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In statistics, the hat matrix, ''H'', sometimes also called the influence matrix〔(Data Assimilation: Observation influence diagnostic of a data assimilation system )〕 or projection matrix, maps the vector of response values (dependent variable values) to the vector of fitted values (or predicted values). It describes the influence each response value has on each fitted value.〔 〕 The diagonal elements of the hat matrix are the leverages, which describe the influence each response value has on the fitted value for that same observation. If the vector of response values is denoted by y and the vector of fitted values by ŷ, : As ŷ is usually pronounced "y-hat", the hat matrix is so named as it "puts a hat on y". The formula for the vector of residuals r can also be expressed compactly using the hat matrix: : Moreover, the element in the ''i''th row and ''j''th column of ''H'' is equal to the covariance between the ''j''th response value and the ''i''th fitted value, divided by the variance of the former: : The covariance matrix of the residuals is therefore, by error propagation, equal to , where Σ is the covariance matrix of the error vector (and by extension, the response vector as well). For the case of linear models with independent and identically distributed errors in which Σ = ''σ''2''I'', this reduces to (''I'' − ''H'')''σ''2.〔 Many types of models and techniques are subject to this formulation. A few examples are: * Linear model / linear least squares * Smoothing splines * Regression splines * Local regression * Kernel regression * Linear filtering == Linear model == Suppose that we wish to estimate a linear model using linear least squares. The model can be written as : where ''X'' is a matrix of explanatory variables (the design matrix), ''β'' is a vector of unknown parameters to be estimated, and ''ε'' is the error vector. 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「hat matrix」の詳細全文を読む スポンサード リンク
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