Section 20 Principle of Least Squares: Mathematical form

20.0.1 Null Model

\[ \large y_{i} = \beta_0 + \epsilon_{i} \]

20.0.2 Regression Model

\[ \large y_{i} = \beta_0 + \beta_1 x_{i} + \epsilon_{i} \]

20.1 Prediction

\[ \large \hat y_{i} = \hat \beta_0 + \hat \beta_1 x_{i} \]

20.2 Residual

\[ \large \hat\epsilon_{i} = y_i - \hat y_{i} \]

\[ \large \hat\epsilon_{i} = y_i - \hat \beta_0 - \hat \beta_1 x_{i} \]

20.3 Residual Sum of Squares

\[ \large RSS = \hat\epsilon_{1}^2 + \hat\epsilon_{2}^2 + ... + \hat\epsilon_{n}^2 \]

\[ \large RSS = \sum\limits_{i=1}^{n} \hat\epsilon_{i}^2 \]