山脊回归是一个method for estimating coefficients of linear models that include linearly correlated predictors.

Coefficient estimates for multiple linear regression models rely on the independence of the model terms. When terms are correlated and the columns of the design matrixX具有近似线性依赖性，矩阵（X^TX)^–1接近单数。因此，最小二乘的估计值

is highly sensitive to random errors in the observed responsey，产生较大的差异。例如，当您在没有实验设计的情况下收集数据时，可能会出现多重共线性的这种情况。

脊回归通过使用回归系数估算回归系数来解决多重共线性的问题

在哪里k是山脊参数，Iis the identity matrix. Small, positive values ofkimprove the conditioning of the problem and reduce the variance of the estimates. While biased, the reduced variance of ridge estimates often results in a smaller mean squared error when compared to least-squares estimates.

脊回归模型的系数估计值的缩放取决于scaledinput argument.

假设山脊参数k等于0。ridge， 什么时候scaledis equal to1, are estimates of theb_i¹in the multilinear model

在哪里z_i= (x_i–μ_i)/σ_iare the centered and scaled predictors,y–μ_yis the centered response, andε是一个n error term. You can rewrite the model as

with $$b_0^0={\mu }_y-{\displaystyle \underset{i=1}{\overset{p}{\sum }}\frac{b_i^1{\mu }_i}{{\sigma }_i}}$$and $$b_i^0=\frac{b_i^1}{{\sigma }_i}$$。这b_i⁰术语对应于返回的系数ridge什么时候scaledis equal to0。

More generally, for any value ofk, ifB1 =山脊（Y，X，K，1）, then

m = mean(X); s = std(X,0,1)'; B1_scaled = B1./s; B0 = [mean(y)-m*B1_scaled; B1_scaled]

在哪里B0 =山脊（Y，X，K，0）。