Abstract: In this paper,a method for time series,based on wavelet-domain hidden Markov model(WHMM),is proposed.In the first,after introduction of discrete wavelet transform briefly,we use the Gaussian mixture model(GMM) to describe the non-Gaussian feature of an individual wavelet coefficient.To capture the key statistical dependency and persistence property of the joint probability density in the whole wavelet coefficients of real-world signals,the hidden Markov tree(HMT) structure is adopted.The mo del trai山ng and the likelihood determination associated with the WHMM have been thoroughly studied.Then,from the Bayesian viewpoint and under the maximum a posteriori(MAP) probability estimation framework,we develop a model that deals with smoothing,interpolation and prediction of time series using WHMM as the prior knowledge.Thirdly,the Euler-Lagrange equation of the time series and the differential of log-likelihoodfunction have been deduced in detail by means of orthogonal wavelet transform and differential principle.Finally,a concise linear equation about smoothing,interpolation and prediction of time series is obtained and the expectation maximization(EM) algorithm and conjugate gradient(CG) algorithm are adopted to compute the WHMM parameters and reconstruct the time series alternately.Experimental results are so pleasant that WHMM can be applied in the time series of economic sphere.

Key words: time series, wavelet transform, hidden markov model, expectation maximization algorithm, conjugate gradient algorithm