""" Author: Roger Labbe Copyright: 2014 This code performs various basic statistics functions for the Kalman and Bayesian Filters in Python book. Much of this code is non-optimal; production code should call the equivalent scipy.stats functions. I wrote the code in this form to make explicit how the computations are done. The scipy.stats module has many more useful functions than what I have written here. In some cases, however, my code is significantly faster, at least on my machine. For example, gaussian average 794 ns, whereas stats.norm(), using the frozen form, averages 116 microseconds per call. """ from __future__ import (absolute_import, division, print_function, unicode_literals) import math import numpy as np import numpy.linalg as linalg import matplotlib.pyplot as plt import scipy.sparse as sp import scipy.sparse.linalg as spln import scipy.stats from matplotlib.patches import Ellipse _two_pi = 2*math.pi def gaussian(x, mean, var): """returns normal distribution (pdf) for x given a Gaussian with the specified mean and variance. All must be scalars. gaussian (1,2,3) is equivalent to scipy.stats.norm(2,math.sqrt(3)).pdf(1) It is quite a bit faster albeit much less flexible than the latter. @param x test Parameters ---------- x : scalar The value for which we compute the probability mean : scalar Mean of the Gaussian var : scalar Variance of the Gaussian Returns ------- probability : float probability of x for the Gaussian (mean, var). E.g. 0.101 denotes 10.1%. """ return math.exp((-0.5*(x-mean)**2)/var) / math.sqrt(_two_pi*var) def mul (mean1, var1, mean2, var2): """ multiply Gaussians (mean1, var1) with (mean2, var2) and return the results as a tuple (mean,var). var1 and var2 are variances - sigma squared in the usual parlance. """ mean = (var1*mean2 + var2*mean1) / (var1 + var2) var = 1 / (1/var1 + 1/var2) return (mean, var) def add (mean1, var1, mean2, var2): """ add the Gaussians (mean1, var1) with (mean2, var2) and return the results as a tuple (mean,var). var1 and var2 are variances - sigma squared in the usual parlance. """ return (mean1+mean2, var1+var2) def multivariate_gaussian(x, mu, cov): """ This is designed to replace scipy.stats.multivariate_normal which is not available before version 0.14. You may either pass in a multivariate set of data: multivariate_gaussian (array([1,1]), array([3,4]), eye(2)*1.4) multivariate_gaussian (array([1,1,1]), array([3,4,5]), 1.4) or unidimensional data: multivariate_gaussian(1, 3, 1.4) In the multivariate case if cov is a scalar it is interpreted as eye(n)*cov The function gaussian() implements the 1D (univariate)case, and is much faster than this function. equivalent calls: multivariate_gaussian(1, 2, 3) scipy.stats.multivariate_normal(2,3).pdf(1) Parameters ---------- x : scalar, or np.array-like Value to compute the probability for. May be a scalar if univariate, or any type that can be converted to an np.array (list, tuple, etc). np.array is best for speed. mu : scalar, or np.array-like mean for the Gaussian . May be a scalar if univariate, or any type that can be converted to an np.array (list, tuple, etc).np.array is best for speed. cov : scalar, or np.array-like Covariance for the Gaussian . May be a scalar if univariate, or any type that can be converted to an np.array (list, tuple, etc).np.array is best for speed. Returns ------- probability : float probability for x for the Gaussian (mu,cov) """ scipy.stats.multivariate_normal # force all to numpy.array type x = np.array(x, copy=False, ndmin=1) mu = np.array(mu,copy=False, ndmin=1) nx = len(mu) cov = _to_cov(cov, nx) norm_coeff = nx*math.log(2*math.pi) + np.linalg.slogdet(cov)[1] err = x - mu if (sp.issparse(cov)): numerator = spln.spsolve(cov, err).T.dot(err) else: numerator = np.linalg.solve(cov, err).T.dot(err) return math.exp(-0.5*(norm_coeff + numerator)) def plot_gaussian(mean, variance, mean_line=False, xlim=None, xlabel=None, ylabel=None): """ plots the normal distribution with the given mean and variance. x-axis contains the mean, the y-axis shows the probability. mean_line : draws a line at x=mean xlim: optionally specify the limits for the x axis as tuple (low,high). If