The autoregressive-integrated moving average (AMIRA) process plays an essential role in time series models. Classical method of finding the maximum likelihood (ML) estimate of the parameters in an ARIMA(p; d; q) model consists of evaluating the likelihood function through the Box-Jenkins approach or the Innovations Algorithm and optimizing it by numerical methods such as the quasi-Newton algorithms. However, these approaches have several drawbacks. The quasi-Newton methods tend to be unstable when the likelihood function is highly nonlinear. In this paper, we consider a state space representation of the ARIMA(p; d; q) process. The likelihood function can be easily expressed by the Kalman filter and the ML estimates can be obtained through a combination of Kalman smoother and the EM Algorithm. The updating equations in the EM algorithm possess a simple analytical form. A quasi-Newton scheme has also been implemented to accelerate the convergence of the EM Algorithm. The simulations studies show that the EM algorithm is more robust to starting values and the number of parameters, and the quasi-Newton acceleration scheme significantly improves the convergence rate of the EM algorithm. / 差分自回歸移動平均(AMIRA)模型在時間序列模型中有著重要地位。ARIMA模型的傳統極大似然估計方法通過Box-Jenkins方法或者新息算法(Innovations Algorithm)計算出似然函數，再通過擬牛頓(quasi-Newton)法等數值方法將之極大化，從而得到參數的極大似然估計。然而，此類方法在一定條件下存在缺陷。例如，當似然函數高度非線性時，擬牛頓法表現出不穩定的現象。本文考慮ARIMA模型的一種狀態空間(state-space)模型表示。在此表示下，參數的似然函數可以通過卡爾曼濾波算法計算，而參數的極大似然估計可以通過卡爾曼平滑和EM算法簡單得出。本問題中EM算法的迭代公式有簡潔的解析形式。同時，我們進一步考慮了一個擬牛頓加速算法來加快EM算法的收斂速度。通過模擬實驗我們發現，對於不同的初始值和參數個數，EM算法比擬牛頓法更為穩健。同時，擬牛頓的加速算法可以顯著加快EM算法的收斂速度。 / Huang, Rui. / Thesis M.Phil. Chinese University of Hong Kong 2015. / Includes bibliographical references (leaves 57-58). / Abstracts also in Chinese. / Title from PDF title page (viewed on 06, October, 2016). / Detailed summary in vernacular field only.
| Identifer | oai:union.ndltd.org:cuhk.edu.hk/oai:cuhk-dr:cuhk_1291482 |
| Date | January 2015 |
| Contributors | Huang, Rui , active 2015 (author.), Chan, Ngai Hang (thesis advisor.), Chinese University of Hong Kong Graduate School. Division of Risk Management Science. (degree granting institution.) |
| Source Sets | The Chinese University of Hong Kong |
| Language | English, Chinese |
| Detected Language | English |
| Type | Text, bibliography, text |
| Format | electronic resource, electronic resource, remote, 1 online resource (vii, 58 leaves), computer, online resource |
| Rights | Use of this resource is governed by the terms and conditions of the Creative Commons "Attribution-NonCommercial-NoDerivatives 4.0 International" License (http://creativecommons.org/licenses/by-nc-nd/4.0/) |
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