基于小波域隐马尔可夫模型的时间序列分析-平滑、插值和预测  

Time Series Analysis Based on Wavelet-Domain HMM-Smoothing,Interpolation and Prediction

作　　者：张冬青[1] 韩玉兵[2] 宁宣熙[1] 刘雪妮[1] 

机构地区：[1]南京航空航天大学经济与管理学院,江苏南京210016 [2]南京理工大学电光学院,江苏南京210094

出　　处：《中国管理科学》2008年第2期122-127,共6页Chinese Journal of Management Science

基　　金：国家自然科学基金资助项目(70571037)

摘　　要：提出一种基于小波域隐马尔可夫模型的时间序列分析方法。首先介绍了离散小波变换;并针对小波系数进行统计建模,分别讨论了单个小波系数的混合高斯模型、不同尺度小波系数之间的隐马尔可夫树结构、模型训练及似然计算等问题;其次,提出了关于时间序列插值、平滑和预测的统一数学模型,并运用极大后验概率估计和贝叶斯原理,将小波域隐马尔可夫模型作为先验知识给出了一种分析时间序列的新方法;然后,详细推导了时间序列重建问题的Euler-Lagrange方程及对数似然的导数计算,将时间序列的插值、平滑和预测归结为一个简单线性方程的求解;最后通过期望极大化(EM)算法和共扼梯度算法进行交替迭代来计算小波域隐马尔可夫模型参数和重建时间序列。实验结果表明该方法在经济领域时间序列分析中的有效性。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 model training 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 u- sing WHMM as the prior knowledge. Thirdly, the Euler-Lagrange equation of the time series and the differential of log-likelihood function 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.

关 键 词：时间序列 小波变换 隐马尔可夫模型 EM算法 共扼梯度算法 

分 类 号：O211.61[理学—概率论与数理统计]