主成分選取與因子選取在費雪區別分析上的探討

Abstract

當我們的資料變數很多時，我們通常會使用主成分\n或因子來降低資料變數；\n在選取主成分與因子時，我們通常會以特徵值來做選擇，\n然而變異數大(亦即特徵值大)的主成分或因子雖然解釋了大部分變異，\n但卻不一定保留了最多後續要分析的資訊，\n例如利用由特徵值所選取出來最好的主成分或因子\n來當做區別資料之變數，所得結果不一定理想。\n在此我們假設資料是來自於兩個多維常態母體，\n我們將分別利用由Mardia等人 (1979) 和Chang (1983) 所提出的兩種方法\n來選取出具區別能力的主成分，將其區別結果與由特徵值所選取出最好的主成分\n之區別結果作一比較；並且將此二方法應用在選取因子上。\n同時我們也證明Mardia等人 (1979) 和Chang (1983)的方法對於\n主成分及因子(利用主成分方法轉換)有相同的選取順序。\n本文更進一步地將Mardia等人\n所提出之方法運用至三群資料上，探討當資料來自於三個\n多維常態母體時，我們該如何利用此方法來選取具區別能力之變數。

Principal component analysis or factor analysis are often used\nto reduce the dimensionality of the original variables.\nHowever, the principal component or factor, which has\nlarger variance (i.e eigenvalue) explaining larger proportion of total sample\nvariance, may not retain the most information for other analyses later.\nFor example, using the first few principal components or factors\nhaving the largest corresponding eigenvalues as\ndiscriminant variables, the discriminant result\nmay not be good or even appropriate.\n\n\\hspace{2.05em}We first discuss two methods, given by Mardia et al. (1979) and Chang (1983)\nfor choosing discriminant variables when data are randomly obtained from\na mixture of two multivariate normal distributions.\nWe then use the discriminant result (or classification error rates)\nto compare these two methods and the traditional method of using the\nprincipal components, which have the larger corresponding eigenvalues,\nas discriminant variables. We also prove that the both the two methods\nhave the same selection order on principal components and factor (obtained\nby the principal component method).\nFurthermore, we use the method of\nMardia et al. to select appropriate discriminators when data is from\nthree populations.