# Questions tagged [ridge-regression]

A regularization method for regression models that shrinks coefficients towards zero.

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62k views

### How to derive the ridge regression solution?

I am having some issues with the derivation of the solution for ridge regression. I know the regression solution without the regularization term: $$\beta = (X^TX)^{-1}X^Ty.$$ But after adding the ...
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### Why does shrinkage work?

In order to solve problems of model selection, a number of methods (LASSO, ridge regression, etc.) will shrink the coefficients of predictor variables towards zero. I am looking for an intuitive ...
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### What problem do shrinkage methods solve?

The holiday season has given me the opportunity to curl up next to the fire with The Elements of Statistical Learning. Coming from a (frequentist) econometrics perspective, I'm having trouble grasping ...
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### The proof of shrinking coefficients using ridge regression through “spectral decomposition”

I have understood how ridge regression shrinks coefficients towards zero geometrically. Moreover I know how to prove that in the special "Orthonormal Case," but I am confused how that works in the ...
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### Why is ridge regression called “ridge”, why is it needed, and what happens when $\lambda$ goes to infinity?

Ridge regression coefficient estimate $\hat{\beta}^R$ are the values that minimize the $$\text{RSS} + \lambda \sum_{j=1}^p\beta_j^2.$$ My questions are: If $\lambda = 0$, then we see that the ...
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### Lasso and Ridge tuning parameter scope

In ridge and lasso linear regression, an important step is to choose the tuning parameter lambda, often I use grid search on log scale from -6->4, it works well on ridge, but on lasso, should I take ...
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### How can I estimate coefficient standard errors when using ridge regression?

I am using ridge regression on highly multicollinear data. Using OLS I get large standard errors on the coefficients due to the multicollinearity. I know ridge regression is a way to deal with this ...
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### L1 regression estimates median whereas L2 regression estimates mean?

So I was asked a question on which central measures L1 (i.e., lasso) and L2 (i.e., ridge regression) estimated. The answer is L1=median and L2=mean. Is there any type of intuitive reasoning to this? ...
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### Using regularization when doing statistical inference

I know about the benefits of regularization when building predictive models (bias vs. variance, preventing overfitting). But, I'm wondering if it is a good idea to also do regularization (lasso, ...
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### Ridge penalized GLMs using row augmentation?

I've read that ridge regression could be achieved by simply adding rows of data to the original data matrix, where each row is constructed using 0 for the dependent variables and the square root of $k$...
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### Estimating R-squared and statistical significance from penalized regression model

I am using the R package penalized to obtain shrunken estimates of coefficients for a dataset where I have lots of predictors and little knowledge of which ones are important. After I've picked tuning ...
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### What are the implications of scaling the features to xgboost?

Doing research about the xgboost algorithm I went through the documentation. I have heard that xgboost does not care much about the scale of the input features In this approach trees are ...
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### Grid fineness and overfitting when tuning $\lambda$ in LASSO, ridge, elastic net

I wonder about the optimal grid fineness and what the relation between grid fineness and overfitting is in regularization methods such as LASSO, ridge regression or elastic net. Suppose I want ...
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### Is there a “fused” version Ridge regression?

we know there is a fused version of LASSO. Fused LASSO adds a further regularizer demanding the smoothness of \beta. More details could be found here I am wondering why I cannot find something ...
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### Ridge regression formulation as constrained versus penalized: How are they equivalent?

I seem to be misunderstanding a claim about linear regression methods that I've seen in various places. The parameters of the problem are: Input: $N$ data samples of $p+1$ quantities each ...
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### Ridge & LASSO norms

This post follows this one: Why does ridge estimate become better than OLS by adding a constant to the diagonal? Here is my question: As far as I know, ridge regularization uses a $\ell_2$-norm (...
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### What are the leverage values for Ridge regression?

In linear least squares the parameter estimates are: $\hat{\beta} = \left(X^{\top}X\right)^{-1}X^{\top}y$. In Ridge regression the standardized parameter estimates are given by \$\hat{\beta}_{\Gamma} = ...