現今，驗證分群結果較主流的方法是透過計算各種cluster validation index來檢驗，但是這些指數在類別變數很多的資料時卻不一定能得到合理的答案，因此，本研究利用階層式分群對目標變數建立分群樹，對另一變數則利用歐式距離建立分群樹，再根據兩分群樹繪製熱力圖，從熱力圖的顏色區塊找出資料幾何較相關的群體；接著，利用ANOVA的概念模擬原始資料，並以模擬資料的分群編碼繪製信度直方圖，以呈現群體相似度，進一步驗證階層式分群結果的正確性及穩固性；若信度直方圖所呈現的趨勢與原始分群結果符合，則可判斷分群的結果正確；本研究方法與cluster validation index的差異是我們可以依據熱力圖所呈現的資料幾何結構，在分群樹上的不同高度做切割，找出相關性高的群組，提出檢驗階層式分群結果的信度指標。

Nowadays, the most popular method of validating clustering results is to verify through various cluster validation indexes. However, these indexes may not get reasonable answers whenever data with a lot of categorical variables. This study aims to provide a stable method to detect the homogeneity and stability of Hierarchical Clustering (HC). Multiple HC trees based on simulated data are built, and the path to each category in a tree is recorded. Histogram based on the coding path of simulated data is built to validate the reliability and stability of the clustering results from HC. The difference between the proposed method and the common cluster validation indexes is that we can rely on the clustering results presented by the heatmap, cut at different heights on the dendrogram to find reasonable and highly relevant groups, and increase the flexibility of the clustering.

一、 中文參考文獻
[1] 張順全 (1999) 類別資料結構的訊息視覺化

二、 英文參考文獻
[1] Balcan, M. F., Liang, Y., & Gupta, P. (2014). Robust hierarchical clustering. The Journal of Machine Learning Research, 15(1), 3831-3871.
[2] Ben-Hur, A., Elisseeff, A., & Guyon, I. (2001). A stability based method for discovering structure in clustered data. In Biocomputing 2002 (pp. 6-17).
[3] Brock, G., Pihur, V., Datta, S., & Datta, S. (2011). clValid, an R package for cluster validation. Journal of Statistical Software (Brock et al., March 2008).
[4] Caliński, T., & Harabasz, J. (1974). A dendrite method for cluster analysis. Communications in Statistics-theory and Methods, 3(1), 1-27.
[5] Carlsson, G. E., & Mémoli, F. (2010). Characterization, stability and convergence of hierarchical clustering methods. J. Mach. Learn. Res., 11(Apr), 1425-1470.
[6] Chou, E., McVey, C., Hsieh, Y. C., Enriquez, S., & Hsieh, F. (2020). Extreme-K categorical samples problem. arXiv preprint arXiv:2007.15039.
[7] Dunn, J. C. (1974). A graph theoretic analysis of pattern classification via Tamura's fuzzy relation. IEEE Transactions on Systems, Man, and Cybernetics, (3), 310-313.
[8] Dunn, J. C. (1974). Well-separated clusters and optimal fuzzy partitions. Journal of cybernetics, 4(1), 95-104.
[9] Fushing, H., & Roy, T. (2018). Complexity of possibly gapped histogram and analysis of histogram. Royal Society open science, 5(2), 171026.
[10] Goodman, L. A., & Kruskal, W. H. (1979). Measures of association for cross classifications. Measures of association for cross classifications, 2-34.
[11] Rendón, E., Abundez, I., Arizmendi, A., & Quiroz, E. M. (2011). Internal versus external cluster validation indexes. International Journal of computers and communications, 5(1), 27-34.
[12] Shannon, C. E. (1948). A mathematical theory of communication. The Bell system technical journal, 27(3), 379-423.
[13] Smith, S. P., & Dubes, R. (1980). Stability of a hierarchical clustering. Pattern Recognition, 12(3), 177-187.