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329 lines
9.5 KiB
Python
329 lines
9.5 KiB
Python
"""
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Functions which are common and require SciPy Base and Level 1 SciPy
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(special, linalg)
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"""
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from numpy import arange, newaxis, hstack, prod, array, frombuffer, load
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__all__ = ['central_diff_weights', 'derivative', 'ascent', 'face',
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'electrocardiogram']
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def central_diff_weights(Np, ndiv=1):
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"""
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Return weights for an Np-point central derivative.
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Assumes equally-spaced function points.
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If weights are in the vector w, then
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derivative is w[0] * f(x-ho*dx) + ... + w[-1] * f(x+h0*dx)
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Parameters
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----------
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Np : int
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Number of points for the central derivative.
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ndiv : int, optional
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Number of divisions. Default is 1.
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Returns
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-------
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w : ndarray
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Weights for an Np-point central derivative. Its size is `Np`.
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Notes
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-----
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Can be inaccurate for a large number of points.
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Examples
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--------
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We can calculate a derivative value of a function.
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>>> from scipy.misc import central_diff_weights
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>>> def f(x):
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... return 2 * x**2 + 3
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>>> x = 3.0 # derivative point
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>>> h = 0.1 # differential step
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>>> Np = 3 # point number for central derivative
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>>> weights = central_diff_weights(Np) # weights for first derivative
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>>> vals = [f(x + (i - Np/2) * h) for i in range(Np)]
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>>> sum(w * v for (w, v) in zip(weights, vals))/h
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11.79999999999998
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This value is close to the analytical solution:
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f'(x) = 4x, so f'(3) = 12
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References
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----------
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.. [1] https://en.wikipedia.org/wiki/Finite_difference
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"""
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if Np < ndiv + 1:
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raise ValueError("Number of points must be at least the derivative order + 1.")
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if Np % 2 == 0:
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raise ValueError("The number of points must be odd.")
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from scipy import linalg
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ho = Np >> 1
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x = arange(-ho,ho+1.0)
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x = x[:,newaxis]
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X = x**0.0
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for k in range(1,Np):
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X = hstack([X,x**k])
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w = prod(arange(1,ndiv+1),axis=0)*linalg.inv(X)[ndiv]
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return w
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def derivative(func, x0, dx=1.0, n=1, args=(), order=3):
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"""
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Find the nth derivative of a function at a point.
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Given a function, use a central difference formula with spacing `dx` to
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compute the nth derivative at `x0`.
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Parameters
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----------
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func : function
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Input function.
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x0 : float
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The point at which the nth derivative is found.
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dx : float, optional
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Spacing.
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n : int, optional
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Order of the derivative. Default is 1.
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args : tuple, optional
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Arguments
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order : int, optional
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Number of points to use, must be odd.
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Notes
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-----
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Decreasing the step size too small can result in round-off error.
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Examples
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--------
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>>> from scipy.misc import derivative
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>>> def f(x):
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... return x**3 + x**2
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>>> derivative(f, 1.0, dx=1e-6)
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4.9999999999217337
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"""
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if order < n + 1:
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raise ValueError("'order' (the number of points used to compute the derivative), "
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"must be at least the derivative order 'n' + 1.")
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if order % 2 == 0:
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raise ValueError("'order' (the number of points used to compute the derivative) "
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"must be odd.")
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# pre-computed for n=1 and 2 and low-order for speed.
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if n == 1:
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if order == 3:
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weights = array([-1,0,1])/2.0
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elif order == 5:
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weights = array([1,-8,0,8,-1])/12.0
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elif order == 7:
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weights = array([-1,9,-45,0,45,-9,1])/60.0
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elif order == 9:
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weights = array([3,-32,168,-672,0,672,-168,32,-3])/840.0
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else:
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weights = central_diff_weights(order,1)
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elif n == 2:
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if order == 3:
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weights = array([1,-2.0,1])
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elif order == 5:
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weights = array([-1,16,-30,16,-1])/12.0
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elif order == 7:
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weights = array([2,-27,270,-490,270,-27,2])/180.0
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elif order == 9:
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weights = array([-9,128,-1008,8064,-14350,8064,-1008,128,-9])/5040.0
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else:
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weights = central_diff_weights(order,2)
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else:
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weights = central_diff_weights(order, n)
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val = 0.0
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ho = order >> 1
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for k in range(order):
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val += weights[k]*func(x0+(k-ho)*dx,*args)
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return val / prod((dx,)*n,axis=0)
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def ascent():
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"""
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Get an 8-bit grayscale bit-depth, 512 x 512 derived image for easy use in demos
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The image is derived from accent-to-the-top.jpg at
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http://www.public-domain-image.com/people-public-domain-images-pictures/
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Parameters
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----------
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None
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Returns
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-------
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ascent : ndarray
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convenient image to use for testing and demonstration
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Examples
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--------
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>>> import scipy.misc
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>>> ascent = scipy.misc.ascent()
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>>> ascent.shape
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(512, 512)
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>>> ascent.max()
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255
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>>> import matplotlib.pyplot as plt
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>>> plt.gray()
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>>> plt.imshow(ascent)
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>>> plt.show()
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"""
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import pickle
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import os
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fname = os.path.join(os.path.dirname(__file__),'ascent.dat')
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with open(fname, 'rb') as f:
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ascent = array(pickle.load(f))
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return ascent
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def face(gray=False):
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"""
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Get a 1024 x 768, color image of a raccoon face.
