Abstract
Problems of statistical analysis of discrete-valued time series are considered. Two approaches for construction of parsimonious (small-parametric) models for observed discrete data are proposed based on high-order Markov chains.Consistent statistical estimators for parameters of the developed models and some known models, and also statistical tests on the values of parameters are constructed. Probabilistic properties of the constructed statistical inferences are given. The developed theory is also applied for statistical analysis of spatio-temporal data. Theoretical results are illustrated by computer experiments on real statistical data.
Highlights
Time series analysis is deep developed (Anderson 1971) for “continuous” data when the observation space A is some Euclidean space or its subspace of nonzero Lebesque measure: A⊆Rm, mes(A) > 0
Number of independent parameters for the MC(s) model increases exponentially w.r.t. the memory depth s: DMC(s) = N s(N − 1). To identify this model (1) we need to have huge data set and the computation work of size O N s+1. To avoid this “curse of dimensionality” we propose to use the parsimonious (“small-parametric”) models of high-order Markov chains that are determined by small number of parameters d DMC(s) (Kharin 2013)
Frequencies Based Estimator (FBE) (17) has the following significant advantages w.r.t. the maximum likelihood estimator (MLE): 1) explicit expression of FBE w.r.t. the iterative computation of MLE; 2) fast iterative computation of FBE if we extend the basis {ψi(·)}; 3) possibility to control the computational complexity of FBE by variation of the matrix H
Summary
Time series analysis is deep developed (Anderson 1971) for “continuous” data when the observation space A is some Euclidean space or its subspace of nonzero Lebesque measure: A⊆Rm, mes(A) > 0. An universal base model for discrete-valued time series xt ∈ A is the homogeneous Markov chain MC(s) of some order s ∈ N0, determined by the generalized Markov property (t > s): P{xt=it|xt−1=it−1, . Number of independent parameters for the MC(s) model increases exponentially w.r.t. the memory depth s: DMC(s) = N s(N − 1) To identify this model (1) we need to have huge data set and the computation work of size O N s+1. To avoid this “curse of dimensionality” we propose to use the parsimonious (“small-parametric”) models of high-order Markov chains that are determined by small number of parameters d DMC(s) (Kharin 2013)
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