時間序列類型的資料在日常生活的多種應用領域皆無所不在，其中應用於像是金融經
濟相關領域，以及自然環境科學等領域當中。而在單變量的時間序列的模型建構上，移動
平均自迴歸模型(ARMA) 因具有許多良好的性質，而成為單變量時間序列建模中的熱門方
式。這類模型還可以近一步與季節性整合運算結合，發展成更強而有力的架構，季節性整合
移動平均自迴歸模型(SARIMA). 在多變量時間序列上，向量自迴歸模型為實際應用上最受
眾人使用的模型之一，因為它可以以迴歸模型的形式示人外，其亦可以以解析解的方式被
進行推論。這篇論文主要討論關於季節性整合移動平均自迴歸模型以及向量自迴歸模型配
適流程的探討外，其中更特別著重於各類模型在R 語言當中的自動化配適函數的開發。與
現在已存在於套件forecast 的自動化配適函數auto.arima 相比，這篇論文是透過交叉驗證
(cross-validation) 的方法去決定模型內的架構參數，以及滯後順序選擇的使用，來發展模型
的自動化配適。而在數值模擬實驗當中，此論文所開發的自動化配適函數，可以獲得較過去
已開發函數更為有效的模型配適架構，其所配適的模型可以更為正確得篩選到正確的模型架
構，且在測試集上有更佳的預測表現。但此論文中所開發的配適函數在資料量較大時，則需
要花上更多的計算時間。最後將這個新方法套用於香水百合銷量以及銷價的資料集中，可以
發現不僅僅在整個資料集上都有較佳的表現性能外，在信息標準(information criterion) 的評
量下也可以得到更好的計算成果。此外在多變量模型向量自迴歸模型的自動配適上，此論
文開發函數var.auto 在向量自迴歸模型於R 語言中，提供了自動配適的想法以及滾動窗口
(rolling windows) 預測的開發。而此函數在數值模擬實驗與實際應用資料集中也都有相當理
想的表現成績。

Time series data are ubiquitous with important applications in finance and economicrelated
fields, environmental science, and many other fields. For modeling univariate time
series, the autoregressive and moving average (ARMA) models are the most popular model
class with good properties. This family can further integrate with seasonal operators and integrations
to form a powerful model class: integrated seasonal ARMA (SARIMA) models. For
modeling multivariate time series, the vector autoregression (VAR) is the most popular model
used in practical applications due to its convenient representation in a regression form such
that the inferences can be done with closed forms. This thesis discusses the fitting procedures
for univariate SARIMA models and VAR models, in particular the autofitting algorithms are
developed in R. In contrast to the existing autofit function auto.arima in R package forecast
Hyndman and Khandakar (2008), this study adopts a cross-validation approach for model structure
determination and order selection in the auto fitting procedures. Numerical studies show
that the proposed algorithm performs effectively compared to the existing autofit algorithm in
terms of the correctness of the model structure and smaller prediction errors for test data. But,
the new algorithm requires more computation time, for a series with large data size. Finally,
the methodology was applied to the lily data set and found that the proposed procedure not
only achieved better performance on prediction, but also achieve smaller information criterion
values under both univariate and multivariate settings. Besides univariate ARIMA model auto
fitting, the R function var.auto implements automatic fitting VAR and performs rolling windows
prediction. This methodology has satisfactory performance shown in simulation studies and
the application data for lily data.

摘要i
Abstract ii
List of Figures v
List of Tables vii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Overview on SARIMA Model 7
2.1 Model Specification for Univariate Time Series . . . . . . . . . . . . . . . . . 7
2.2 Model Specification for Multivariate Time Series . . . . . . . . . . . . . . . 8
2.3 Cross Validation on Time Series . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Forecast on Rolling Windows . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Automatic Fitting of Univariate SARIMA Models 11
3.1 Fitting Procedure of cv.arima . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Comparisons between R functions: cv.arima v.s. auto.arima . . . . . . . . . 15
3.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Automatic Fitting of VAR Models 29
4.1 Fitting Procedure of var.auto . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Function Table of var.auto . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5 Application 39
5.1 Univariate Time Series Autofit . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Multivariate Time Series Autofit . . . . . . . . . . . . . . . . . . . . . . . . 45
6 Conclusion 48
References 49
Appendix I : Coefficient Setting in Univariate SARIMA Simulation Study 52
Appendix II : Demonstrate the Automatic Fitting Processes 54

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