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Instatistics, theresidual sum of squares (RSS), also known as thesum of squared residuals (SSR) or thesum of squared estimate of errors (SSE), is thesum of thesquares ofresiduals (deviations predicted from actual empirical values of data). It is a measure of the discrepancy between the data and an estimation model, such as alinear regression. A small RSS indicates a tight fit of the model to the data. It is used as anoptimality criterion in parameter selection andmodel selection.

In general,total sum of squares =explained sum of squares + residual sum of squares. For a proof of this in the multivariateordinary least squares (OLS) case, seepartitioning in the general OLS model.

In a model with a single explanatory variable, RSS is given by:^[1]

$${\displaystyle \mathrm{RSS}=\sum _{i=1}^n{\left(y_i-f(x_i)\right)}^2}$$

wherey_i is thei^th value of the variable to be predicted,x_i is thei^th value of the explanatory variable, and${\displaystyle f(x_i)}$ is the predicted value ofy_i (also termed${\displaystyle \widehat{y_i}}$).In a standard linear simpleregression model,${\displaystyle y_i=\alpha +\beta x_i+{\epsilon }_i}$, where${\displaystyle \alpha }$ and${\displaystyle \beta }$ arecoefficients,y andx are theregressand and theregressor, respectively, and ε is theerror term. The sum of squares of residuals is the sum of squares of${\displaystyle {\hat{\epsilon }}_i}$; that is

$${\displaystyle \mathrm{RSS}=\sum _{i=1}^n{\left({\hat{\epsilon }}_i\right)}^2=\sum _{i=1}^n{\left(y_i-(\hat{\alpha }+\hat{\beta }{x}_i)\right)}^2}$$

where${\displaystyle \hat{\alpha }}$ is the estimated value of the constant term${\displaystyle \alpha }$ and${\displaystyle \hat{\beta }}$ is the estimated value of the slope coefficient${\displaystyle \beta }$.

The general regression model withn observations andk explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is

$${\displaystyle y=X\beta +e}$$

wherey is ann × 1 vector of dependent variable observations, each column of then ×k matrixX is a vector of observations on one of thek explanators,${\displaystyle \beta }$ is ak × 1 vector of true coefficients, ande is ann× 1 vector of the true underlying errors. Theordinary least squares estimator for${\displaystyle \beta }$ is

The residual vector${\displaystyle \hat{e}=y-X\hat{\beta }=y-X(X^{\mathrm{T}}X{)}^{-1}{X}^{\mathrm{T}}y}$; so the residual sum of squares is:

$${\displaystyle \mathrm{RSS}={\hat{e}}^{\mathrm{T}}\hat{e}={\Vert \hat{e}\Vert }^2,}$$

(equivalent to the square of thenorm of residuals). In full:

whereH is thehat matrix, or the projection matrix in linear regression.

Theleast-squares regression line is given by

$${\displaystyle y=ax+b,}$$

where${\displaystyle b=\overline{y}-a\overline{x}}$ and${\displaystyle a=\frac{S_{xy}}{S_{xx}}}$, where${\displaystyle S_{xy}=\sum _{i=1}^n(\overline{x}-x_i)(\overline{y}-y_i)}$ and${\displaystyle S_{xx}=\sum _{i=1}^n(\overline{x}-x_i{)}^2.}$

Therefore,

where${\displaystyle S_{yy}=\sum _{i=1}^n(\overline{y}-y_i{)}^2.}$

ThePearson product-moment correlation is given by${\displaystyle r=\frac{S_{xy}}{\sqrt{S_{xx}{S}_{yy}}};}$ therefore,${\displaystyle \mathrm{RSS}=S_{yy}(1-r^2).}$

• Draper, N.R.; Smith, H. (1998).Applied Regression Analysis (3rd ed.). John Wiley.ISBN 0-471-17082-8.

• Least squares
• Errors and residuals

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