隱馬爾可夫模型中的擬最大概似估計

Pseudo Maximum Likelihood in Hidden Markov Model

石儲輔(Chu-Fu Shih)

指導教授 ： 陳宏

國立臺灣大學/理學院/應用數學科學研究所/碩士(2016年)

https://doi.org/10.6342/NTU201600665

摘要

隱馬爾可夫模型是應用統計學的基本工具，計量經濟學，和機器學習，把數據當作從多個亞群產生的資料。當觀測的序列是從一個離散時間，有限狀態隱馬爾可夫模型，估計這些模型的參數，目前的做法依賴於局部搜索演算法如EM算法。基於觀察的轉變，轉移計數是隱藏的轉移矩陣的充分統計，並且由觀測形成的適當的統計信息可以作為一個替代，一命名為配對方法的新方法擬作為過渡矩陣和參數在隱馬爾可夫模型的初始估計。在適當規律性的條件下，它可以證明，EM由下式給出一個合適的初始估計，可導致最大概似估計。然而，目前沒有在隱馬爾可夫模型尋找合適的初始點的方法。配對方法可以提供一個良好的初始參數估計可加快EM在計算時間方面。當底層狀態轉移矩陣不考慮，邊際分佈將一個混合分佈，而在狀態轉移矩陣只有有限的信息保存為推論。為了萃取包含在過渡矩陣中的數據的完整信息，我們通過擴大馬爾可夫鏈，萃取動態轉移矩陣的信息。
提供隱藏轉移矩陣的一致性和漸近常態的估計。

關鍵字

隱馬爾可夫鏈 ； 狀態轉換 ； 混合模型 ； 混合狀態 ； EM算法 ； 最大概似估計

並列摘要

Hidden Markov models are a fundamental tool in applied statistics, econometrics, and machine learning for treating data taken from multiple subpopulations. When the sequence of observations is from a discrete-time, finite-state hidden Markov model, the current practice for estimating the parameters of such models relies on local search heuristics such as the EM algorithm. A new method named as pairing method is proposed to serve as an initial estimate of the transition matrix and parameters in hidden Markov models. Under regularity conditions, it can be shown that EM leads to the maximum likelihood estimator by given a suitable initial estimate. However, there is no method of finding suitable initial points in hidden Markov model. Pairing method can provide a good initial parameter estimate which can expedite EM in terms of computing time.When the underlying state transition matrix is not taken into consideration, the marginal distribution will be
a mixture distribution while only limited information on state transition matrix is kept for inference. In order to recover full information contained in the data on transition matrix, we utilize characteristics of stochastic matrix by enlarging the Markov chain to recover information governing dynamic of transition matrix. Consistent and asymptotic normal estimators of hidden transition matrix are provided.

並列關鍵字

EM algorithm ； hidden ； Markov chain ； mixing condition ； maximum likelihood ； mixture model ； regime switching

參考文獻

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