Ordinary Least Squares and Ridge Regression Variance¶. Due to the few points in each dimension and the straight line that linear regression uses to follow these points as well as it can, noise on the observations will cause great variance as shown in the first plot.
Ordinary least squares (OLS) regression produces regression coefficients that are unbiased estimators of the corresponding population coefficients with the least variance. This is the Gauss-Markov Theorem. In most situations, this is exactly what we want.However, there may be a model with less variance (i.e. Smaller SSE), but at the cost of added bias. This may be desirable, for example, when some of the data is highly correlated. This occurs in the following situations:. There are many independent variables, especially when there are more variables than observations.
Data is close to multicollinearity, in which case small changes to X can result in large changes to the regression coefficients.Here the OLS model over-fits the data, i.e. It captures the data well, but is not so good at forecasting based on new data.
In these cases, Ridge and LASSO Regression can produce better models by reducing the variance at the expense of adding bias.Topics.
. Overview - Ridge Regression. Training Ridge Regression Model. Choosing Optimal Lambda Value - k-Cross Validation. Bias and variance of ridge regression. Assumptions of Ridge RegressionsOverview - Ridge RegressionRidge regression is a parsimonious model which performs L2 regularization.
The L2 regularization adds a penality equivalent to the square of the maginitude of regression coefficients and tries to minimize them. The equation of rigde regression looks like as given below. LS Obj + λ (sum of the square of coefficients)Here the objective is as follows:.If λ = 0, the output is similar to simple linear regression.If λ = very large, the coefficients will become zero.The following diagram is the visual interpretation comparing OLS and ridge regression.Training Ridge Regression ModelTo build the ridge regression in r we use glmnetfunction from glmnet package in R. Let’s use ridge regression to predict the mileage of the car using mtcars dataset. # Loaging the library library (glmnet ) # Getting the independent variablexvar. # here x is the test datasetpred.