## Introduction

In statistics, ridge regression, also known as Tikhonov regularization, is a type of regularized linear regression where the coefficient estimates are shrunk towards zero by imposing a penalty on their size. The ridge coefficients minimize a penalized residual sum of squares,

RSS(β)+λ∑j=1pβj2

where β=(β1,…,βp) are the coefficients to be estimated, RSS is the residual sum of squares, and λ>0 is the penalty parameter. Ridge regression is an example of a shrinkage method, where data-driven estimators have their coefficient estimates shrunken towards zero relative to the least squares estimates. The amount of shrinkage is set by the penalty parameter λ. Hence more data generally leads to estimators with less shrinkage (that is, closer to the least squares estimates), whereas if few data are available relatively more shrinkage is desired. In certain circumstances optimal choice ofλ can be determined using cross-validation or generalized cross-validation.

## The penalty term

The penalty term is a term in ridge regression that is used to help avoid overfitting. The penalty term is also known as the regularization term. The penalty term is added to the loss function. The penalty term is used to shrink the coefficients. The penalty term is also used to reduce the variance of the estimates.

### Lasso regression

Lasso regression is a type of linear regression that uses a regularization term in order to reduce the number of variables that are used in the model. The penalty term is used to shrink the coefficients of the features that are not important, and it can be used to select the features that are most important in the model.

### Ridge regression

Ridge regression is an approach used when the data shows multicollinearity (correlation between independent variables). When multicollinearity occurs, least squares estimates (of the regression coefficients 𝛽) tend to be biased, inefficient (inconsistent), and difficult to interpret because of unstable variance. These problems associated with OLS are more pronounced in datasets with small samples. Ridge regression addresses these issues by adding a penalty term to the objective function that is proportional to the sum of the squares of the coefficient estimates; this forces the coefficient estimates to be relatively small even though they are estimated with optimization methods that do not explicitly impose such a constraint. The penalty term is identified by lambda ().

### Elastic net

Elastic net is a regularization technique used in machine learning to penalize models with too many features. It is a combination of L1 and L2 regularization, where L1 regularization encourages sparsity (fewer features) and L2 encourages small weights. Elastic net is useful because it can prevent overfitting while still allowing the model to learn from all of the features.

## Why use ridge regression?

Ridge regression is a type of linear regression that includes a penalty term. The penalty term is used to reduce the complexity of the model and prevent overfitting. Overfitting is when a model performs well on the training data but does not generalize well to new data. Ridge regression is less likely to overfit than other types of linear regression.

### Bias-variance tradeoff

Ridge regression is a type of linear regression that is used when there is a concern about multicollinearity. Multicollinearity occurs when there are high correlations between independent variables. This can cause problems with the interpretability of the coefficients and can lead to overfitting of the data. Ridge regression addresses these problems by penalizing the size of the coefficients, which shrinks them towards zero. This reduces the variance of the model but increases the bias. The penalty term in ridge regression is typically denoted by λ (lambda). A larger value for lambda results in more shrinkage and a smaller value results in less shrinkage.

### Improved interpretability

Ridge regression is a type of regression analysis that creates models that are more interpretable by reducing the variance of the coefficients. This is done by adding a penalty term to the cost function that shrinks the coefficients towards zero. The penalty term is usually represented as λ (lambda), and the higher the value of λ, the more the coefficients are shrunk.

The trade-off with ridge regression is that interpretability is improved at the expense of model accuracy. In other words,ridge regression will not produce the best predictions, but it will produce models that are easier to understand.

There are situations where interpretability is more important than accuracy, such as in medical research or other fields where decisions need to be made based on the model results. In these cases,ridge regression can be a useful tool.

### Reduced overfitting

Ridge regression is an approach used when the data shows evidence of multicollinearity, which is when there is a high correlation among independent variables. In this situation, the coefficient estimates can be very sensitive to small changes in the data. Ridge regression reduces the magnitude of the coefficient estimates, and it helps to reduce overfitting.

