\[\hat{\beta^{ridge}} = \underset{\beta^{ridge}}{\operatorname{argmin}} \sum_{i=1}^{n} (y-x^T\beta^{ridge})^2 + \lambda\sum_{j=1}^{k}(\beta_j)^2\]
It can also be expressed as \[\hat{\beta^{ridge}} = \underset{\beta^{ridge}}{\operatorname{argmin}} \sum_{i=1}^{n} (y-x^T\beta^{ridge})^2 \]
subjected to \(\sum_{j=1}^{k}(\beta_j)^2 \leq t\). There is one-to-one mapping between the \(t\) and \(\lambda\).
To ensure that variables’ coefficients are penalized fairly, standardization of the variables are required before running the regression.