:: 1.6.0
|
|
|
|
This is equivalent to the `density` argument, but produces incorrect
|
|
results for unequal bin widths. It should not be used.
|
|
|
|
.. versionchanged:: 1.15.0
|
|
DeprecationWarnings are actually emitted.
|
|
|
|
weights : array_like, optional
|
|
An array of weights, of the same shape as `a`. Each value in
|
|
`a` only contributes its associated weight towards the bin count
|
|
(instead of 1). If `density` is True, the weights are
|
|
normalized, so that the integral of the density over the range
|
|
remains 1.
|
|
density : bool, optional
|
|
If ``False``, the result will contain the number of samples in
|
|
each bin. If ``True``, the result is the value of the
|
|
probability *density* function at the bin, normalized such that
|
|
the *integral* over the range is 1. Note that the sum of the
|
|
histogram values will not be equal to 1 unless bins of unity
|
|
width are chosen; it is not a probability *mass* function.
|
|
|
|
Overrides the ``normed`` keyword if given.
|
|
|
|
Returns
|
|
-------
|
|
hist : array
|
|
The values of the histogram. See `density` and `weights` for a
|
|
description of the possible semantics.
|
|
bin_edges : array of dtype float
|
|
Return the bin edges ``(length(hist)+1)``.
|
|
|
|
|
|
See Also
|
|
--------
|
|
histogramdd, bincount, searchsorted, digitize, histogram_bin_edges
|
|
|
|
Notes
|
|
-----
|
|
All but the last (righthand-most) bin is half-open. In other words,
|
|
if `bins` is::
|
|
|
|
[1, 2, 3, 4]
|
|
|
|
then the first bin is ``[1, 2)`` (including 1, but excluding 2) and
|
|
the second ``[2, 3)``. The last bin, however, is ``[3, 4]``, which
|
|
*includes* 4.
|
|
|
|
|
|
Examples
|
|
--------
|
|
>>> np.histogram([1, 2, 1], bins=[0, 1, 2, 3])
|
|
(array([0, 2, 1]), array([0, 1, 2, 3]))
|
|
>>> np.histogram(np.arange(4), bins=np.arange(5), density=True)
|
|
(array([0.25, 0.25, 0.25, 0.25]), array([0, 1, 2, 3, 4]))
|
|
>>> np.histogram([[1, 2, 1], [1, 0, 1]], bins=[0,1,2,3])
|
|
(array([1, 4, 1]), array([0, 1, 2, 3]))
|
|
|
|
>>> a = np.arange(5)
|
|
>>> hist, bin_edges = np.histogram(a, density=True)
|
|
>>> hist
|
|
array([0.5, 0. , 0.5, 0. , 0. , 0.5, 0. , 0.5, 0. , 0.5])
|
|
>>> hist.sum()
|
|
2.4999999999999996
|
|
>>> np.sum(hist * np.diff(bin_edges))
|
|
1.0
|
|
|
|
.. versionadded:: 1.11.0
|
|
|
|
Automated Bin Selection Methods example, using 2 peak random data
|
|
with 2000 points:
|
|
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> rng = np.random.RandomState(10) # deterministic random data
|
|
>>> a = np.hstack((rng.normal(size=1000),
|
|
... rng.normal(loc=5, scale=2, size=1000)))
|
|
>>> _ = plt.hist(a, bins='auto') # arguments are passed to np.histogram
|
|
>>> plt.title("Histogram with 'auto' bins")
|
|
Text(0.5, 1.0, "Histogram with 'auto' bins")
|
|
>>> plt.show()
|
|
|
|
"""
|
|
a, weights = _ravel_and_check_weights(a, weights)
|
|
|
|
bin_edges, uniform_bins = _get_bin_edges(a, bins, range, weights)
|
|
|
|
# Histogram is an integer or a float array depending on the weights.
|
|
if weights is None:
|
|
ntype = np.dtype(np.intp)
|
|
else:
|
|
ntype = weights.dtype
|
|
|
|
# We set a block size, as this allows us to iterate over chunks when
|
|
# computing histograms, to minimize memory usage.
|
|
BLOCK = 65536
|
|
|
|
# The fast path uses bincount, but that only works for certain types
|
|
# of weight
|
|
simple_weights = (
|
|
weights is None or
|
|
np.can_cast(weights.dtype, np.double) or
|
|
np.can_cast(weights.dtype, complex)
|
|
)
|
|
|
|
if uniform_bins is not None and simple_weights:
