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# Ridge Regression

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by Putcha Vaishnavi

## What is Ridge regression?

The Ridge regression is a technique which is specialized to analyze multiple regression data which is multicollinearity in nature. Though linear regression and logistic regression are the most beloved members of the regression family, according to a record-talk at NYC DataScience Academy, you must be familiar with using regression without regularization.

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Ridge regression is one of the most fundamental regularization techniques which is not used by many due to the complex science behind it. If you have an overall idea about the concept of multiple regression, it’s not so difficult to explore the science behind Ridge regression in r. Regression is the same, what makes regularization different is that the way how the model coefficients are determined. To understand this you need to know about multicollinearity in detail.

### What is multicollinearity?

The term multicollinearity also refers to collinearity concept in statistics. In this phenomenon, one predicted value in multiple regression models is linearly predicted with others to attain a certain level of accuracy.

The concept multicollinearity occurs when there are high correlations between more than two predicted variables.

The following is the ridge regression in r formula with an example:

For example, a person’s height, weight, age, annual income, etc.

The following are two regularization techniques for creating parsimonious models with a large number of features, the practical use, and the inherent properties are completely different.

• Ridge regression
• Lasso regression

Ridge regression is used to create a parsimonious model in the following scenarios:

• The number of predictor variables in a given set exceeds the number of observations.
• The dataset has multicollinearity (correlations between predictor variables).

The regularization techniques are as follows:

• Penalize the magnitude of coefficients of features
• Minimize the error between the actual and predicted observations

Now let us understand how the ridge regression model actually works:

Ridge regression performs L2 regularization. Here the penalty equivalent is added to the square of the magnitude of coefficients. The minimization objective is as followed.

Taking a response vector y ∈ Rn and a predictor matrix X ∈ Rn×p, the ridge regression coefficients are defined as:

• Here λ is the turning factor that controls the strength of the penalty term.
• If λ = 0, the objective becomes similar to simple linear regression. So we get the same coefficients as simple linear regression.
• If λ = ∞, the coefficients will be zero because of infinite weightage on the square of coefficients as anything less than zero makes the objective infinite.
• If 0 < λ < ∞, the magnitude of λ decides the weightage given to the different parts of the objective.
• In simple terms, the minimization objective = LS Obj + λ (sum of the square of coefficients)
• Where LS Obj is Least Square Objective that is the linear regression objective without regularization.

As ridge regression in r shrinks the coefficients towards zero, it introduces some bias. But it can reduce the variance to a great extent which will result in a better mean-squared error. The amount of shrinkage is controlled by λ which multiplies the ridge penalty. As large λ means more shrinkage, we can get different coefficient estimates for the different values of λ.

### Ridge Regression Example:

For example, ridge regression can be used for the analysis of prostate-specific antigen and clinical measures among people who were about to have their prostates removed. The performance of ridge regression is good when there is a subset of true coefficients which are small or even zero. But it doesn’t give good results when all the true coefficients are moderately large. However, it can still perform linear regression over a narrow range of (small) λ values.

There are few other regressions that would help in analyzing and predicting the results accurate. They are as follows:

• Linear Regression
• Logistic Regression
• Polynomial Regression
• Stepwise Regression
• Lasso Regression
• Regression Analysis
• Multiple Regression

Linear Regression: It is the easiest and popular statistical technique for widely used in predictive modeling. It gives an equation, which features as independent variables that depend on the target variable. Read the article on Linear Regression to learn more.

Logistic Regression: It is one among the types of the regression model in which regression analysis is done based on the dependent binary variable. This method is used to explain the relationship between the ratio-level independent variable, binary independent variables. Have a look at the following article to know more about Logistic Regression

Polynomial Regression: This is another form of regression where the maximum power of the independent variable is more than one. The second order polynomial equation is given below:

``Y =Θ1 +Θ2*x +Θ3*x2 `` 