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Concept
Projection matrix
Summary
In statistics, the projection matrix , sometimes also called the influence matrix or hat 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 projection 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 and the vector of fitted values by ,
As is usually pronounced "y-hat", the projection matrix is also named hat matrix as it "puts a hat on ".
The element in the ith row and jth column of is equal to the covariance between the jth response value and the ith fitted value, divided by the variance of the former:
The formula for the vector of residuals can also be expressed compactly using the projection matrix:
where is the identity matrix. The matrix is sometimes referred to as the residual maker matrix or the annihilator matrix.
The covariance matrix of the residuals , by error propagation, equals
where 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 , this reduces to:
From the figure, it is clear that the closest point from the vector onto the column space of , is , and is one where we can draw a line orthogonal to the column space of . A vector that is orthogonal to the column space of a matrix is in the nullspace of the matrix transpose, so
From there, one rearranges, so
Therefore, since is on the column space of , the projection matrix, which maps onto is just , or
Suppose that we wish to estimate a linear model using linear least squares. The model can be written as
where 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.
Many types of models and techniques are subject to this formulation.
Official source
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