In statistics, the hat matrix, H, sometimes also called projection matrix, maps the vector of observed values to the vector of fitted values. It describes the influence each observed value has on each fitted value. The diagonal elements of the hat matrix are the leverages, which describe the influence each observed value has on the fitted value for that same observation.
If the vector of observed 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".
Suppose that we wish to solve 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.
Read more about Hat Matrix: Uncorrelated Errors, Correlated Errors, Blockwise Formula
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