# Residual sum of squares

In statistics, the **residual sum of squares (RSS)**, also known as the **sum of squared residuals (SSR)** or the **sum of squared estimate of errors (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.

## One explanatory variable

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

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 is the predicted value of *y*_{i} (also termed ).
In a standard linear simple regression model, , 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

where is the estimated value of the constant term and is the estimated value of the slope coefficient *b*.

## Matrix expression for the OLS residual sum of squares

The general regression model with n observations and k explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is

where y is an *n* × 1 vector of dependent variable observations, each column of the *n* × *k* matrix X is a vector of observations on one of the *k* explanators, is a *k* × 1 vector of true coefficients, and e is an *n*× 1 vector of the true underlying errors. The ordinary least squares estimator for is

The residual vector = ; so the residual sum of squares is:

- ,

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

- ,

where H is the hat matrix, or the projection matrix in linear regression.

## Relation with Pearson's product-moment correlation

The least-squares regression line is given by

- ,

where and , where and

Therefore,

where

The Pearson product-moment correlation is given by therefore,

## See also

## References

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

- Archdeacon, Thomas J. (1994).
*Correlation and regression analysis : a historian's guide*. University of Wisconsin Press. pp. 161–162. ISBN 0-299-13650-7. OCLC 27266095.