|
|
# Fast algorithm for equal bins
|
|
# We now convert values of a to bin indices, under the assumption of
|
|
# equal bin widths (which is valid here).
|
|
first_edge, last_edge, n_equal_bins = uniform_bins
|
|
|
|
# Initialize empty histogram
|
|
n = np.zeros(n_equal_bins, ntype)
|
|
|
|
# Pre-compute histogram scaling factor
|
|
norm = n_equal_bins / _unsigned_subtract(last_edge, first_edge)
|
|
|
|
# We iterate over blocks here for two reasons: the first is that for
|
|
# large arrays, it is actually faster (for example for a 10^8 array it
|
|
# is 2x as fast) and it results in a memory footprint 3x lower in the
|
|
# limit of large arrays.
|
|
for i in _range(0, len(a), BLOCK):
|
|
tmp_a = a[i:i+BLOCK]
|
|
if weights is None:
|
|
tmp_w = None
|
|
else:
|
|
tmp_w = weights[i:i + BLOCK]
|
|
|
|
# Only include values in the right range
|
|
keep = (tmp_a >= first_edge)
|
|
keep &= (tmp_a <= last_edge)
|
|
if not np.logical_and.reduce(keep):
|
|
tmp_a = tmp_a[keep]
|
|
if tmp_w is not None:
|
|
tmp_w = tmp_w[keep]
|
|
|
|
# This cast ensures no type promotions occur below, which gh-10322
|
|
# make unpredictable. Getting it wrong leads to precision errors
|
|
# like gh-8123.
|
|
tmp_a = tmp_a.astype(bin_edges.dtype, copy=False)
|
|
|
|
# Compute the bin indices, and for values that lie exactly on
|
|
# last_edge we need to subtract one
|
|
f_indices = _unsigned_subtract(tmp_a, first_edge) * norm
|
|
indices = f_indices.astype(np.intp)
|
|
indices[indices == n_equal_bins] -= 1
|
|
|
|
# The index computation is not guaranteed to give exactly
|
|
# consistent results within ~1 ULP of the bin edges.
|
|
decrement = tmp_a < bin_edges[indices]
|
|
indices[decrement] -= 1
|
|
# The last bin includes the right edge. The other bins do not.
|
|
increment = ((tmp_a >= bin_edges[indices + 1])
|
|
& (indices != n_equal_bins - 1))
|
|
indices[increment] += 1
|
|
|
|
# We now compute the histogram using bincount
|
|
if ntype.kind == 'c':
|
|
n.real += np.bincount(indices, weights=tmp_w.real,
|
|
minlength=n_equal_bins)
|
|
n.imag += np.bincount(indices, weights=tmp_w.imag,
|
|
minlength=n_equal_bins)
|
|
else:
|
|
n += np.bincount(indices, weights=tmp_w,
|
|
minlength=n_equal_bins).astype(ntype)
|
|
else:
|
|
# Compute via cumulative histogram
|
|
cum_n = np.zeros(bin_edges.shape, ntype)
|
|
if weights is None:
|
|
for i in _range(0, len(a), BLOCK):
|
|
sa = np.sort(a[i:i+BLOCK])
|
|
cum_n += _search_sorted_inclusive(sa, bin_edges)
|
|
else:
|
|
zero = np.zeros(1, dtype=ntype)
|
|
for i in _range(0, len(a), BLOCK):
|
|
tmp_a = a[i:i+BLOCK]
|
|
tmp_w = weights[i:i+BLOCK]
|
|
sorting_index = np.argsort(tmp_a)
|
|
sa = tmp_a[sorting_index]
|
|
sw = tmp_w[sorting_index]
|
|
cw = np.concatenate((zero, sw.cumsum()))
|
|
bin_index = _search_sorted_inclusive(sa, bin_edges)
|
|
cum_n += cw[bin_index]
|
|
|
|
n = np.diff(cum_n)
|
|
|
|
# density overrides the normed keyword
|
|
if density is not None:
|
|
if normed is not None:
|
|
# 2018-06-13, numpy 1.15.0 (this was not noisily deprecated in 1.6)
|
|
warnings.warn(
|
|
"The normed argument is ignored when density is provided. "
|
|
"In future passing both will result in an error.",
|
|
DeprecationWarning, stacklevel=3)
|
|
normed = None
|
|
|
|
if density:
|
|
db = np.array(np.diff(bin_edges), float)
|
|
return n/db/n.sum(), bin_edges
|
|
elif normed:
|
|
# 2018-06-13, numpy 1.15.0 (this was not noisily deprecated in 1.6)
|
|
warnings.warn(
|
|
"Passing `normed=True` on non-uniform bins has always been "
