ML parameter estimation of a multiscale stochastic process using the EM algorithm

Abstract

An algorithm for estimation of the parameters of a multiscale stochastic process based on scale-recursive dynamics on trees is presented. The expectation-maximization algorithm is used to provide maximum likelihood estimates for the general case of a nonhomogeneous tree with no fixed structure for the process dynamics. Experimental results are presented using synthetic data.

Use of Scale Recursive Estimation for assimilation of precipitation data from TRMM (PR and TMI) and NEXRAD

“…In the upward sweep at the leaf nodes, the innovation (suffix I) error variance is given by The EM algorithm is based on two main steps, here shown in their very core for their application in the case of the multi-scale Kalman filter. The reader is referred for a deeper insight to Chou et al [14] and Kannan et al [40]. The ''Estimation'' step calculates some expected quantities required for the maximization of the Likelihood function, written in terms of the observed and estimated quantities.…”

“…This is based on two main steps, here briefly commented (the reader is referred for a deeper insight to [24,15,40]). The ''Expectation'' step, which is carried out after the SRE, calculates some expected statistics on the basis of the filtered estimates and the measurements.…”

“…For the lognormal random cascade model, the log-likelihood function, L log , or logarithmic expected probability of the observations, can be written (see e.g. [40,28])…”

“…Gupta et al [34] showed the use of Maximum Likelihood Estimation (henceforth, MLE), using an estimation maximization (henceforth, EM) approach (e.g. [40]) for system identification. Bocchiola and Rosso [10] applied SRE coupled with EM to the analysis of rainfall time series from single rain gages under the hypothesis of random cascade in the time domain (e.g.…”

Unsupervised image classification with a hierarchical EM algorithm

“…Then the EM algorithm can be conducted without any PL and Monte Carlo approximation ( [10,14] for discrete cases, [8,9] for continuous Gaussian models).…”