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Questions tagged [ridge-regression]

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

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71 votes
5 answers
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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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32 votes
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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 ...
jeza's user avatar
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62 votes
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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 ...
aspiringstatistician's user avatar
94 votes
3 answers
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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 ...
cgo's user avatar
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71 votes
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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 ...
Charlie's user avatar
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156 votes
8 answers
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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 ...
Yongwei Xing's user avatar
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72 votes
3 answers
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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 = \...
Heisenberg's user avatar
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97 votes
2 answers
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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 ...
NPE's user avatar
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209 votes
4 answers
174k 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 ...
Larry Wang's user avatar
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26 votes
2 answers
17k 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 ...
renakre's user avatar
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40 votes
2 answers
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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?
Carl's user avatar
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70 votes
6 answers
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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:...
amoeba's user avatar
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81 votes
5 answers
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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. ...
amoeba's user avatar
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22 votes
1 answer
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Regularization for ARIMA models

I am aware of LASSO, ridge and elastic-net type of regularization in linear regression models. Question: Can this (or a similar) kind of penalized estimation be applied to ARIMA modelling (with a non-...
Richard Hardy's user avatar
26 votes
6 answers
19k 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 ...
yliueagle's user avatar
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57 votes
2 answers
40k views

When will L1 regularization work better than L2 and vice versa?

Note: I know that L1 has feature selection property. I am trying to understand which one to choose when feature selection is completely irrelevant. How to decide which regularization (L1 or L2) to ...
GeorgeOfTheRF's user avatar
45 votes
4 answers
26k 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? ...
Bstat's user avatar
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24 votes
3 answers
16k 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 $\...
Jose G's user avatar
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22 votes
1 answer
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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 ($p$)...
Ram Sharma's user avatar
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13 votes
2 answers
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Ridge regression in R with p values and goodness of fit [closed]

Doing ridge regression in R I have discovered linearRidge in the ridge package - which fits a model, reports coefficients and p ...
Sideshow Bob's user avatar
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44 votes
1 answer
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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 ...
amoeba's user avatar
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33 votes
2 answers
6k 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 ...
John's user avatar
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26 votes
1 answer
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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 ...
figs_and_nuts's user avatar
17 votes
1 answer
3k 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 + \...
Benoit Sanchez's user avatar
15 votes
3 answers
3k 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 ...
jeza's user avatar
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14 votes
4 answers
7k 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, ...
The Baron's user avatar
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10 votes
4 answers
1k views

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 ...
Richard Hardy's user avatar
46 votes
2 answers
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 ...
Oliver Angelil's user avatar
45 votes
4 answers
40k 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.
user3269's user avatar
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44 votes
2 answers
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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?
GeorgeOfTheRF's user avatar
38 votes
3 answers
69k 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 ...
John 's user avatar
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25 votes
3 answers
5k 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, ...
user162381's user avatar
25 votes
3 answers
16k 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 ...
James Davison's user avatar
20 votes
1 answer
5k 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 ...
amoeba's user avatar
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15 votes
3 answers
8k 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?
Minaj's user avatar
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14 votes
3 answers
4k 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 $k$...
Snowflake's user avatar
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13 votes
2 answers
9k views

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 consisting ...
user101311's user avatar
8 votes
1 answer
2k 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)^...
skd's user avatar
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8 votes
2 answers
5k 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 ...
user3450805's user avatar
49 votes
1 answer
29k 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: ...
stackoverflowuser2010's user avatar
33 votes
2 answers
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 ...
user10024395's user avatar
24 votes
1 answer
18k 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 ...
TrynnaDoStat's user avatar
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23 votes
2 answers
2k views

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 ...
amoeba's user avatar
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20 votes
2 answers
5k views

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 ...
Richard Hardy's user avatar
20 votes
2 answers
10k 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 ...
Stephen Turner's user avatar
11 votes
1 answer
10k views

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 ...
Zakaria Al-Jammal's user avatar
9 votes
2 answers
3k 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) ...
Chris Taylor's user avatar
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8 votes
1 answer
2k 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 ...
Richard Hardy's user avatar
3 votes
2 answers
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 ...
Deepak Tatyaji Ahire's user avatar
3 votes
1 answer
2k 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 ...
Clarinetist's user avatar
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