What is a residual?

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\def\rvw{\vec{W}} \def\rvx{\vec{X}} \def\rvy{\vec{Y}} \def\rvz{\vec{Z}} \def\seta{A} \def\setb{B} \def\setc{C} \def\setd{D} \def\sete{E} \def\setf{F} \def\setg{G} \def\seth{H} \def\seti{I} \def\setj{J} \def\setk{K} \def\setl{L} \def\setm{M} \def\setn{N} \def\seto{O} \def\setp{P} \def\setq{Q} \def\setr{R} \def\sets{S} \def\sett{T} \def\setu{U} \def\setv{V} \def\setw{W} \def\setx{X} \def\sety{Y} \def\setz{Z} \def\fa{a} \def\fb{b} \def\fc{c} \def\fd{d} \def\fe{e} \def\ff{f} \def\fg{g} \def\fh{h} \def\fi{i} \def\fj{j} \def\fk{k} \def\fl{l} \def\fm{m} \def\fn{n} \def\fo{o} \def\fp{p} \def\fq{q} \def\fr{r} \def\fs{s} \def\ft{t} \def\fu{u} \def\fv{v} \def\fw{w} \def\fx{x} \def\fy{y} \def\fz{z} \def\fA{A} \def\fB{B} \def\fC{C} \def\fD{D} \def\fE{E} \def\fF{F} \def\fG{G} \def\fH{H} \def\fI{I} \def\fJ{J} \def\fK{K} \def\fL{L} \def\fM{M} \def\fN{N} \def\fO{O} \def\fP{P} \def\fQ{Q} \def\fR{R} \def\fS{S} \def\fT{T} \def\fU{U} \def\fV{V} \def\fW{W} \def\fX{X} \def\fY{Y} \def\fZ{Z} \def\ma{A} 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The residual $\ve$ is the error vector between the true output vector $\vy$ and its estimate $\hat{\vy}$ :

$\ve = \vy - \hat{\vy}$

Residuals for linear regressions

When $\hat{\vy}$ is produced by a linear model, the residual represent the aspects of $\vy$ that cannot be explained by the columns of the design matrix $\mx$ :

$\ve = \vy - \mx\hat{\vw}$

Since $\hat{vy} = \mx\hat{\vw}$ is a vector in the linear space spanned by the columns of $\mx$ , the residual is a vector outside of this linear space, pointing towards $\vy$ .

Note: and when the MSE-loss is used, the residual is orthogonal to this linear space.

Let $\mathcal{M}(\mx)$ be the column space of $\mx$ . The vector $\hat{\vy}$ is in this linear subspace while $\vy$ isn’t. This is illustrated on the picture below.

Residual geometry .

Residuals for OLS

The parameter vector $\hat{\vw}$ can be expressed using a closed-form formula in the case of an OLS regression;

$\hat{vw} = (\mx^{\top}\mx)^{-1}\mx^{\top}\vy$

Pluging this formula into the definition of a residual, we get:

$\ve = \vy - \underbrace{\mx(\mx^{\top}\mx)^{-1}\mx^{\top}}_{\mh}\vy$

$\mh$ is the hat-matrix (because it puts a hat on $\vy$ ). Rewriting the equality with $\vy$ as factor, we get:

$\ve = (\mi_\sn - \mh)\vy$

As mentioned above, for OLS regressions, the residual is a vector orthogonal to the column space of $\mx$ .