|
|
"broken, and computes neither the probability density "
|
|
"function nor the probability mass function. "
|
|
"The result is only correct if the bins are uniform, when "
|
|
"density=True will produce the same result anyway. "
|
|
"The argument will be removed in a future version of "
|
|
"numpy.",
|
|
np.VisibleDeprecationWarning, stacklevel=3)
|
|
|
|
# this normalization is incorrect, but
|
|
db = np.array(np.diff(bin_edges), float)
|
|
return n/(n*db).sum(), bin_edges
|
|
else:
|
|
if normed is not None:
|
|
# 2018-06-13, numpy 1.15.0 (this was not noisily deprecated in 1.6)
|
|
warnings.warn(
|
|
"Passing normed=False is deprecated, and has no effect. "
|
|
"Consider passing the density argument instead.",
|
|
DeprecationWarning, stacklevel=3)
|
|
return n, bin_edges
|
|
|
|
|
|
def _histogramdd_dispatcher(sample, bins=None, range=None, normed=None,
|
|
weights=None, density=None):
|
|
if hasattr(sample, 'shape'): # same condition as used in histogramdd
|
|
yield sample
|
|
else:
|
|
yield from sample
|
|
with contextlib.suppress(TypeError):
|
|
yield from bins
|
|
yield weights
|
|
|
|
|
|
@array_function_dispatch(_histogramdd_dispatcher)
|
|
def histogramdd(sample, bins=10, range=None, normed=None, weights=None,
|
|
density=None):
|
|
"""
|
|
Compute the multidimensional histogram of some data.
|
|
|
|
Parameters
|
|
----------
|
|
sample : (N, D) array, or (D, N) array_like
|
|
The data to be histogrammed.
|
|
|
|
Note the unusual interpretation of sample when an array_like:
|
|
|
|
* When an array, each row is a coordinate in a D-dimensional space -
|
|
such as ``histogramgramdd(np.array([p1, p2, p3]))``.
|
|
* When an array_like, each element is the list of values for single
|
|
coordinate - such as ``histogramgramdd((X, Y, Z))``.
|
|
|
|
The first form should be preferred.
|
|
|
|
bins : sequence or int, optional
|
|
The bin specification:
|
|
|
|
* A sequence of arrays describing the monotonically increasing bin
|
|
edges along each dimension.
|
|
* The number of bins for each dimension (nx, ny, ... =bins)
|
|
* The number of bins for all dimensions (nx=ny=...=bins).
|
|
|
|
range : sequence, optional
|
|
A sequence of length D, each an optional (lower, upper) tuple giving
|
|
the outer bin edges to be used if the edges are not given explicitly in
|
|
`bins`.
|
|
An entry of None in the sequence results in the minimum and maximum
|
|
values being used for the corresponding dimension.
|
|
The default, None, is equivalent to passing a tuple of D None values.
|
|
density : bool, optional
|
|
If False, the default, returns the number of samples in each bin.
|
|
If True, returns the probability *density* function at the bin,
|
|
``bin_count / sample_count / bin_volume``.
|
|
normed : bool, optional
|
|
An alias for the density argument that behaves identically. To avoid
|
|
confusion with the broken normed argument to `histogram`, `density`
|
|
should be preferred.
|
|
weights : (N,) array_like, optional
|
|
An array of values `w_i` weighing each sample `(x_i, y_i, z_i, ...)`.
|
|
Weights are normalized to 1 if normed is True. If normed is False,
|
|
the values of the returned histogram are equal to the sum of the
|
|
weights belonging to the samples falling into each bin.
|
|
|
|
Returns
|
|
-------
|
|
H : ndarray
|
|
The multidimensional histogram of sample x. See normed and weights
|
|
for the different possible semantics.
|
|
edges : list
|
|
A list of D arrays describing the bin edges for each dimension.