## When to use ridge regression

Ridge regression is a type of linear regression that is used to create predictive models. It is used when there is a need for prediction and there is a possibility of overfitting. Ridge regression is also used when multicollinearity is present.

### Linear models with many features

Ridge regression is a linear model that uses L2 regularization. This technique shrinks the coefficients by penalizing them with a penalty term defaulted to 1/N where N is the number of features. When to use this model? If your training set has many features and you want to reduce overfitting, you can use ridge regression.

Ridge regression is a good choice when you have many features, especially if some of those features are correlated with each other. By adding the L2 penalty, ridge regression pulls these correlated features away from each other, reducing their correlation and thus multicollinearity. This in turn can improve your model’s accuracy.

### Non-linear models with few features

Ridge regression is particularly useful when you have a lot of features (variables) in your data set, and you want to avoid overfitting. Overfitting occurs when your model is too complex for the data set and results in a model that does not generalize well to new data. Ridge regression helps to avoid overfitting by adding a penalty term to the regression equation that shrinks the coefficients of the less important variables. This results in a model that is better able to generalize to new data.

Ridge regression is also useful when you have non-linear relationships in your data. This can be due to interactions between variables, or because the dependent variable itself is not linear. In these cases, adding a penalty term can help to improve the fit of the model.

## How to use ridge regression

Ridge regression is a technique used when there is multicollinearity in the data. When multicollinearity exists, the coefficient estimates can be very large and unreliable. Ridge regression solves this problem by adding a penalty term to the regression that shrinks the coefficient estimates.

### Choosing the penalty term

Choosing the penalty term

The penalty term in ridge regression is L2 regularization. This means that the penalty term is the sum of the squares of the coefficients. The penalty term is added to the cost function, so the cost function for ridge regression is:

J(w) = RSS + λ||w||2

where ||w||2 is the sum of the squares of the coefficients, and λ is the regularization parameter.

The value of λ determines how much weight is given to the penalty term. If λ is large, then the penalty term has a large influence on the cost function, and the coefficients will be close to 0. If λ is small, then the penalty term has a small influence on the cost function, and the coefficients will be close to their least squares estimates.

### Fitting the model

The penalty term in ridge regression is

fit the model:

minimize ∑(yi−(mx+b))2+λ(am2+bm2)

subject to:

∑ai=0,∑bi=0,i=1…n,a≥0,b≥0.

### Interpreting the results

Now that you understand how the penalty term affects the results of ridge regression, let’s take a look at how to interpret the results.

The first thing to note is that, in general, the larger the value of alpha, the lower the value of R-squared. This makes sense, since alpha is penalizing coefficients, and thus shrinking them. With a large alpha, coefficients are shrunk more, resulting in a model with a lower R-squared.

Secondly, you’ll notice that as alpha increases, the magnitude of the coefficients also decrease. This again makes sense, since we are penalizing larger coefficients.

Finally, one important thing to note is thatridge regression will always give you a better R-squared value than linear regression. This is because ridge regression shrinks coefficients towards 0 (but never actually reaches 0). So even though linear regression might give you a slightly higher R-squared value, it also has the potential to give you coefficient values that are much too large. Therefore, ridge regression should be your go-to when performing linear Regression with L2 regularization.

## Conclusion

The penalty term in ridge regression is designed to prevent overfitting by adding a penalty for complex models. The penalty term is typically a function of the number of coefficients in the model. The penalty term is added to the error function, and the model is fit by minimizing the sum of the error function and the penalty term.

The penalty term can have a significant impact on the fit of the model. In general, models with higher penalties will be less flexible and will have poorer fits to the data. However, it is important to find the right balance between flexibility and parsimony. If the penalty term is too high, then the model will be too simple and will not be able to capture all of the patterns in the data. If the penalty term is too low, then the model will be too complex and will overfit the data.