The Expectation Maximization(EM) algorithm estimates the parameters of the multivariate probability density function in the form of a Gaussian mixture distribution with a specified number of mixtures.
Bilmes98 J. A. Bilmes. *A Gentle Tutorial of the EM Algorithm and its Application to Parameter Estimation for Gaussian Mixture and Hidden Markov Models*. Technical Report TR-97-021, International Computer Science Institute and Computer Science Division, University of California at Berkeley, April 1998.
Parameters of the EM algorithm. All parameters are public. You can initialize them by a constructor and then override some of them directly if you want.
..ocv:function:: CvEMParams::CvEMParams( int nclusters, int cov_mat_type=CvEM::COV_MAT_DIAGONAL, int start_step=CvEM::START_AUTO_STEP, CvTermCriteria term_crit=cvTermCriteria(CV_TERMCRIT_ITER+CV_TERMCRIT_EPS, 100, FLT_EPSILON), const CvMat* probs=0, const CvMat* weights=0, const CvMat* means=0, const CvMat** covs=0 )
:param nclusters:The number of mixture components in the gaussian mixture model. Some of EM implementation could determine the optimal number of mixtures within a specified value range, but that is not the case in ML yet.
:param cov_mat_type:Constraint on covariance matrices which defines type of matrices. Possible values are:
***CvEM::COV_MAT_SPHERICAL** A scaled identity matrix :math:`\mu_k * I`. There is the only parameter :math:`\mu_k` to be estimated for earch matrix. The option may be used in special cases, when the constraint is relevant, or as a first step in the optimization (for example in case when the data is preprocessed with PCA). The results of such preliminary estimation may be passed again to the optimization procedure, this time with ``cov_mat_type=CvEM::COV_MAT_DIAGONAL``.
***CvEM::COV_MAT_DIAGONAL** A diagonal matrix with positive diagonal elements. The number of free parameters is ``d`` for each matrix. This is most commonly used option yielding good estimation results.
***CvEM::COV_MAT_GENERIC** A symmetric positively defined matrix. The number of free parameters in each matrix is about :math:`d^2/2`. It is not recommended to use this option, unless there is pretty accurate initial estimation of the parameters and/or a huge number of training samples.
***CvEM::START_E_STEP** Start with Expectation step. You need to provide means :math:`a_k` of mixture components to use this option. Optionally you can pass weights :math:`\pi_k` and covariance matrices :math:`S_k` of mixture components.
***CvEM::START_M_STEP** Start with Maximization step. You need to provide initial probabilites :math:`p_{i,k}` to use this option.
***CvEM::START_AUTO_STEP** Start with Expectation step. You need not provide any parameters because they will be estimated by the k-means algorithm.
:param term_crit:The termination criteria of the EM algorithm. The EM algorithm can be terminated by the number of iterations ``term_crit.max_iter`` (number of M-steps) or when relative change of likelihood logarithm is less than ``term_crit.epsilon``.
:param probs:Initial probabilities :math:`p_{i,k}` of sample :math:`i` to belong to mixture component :math:`k`. It is a floating-point matrix of :math:`nsamples \times nclusters` size. It is used and must be not NULL only when ``start_step=CvEM::START_M_STEP``.
:param weights:Initial weights :math:`\pi_k` of mixture components. It is a floating-point vector with :math:`nclusters` elements. It is used (if not NULL) only when ``start_step=CvEM::START_E_STEP``.
:param means:Initial means :math:`a_k` of mixture components. It is a floating-point matrix of :math:`nclusters \times dims` size. It is used used and must be not NULL only when ``start_step=CvEM::START_E_STEP``.
:param covs:Initial covariance matrices :math:`S_k` of mixture components. Each of covariance matrices is a valid square floating-point matrix of :math:`dims \times dims` size. It is used (if not NULL) only when ``start_step=CvEM::START_E_STEP``.
With another contstructor it is possible to override a variety of parameters from a single number of mixtures (the only essential problem-dependent parameter) to initial values for the mixture parameters.
CvEM
----
..ocv:class:: CvEM
The class implements the EM algorithm as described in the beginning of this section.
:param samples:Samples from which the Gaussian mixture model will be estimated.
:param sample_idx:Mask of samples to use. All samples are used by default.
:param params:Parameters of the EM algorithm.
:param labels:The optional output "class label" for each sample: :math:`\texttt{labels}_i=\texttt{arg max}_k(p_{i,k}), i=1..N` (indices of the most probable mixture component for each sample).
Unlike many of the ML models, EM is an unsupervised learning algorithm and it does not take responses (class labels or function values) as input. Instead, it computes the
*Maximum Likelihood Estimate* of the Gaussian mixture parameters from an input sample set, stores all the parameters inside the structure:
:math:`p_{i,k}` in ``probs``,
:math:`a_k` in ``means`` ,
:math:`S_k` in ``covs[k]``,
:math:`\pi_k` in ``weights`` , and optionally computes the output "class label" for each sample:
For each training sample :math:`i` (that have been passed to the constructor or to :ocv:func:`CvEM::train`) returns probabilites :math:`p_{i,k}` to belong to a mixture component :math:`k`.
:param fs:A file storage with parameters of the EM algorithm.
:param node:The parent map. If it is NULL, the function searches a node with parameters in all the top-level nodes (streams), starting with the first one.
The function reads EM parameters from the specified file storage node. For example of clustering random samples of multi-Gaussian distribution using EM see em.cpp sample in OpenCV distribution.
