Abstract:
The formulation and method of solving the problem of finding an estimate of the maximum likelihood of the parameters of a mixture of normal multidimensional distributions that allow excluding the occurrence of “infinitely” large values of the likelihood function are considered. It is proposed to condition the set of possible covariance matrices so that they do not become singular. To do this, a boundary $\lambda_0$ is introduced for the eigenvalues of the estimated covariance matrices of the elements of the mixture. Correction of small eigenvalues, smaller than $\lambda_0$, covariance matrices of the elements of the mixture makes it possible to neutralize the arising computational problems with unlimited likelihood, however, at the cost of a rather laborious procedure for replacing the eigenvalues of covariance matrices, if necessary, at each step of the iterative EM (expectation-minimization) algorithm. The justification of the legitimacy of the proposed actions allows not only to obtain a detailed description of the corresponding actions but also to offer options for choosing a new additional parameter of the mixture elements, $\lambda_0$. Estimating mixture parameters under constraints primarily solves computational problems but can affect the efficiency of using mixture estimation. This is illustrated by the example of a data classification problem.
Keywords:mixture of normal multivariate distributions, maximum likelihood estimate, EM-algorithm.
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