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題名:基於馬氏距離的模糊分群演算法
作者:易正明 引用關係
作者(外文):JENG-MING YIH
校院名稱:國立臺中教育大學
系所名稱:教育測驗統計研究所
指導教授:劉湘川
許天維
學位類別:博士
出版日期:2009
主題關鍵詞:模糊平均數分群演算法GK分群演算法GG分群演算法FCM-M演算法FCM-CM演算法Fuzzy C-Means algorithmGK-algorithmGG-algorithmFCM-M algorithmFCM-CM algorithm
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模糊平均數分群演算法(Fuzzy C-Means algorithm)的距離計算以歐基里德距離(Euclidean distanc)進行數量的計算,用來辨識資料結構為球形(spherical)的分群。對於非球形結構資料的分類,有GK分群演算法(Gustafson -Kessel clustering algorithm)及GG分群演算法(Gath and Geva clustering algorithm),此兩種演算法分別用來處理非球形結構資料的分類。然而GK分群演算法搭配模糊共變數矩陣,其目標函數受限於經由此模糊共變數矩陣計算而得的馬氏(Mahalanobis)距離;GG分群演算法適用於資料分佈為多變量常態高斯(Gaussian)分佈。
參照GK與GG分群演算法所採用馬氏距離的概念,應用於模糊分群演算法,將其中的歐基里德距離以馬氏距離取代,拓展GK與GG分群演算法的限制,重要的是關係式能夠直接由本身的運算式導出,如此的目標函數是動態性,必能靈敏的反應樣本點的特質並獲得較佳的分類結果,我們稱此種演算法為馬氏距離為基礎的模糊分群演算法,簡稱FCM-M演算法(A Fuzzy C-Means algorithm based on Mahalanobis distance)。FCM-M演算法相當靈敏,對於各群資料的共變數矩陣不同,均能靈敏的呈現資料的分類結果。當各群資料的共變數矩陣不同,亦能穩健地呈現資料的分類結果,提出以各群資料共同的共變數矩陣經由來馬氏距離計算而得的目標函數值,由實證資料獲得較高的分類正確率,我們稱此種演算法為具共同的共變數矩陣的馬氏距離模糊分群演算法,簡稱FCM-CM演算法(Fuzzy C-Means algorithm based on Common Mahalanobis distances),FCM-CM演算法在包括教育評量的實證資料,均能獲得最高的正確率。
Some of the well-known fuzzy clustering algorithms are based on Euclidean distance function, which can only be used to detect spherical structural clusters. Gustafson-Kessel (GK) clustering algorithm and Gath-Geva (GG) clustering algorithm were developed to detect non-spherical structural clusters. However, GK algorithm needs added constraint of fuzzy covariance matrix, GK algorithm can only be used for the data with multivariate Gaussian distribution.
A Fuzzy C-Means algorithm based on Mahalanobis distance (FCM-M) was proposed by our previous work to improve those limitations of GG and GK algorithms, but it is not stable enough when some of its covariance matrices are not equal. In this paper, A improved Fuzzy C-Means algorithm based on Common Mahalanobis distances (FCM-CM) is proposed The experimental results of real data sets show that the performance of our proposed FCM-CM algorithm is better than those of the FCM, GG, GK and FCM-M algorithms.
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