Residual Sum of Square (from $\boldsymbol{\hat{\varepsilon}}$)

Calculates the residual sum of squares $\left( \mathrm{RSS} \right)$ using $$ \mathrm{RSS} = \sum_{i = 1}^{n} \left( Y_i - \hat{Y}_i \right)^2 \\ = \sum_{i = 1}^{n} \left( Y_i - \left[ \hat{\beta}_{1} + \hat{\beta}_{2} X_{2i} + \hat{\beta}_{3} X_{3i} + \dots + \hat{\beta}_{k} X_{ki} \right] \right)^2 \\ = \sum_{i = 1}^{n} \left( Y_i - \hat{\beta}_{1} - \hat{\beta}_{2} X_{2i} - \hat{\beta}_{3} X_{3i} - \dots - \hat{\beta}_{k} X_{ki} \right)^2 . $$ In matrix form $$ \mathrm{RSS} = \sum_{i = 1}^{n} \left( \mathbf{y} - \mathbf{\hat{y}} \right)^{2} \\ = \sum_{i = 1}^{n} \left( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} \right)^{2} \\ = \left( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} \right)^{\prime} \left( \mathbf{y} - \mathbf{X} \boldsymbol{\hat{\beta}} \right) . $$ Or simply $$ \mathrm{RSS} = \sum_{i = 1}^{n} \boldsymbol{\hat{\varepsilon}}_{i}^{2} = \boldsymbol{\hat{\varepsilon}}^{\prime} \boldsymbol{\hat{\varepsilon}} $$ where $\boldsymbol{\hat{\varepsilon}}$ is an $n \times 1$ vector of residuals, that is, the difference between the observed and predicted value of $\mathbf{y}$ $\left( \boldsymbol{\hat{\varepsilon}} = \mathbf{y} - \mathbf{\hat{y}} \right)$. Equivalent computational matrix formula $$ \mathrm{RSS} = \mathbf{y}^{\prime} \mathbf{y} - 2 \boldsymbol{\hat{\beta}} \mathbf{X}^{\prime} \mathbf{y} + \boldsymbol{\hat{\beta}}^{\prime} \mathbf{X}^{\prime} \mathbf{X} \boldsymbol{\hat{\beta}}. $$ Note that $$ \mathrm{TSS} = \mathrm{ESS} + \mathrm{RSS}. $$

.RSS(epsilonhat = NULL, X, y, betahat = NULL)

Arguments

epsilonhat	Numeric vector of length `n` or `n` by 1 numeric matrix. $n \times 1$ vector of residuals.
X	`n` by `k` numeric matrix. The data matrix $\mathbf{X}$ (also known as design matrix, model matrix or regressor matrix) is an $n \times k$ matrix of $n$ observations of $k$ regressors, which includes a regressor whose value is 1 for each observation on the first column.
y	Numeric vector of length `n` or `n` by `1` matrix. The vector $\mathbf{y}$ is an $n \times 1$ vector of observations on the regressand variable.
betahat	Numeric vector of length `k` or `k` by `1` matrix. The vector $\boldsymbol{\hat{\beta}}$ is a $k \times 1$ vector of estimates of $k$ unknown regression coefficients.

Value

Returns residual sum of squares $\left( \mathrm{RSS} \right)$.

Details

If epsilonhat = NULL, $\left( \mathrm{RSS} \right)$ is computed with X and y as required arguments and betahat as an optional argument.

References

Wikipedia: Residual Sum of Squares

Wikipedia: Explained Sum of Squares

Wikipedia: Total Sum of Squares

Wikipedia: Coefficient of Determination

Author

Ivan Jacob Agaloos Pesigan

epsilonhat	Numeric vector of length `n` or `n` by 1 numeric matrix. \(n \times 1\) vector of residuals.
X	`n` by `k` numeric matrix. The data matrix \(\mathbf{X}\) (also known as design matrix, model matrix or regressor matrix) is an \(n \times k\) matrix of \(n\) observations of \(k\) regressors, which includes a regressor whose value is 1 for each observation on the first column.
y	Numeric vector of length `n` or `n` by `1` matrix. The vector \(\mathbf{y}\) is an \(n \times 1\) vector of observations on the regressand variable.
betahat	Numeric vector of length `k` or `k` by `1` matrix. The vector \(\boldsymbol{\hat{\beta}}\) is a \(k \times 1\) vector of estimates of \(k\) unknown regression coefficients.

Residual Sum of Square (from \(\boldsymbol{\hat{\varepsilon}}\))

Arguments

Value

Details

References

See also

Author