951938802c
Added text in book about multivariate Gaussian. Wrote tests to confirm that my multivariate function yields the same results as numpy's version, and made everything compliant with py.test.
306 lines
8.3 KiB
Python
306 lines
8.3 KiB
Python
"""
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Author: Roger Labbe
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Copyright: 2014
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This code performs various basic statistics functions for the
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Kalman and Bayesian Filters in Python book. Much of this code
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is non-optimal; production code should call the equivalent scipy.stats
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functions. I wrote the code in this form to make explicit how the
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computations are done. The scipy.stats module has many more useful functions
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than what I have written here. In some cases, however, my code is significantly
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faster, at least on my machine. For example, gaussian average 794 ns, whereas
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stats.norm(), using the frozen form, averages 116 microseconds per call.
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"""
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from __future__ import (absolute_import, division, print_function,
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unicode_literals)
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import math
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import numpy as np
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import numpy.linalg as linalg
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import matplotlib.pyplot as plt
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import scipy.sparse as sp
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import scipy.sparse.linalg as spln
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import scipy.stats
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from matplotlib.patches import Ellipse
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_two_pi = 2*math.pi
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def gaussian(x, mean, var):
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"""returns normal distribution for x given a gaussian with the specified
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mean and variance. All must be scalars.
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gaussian (1,2,3) is equivalent to scipy.stats.norm(2,math.sqrt(3)).pdf(1)
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It is quite a bit faster albeit much less flexible than the latter.
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"""
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return math.exp((-0.5*(x-mean)**2)/var) / math.sqrt(_two_pi*var)
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# return scipy.stats.norm(mean, math.sqrt(var)).pdf(x)
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def mul (mean1, var1, mean2, var2):
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""" multiply Gaussians (mean1, var1) with (mean2, var2) and return the
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results as a tuple (mean,var).
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var1 and var2 are variances - sigma squared in the usual parlance.
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"""
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mean = (var1*mean2 + var2*mean1) / (var1 + var2)
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var = 1 / (1/var1 + 1/var2)
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return (mean, var)
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def add (mean1, var1, mean2, var2):
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""" add the Gaussians (mean1, var1) with (mean2, var2) and return the
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results as a tuple (mean,var).
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var1 and var2 are variances - sigma squared in the usual parlance.
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"""
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return (mean1+mean2, var1+var2)
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def multivariate_gaussian(x, mu, cov):
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""" This is designed to replace scipy.stats.multivariate_normal
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which is not available before version 0.14. You may either pass in a
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multivariate set of data:
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multivariate_gaussian (array([1,1]), array([3,4]), eye(2)*1.4)
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multivariate_gaussian (array([1,1,1]), array([3,4,5]), 1.4)
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or unidimensional data:
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multivariate_gaussian(1, 3, 1.4)
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In the multivariate case if cov is a scalar it is interpreted as eye(n)*cov
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The function gaussian() implements the 1D (univariate)case, and is much
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faster than this function.
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equivalent calls:
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multivariate_gaussian(1, 2, 3)
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scipy.stats.multivariate_normal(2,3).pdf(1)
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"""
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# force all to numpy.array type
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x = np.array(x, copy=False, ndmin=1)
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mu = np.array(mu,copy=False, ndmin=1)
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nx = len(mu)
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cov = _to_cov(cov, nx)
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norm_coeff = nx*math.log(2*math.pi) + np.linalg.slogdet(cov)[1]
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err = x - mu
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if (sp.issparse(cov)):
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numerator = spln.spsolve(cov, err).T.dot(err)
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else:
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numerator = np.linalg.solve(cov, err).T.dot(err)
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return math.exp(-0.5*(norm_coeff + numerator))
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def plot_gaussian(mean, variance,
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mean_line=False,
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xlim=None,
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xlabel=None,
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ylabel=None):
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""" plots the normal distribution with the given mean and variance. x-axis
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contains the mean, the y-axis shows the probability.
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mean_line : draws a line at x=mean
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xlim: optionally specify the limits for the x axis as tuple (low,high).
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If not specified, limits will be automatically chosen to be 'nice'
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xlabel : optional label for the x-axis
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ylabel : optional label for the y-axis
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"""
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sigma = math.sqrt(variance)
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n = scipy.stats.norm(mean, sigma)
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if xlim is None:
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min_x = n.ppf(0.001)
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max_x = n.ppf(0.999)
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else:
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min_x = xlim[0]
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max_x = xlim[1]
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xs = np.arange(min_x, max_x, (max_x - min_x) / 1000)
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plt.plot(xs,n.pdf(xs))
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plt.xlim((min_x, max_x))
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if mean_line:
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plt.axvline(mean)
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if xlabel:
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plt.xlabel(xlabel)
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if ylabel:
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plt.ylabel(ylabel)
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def covariance_ellipse(P, deviations=1):
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""" returns a tuple defining the ellipse representing the 2 dimensional
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covariance matrix P.
