K中心函數型分群法於多變量函數型資料之應用

K-centres Functional Clustering of Multivariate Functional Data

郭令証(Ling-Cheng Kuo)

指導教授 ： 李百靈

淡江大學/商管學院/統計學系碩士班/碩士(2015年)

https://doi.org/10.6846/TKU.2015.00158

摘要

近年來函數型資料分析逐漸受到重視，而函數型資料之分群分析在實務應用上更是重要的議題。本研究主要是想探討多變量函數型資料的分群問題，當樣本中的每個觀察對象均具有多個變量的函數型資料時，如何找到所有觀測對象的合理分群結構是個尚待解決的問題。本研究將提出一套新的 K 中心多變量函數型分群演算法，其根據 Chiou 與 Li (2007) 對單變量函數型資料所提出的 K 中心函數型分群之概念，利用變量本身的變異程度與不同變量間的相關性來決定各變量適當的權數，並以各變量之加權距離來做分群。本文所提出的方法將可於分群時同時考慮到各變量的平均函數與共變異函數等特徵，並根據各變量所包含的分群訊息來調整各變量對於分群的重要性，最後可對多變量函數型資料得到一個單一的分群結果。從數值模擬研究中可發現，K中心多變量函數型分群演算法不僅能改善單變量函數型分群方法的分群正確率，相較於其它現有的多變量函數型分群方法也有較好的表現。此外，在大部份的情況下，給予權數時若能考慮變量本身的變異程度或變量間的相關性，亦可提高分群正確率。本研究進一步提供兩種指標，以作為在實務分析上挑選最佳權數的方法。

關鍵字

分群分析 ； 多變量函數型資料 ； 函數型主成份分析

並列摘要

Cluster analysis of multivariate functional data is an important issue in real applications. In this study, we propose a novel k-centres multivariate functional clustering (mKCFC) algorithm for the multivariate functional data. The proposed approach is an extension of the k-centres functional clustering method (Chiou and Li, 2007), which is proposed for the univariate functional data, and can take the means and modes of variation differentials among clusters of each variable into account simultaneously. The mKCFC approach adopts a weighted distance for clustering. The weight of each variable represents the importance of a variable to the cluster information and is determined by the within-variable variation or the between-variable correlations. The numerical results of simulations show that the mKCFC method outperforms the other multivariate functional clustering approaches in most cases. Moreover, the weight of mKCFC is flexible and can be chosen by the objective of clustering, and we provide two indices for selecting the optimal weight in this study.

並列關鍵字

Clustering analysis ； Multivariate functional data ； Functional principal components analysis

參考文獻

[3] Bruno, F. and Cocchi, D. (2002) A unified strategy for building simple air quality indices. Environmetrics, 3:243-261.

[4] Cali´ nski, T. and Harabasz, J. (1974) A dendrite method for cluster analysis. Communications in Statistics, 3:1-27.

[5] Chiou, J.-M. and Li, P.-L. (2007) Functional clustering and identifying substructures of longitudinal data. Journal of the Royal Statistical Society Series, 69:679-699.

[6] Chiou, J.-M., Chen, Y.-T. and Yang, Y.-F. (2014) Multivariate functional principal component analysis: a normalization approach. Statistica Sinica, 24:1571-1596.

[7] Diakoulaki, D., Mavrotas, G. and Papayannakis, L. (1995) Determining objective weights in multiple criteria problems: the critic method. Computers and