Questions tagged [ridge-regression]

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

Filter by
Sorted by
Tagged with
42
votes
5answers
52k 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 ...
56
votes
2answers
8k views

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 ...
63
votes
5answers
14k views

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 ...
73
votes
2answers
19k views

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 ...
20
votes
1answer
7k views

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 ...
60
votes
3answers
8k views

Why does ridge estimate become better than OLS by adding a constant to the diagonal?

I understand that the ridge regression estimate is the $\beta$ that minimizes residual sum of square and a penalty on the size of $\beta$ $$\beta_\mathrm{ridge} = (\lambda I_D + X'X)^{-1}X'y = \...
83
votes
2answers
38k views

When to use regularization methods for regression?

In what circumstances should one consider using regularization methods (ridge, lasso or least angles regression) instead of OLS? In case this helps steer the discussion, my main interest is improving ...
174
votes
3answers
127k views

When should I use lasso vs ridge?

Say I want to estimate a large number of parameters, and I want to penalize some of them because I believe they should have little effect compared to the others. How do I decide what penalization ...
104
votes
6answers
50k views

Why L1 norm for sparse models

I am reading the books about linear regression. There are some sentences about the L1 and L2 norm. I know them, just don't understand why L1 norm for sparse models. Can someone use give a simple ...
33
votes
2answers
9k views

Is Tikhonov regularization the same as Ridge Regression?

Tikhonov regularization and ridge regression are terms often used as if they were identical. Is it possible to specify exactly what the difference is?
65
votes
5answers
4k views

Unified view on shrinkage: what is the relation (if any) between Stein's paradox, ridge regression, and random effects in mixed models?

Consider the following three phenomena. Stein's paradox: given some data from multivariate normal distribution in $\mathbb R^n, \: n\ge 3$, sample mean is not a very good estimator of the true mean. ...
18
votes
2answers
5k views

Why Lasso or ElasticNet perform better than Ridge when the features are correlated

I have a set of 150 features, and many of them are highly correlated with each other. My goal is to predict the value of a discrete variable, whose range is 1-8. My sample size is 550, and I am using ...
15
votes
1answer
8k views

Regression in $p>n$ setting: how to choose regularization method (Lasso, PLS, PCR, ridge)?

I am trying see whether to go for ridge regression, LASSO, principal component regression (PCR), or Partial Least Squares (PLS) in a situation where there are large number of variables / features ($p$)...
21
votes
5answers
9k views

Reason for not shrinking the bias (intercept) term in regression

For a linear model $y=\beta_0+x\beta+\varepsilon$, the shrinkage term is always $P(\beta) $. What is the reason that we do not shrink the bias (intercept) term $\beta_0$? Should we shrink the bias ...
19
votes
3answers
7k views

Relationship between ridge regression and PCA regression

I remember having read somewhere on the web a connection between ridge regression (with $\ell_2$ regularization) and PCA regression: while using $\ell_2$-regularized regression with hyperparameter $\...
11
votes
2answers
609 views

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 ...
9
votes
3answers
3k views

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, ...
52
votes
6answers
4k views

Is ridge regression useless in high dimensions ($n \ll p$)? How can OLS fail to overfit?

Consider a good old regression problem with $p$ predictors and sample size $n$. The usual wisdom is that OLS estimator will overfit and will generally be outperformed by the ridge regression estimator:...
36
votes
3answers
59k views

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 ...
35
votes
4answers
24k views

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.
15
votes
1answer
3k views

Lagrangian relaxation in the context of ridge regression

In "The Elements of Statistical Learning" (2nd ed), p63, the authors give the following two formulations of the ridge regression problem: $$ \hat{\beta}^{ridge} = \underset{\beta}{\operatorname{...
14
votes
3answers
3k views

How to find regression coefficients $\beta$ in ridge regression?

