透過函數型自編碼器實現K-中心函數分群法的非線性擴展

Abstract

本研究提出傳統 K-中心函數分群法（KCFC）的非線性延伸，建立一種嶄新的函數型資料分群方法：K-中心函數型自編碼器分群法（KCFAEC）。本方法將 KCFC 中的函數主成分分析（FPCA）替換為非線性的函數型自編碼器（FAE），後者將傳統的自編碼器架構延伸至函數型資料上。與 KCFC 相同，本方法以重建誤差作為重新指派群集的準則，透過迭代更新各群集的模型與樣本的群集標籤，達到分群目的。在此架構下，每個群集可學習自身的非線性表徵空間，藉以捕捉群集間潛在的結構差異。模擬研究與真實資料實驗（包含音素及腦電波資料）顯示，本方法的分群表現優於或相當於現有傳統方法，尤其在群集間具有明顯非線性結構差異的情境中，更能凸顯 KCFAEC 的效能，證實了本方法在實務上的價值與應用潛力。

This study proposes a nonlinear extension of the traditional K-centres functional clustering (KCFC) method, introducing a novel clustering approach for functional data: the K-Centres Functional Autoencoder Clustering (KCFAEC). In the proposed method, functional principal component analysis (FPCA) used in KCFC is replaced with a nonlinear functional autoencoder (FAE), which extends the autoencoder network to functional data. Similar to KCFC, the reconstruction loss is used for cluster reassignment, and the clustering process alternates between updating cluster-specific models and cluster labels. This process enables each cluster to possess its own nonlinear representation space, capturing latent structural differences across clusters. Simulation studies and real data examples—including the Phoneme and EEG datasets—demonstrate that KCFAEC consistently outperforms or matches other traditional methods in terms of clustering performance, especially in scenarios where clusters exhibit distinct nonlinear substructures, confirming the effectiveness and practical value of the proposed method.