Fuzzy Canonical Discriminant Analysis

模糊典型判別分析

鄭宇庭(Yu-Ting Cheng)

《Journal of Data Analysis》 6卷1期 (2011/02) Pp. 115-133

https://doi.org/10.6338/JDA.201102_6(1).0008

摘要

判別分析主要在於利用已知群組之樣本點，對未知樣本點做群組歸屬判斷。傳統判別分析對於已知樣本點，只能限制其完全屬於或完全不屬於某一群組，因此常會失去一些原本資料所給的訊息。本文嘗試利用模糊數學的多值邏輯理論，對於群組界限不是很明確的樣本點，給定其屬於各群組的隸屬度，以此做為已知資料，對未知樣本點做群組歸屬判斷，而判定結果也以樣本點屬於各群組的隸屬度表示。模糊判別分析方法主要是將未知資料點納入資料矩陣之中，以樣本點屬於各群的隸屬度為權數大小，並以拉氏乘數（Lagrange multiplier）法，找出使群間加權離均平方和與群內加權離均平方和比值為最大的各參數相對關係。給定一初始值之後，採用遞迴（iterative）運算的方式，求出各欲估參數的值。將本文推導所得之模糊判別分析方法，應用於鳶尾花品種判別分析。以此結果與傳統判別分析方法做比較，發現在建立模式的已知樣本數較少時，採用模糊判別分析方法可以改善部份誤判情形。

關鍵字

判別分析 ； 模糊 ； 典型 ； 分類問題

並列摘要

The main purpose of discriminant analysis is to apply a set of known observations to classify the observation of unknown groups into pre-defined groups. In traditional discriminant analysis, the classification of the data is limited to either belong or not belong to a specific set. As such some of the information contained in the data might have been ignored. In this paper, we propose fuzzy canonical discriminant analysis, a new classification method, to classify groups of known observations and determine the membership function of each set. This membership function is then taken to apply on the unknown observations. The fuzzy canonical discriminant analysis takes in data matrices with unknown observations which are weighted by membership degrees. To find out the correlation between parameters, this paper maximizes the ratio of the weighted sum of square between the ”between groups” and the ”within groups” by Lagrange Multiplier method. The initial value is given and then iterative algorithm is applied to calculate the estimation of the parameters.We compare fuzzy discriminant analysis with canonical discrimination, based on the example from three species of Iris. We found that it improves the accuracy of discriminant analysis when the sample size is small.

並列關鍵字

Fuzzy canonical discriminant analysis ； Fisher canonical discrimination ； Classification

參考文獻

Amo, A.,Montero, J.,Cutello, V.(1999).On the principles of fuzzy classification.Proceedings 18th International Conference of the North American, Fuzzy Information Processing Society Conference.(Proceedings 18th International Conference of the North American, Fuzzy Information Processing Society Conference).

Amo, A.,Montero, J.,Biging, G.(2000).Classifying pixels by means of fuzzy relations.International Journal of General Systems.29,605-621.

Amo, A.,Montero, J.,Biging, G.,Cutello, V.(2004).Fuzzy classification systems.European Journal of Operational Research.156(2),495-507.

Bezdek, J. C.(1980).A Convergence Theorem for the Fuzzy isodata Clustering Algorithms.IEEE Trans. Pattern Anal..PAMI-2(1),1-8.

Chen, Z. P.,Jiang, J. H.,Li, Y.,Liang, Y. Z.,Yu, R. Q.(1999).Fuzzy linear discriminant analysis for chemical data sets.Chemometrics and Intelligent Laboratory Systems.45(1-2),295-302.