In ridge regression, the objective function to be minimized is: $$\text{RSS}+\lambda \sum\beta_j^2.$$ Can this be optimized using the Lagrange multiplier method? Or is it straight differentiation?
8
votes
2answers
8k views

Ridge regression in R with p values and goodness of fit

Doing ridge regression in R I have discovered linearRidge in the ridge package - which fits a model, reports coefficients and p ...
7
votes
2answers
2k views

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 ...
36
votes
1answer
6k views

When is nested cross-validation really needed and can make a practical difference?

When using cross-validation to do model selection (such as e.g. hyperparameter tuning) and to assess the performance of the best model, one should use nested cross-validation. The outer loop is to ...
35
votes
2answers
31k views

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?
16
votes
1answer
3k views

Under exactly what conditions is ridge regression able to provide an improvement over ordinary least squares regression?

Ridge regression estimates parameters $\boldsymbol \beta$ in a linear model $\mathbf y = \mathbf X \boldsymbol \beta$ by $$\hat{\boldsymbol \beta}_\lambda = (\mathbf X^\top \mathbf X + \lambda \mathbf ...
21
votes
2answers
5k views

What are the assumptions of ridge regression and how to test them?

Consider the standard model for multiple regression $$Y=X\beta+\varepsilon$$ where $\varepsilon \sim \mathcal N(0, \sigma^2I_n)$, so normality, homoscedasticity and uncorrelatedness of errors all hold....
18
votes
3answers
11k views

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 ...
28
votes
2answers
4k views

Why is glmnet ridge regression giving me a different answer than manual calculation?

I'm using glmnet to calculate ridge regression estimates. I got some results that made me suspicious in that glmnet is really doing what I think it does. To check this I wrote a simple R script where ...
18
votes
3answers
2k views

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, ...
12
votes
2answers
2k views

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 $k$...
20
votes
2answers
9k views

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 ...
12
votes
1answer
872 views

Understanding negative ridge regression

I'm looking for literature about negative ridge regression. In short, it is a generalization of linear ridge regression using negative $\lambda$ in the estimator formula: $$\hat\beta = ( X^\top X + \...
8
votes
1answer
6k views

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 ...
8
votes
1answer
1k views

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 ...
6
votes
1answer
776 views

Equivalence between Elastic Net formulations

According to Hastie's paper, the elastic net has two equivalent formulations: $$\hat{\beta} = \underset{\beta}{\operatorname{argmin}} \left\{ \sum_{i=1}^N\left(y_i-\sum_{j=1}^p x_{ij} \beta_j\right)^...
1
vote
1answer
827 views

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 ...
34
votes
1answer
19k views

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: ...
22
votes
1answer
3k views

Bridge penalty vs. Elastic Net regularization

Some penalty functions and approximations are well studied, such as the LASSO ($L_1$) and the Ridge ($L_2$) and how these compare in regression. I've been reading about the Bridge penalty, which is ...
24
votes
4answers
12k views

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? ...
13
votes
1answer
7k views

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 ...
8
votes
1answer
2k views

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) ...
4
votes
1answer
459 views

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 ...
12
votes
5answers
6k views

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 (...
6
votes
2answers
6k views

Can ridge regression be used in the presence of categorical predictors?

I have a regression problem and I am thinking of using ridge regression. One of the predictors is subject's gender, which is a categorical variable. How to take care of this variable for ridge ...
3
votes
1answer
464 views

Implementing linear regression with standardization

I have this confusion related to implementing linear regression with normalization. Let's say I have a training set trainX and ...
8
votes
2answers
5k views

Confused by MATLAB's implementation of ridge

I have two different implementations of ridge in MATLAB. One is simply $\mathbf x = (\mathbf{A}'\mathbf{A}+\mathbf{I}\lambda)^{-1}\mathbf{A}'\mathbf b$ (as seen ...
1
vote
2answers
822 views

Lasso regression feature selection

I have been reading many articles on LASSO regression, and everyone claims that LASSO address multicollinearity showing contour plots of cost function touching the corner of the diamond(x1+x2). In the ...
17
votes
2answers
11k 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 ...