stats¶

A collection of functions used to compute and plot statistics relevant to Bayesian filters.

FilterPy library. http://github.com/rlabbe/filterpy

Documentation at: https://filterpy.readthedocs.org

Supporting book at: https://github.com/rlabbe/Kalman-and-Bayesian-Filters-in-Python

This is licensed under an MIT license. See the readme.MD file for more information.

filterpy.stats.gaussian(x, mean, var, normed=True)[source]¶

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.

Parameters:

xscalar or array-like: The value for which we compute the probability
meanscalar: Mean of the Gaussian
varscalar: Variance of the Gaussian
normbool, default True: Normalize the output if the input is an array of values.

Returns:

probabilityfloat: probability of x for the Gaussian (mean, var). E.g. 0.101 denotes 10.1%.

Examples

>>> gaussian(8, 1, 2)
1.3498566943461957e-06

>>> gaussian([8, 7, 9], 1, 2)
array([1.34985669e-06, 3.48132630e-05, 3.17455867e-08])

filterpy.stats.mul(mean1, var1, mean2, var2)[source]¶

Multiply Gaussian (mean1, var1) with (mean2, var2) and return the results as a tuple (mean, var).

Strictly speaking the product of two Gaussian PDFs is a Gaussian function, not Gaussian PDF. It is, however, proportional to a Gaussian PDF, so it is safe to treat the output as a PDF for any filter using Bayes equation, which normalizes the result anyway.

Parameters:

mean1scalar: mean of first Gaussian
var1scalar: variance of first Gaussian
mean2scalar: mean of second Gaussian
var2scalar: variance of second Gaussian

Returns:

meanscalar: mean of product
varscalar: variance of product

References

Bromily. “Products and Convolutions of Gaussian Probability Functions”, Tina Memo No. 2003-003. http://www.tina-vision.net/docs/memos/2003-003.pdf

Examples

>>> mul(1, 2, 3, 4)
(1.6666666666666667, 1.3333333333333333)

filterpy.stats.add(mean1, var1, mean2, var2)[source]¶

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.

filterpy.stats.log_likelihood(z, x, P, H, R)[source]¶: Returns log-likelihood of the measurement z given the Gaussian posterior (x, P) using measurement function H and measurement covariance error R

filterpy.stats.likelihood(z, x, P, H, R)[source]¶: Returns likelihood of the measurement z given the Gaussian posterior (x, P) using measurement function H and measurement covariance error R

filterpy.stats.logpdf(x, mean=None, cov=1, allow_singular=True)[source]¶

Computes the log of the probability density function of the normal N(mean, cov) for the data x. The normal may be univariate or multivariate.

Wrapper for older versions of scipy.multivariate_normal.logpdf which don’t support support the allow_singular keyword prior to verion 0.15.0.

If it is not supported, and cov is singular or not PSD you may get an exception.

x and mean may be column vectors, row vectors, or lists.

filterpy.stats.multivariate_gaussian(x, mu, cov)[source]¶

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:

xfloat, 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.
mufloat, 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.
covfloat, 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:

probabilityfloat: probability for x for the Gaussian (mu,cov)

filterpy.stats.multivariate_multiply(m1, c1, m2, c2)[source]¶

Multiplies the two multivariate Gaussians together and returns the results as the tuple (mean, covariance).

Parameters:

m1array-like

Mean of first Gaussian. Must be convertable to an 1D array via numpy.asarray(), For example 6, [6], [6, 5], np.array([3, 4, 5, 6]) are all valid.

c1matrix-like

Covariance of first Gaussian. Must be convertable to an 2D array via numpy.asarray().

m2array-like: Mean of second Gaussian. Must be convertable to an 1D array via numpy.asarray(), For example 6, [6], [6, 5], np.array([3, 4, 5, 6]) are all valid.

c2matrix-like

Covariance of second Gaussian. Must be convertable to an 2D array via numpy.asarray().

