Statistical separation of mixtures in the problem of reconstructing the coefficients of an Itô stochastic process-type model of the interplanetary magnetic flux density: <i>ℓ</i> ₂-distance minimization vs likelihood maximization

Abstract Two approaches to the problem of statistical separation of finite mixtures of probability distributions are discussed. The first of them consists in finding maximum likelihood estimates of the parameters of the mixture by the EM-algorithm, whereas the second approach consists in finding the values of the parameters that deliver minimum to the distance between the theoretical and empirical distribution functions. It is demonstrated that the second approach is preferable at least in the problem of statistical reconstruction of the coefficients of an Itô stochastic process that requires dynamic separation of finite normal mixtures in the moving window mode. For this problem, the performance of the numerical procedures is critical. A combination of numerical procedures is described that provides (almost) the same value of the likelihood function for the second approach that is attained by the EM-algorithm, but ensures multiple decrease of the ℓ 2 -distance between the theoretical mixture and the empirical distribution function while demonstrating better performance. A kind of ‘metric’ regularization of the problem of likelihood maximization is proposed. The proposed techniques are illustrated by adjusting the Itô process model to the time series of the interplanetary magnetic field (magnetic flux density) registered by the Global Geospace Science (GGS) Wind apparatus (the spacecraft placed in the Lagrange point between the Earth and the Sun).