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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Mahalanobis distance ： ウィキペディア英語版 Mahalanobis distance The Mahalanobis distance is a measure of the distance between a point P and a distribution D, introduced by P. C. Mahalanobis in 1936. It is a multi-dimensional generalization of the idea of measuring how many standard deviations away P is from the mean of D. This distance is zero if P is at the mean of D, and grows as P moves away from the mean: along each principal component axis, it measures the number of standard deviations from P to the mean of D. If each of these axes is rescaled to have unit variance, then Mahalanobis distance corresponds to standard Euclidean distance in the transformed space. Mahalanobis distance is thus unitless and scale-invariant, and takes into account the correlations of the data set. ==Definition and properties== The Mahalanobis distance of an observation $\backslash underline\; =\; (\; x\_1,\; x\_2,\; x\_3,\; \backslash dots,\; x\_N\; )^T$ from a set of observations with mean $\backslash underline\; =\; (\; \backslash mu\_1,\; \backslash mu\_2,\; \backslash mu\_3,\; \backslash dots\; ,\; \backslash mu\_N\; )^T$ and covariance matrix ''S'' is defined as: :$D\_M(\backslash underline)\; =\; \backslash sqrt)^T\; S^\; (\backslash underline-\backslash underline)\}.\backslash ,$〔De Maesschalck, Roy; Jouan-Rimbaud, Delphine; and Massart, Désiré L. (2000); ''The Mahalanobis distance'', Chemometrics and Intelligent Laboratory Systems 50:1–18〕 Mahalanobis distance (or "generalized squared interpoint distance" for its squared value〔Gnanadesikan, Ramanathan; and Kettenring, John R. (1972); ''Robust estimates, residuals, and outlier detection with multiresponse data'', Biometrics 28:81–124〕) can also be defined as a dissimilarity measure between two random vectors $\backslash underline$ and $\backslash underline$ of the same distribution with the covariance matrix ''S'': :$d(\backslash underline,\backslash underline)=\backslash sqrt)^T\; S^\; (\backslash underline-\backslash underline)\}.\backslash ,$ If the covariance matrix is the identity matrix, the Mahalanobis distance reduces to the Euclidean distance. If the covariance matrix is diagonal, then the resulting distance measure is called a ''normalized Euclidean distance'': :$d(\backslash underline,\backslash underline)=$ \sqrt}}, where ''si'' is the variance of the ''xi'' and ''yi'' over the sample set. Mahalanobis distance is preserved under full-rank linear transformations of the space spanned by the data. This means that if the data has a nontrivial nullspace, Mahalanobis distance can be computed after projecting the data (non-degenerately) down onto any space of the appropriate dimension for the data. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Mahalanobis distance」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.