一個有外生變數 ARMA 模型的建立以及利用卡爾曼濾波器對其模型之預測做修正

Abstract

研究動機: 卡爾曼濾波器 (Kalman Filter) 是可以隨者新增的訊息來持續修 正估計值的演算法，由於其計算便捷，所以過去已被廣泛應用於不同類型 時間序列的資料的分析。而對於 ARMA 模型可以辨識的時間序列中，卡爾 曼濾波器也可以對 ARMA 模型的預測值做修正，此外，卡爾曼濾波器也有 者配合增加外生變數 (Exogenous variable) 的彈性。\n研究目的: 本文探討如何以有外生變數的 ARMA 模型對這類的資料做建 構，及如何以卡爾曼濾波器對該模型的預測值做修正。\n研究創新: 對於有外生變數的 ARMA 模型的建立，我們提出按順序先分離外部影響的方法，除了提高 ARMA 模型辨識能力，也在後續更方便建立 有外生變數之 ARMA 模型。\n研究方法: 本文利用 ARIMA 模型以及狀態空間模型(State-space model) 以及卡爾曼濾波器，來進一步討論新模型的設計及預測修正。\n研究結果:部分 ARIMA 模型在無法被辨識的時間序列資料，利用本研 究提出之分離外部影響方法後，即出現可辨識的 ARMA 模型，因此可以新 模型作替代。另外，比較卡爾曼濾波器在新模型以及 ARIMA 模型的預測修 正，卡爾曼濾波器在新模型的預測修正取得較小的誤差。

Motivation: Kalman Filter is an algorithm which can update the estimate with adding current information. Due to computing conveniently, the filter has been broadly applied in various type of time series data. Kalman Filter can also applied in data identified by ARMA model, directly. In addition, the filter is flexible for extending exogenous variable.\nObjective: In some time series data with exogenous factors, the applications of Kalman filter are still limited. Therefore, the research focus on how to construct ARMA model with exogenous variables and how to applied by Kalman Filter.\nInnovation: For construction of ARMA model with exogenous variable, we propose orderly steps about method of separating the exogenous influence, im- proved the recognizing ability of model, and integrating both ARMA model and exogenous influence. In addition, we analyze the application of Kalman filter in the model.\nMethod: The research use Kalman Filter, ARIMA model and State-space Representation for design new model and applied by Kalman Filter.\nConclusion: Some time series data which failed to be identified by ARIMA model can be identified by ARMA model after process of decomposing exogenous influence method. Moreover, comparing with Kalman filter applied in ARIMA model, the predction of Kalman filter applied in new model get smaller error.