Distributions

Distributions

A series of classes for modeling distributions and stochastic processes.

If you use these classes, please cite the following paper,

• J. Early and A. Sykulski. Smoothing and interpolating noisy GPS data with smoothing splines. IEEE Transactions on Signal Processing. In prep.

Table of contents

1. Quick start
2. Distribution class

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Quick start

The Distribution class is an abstract class, so in practice one only uses its subclasses such as NormalDistribution directly. For example, to create a Gaussian distribution,

sigma = 1;
distribution = NormalDistribution(sigma);

and now you can create some random values,

n = 1000;
z = distribution.rand([n 1]);

and plot those values,

figure
histogram(z,'Normalization','pdf')

which can be compared against the expected pdf,

zdist = linspace(min(z),max(z),100)';
hold on
plot(zdist,distribution.pdf(zdist))

All distribution subclasses include the cdf as a function, as well as the total variance of the process.

Correlated values

It is also sometimes useful to correlate the random values in order to create a stochastic process. Let's define an autocorrelation sequence,

tau = 10;
distribution.rho = @(z) exp(-(z/tau).^2);

and now generate a signal,

t = (0:500).';
z = distribution.noise(t);

The signal now looks smooth on time scales less than O(tau).

Overview

The Distribution class is an abstract class, but most subclasses only need to define a pdf and cdf in order to gain most of the useful functionality, such as random number generation, stochastic process generation, as well as error metrics like the Anderson-Darling and Kolmogorov-Smirnov errors.

There are currently five built-in distribution subclasses,

• NormalDistribution
• StudentTDistribution
• RayleighDistribution
• AddedDistribution
• TwoDimDistanceDistribution

Distribution class

The distribution class represents a univariate probability distribution. As such the class requires the following properties,

• pdf, the probability density function defined as a function, e.g., @(z) exp(-(z.*z)/(2*sigma*sigma))/(sigma*sqrt(2*pi)).
• cdf, the cumulative distribution function defined as a function, e.g., @(z) 0.5*(1 + erf(z/(sigma*sqrt(2)))).

Optionally, one can also define

• rho, the autocorrelation function of the process.

When the cdf is defined, the following functions can be used,

• rand(), generates random numbers with the cdf.
• locationOfCDFPercentile() returns the location of a given cdf percentile.
• varianceInRange() returns the variance between two cdf percentiles.
• varianceInPercentileRange() returns the total variance in a given percentile range.
• kolmogorovSmirnovError(), returns the Kolmogorov-Smirnov test statistic for a given set of data.
• andersonDarlingError(), returns the Anderson-Darling test statistic for a given set of data.

Once rho is also defined,

• noise() returns a signal with increments following rand and autocorrelation sequence following rho.

Normal distribution

The NormalDistribution does the obvious thing and adds a single additional property, sigma and implements a normal distribution.

Student's t-distribution

The StudentTDistribution has additional two properties, sigma and nu, and implements a Student's t-distribution.

Added distribution

The AddedDistribution is a univariate mixture distribution, essentially the sum of two (or more) distributions.

The class is initialized with AddedDistribution(scalings,distribution1,distribution2,varargin) where scalings is a vector that must sum to 1, followed by two (or more) distributions

Two dimensional distance distribution

The TwoDimDistanceDistribution takes a single one-dimensional distribution and returns the radial distribution of two independent distribution on x-y. Thus, for a normal distribution this gives you an emirically computed Rayleigh distribution.