基於最佳正交近似轉換之多變量時間序列解析建模

Abstract

VARMA模型為多變量時間序列分析中最廣泛使用的模型之一。當時間序列數目增加時，需估計的參數也隨之大量增加。即使在較少的時間序列數時，若鑑定出高階數的自相關模型，也會導致參數估計量增加，且過多的參數矩陣讓模型解釋難度大幅上升。因此，將屬於VARMA模型中多變量時間序列降維轉換後再估計，一直是多變量時間序列分析中的重要問題。若能將多變量時間序列錯綜複雜的交互影響關係轉換為多個獨立單變量時間序列後，再進行預測、還原，即使時間序列個數沒有減少，仍能大量降低需估計的參數，亦提升模型解釋的能力。 本論文以二元VARMA模型為研究基礎，透過延遲其中一序列至不同期數的技巧，觀察二元時間序列間之關聯性變化。並推導出在使用最佳正交近似轉換將時間序列轉換成兩獨立時間序列後，對轉換後序列進行模型估計、預測再轉換回原空間，模型預測品質的變化方程式。而在轉換方式主要分為兩種，而第二種方法可再細分為二。在方法一中，透過將其中一序列延遲k期，消除序列一與序列二在延遲k的相關性。第二種方法中，透過在資料矩陣加入序列二延遲0,…,k期後的序列，消除多個延遲期數的相關性。此方法中再以擬合分作方法二及方法三。方法二，將轉換後的k+2個序列各自進行擬合。而方法三，僅對為延遲的序列一、序列二進行擬合，其餘未擬合序列則以序列二擬合而得的參數進行擬合。除針對不同方法下的變化進行性質比較、探討外，亦以數值模擬分析進行驗證，在低、中、高度相關的情境下進行探討。在低度相關的情況下，因為本就低度相關，因此以原序列各自擬合ARIMA即可有不錯的結果，但以轉換消除相關性結果仍比直接忽視相關性好。在中度相關的情境中，雖然轉換對序列的影響較為明顯，但轉換間造成的誤差也較弱相關時多，因此結果較為浮動。在高度相關的情境下，方法一適合選用較少的期數，方法二、三則適合選用較多期數。

The VARMA model is one of the most widely used models in multivariate time series analysis. However, as the number of time series increases, the number of parameters to be estimated also increases significantly. Even in cases with a lower number of time series in lower dimensions, identifying high-order autocorrelations can lead to the complexity of model interpretation. Therefore, reducing the dimensionality and/or transforming multivariate time series into orthogonal basis has been a crucial challenge in multivariate time series analysis. Complex interdependencies among multiple time series can be simplified through proper transformation. Not only the number of parameters to be estimated can be reduced, but also the model interpretability is enhanced. This research employs the bivariate VARMA model as the study vehicle. Through delaying one of the series with different lags, the interrelationships between the bivariate time series over different lag periods can be reduced or even cancelled out. The minimal transformation to orthonormality is used to convert the time series into two independent time series. The transformed sequences are then estimated and predicted, followed by the reconstruction to the original space. Three transformation frameworks are proposed, and the consequent derivation is performed for the numerical analytics on the impacts of different parameter settings. In situations of low correlation, due to the inherently low correlation, obtaining decent results can be achieved by individually fitting ARIMA models to the original series. However, employing transformations to eliminate correlation still yields better results compared to ignoring the correlation entirely. In moderately correlated scenarios, while the impact of transformations on the sequences is more pronounced, the errors introduced by the transformations are also more significant when the correlation is weaker. As a result, the outcomes tend to fluctuate more. In highly correlated scenarios, the first method is suitable for selecting a smaller number of periods, while the second and third methods are more suitable for choosing a larger number of periods.