Sum of squared residuals

In statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared errors of prediction (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepancy between the data and an estimation model. A small RSS indicates a tight fit of the model to the data. It is used as an optimality criterion in parameter selection and model selection.

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

In a model with a single explanatory variable, RSS is given by:

where y_i is the i ^th value of the variable to be predicted, x_i is the i ^th value of the explanatory variable, and ${\displaystyle f(x_{i})}$ is the predicted value of y_i (also termed ${\displaystyle {\hat {y_{i}}}}$). In a standard linear simple regression model, ${\displaystyle y_{i}=a+bx_{i}+\varepsilon _{i}\,}$, where a and b are coefficients, y and x are the regressand and the regressor, respectively, and ε is the error term. The sum of squares of residuals is the sum of squares of estimates of ε_i; that is

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