# Detailed comparison of two methods for obtaining the ridge regression solution

I have come across two different ways of obtaining the ridge regression solution, which are as follows:

Method1:-(obtained from here)
$$RSS(\beta) = (Y-X\beta)^T\cdot(Y-X\beta)+\lambda\beta^T\Omega\beta$$
In this case,

1. $$X$$ and $$Y$$ need not be centered
2. $$X$$ contains the column of ones. i.e. $$X$$ has dimensions $$N\times (p+1)$$, where $$N$$ is the number of data points and $$p$$ is the dimension of each data point.
3. $$\beta=[ \space\space\beta_0 \space\space\space \beta_1 \space\space\space \beta_2 \space\space\space \cdots\space\space\space \beta_p\space\space]^T \text{ and } \Omega=\text{diag}(0,1,1,\cdots ,1)\implies \beta^T\Omega\beta=\sum_{i=1}^p\beta_i^2$$

$$\implies RSS(\beta)=Y^TY-2Y^TX\beta+\beta^T(X^TX+\lambda\Omega)\beta$$

$$\implies \dfrac{\partial}{\partial\beta}RSS(\beta)=0\implies\hat{\beta}_{ridge}=(X^TX+\lambda\Omega)^{-1}X^TY=[ \space\space\hat{\beta_0} \space\space\space \hat{\beta_1} \space\space\space \hat{\beta_2} \space\space\space \cdots\space\space\space \hat{\beta_p} \space\space]^T$$

Method2:-
$$RSS(\beta) = (Y-X\beta)^T\cdot(Y-X\beta)+\lambda\beta^T\beta$$
In this case,

1. $$X$$ and $$Y$$ need to be centered
2. $$X$$ does not contains the column of ones. i.e. $$X$$ has dimensions $$N\times p$$, where $$N$$ is the number of data points and $$p$$ is the dimension of each data point.
3. $$\beta=[ \space\space \beta_1 \space\space\space \beta_2 \space\space\space \cdots\space\space\space \beta_p\space\space]^T \implies \beta^T\beta=\sum_{i=1}^p\beta_i^2$$

$$\implies RSS(\beta)=Y^TY-2Y^TX\beta+\beta^T(X^TX+\lambda I)\beta$$

$$\implies \dfrac{\partial}{\partial\beta}RSS(\beta)=0\implies\hat{\beta}_{ridge}=(X^TX+\lambda I)^{-1}X^TY=[ \space\space \hat{\beta_1} \space\space\space \hat{\beta_2} \space\space\space \cdots\space\space\space \hat{\beta_p} \space\space]^T$$ and $$\hat{\beta_0}=\bar{Y}$$

now, even though method 1 seems more natural, most books follow method 2. why so? What advantage does method 2 have over method 1?