Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Multiple linear regression ▷ Explained sum of squares

Definition: Let there be a multiple linear regression with independent observations using measured data $y$ and design matrix $X$:

$\label{eq:mlr} y = X\beta + \varepsilon, \; \varepsilon_i \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; .$

Then, the explained sum of squares (ESS) is defined as the sum of squared deviations of the fitted signal from the average signal:

$\label{eq:ess} \mathrm{ESS} = \sum_{i=1}^n (\hat{y}_i - \bar{y})^2 \quad \text{where} \quad \hat{y} = X \hat{\beta} \quad \text{and} \quad \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i$

with estimated regression coefficients $\hat{\beta}$, e.g. obtained via ordinary least squares.

Sources:

Metadata: ID: D38 | shortcut: ess | author: JoramSoch | date: 2020-03-21, 21:57.