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Partial least squares regression (PLS regression) is a statistical method that bears some relation to principal components regression; instead of finding hyperplanes of minimum variance between the response and independent variables, it finds a linear regression model by projecting the predicted variables and the observable variables to a new space. Because both the ''X'' and ''Y'' data are projected to new spaces, the PLS family of methods are known as bilinear factor models. Partial least squares Discriminant Analysis (PLS-DA) is a variant used when the Y is categorical. PLS is used to find the fundamental relations between two matrices (''X'' and ''Y''), i.e. a latent variable approach to modeling the covariance structures in these two spaces. A PLS model will try to find the multidimensional direction in the ''X'' space that explains the maximum multidimensional variance direction in the ''Y'' space. PLS regression is particularly suited when the matrix of predictors has more variables than observations, and when there is multicollinearity among ''X'' values. By contrast, standard regression will fail in these cases (unless it is regularized). The PLS algorithm is employed in partial least squares path modeling, a method of modeling a "causal" network of latent variables (causes cannot be determined without experimental or quasi-experimental methods, but one typically bases a latent variable model on the prior theoretical assumption that latent variables cause manifestations in their measured indicators). This technique is a form of structural equation modeling, distinguished from the classical method by being component-based rather than covariance-based. Partial least squares was introduced by the Swedish statistician Herman Wold, who then developed it with his son, Svante Wold. An alternative term for PLS (and more correct according to Svante Wold) is ''projection to latent structures'', but the term ''partial least squares'' is still dominant in many areas. Although the original applications were in the social sciences, PLS regression is today most widely used in chemometrics and related areas. It is also used in bioinformatics, sensometrics, neuroscience and anthropology. In contrast, PLS path modeling is most often used in social sciences, econometrics, marketing and strategic management. ==Underlying model== The general underlying model of multivariate PLS is : : where is an matrix of predictors, is an matrix of responses; and are matrices that are, respectively, projections of (the ''X score'', ''component'' or ''factor'' matrix) and projections of (the ''Y scores''); and are, respectively, and orthogonal ''loading'' matrices; and matrices and are the error terms, assumed to be independent and identically distributed random normal variables. The decompositions of and are made so as to maximise the covariance between and . 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「Partial least squares regression」の詳細全文を読む スポンサード リンク
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