mvmdist-master.zip_MVMD-034_finisho8e_von Mises 分布

# mvmdist A Matlab package for probabilistic modeling of circular data with mixtures of von Mises distributions. This package provides a class-based interface, similar to MATLAB's build-in functions for handling [Gaussian mixture models](https://de.mathworks.com/help/stats/gmdistribution.html). For further information on circular probability distributions and von Mises mixture models in particular, these papers give a comprehensive overview on the general concepts: * [J. Bentley (2006), "Modelling Circular Data using a Mixture of von Mises and Uniform Distributions"](https://www.stat.sfu.ca/content/dam/sfu/stat/alumnitheses/MiscellaniousTheses/Bentley-2006.pdf) * [I. S. Dhillon and S. Sra (2003), "Modeling Data using Directional Distributions"](http://www.cs.utexas.edu/users/inderjit/public_papers/tr03-06.pdf) ## Installation 1. Clone or download this repository and add it to your MATLAB search path. 2. Compile the *.mex-file located at `@VonMisesMixture/private` by browsing to that directory in MATLAB and running ```mex sampleVonMisesMex.c``` from the command line. ## Examples ### Constructing a von Mises mixture model The following code creates a von Mises mixture model with two mixture components. ```matlab p = [0.5; 0.5]; % Mixture weights. mu = [-pi/2; pi/2]; % Component means. kappa = [5; 10]; % Concentration parameters of components. vmm = VonMisesMixture(p, mu, kappa); ``` ### Plotting the probability density function Given a vector of angular values, the probability density function of a von Mises mixture model can be computed with the ```pdf()``` method provided by the ```VonMisesMixture``` class. The following code example plots the probability density function of a ```VonMisesMixture``` class instance. ```matlab angles = linspace(-pi, pi, 1000)'; % The pdf() function expects a column-vector as input. likelihoods = vmm.pdf(angles); plot(angles, likelihoods); grid on; ``` ### Drawing samples from a von Mises mixture model This implementation uses the method introduced by [Barabesi (2005)](http://sa-ijas.stat.unipd.it/sites/sa-ijas.stat.unipd.it/files/417-426.pdf) to generate samples from a von Mises distribution. To speed up the sampling process, a mex-function is used by default which is significantly faster than the plain MATLAB implementation (especially for a large number of samples). ```matlab nSamples = 10000; samples = vmm.random(nSamples); % Generate samples using *.mex-function. ``` If the plain MATLAB sampling method should be used, a second argument has to be passed to the ```random()``` function which sets the internal ```useMex``` flag to ```false```: ```matlab samples = vmm.random(nSamples, false); % Generate samples using MATLAB implementation. ``` ### Maximum likelihood parameter estimation Parameter estimation is conducted via an Expectation-Maximization (EM) approach described in [Hung et al. (2012)](http://www.tandfonline.com/doi/abs/10.1080/02664763.2012.706268). The following example shows how a model can be fitted on a set of data samples drawn from an initial von Mises mixture distribution with three components. ```matlab p = [0.3; 0.4; 0.3]; % Mixture weights. mu = [-pi/2; 0; pi/3]; % Component means. kappa = [5; 10; 2.5]; % Concentration parameters of components. vmm = VonMisesMixture(p, mu, kappa); % Initialize model. samples = vmm.random(10000); % Draw 10000 samples. % Fit new model on data samples assuming 3 mixture components. nComponents = 3; fittedVmm = fitmvmdist(samples, nComponents, ... 'MaxIter', 250); % Set maximum number of EM iterations to 250 % Plot initial and fitted distributions. angles = linspace(-pi, pi, 1000)'; likelihoodsInitial = vmm.pdf(angles); likelihoodsFitted = fittedVmm.pdf(angles); plot(angles, likelihoodsInitial); hold on; plot(angles, likelihoodsFitted); grid on; axis([-pi, pi, 0, 1]); ```