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raccoon-procyon-lotor.jpg at http://www.public-domain-image.com
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Parameters
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----------
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gray : bool, optional
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If True return 8-bit grey-scale image, otherwise return a color image
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Returns
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-------
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face : ndarray
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image of a racoon face
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Examples
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--------
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>>> import scipy.misc
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>>> face = scipy.misc.face()
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>>> face.shape
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(768, 1024, 3)
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>>> face.max()
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255
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>>> face.dtype
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dtype('uint8')
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>>> import matplotlib.pyplot as plt
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>>> plt.gray()
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>>> plt.imshow(face)
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>>> plt.show()
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"""
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import bz2
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import os
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with open(os.path.join(os.path.dirname(__file__), 'face.dat'), 'rb') as f:
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rawdata = f.read()
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data = bz2.decompress(rawdata)
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face = frombuffer(data, dtype='uint8')
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face.shape = (768, 1024, 3)
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if gray is True:
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face = (0.21 * face[:,:,0] + 0.71 * face[:,:,1] + 0.07 * face[:,:,2]).astype('uint8')
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return face
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def electrocardiogram():
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"""
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Load an electrocardiogram as an example for a 1-D signal.
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The returned signal is a 5 minute long electrocardiogram (ECG), a medical
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recording of the heart's electrical activity, sampled at 360 Hz.
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Returns
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-------
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ecg : ndarray
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The electrocardiogram in millivolt (mV) sampled at 360 Hz.
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Notes
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-----
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The provided signal is an excerpt (19:35 to 24:35) from the `record 208`_
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(lead MLII) provided by the MIT-BIH Arrhythmia Database [1]_ on
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PhysioNet [2]_. The excerpt includes noise induced artifacts, typical
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heartbeats as well as pathological changes.
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.. _record 208: https://physionet.org/physiobank/database/html/mitdbdir/records.htm#208
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.. versionadded:: 1.1.0
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References
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----------
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.. [1] Moody GB, Mark RG. The impact of the MIT-BIH Arrhythmia Database.
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IEEE Eng in Med and Biol 20(3):45-50 (May-June 2001).
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(PMID: 11446209); :doi:`10.13026/C2F305`
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.. [2] Goldberger AL, Amaral LAN, Glass L, Hausdorff JM, Ivanov PCh,
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Mark RG, Mietus JE, Moody GB, Peng C-K, Stanley HE. PhysioBank,
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PhysioToolkit, and PhysioNet: Components of a New Research Resource
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for Complex Physiologic Signals. Circulation 101(23):e215-e220;
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:doi:`10.1161/01.CIR.101.23.e215`
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Examples
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--------
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>>> from scipy.misc import electrocardiogram
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>>> ecg = electrocardiogram()
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>>> ecg
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array([-0.245, -0.215, -0.185, ..., -0.405, -0.395, -0.385])
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>>> ecg.shape, ecg.mean(), ecg.std()
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((108000,), -0.16510875, 0.5992473991177294)
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As stated the signal features several areas with a different morphology.
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E.g., the first few seconds show the electrical activity of a heart in
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normal sinus rhythm as seen below.
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>>> import matplotlib.pyplot as plt
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>>> fs = 360
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>>> time = np.arange(ecg.size) / fs
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>>> plt.plot(time, ecg)
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>>> plt.xlabel("time in s")
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>>> plt.ylabel("ECG in mV")
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>>> plt.xlim(9, 10.2)
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>>> plt.ylim(-1, 1.5)
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>>> plt.show()
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After second 16, however, the first premature ventricular contractions, also
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called extrasystoles, appear. These have a different morphology compared to
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typical heartbeats. The difference can easily be observed in the following
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plot.
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>>> plt.plot(time, ecg)
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>>> plt.xlabel("time in s")
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>>> plt.ylabel("ECG in mV")
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>>> plt.xlim(46.5, 50)
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>>> plt.ylim(-2, 1.5)
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>>> plt.show()
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At several points large artifacts disturb the recording, e.g.:
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>>> plt.plot(time, ecg)
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>>> plt.xlabel("time in s")
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>>> plt.ylabel("ECG in mV")
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>>> plt.xlim(207, 215)
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>>> plt.ylim(-2, 3.5)
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>>> plt.show()
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Finally, examining the power spectrum reveals that most of the biosignal is
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made up of lower frequencies. At 60 Hz the noise induced by the mains
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electricity can be clearly observed.
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>>> from scipy.signal import welch
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>>> f, Pxx = welch(ecg, fs=fs, nperseg=2048, scaling="spectrum")
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>>> plt.semilogy(f, Pxx)
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>>> plt.xlabel("Frequency in Hz")
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>>> plt.ylabel("Power spectrum of the ECG in mV**2")
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>>> plt.xlim(f[[0, -1]])
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>>> plt.show()
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"""
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import os
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file_path = os.path.join(os.path.dirname(__file__), "ecg.dat")
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with load(file_path) as file:
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ecg = file["ecg"].astype(int) # np.uint16 -> int
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# Convert raw output of ADC to mV: (ecg - adc_zero) / adc_gain
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ecg = (ecg - 1024) / 200.0
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return ecg
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