一種多變量函數主成分分析的混合方法

Abstract

近來，多變量函數數據愈來愈普遍。由主成分分析擴展而成的函數主成分分析 (FPCA) 是函數數據分析的基礎工具。儘管 FPCA 已經擴展到處理多變量函數數據，在處理多個函數中具有不同值域的稀疏觀測資料時仍存在一些問題。對於多變量函數主成分 (MFPCs) 的估計以及多變量函數主成分分數 (MFPC scores) 的預測，我們整合了先前研究的想法，提出了一個混合的方法。具體上，我們修改Happ and Greven (2018) 中一個連結 FPCA 和多變量 FPCA 的矩陣之元素，利用 FPCA 中的特徵值估計作為對角元素，而非對角區塊內的元素由交叉共變異函數的傅立葉係數與個別單變量 FPCA 的估計特徵函數所組成。對於 MFPC scores 的預測，我們考慮了兩種方法；其一利用上述修改後的矩陣並遵循了Happ and Greven (2018) 的流程，其二採用了Chiou et al. (2014) 的加權最小平方法 (WLS) 的想法。我們所提出的方法在各個實驗設定下至少會與Happ and Greven (2018) 和Chiou et al. (2014) 兩者中的最佳方法一樣好。我們觀察到，以 WLS 估計 MFPC scores 在稀疏資料或是資料的各變數尺度差異大之情況下是適合的，而另一個提出的方法則對於各變數的觀測值數量較不敏感。

Data with multiple functional observations has become prevalent recently. Functional principal component analysis (FPCA), extended from principal component analysis, is a fundamental tool in functional data analysis. Although FPCA has been further extended to handle multivariate functional data, issues persist with sparse observations in multivariate functions over different domains. To estimate multivariate functional principal components (MFPCs) and to predict MFPC scores, we integrated the ideas in existing work and proposed a hybrid strategy. Specifically, we modified the elements in the matrix linking FPCA with multivariate FPCA in Happ and Greven (2018), and utilized the estimated eigenvalues in FPCA for the diagonal elements while the off-diagonal blocks consist of the Fourier coefficients of the estimated cross covariance functions represented with the estimated eigenfunctions of each univariate FPCA. To predict the MFPC scores, we considered two approaches; first, we followed Happ and Greven (2018) with the modified matrix, and second, we employed the weighted least squares (WLS) method by Chiou et al. (2014). Our proposed approaches perform at least as the best between Chiou et al. (2014) and Happ and Greven (2018) under various simulation settings. Further, predicting the MFPC scores with the WLS method is more appropriate for sparse data or data with significant different scales between variables, while the other proposed method is less sensitive to the data where the scales of number of observations vary with variables.