Residual Error

We have defined the residual error as

$$\boldsymbol{\epsilon}=\boldsymbol{y}-\tilde{\boldsymbol{y}}=\left[\boldsymbol{I}-\boldsymbol{X}\left(\boldsymbol{X}^T\boldsymbol{X}\right)^{-1}\boldsymbol{X}^T\right]\boldsymbol{y}.$$

The residual errors are then the projections of $\boldsymbol{y}$ onto the orthogonal component of the space defined by the column vectors of $\boldsymbol{X}$.

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