not specified, limits will be automatically chosen to be 'nice' xlabel : optional label for the x-axis ylabel : optional label for the y-axis """ sigma = math.sqrt(variance) n = scipy.stats.norm(mean, sigma) if xlim is None: min_x = n.ppf(0.001) max_x = n.ppf(0.999) else: min_x = xlim[0] max_x = xlim[1] xs = np.arange(min_x, max_x, (max_x - min_x) / 1000) plt.plot(xs,n.pdf(xs)) plt.xlim((min_x, max_x)) if mean_line: plt.axvline(mean) if xlabel: plt.xlabel(xlabel) if ylabel: plt.ylabel(ylabel) def covariance_ellipse(P, deviations=1): """ returns a tuple defining the ellipse representing the 2 dimensional covariance matrix P. Parameters ---------- P : nd.array shape (2,2) covariance matrix deviations : int (optional, default = 1) # of standard deviations. Default is 1. Returns (angle_radians, width_radius, height_radius) """ U,s,v = linalg.svd(P) orientation = math.atan2(U[1,0],U[0,0]) width = deviations*math.sqrt(s[0]) height = deviations*math.sqrt(s[1]) assert width >= height return (orientation, width, height) def is_inside_ellipse(x,y, ex, ey, orientation, width, height): co = np.cos(orientation) so = np.sin(orientation) xx = x*co + y*so yy = y*co - x*so return (xx / width)**2 + (yy / height)**2 <= 1. return ((x-ex)*co - (y-ey)*so)**2/width**2 + \ ((x-ex)*so + (y-ey)*co)**2/height**2 <= 1 def plot_covariance_ellipse(mean, cov=None, variance = 1.0, ellipse=None, title=None, axis_equal=True, facecolor='none', edgecolor='#004080', alpha=1.0): """ plots the covariance ellipse where mean is a (x,y) tuple for the mean of the covariance (center of ellipse) cov is a 2x2 covariance matrix. variance is the normal sigma^2 that we want to plot. If list-like, ellipses for all ellipses will be ploted. E.g. [1,2] will plot the sigma^2 = 1 and sigma^2 = 2 ellipses. ellipse is a (angle,width,height) tuple containing the angle in radians, and width and height radii. You may provide either cov or ellipse, but not both. plt.show() is not called, allowing you to plot multiple things on the same figure. """ assert cov is None or ellipse is None assert not (cov is None and ellipse is None) if cov is not None: ellipse = covariance_ellipse(cov) if axis_equal: plt.axis('equal') if title is not None: plt.title (title) if np.isscalar(variance): variance = [variance] ax = plt.gca() angle = np.degrees(ellipse[0]) width = ellipse[1] * 2. height = ellipse[2] * 2. for var in variance: sd = np.sqrt(var) e = Ellipse(xy=mean, width=sd*width, height=sd*height, angle=angle, facecolor=facecolor, edgecolor=edgecolor, alpha=alpha, lw=2) ax.add_patch(e) plt.scatter(mean[0], mean[1], marker='+') # mark the center def _to_cov(x,n): """ If x is a scalar, returns a covariance matrix generated from it as the identity matrix multiplied by x. The dimension will be nxn. If x is already a numpy array then it is returned unchanged. """ try: x.shape if type(x) != np.ndarray: x = np.asarray(x)[0] return x except: return np.eye(n) * x def do_plot_test(): from numpy.random import multivariate_normal p = np.array([[32, 15],[15., 40.]]) x,y = multivariate_normal(mean=(0,0), cov=p, size=5000).T sd = 2 a,w,h = covariance_ellipse(p,sd) print (np.degrees(a), w, h) count = 0 color=[] for i in range(len(x)): if is_inside_ellipse(x[i], y[i], 0, 0, a, w, h): color.append('b') count += 1 else: color.append('r') plt.scatter(x,y,alpha=0.2, c=color) plt.axis('equal') plot_covariance_ellipse(mean=(0., 0.), cov = p, variance=sd*sd, facecolor='none') print (count / len(x)) if __name__ == '__main__': from scipy.stats import norm do_plot_test() #test_gaussian() # test conversion of scalar to covariance matrix x = multivariate_gaussian(np.array([1,1]), np.array([3,4]), np.eye(2)*1.4) x2 = multivariate_gaussian(np.array([1,1]), np.array([3,4]), 1.4) assert x == x2 # test univarate case rv = norm(loc = 1., scale = np.sqrt(2.3)) x2 = multivariate_gaussian(1.2, 1., 2.3) x3 = gaussian(1.2, 1., 2.3) assert rv.pdf(1.2) == x2 assert abs(x2- x3) < 0.00000001 cov = np.array([[1.0, 1.0], [1.0, 1.1]]) plt.figure() P = np.array([[2,0],[0,2]]) plot_covariance_ellipse((2,7), cov=cov, variance=[1,2], facecolor='g', title='my title', alpha=.2) plt.show() print("all tests passed")