咨询岭回归中因变量与自变量的标准化公式（适配glmnet）

Great question—let’s break down exactly how glmnet handles standardization, since this is a common point of confusion when working with regularized models like ridge regression.

Core Behavior of glmnet's Standardization #

• By default, when you use standardize = TRUE (the default setting for this parameter), only the predictor variables (x) are standardized—the response variable (y) is not modified by this process.
• The exact formula applied to each predictor (x_j) is:

  x_j' = (x_j - mean(x_j)) / sd(x_j)
  

  Here, mean(x_j) and sd(x_j) are calculated using your training dataset. These values are stored by glmnet so they can be reused to standardize new data during prediction.

Why glmnet Doesn't Standardize y #

Ridge regression’s objective is to minimize:

RSS + λ * ||β||²

Where RSS is the residual sum of squares for your original response y. Standardizing y would scale the RSS term, but since glmnet’s regularization targets the coefficients (β), the scale of y doesn’t affect the core regularization logic.

Importantly, even though glmnet fits the model using standardized predictors, it automatically converts the final coefficients back to the original scale of your input predictors. This means you can use raw, unstandardized x values for prediction without extra steps, and the coefficients will be interpretable relative to your original data’s units.

What If I Standardize y Manually? #

If you choose to standardize y before fitting, the model will output coefficients tied to the standardized y scale. You’ll then need to reverse the standardization (multiply predictions by y’s standard deviation and add y’s mean) to get results in your original y units. This extra step is unnecessary for ridge regression in glmnet, since the package handles predictor scaling internally without requiring y to be adjusted.

内容的提问来源于stack exchange，提问作者Majk