# Questions tagged [ridge-regression]

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

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### 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 is given by: $$\beta = (X^\top X)^{-1}X^\top y.$$ But ...
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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 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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### 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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### 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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### Why L1 norm for sparse models

I am reading books about linear regression. There are some sentences about the L1 and L2 norm. I know the formulas, but I don't understand why the L1 norm enforces sparsity in models. Can someone give ...
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### Showing the Equivalence Between the ${L}_{2}$ Norm Regularized Regression and ${L}_{2}$ Norm Constrained Regression Using KKT

According to the references Book 1, Book 2 and paper. It has been mentioned that there is an equivalence between the regularized regression (Ridge, LASSO and Elastic Net) and their constraint ...
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### How to perform non-negative ridge regression?

How to perform non-negative ridge regression? Non-negative lasso is available in scikit-learn, but for ridge, I cannot enforce non-negativity of betas, and indeed, ...
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### Intuition for nonmonotonicity of coefficient paths in ridge regression

Intuitively, why may some of the slope coefficients in ridge regression increase in magnitude when the penalty parameter $\lambda$ is increased? Or in other words, why are the coefficient paths ...
9k views

### If only prediction is of interest, why use lasso over ridge?

On page 223 in An Introduction to Statistical Learning, the authors summarise the differences between ridge regression and lasso. They provide an example (Figure 6.9) of when "lasso tends to ...
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### Ridge, lasso and elastic net

How do ridge, LASSO and elasticnet regularization methods compare? What are their respective advantages and disadvantages? Any good technical paper, or lecture notes would be appreciated as well.
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### What is elastic net regularization, and how does it solve the drawbacks of Ridge ($L^2$) and Lasso ($L^1$)?

Is elastic net regularization always preferred to Lasso & Ridge since it seems to solve the drawbacks of these methods? What is the intuition and what is the math behind elastic net?
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### How to estimate shrinkage parameter in Lasso or ridge regression with >50K variables?

I want to use Lasso or ridge regression for a model with more than 50,000 variables. I want do so using software package in R. How can I estimate the shrinkage parameter ($\lambda$)? Edits: Here is ...
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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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### 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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### 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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### Is regression with L1 regularization the same as Lasso, and with L2 regularization the same as ridge regression? And how to write "Lasso"?

I'm a software engineer learning machine learning, particularly through Andrew Ng's machine learning courses. While studying linear regression with regularization, I've found terms that are confusing: ...
33k views

### Why will ridge regression not shrink some coefficients to zero like lasso?

When explaining LASSO regression, the diagram of a diamond and circle is often used. It is said that because the shape of the constraint in LASSO is a diamond, the least squares solution obtained ...
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### Why does Ridge Regression work well in the presence of multicollinearity?

I am learning about ridge regression and know that ridge regression tends to work better in the presence of multicollinearity. I am wondering why this is true? Either an intuitive answer or a ...
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### The limit of "unit-variance" ridge regression estimator when $\lambda\to\infty$

Consider ridge regression with an additional constraint requiring that $\hat{\mathbf y}$ has unit sum of squares (equivalently, unit variance); if needed, one can assume that $\mathbf y$ has unit sum ...
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### LASSO and ridge from the Bayesian perspective: what about the tuning parameter?

Penalized regression estimators such as LASSO and ridge are said to correspond to Bayesian estimators with certain priors. I guess (as I do not know enough about Bayesian statistics) that for a fixed ...
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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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### Ridge regression results different in using lm.ridge and glmnet

I applied some data to find the best variables solution of regression model using ridge regression in R. I have used lm.ridge and ...
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### PRESS statistic for ridge regression

In ordinary least squares, regressing a target vector $y$ against a set of predictors $X$, the hat matrix is computed as $$H = X (X^tX)^{-1} X^t$$ and the PRESS (predicted residual sum of squares) ...
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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 ...
952 views

### Geometrical interpretation of why can't ridge regression shrink coefficients to 0?

To explain the difference between Ridge and Lasso regression, following diagram is used as it is claimed that Ridge regression cannot shrink the regression coefficients to 0: But my question is, if ...
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### Deriving the Ridge Regression $\boldsymbol{\beta}\mid \mathbf{y}$ distribution

Apparently the estimate $\hat{\boldsymbol{\beta}}$ for ridge regression comes up as the mean or mode of the posterior distribution given by $f_{\boldsymbol{\beta}\mid \mathbf{y}}$. This is the ...
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