Variability in linear models

Variability in linear models

Mathematically, SSE is a Loss function given by:

$SSE={\sum }_{i=1}^N(y_i-{\hat{y}}_i{)}^2$

where $N$ is the number of observations, $y_i$ is the actual value of the dependent variable for observation $i$, and ${\hat{y}}_i$ is the predicted value based on the linear regression model.

The formula for the Regression Sum of Squares (SSR) in the context of linear regression is:

$SSR={\sum }_{i=1}^N({\hat{y}}_i-\bar{y}{)}^2$

Where:

• ${\hat{y}}_i$ is the predicted value of the dependent variable for observation $i$ based on the linear regression model.
• $\bar{y}$ is the mean of the observed values of the dependent variable.
• $N$ is the total number of observations.

SSR measures the amount of variability in the dependent variable that is explained by the independent variables in the model. It reflects how well the regression model captures the relationship between the independent and dependent variables.

Total Sum of Squares (SST) represents the total variability in the dependent variable $y$. The relationship between SST, SSR, and SSE is given by:

$SST=SSR+SSE$

This equation reflects the decomposition of total variability into explained variability (SSR) and unexplained variability (SSE) due to errors.