Polynomial Regression

Definition

Consider explanatory variables $X_i$ and a response variable $Y$ The $k$-variables 2nd order polynomial regression model is defined as $Y={\beta }_0+\sum _{i=1}^k{\beta }_i{X}_i+\underset{\text{quadratic terms}}{\underbrace{\sum _{i=1}^k{\beta }_{ii}{X}_i^2}}+\underset{\text{interaction terms}}{\underbrace{\sum _{i<j}^k{\beta }_{ij}{X}_i{X}_j}}+ϵ$

The single variable $k$-th order polynomial regression model is defined as $Y={\beta }_0+\sum _{i=1}^k{\beta }_i{X}^iϵ$

Facts

The column vectors of the design matrix $\mathbf{X}$ in the polynomial regression are highly correlated, so proper transformation is needed. Among them, orthogonal polynomial is often used.