【OLS vs Ridge Regression】: Performance Comparison between Ordinary Least Squares and Ridge Regression

# 1. Understanding Ordinary Least Squares and Ridge Regression

Ordinary Least Squares (OLS) and Ridge Regression are both common linear regression methods. In practical applications, understanding and mastering the principles and differences between these two methods can better select appropriate models for data modeling and prediction. OLS focuses on minimizing the sum of squared residuals to estimate parameters, while Ridge Regression adds a regularization term on the basis of OLS to deal with multicollinearity issues. Through in-depth study of these two methods, a better understanding of the performance and applicability of linear regression algorithms in different situations can be achieved.

# 2. Principles and Applications of Ordinary Least Squares

### 2.1 What is Ordinary Least Squares
Ordinary Least Squares (OLS) is a common method of linear regression analysis, aiming to fit a linear model that best fits the sample points by observing the data. In OLS, we try to find a straight line such that the sum of the squared vertical distances of all data points to this line is minimized.

### 2.2 Mathematical Principles of Ordinary Least Squares

#### 2.2.1 Minimization of Residual Sum of Squares
In ordinary least squares, our goal is to minimize the sum of squared residuals, that is, the sum of squares of the differences between the actual observed values and the model predicted values. By minimizing the sum of squared residuals, we can obtain the estimated values of the regression coefficients, thereby establishing a linear model.

#### 2.2.2 Derivation of Parameter Estimation
By minimizing the sum of squared residuals, the optimal solution for the regression coefficients can be derived. The derivation of parameter estimation is the core of the OLS method, usually involving mathematical techniques such as matrix operations and differentiation.

#### 2.2.3 Model Evaluation Indicators
In addition to parameter estimation, ***mon model evaluation indicators include Mean Squared Error (MSE), Coefficient of Determination (R-squared), etc., which can help us understand the degree of model fit and predictive ability.

### 2.3 Applications of Ordinary Least Squares
Ordinary least squares are widely used in the fields of statistics and machine learning, especially in linear regression analysis. OLS can yield a concise and intuitive linear model, suitable for situations where there is a strong linear relationship between data features. OLS is also often used for cases with fewer feature variables and lower model complexity.

In practical applications, we can implement the OLS method through Python's Statmodels or other statistical libraries to analyze the linear relationship in a dataset.

This is the principle and application of ordinary least squares. In order to better understand OLS, we will delve into the principles and advantages of ridge regression next.

# 3. Principles and Advantages of Ridge Regression

Ridge Regression is a modified version of the least squares estimation method, which adds a penalty on the absolute values of the coefficients to address the poor performance of ordinary least squares in the presence of multicollinearity. This chapter will delve into the principles, mathematical derivation, and practical advantages of ridge regression.

### 3.1 What is Ridge Regression

Ridge regression is a linear regression algorithm, an改良 version of ordinary least squares. In ordinary least squares, when there is multicollinearity among features (i.e., high correlation between features), it leads to unstable parameter estimates in the model. Ridge regression solves this problem by adding an L2 regularization term.

### 3.2 Mathematical Principles of Ridge Regression

#### 3.2.1 Ridge Regression Regularization Term

The optimization goal of ridge regression is:
\hat{\beta}^{ridge} = argmin_{\beta} ((y - X\beta)^T(y - X\beta) + \alpha\beta^T\beta)
Where, \hat{\beta}^{ridge} is the parameter estimate of ridge regression, y is the dependent variable, X is the matrix of independent variables, \beta is the regression coefficient, \alpha is the hyperparameter, controlling the strength of the regularization term.

#### 3.2.2 Parameter Solution of Ridge Regression

The parameter solution of ridge regression can use the closed-form solution of ordinary least squares, namely:
\hat{\beta}^{ridge} = (X^TX + \alpha I)^{-1}X^Ty
Where, I is the identity matrix.

#### 3.2.3 Comparison between Ridge Regression and Ordinary Least Squares

Compared to ordinary least squares, ridge regression can alleviate the problems caused by multicollinearity, improve the generalization ability of the model, but it will also introduce bias. In cases where the data features have high c