參數式歸屬函數及其應用

Abstract

本文提出一種新的歸屬函數表示方式，稱為參數式歸屬函數(Parametric membership function)，用以取代傳統的歸屬函數定義方式。並且將此方法分別應用於處理包含凹歸屬函數或準凹歸屬函數的模糊目標規劃問題、二群體的模糊區別分析、以及模糊複區別分析。 在模糊目標規劃問題中，模糊目標的歸屬函數被轉換成以參數式表示的歸屬函數，進而得到一個模糊目標規劃問題的一般化輔助模式。由原問題到輔助模式的轉換過程非常直接且容易。在此之前，Hannan的方法只能適用於使用凹函數的模糊目標規劃問題；Nakamura的方法需要重覆的求解過程；Yang等的方法需要使用額外的0-1變數；而Inuiguchi等的方法需要前處理將準凹函數轉成凹函數，並且需要後續處理以求得真正的目標達成度。相較於這些方法，本文所提出的方法整體而言比較容易且有效率。分別以Inuiguchi等及Nakamura的例子測試的結果，本文所提的方法得到的最佳解與這二種方法所得到的結果完全一樣，但是整個過程卻容易許多。 此外，本文也提出一個以參數式歸屬函數為基礎的二群體模糊區別分析方法。藉由將歸屬函數表示成參數式，以及最小化分類誤差的平方和，可以得到各個群體的歸屬函數。參數式歸屬函數可以解決使用傳統歸屬函數所導致的運算困難。利用本文所提出的輔助模式，可以很容易地得到二群體個別的歸屬函數。結果顯示，當二群體之間有重疊時，模糊區別分析的分類誤差小於傳統區別分析方法的誤差。而當二群體之間沒有重疊時，模糊區別分析將退化成一般的區別分析方法。 當參數式歸屬函數應用於模糊複區別分析時，分析結果顯示，當一集群與其他集群間沒有重疊時，模糊複區別分析同樣地可以得到與一般的區別分析方法相同的結果。比較模糊複區別分析與fuzzy c-means的分析結果發現，模糊複區別分析可以得到比fuzzy c-means更清楚的集群結構。此外，最重要的是，模糊複區別分析可以有效辨識異常點(outlier)的存在，這是一般區別分析方法與fuzzy c-means無法辦到的。

This study proposes an alternative expression for membership functions named as parametric membership function. The parametric membership function is applied to solve fuzzy goal programming problems with concave or quasiconcave piecewise linear membership functions and to serve as the basis of two-group fuzzy discriminant analysis and fuzzy multiple discriminant analysis. In solving fuzzy goal programming, parametric membership functions replace the conventional membership functions of fuzzy goals to obtain an auxiliary fuzzy goal programming model. The transform process is easy and straightforward. Compared with other methods for fuzzy goal programming, the proposed method is easier to carry out than the others. A method for two-group fuzzy discriminant analysis is proposed based on the approach of parametric membership function. The parametric membership function overcomes the computational difficulties resulting from using conventional membership function expression. The illustrative examples indicate that the sum of squared error by fuzzy discrimination is less than that by crisp discrimination when the two groups overlap. When there is no overlap between the two groups, fuzzy discrimination degenerates into a crisp discrimination. An auxiliary model is also proposed for fuzzy multiple discriminant analysis. Compared with fuzzy c-means, fuzzy discriminant analysis provides clearer structural information of the clusters than does fuzzy c-means with less fuzzy partition results. The most important advantage of fuzzy multiple discriminant analysis is its ability of identifying outliers, which crisp discriminant analysis and fuzzy c-means are incapable of.