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| In [[statistics]], the '''residual sum of squares (RSS)''' is the [[sum]] of squares of [[errors and residuals in statistics|residuals]]. It is also known as the '''sum of squared residuals (SSR)''' or the '''sum of squared errors of prediction (SSE)'''. 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.
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| 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 [[Explained sum of squares#Partitioning in the general OLS model|partitioning in the general OLS model]].
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| ==One explanatory variable==
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| In a model with a single explanatory variable, RSS is given by
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| :<math>RSS = \sum_{i=1}^n (y_i - f(x_i))^2, </math>
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| where ''y''<sub>''i''</sub> is the ''i'' <sup>th</sup> value of the variable to be predicted, ''x''<sub>''i''</sub> is the ''i'' <sup>th</sup> value of the explanatory variable, and <math>f(x_i)</math> is the predicted value of ''y''<sub>''i''</sub> (also termed <math>\hat{y_i}</math>). | |
| In a standard linear simple [[regression model]], <math>y_i = a+bx_i+\varepsilon_i\,</math>, where ''a'' and ''b'' are [[coefficient]]s, ''y'' and ''x'' are the [[regressand]] and the [[regressor]], respectively, and ε is the [[errors and residuals in statistics|error term]]. The sum of squares of residuals is the sum of squares of [[estimator|estimates]] of ε<sub>''i''</sub>; that is
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| :<math>RSS = \sum_{i=1}^n (\epsilon_i)^2 = \sum_{i=1}^n (y_i - (\alpha + \beta x_i))^2, </math>
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| where <math>\alpha</math> is the estimated value of the constant term <math>a</math> and <math>\beta</math> is the estimated value of the slope coefficient ''b''.
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| ==Matrix expression for the OLS residual sum of squares==
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| 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
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| :<math> y = X \beta + e</math> | |
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| 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, <math>\beta </math> 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 <math>\beta</math> is
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| :<math> \hat \beta = (X^T X)^{-1}X^T y.</math>
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| The residual vector <math>\hat e</math> is <math>y - X \hat \beta = y - X (X^T X)^{-1}X^T y</math>, so the residual sum of squares <math>\hat e ^T \hat e</math> is, after simplification,
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| :<math> RSS = y^T y - y^T X(X^T X)^{-1} X^T y = y^T [I - X(X^T X)^{-1} X^T] y = y^T [I - H] y</math> , where H is the [[hat matrix]], or the prediction matrix in linear regression.
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| ==See also==
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| *[[Sum of squares (statistics)]]
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| *[[Squared deviations]]
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| *[[Errors and residuals in statistics]]
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| *[[Lack-of-fit sum of squares]]
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| *[[Degrees of freedom (statistics)#Sum of squares and degrees of freedom]]
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| *[[Chi-squared distribution#Applications]]
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| ==References==
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| * {{cite book
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| |title = Applied Regression Analysis
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| |edition = 3rd
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| |last1= Draper |first1=N.R. |last2=Smith |first2=H.
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| |publisher = John Wiley
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| |year = 1998
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| |isbn = 0-471-17082-8}}
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| [[Category:Regression analysis]]
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| [[Category:Least squares]]
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