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Parameters
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----------
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P : nd.array shape (2,2)
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covariance matrix
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deviations : int (optional, default = 1)
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# of standard deviations. Default is 1.
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Returns (angle_radians, width_radius, height_radius)
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"""
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U,s,v = linalg.svd(P)
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orientation = math.atan2(U[1,0],U[0,0])
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width = deviations*math.sqrt(s[0])
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height = deviations*math.sqrt(s[1])
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assert width >= height
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return (orientation, width, height)
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def is_inside_ellipse(x,y, ex, ey, orientation, width, height):
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co = np.cos(orientation)
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so = np.sin(orientation)
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xx = x*co + y*so
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yy = y*co - x*so
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return (xx / width)**2 + (yy / height)**2 <= 1.
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return ((x-ex)*co - (y-ey)*so)**2/width**2 + \
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((x-ex)*so + (y-ey)*co)**2/height**2 <= 1
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def plot_covariance_ellipse(mean, cov=None, variance = 1.0,
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ellipse=None, title=None, axis_equal=True,
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facecolor='none', edgecolor='blue'):
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""" plots the covariance ellipse where
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mean is a (x,y) tuple for the mean of the covariance (center of ellipse)
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cov is a 2x2 covariance matrix.
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variance is the normal sigma^2 that we want to plot. If list-like,
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ellipses for all ellipses will be ploted. E.g. [1,2] will plot the
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sigma^2 = 1 and sigma^2 = 2 ellipses.
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ellipse is a (angle,width,height) tuple containing the angle in radians,
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and width and height radii.
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You may provide either cov or ellipse, but not both.
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plt.show() is not called, allowing you to plot multiple things on the
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same figure.
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"""
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assert cov is None or ellipse is None
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assert not (cov is None and ellipse is None)
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if cov is not None:
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ellipse = covariance_ellipse(cov)
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if axis_equal:
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plt.axis('equal')
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if title is not None:
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plt.title (title)
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if np.isscalar(variance):
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variance = [variance]
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ax = plt.gca()
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angle = np.degrees(ellipse[0])
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width = ellipse[1] * 2.
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height = ellipse[2] * 2.
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for var in variance:
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sd = np.sqrt(var)
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e = Ellipse(xy=mean, width=sd*width, height=sd*height, angle=angle,
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facecolor=facecolor,
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edgecolor=edgecolor,
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lw=1)
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ax.add_patch(e)
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plt.scatter(mean[0], mean[1], marker='+') # mark the center
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def _to_cov(x,n):
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""" If x is a scalar, returns a covariance matrix generated from it
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as the identity matrix multiplied by x. The dimension will be nxn.
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If x is already a numpy array then it is returned unchanged.
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"""
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try:
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x.shape
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if type(x) != np.ndarray:
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x = np.asarray(x)[0]
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return x
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except:
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return np.eye(n) * x
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def do_plot_test():
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from numpy.random import multivariate_normal
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p = np.array([[32, 15],[15., 40.]])
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x,y = multivariate_normal(mean=(0,0), cov=p, size=5000).T
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sd = 2
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a,w,h = covariance_ellipse(p,sd)
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print (np.degrees(a), w, h)
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count = 0
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color=[]
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for i in range(len(x)):
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if is_inside_ellipse(x[i], y[i], 0, 0, a, w, h):
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color.append('b')
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count += 1
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else:
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color.append('r')
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plt.scatter(x,y,alpha=0.2, c=color)
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plt.axis('equal')
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plot_covariance_ellipse(mean=(0., 0.),
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cov = p,
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variance=sd*sd)
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print (count / len(x))
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if __name__ == '__main__':
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from scipy.stats import norm
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do_plot_test()
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test_gaussian()
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# test conversion of scalar to covariance matrix
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x = multivariate_gaussian(np.array([1,1]), np.array([3,4]), np.eye(2)*1.4)
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x2 = multivariate_gaussian(np.array([1,1]), np.array([3,4]), 1.4)
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assert x == x2
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# test univarate case
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rv = norm(loc = 1., scale = np.sqrt(2.3))
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x2 = multivariate_gaussian(1.2, 1., 2.3)
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x3 = gaussian(1.2, 1., 2.3)
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assert rv.pdf(1.2) == x2
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assert abs(x2- x3) < 0.00000001
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cov = np.array([[1.0, 1.0],
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[1.0, 1.1]])
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plt.figure()
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P = np.array([[2,0],[0,2]])
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plot_covariance_ellipse((2,7), cov=cov, variance=[1,2], title='my title')
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plt.show()
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print("all tests passed")
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