Returns:

mndarray: mean of the result
cndarray: covariance of the result

Examples

m, c = multivariate_multiply([7.0, 2], [[1.0, 2.0], [2.0, 1.0]],
                             [3.2, 0], [[8.0, 1.1], [1.1,8.0]])

filterpy.stats.plot_gaussian_cdf(mean=0.0, variance=1.0, ax=None, xlim=None, ylim=(0.0, 1.0), xlabel=None, ylabel=None, label=None)[source]¶

Plots a normal distribution CDF with the given mean and variance. x-axis contains the mean, the y-axis shows the cumulative probability.

Parameters:

meanscalar, default 0.: mean for the normal distribution.
variancescalar, default 0.: variance for the normal distribution.
axmatplotlib axes object, optional: If provided, the axes to draw on, otherwise plt.gca() is used.
xlim, ylim: (float,float), optional: specify the limits for the x or y axis as tuple (low,high). If not specified, limits will be automatically chosen to be ‘nice’
xlabelstr,optional: label for the x-axis
ylabelstr, optional: label for the y-axis
labelstr, optional: label for the legend

Returns:

axis of plot

filterpy.stats.plot_gaussian_pdf(mean=0.0, variance=1.0, std=None, ax=None, mean_line=False, xlim=None, ylim=None, xlabel=None, ylabel=None, label=None)[source]¶

Plots a normal distribution PDF with the given mean and variance. x-axis contains the mean, the y-axis shows the probability density.

Parameters:

meanscalar, default 0.: mean for the normal distribution.
variancescalar, default 1., optional: variance for the normal distribution.
std: scalar, default=None, optional: standard deviation of the normal distribution. Use instead of variance if desired
axmatplotlib axes object, optional: If provided, the axes to draw on, otherwise plt.gca() is used.
mean_lineboolean: draws a line at x=mean
xlim, ylim: (float,float), optional: specify the limits for the x or y axis as tuple (low,high). If not specified, limits will be automatically chosen to be ‘nice’
xlabelstr,optional: label for the x-axis
ylabelstr, optional: label for the y-axis
labelstr, optional: label for the legend

Returns:

axis of plot

filterpy.stats.plot_discrete_cdf(xs, ys, ax=None, xlabel=None, ylabel=None, label=None)[source]¶

Plots a normal distribution CDF with the given mean and variance. x-axis contains the mean, the y-axis shows the cumulative probability.

Parameters:

xslist-like of scalars: x values corresponding to the values in y`s. Can be `None, in which case range(len(ys)) will be used.
yslist-like of scalars: list of probabilities to be plotted which should sum to 1.
axmatplotlib axes object, optional: If provided, the axes to draw on, otherwise plt.gca() is used.
xlim, ylim: (float,float), optional: specify the limits for the x or y axis as tuple (low,high). If not specified, limits will be automatically chosen to be ‘nice’
xlabelstr,optional: label for the x-axis
ylabelstr, optional: label for the y-axis
labelstr, optional: label for the legend

Returns:

axis of plot

filterpy.stats.plot_gaussian(mean=0.0, variance=1.0, ax=None, mean_line=False, xlim=None, ylim=None, xlabel=None, ylabel=None, label=None)[source]¶: DEPRECATED. Use plot_gaussian_pdf() instead. This is poorly named, as there are multiple ways to plot a Gaussian.

filterpy.stats.covariance_ellipse(P, deviations=1)[source]¶

Returns a tuple defining the ellipse representing the 2 dimensional covariance matrix P.

Parameters:

Pnd.array shape (2,2): covariance matrix
deviationsint (optional, default = 1): # of standard deviations. Default is 1.
Returns (angle_radians, width_radius, height_radius)

filterpy.stats.plot_covariance_ellipse(mean, cov=None, variance=1.0, std=None, ellipse=None, title=None, axis_equal=True, show_semiaxis=False, facecolor=None, edgecolor=None, fc='none', ec='#004080', alpha=1.0, xlim=None, ylim=None, ls='solid')[source]¶

Deprecated function to plot a covariance ellipse. Use plot_covariance instead.

stats¶

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