|
|
|
|
See Also
|
|
--------
|
|
histogram: 1-D histogram
|
|
histogram2d: 2-D histogram
|
|
|
|
Examples
|
|
--------
|
|
>>> r = np.random.randn(100,3)
|
|
>>> H, edges = np.histogramdd(r, bins = (5, 8, 4))
|
|
>>> H.shape, edges[0].size, edges[1].size, edges[2].size
|
|
((5, 8, 4), 6, 9, 5)
|
|
|
|
"""
|
|
|
|
try:
|
|
# Sample is an ND-array.
|
|
N, D = sample.shape
|
|
except (AttributeError, ValueError):
|
|
# Sample is a sequence of 1D arrays.
|
|
sample = np.atleast_2d(sample).T
|
|
N, D = sample.shape
|
|
|
|
nbin = np.empty(D, int)
|
|
edges = D*[None]
|
|
dedges = D*[None]
|
|
if weights is not None:
|
|
weights = np.asarray(weights)
|
|
|
|
try:
|
|
M = len(bins)
|
|
if M != D:
|
|
raise ValueError(
|
|
'The dimension of bins must be equal to the dimension of the '
|
|
' sample x.')
|
|
except TypeError:
|
|
# bins is an integer
|
|
bins = D*[bins]
|
|
|
|
# normalize the range argument
|
|
if range is None:
|
|
range = (None,) * D
|
|
elif len(range) != D:
|
|
raise ValueError('range argument must have one entry per dimension')
|
|
|
|
# Create edge arrays
|
|
for i in _range(D):
|
|
if np.ndim(bins[i]) == 0:
|
|
if bins[i] < 1:
|
|
raise ValueError(
|
|
'`bins[{}]` must be positive, when an integer'.format(i))
|
|
smin, smax = _get_outer_edges(sample[:,i], range[i])
|
|
edges[i] = np.linspace(smin, smax, bins[i] + 1)
|
|
elif np.ndim(bins[i]) == 1:
|
|
edges[i] = np.asarray(bins[i])
|
|
if np.any(edges[i][:-1] > edges[i][1:]):
|
|
raise ValueError(
|
|
'`bins[{}]` must be monotonically increasing, when an array'
|
|
.format(i))
|
|
else:
|
|
raise ValueError(
|
|
'`bins[{}]` must be a scalar or 1d array'.format(i))
|
|
|
|
nbin[i] = len(edges[i]) + 1 # includes an outlier on each end
|
|
dedges[i] = np.diff(edges[i])
|
|
|
|
# Compute the bin number each sample falls into.
|
|
Ncount = tuple(
|
|
# avoid np.digitize to work around gh-11022
|
|
np.searchsorted(edges[i], sample[:, i], side='right')
|
|
for i in _range(D)
|
|
)
|
|
|
|
# Using digitize, values that fall on an edge are put in the right bin.
|
|
# For the rightmost bin, we want values equal to the right edge to be
|
|
# counted in the last bin, and not as an outlier.
|
|
for i in _range(D):
|
|
# Find which points are on the rightmost edge.
|
|
on_edge = (sample[:, i] == edges[i][-1])
|
|
# Shift these points one bin to the left.
|
|
Ncount[i][on_edge] -= 1
|
|
|
|
# Compute the sample indices in the flattened histogram matrix.
|
|
# This raises an error if the array is too large.
|
|
xy = np.ravel_multi_index(Ncount, nbin)
|
|
|
|
# Compute the number of repetitions in xy and assign it to the
|
|
# flattened histmat.
|
|
hist = np.bincount(xy, weights, minlength=nbin.prod())
|
|
|
|
# Shape into a proper matrix
|
|
hist = hist.reshape(nbin)
|
|
|
|
# This preserves the (bad) behavior observed in gh-7845, for now.
|
|
hist = hist.astype(float, casting='safe')
|
|
|
|
# Remove outliers (indices 0 and -1 for each dimension).
|
|
core = D*(slice(1, -1),)
|
|
hist = hist[core]
|
|
|
|
# handle the aliasing normed argument
|
|
if normed is None:
|
|
if density is None:
|
|
density = False
|
|
elif density is None:
|
|
# an explicit normed argument was passed, alias it to the new name
|
|
density = normed
|
|
else:
|
|
raise TypeError("Cannot specify both 'normed' and 'density'")
|
|
|
|
if density:
|
|
# calculate the probability density function
|
|
s = hist.sum()
|
|
for i in _range(D):
|
|
shape = np.ones(D, int)
|
|
shape[i] = nbin[i] - 2
|
|
hist = hist / dedges[i].reshape(shape)
|
|
hist /= s
|
|
|
|
if (hist.shape != nbin - 2).any():
|
|
raise RuntimeError(
|
|
"Internal Shape Error")
|
|
return hist, edges
|