【Basic】Regression Prediction Models: MATLAB Ridge Regression and Lasso Regression

# 1. Introduction to Regression Prediction Models**

Regression prediction models are statistical learning models designed to forecast the relationship between continuous dependent variables (the dependent variable) and one or more independent variables (the independent variables). These models describe the relationship between the dependent and independent variables by fitting a curve or hyperplane, allowing for predictions of the dependent variable's value based on the independent variables.

Regression prediction models find applications in numerous fields, including:

* Housing price forecasting
* Stock price forecasting
* Medical diagnosis
* Economic forecasting

# 2. Ridge Regression in MATLAB

### 2.1 Principles and Mathematical Derivation of Ridge Regression

Ridge regression is a form of regularized regression that addresses the overfitting problem in ordinary least squares regression by adding a regularization term to the loss function. This term penalizes the absolute values of the coefficients, thereby compelling the model to be simpler and avoiding overfitting.

The loss function of ridge regression is as follows:

J(w) = (1/2) * ||y - Xw||^2 + (lambda/2) * ||w||^2

Where:

* `y` is the target variable
* `X` is the feature matrix
* `w` is the model coefficients
* `lambda` is the regularization parameter

### 2.2 Ridge Regression Parameter Estimation and Regularization Parameter Selection

#### 2.2.1 Ordinary Least Squares Estimation

The coefficients of ridge regression can be estimated by minimizing the loss function. Using ordinary least squares, the estimated coefficients can be represented as:

w = (X^T X + lambda * I)^-1 X^T y

Where:

* `I` is the identity matrix

#### 2.2.2 Regularization Parameter Selection Methods

The choice of the regularization parameter `lambda` is critical. It governs the strength of the regularization term, thus affecting the model'***

***mon methods for selecting the regularization parameter `lambda` include:

* Cross-validation: Split the data into multiple subsets and use different subsets for training and validation, selecting the `lambda` value that performs best on the validation set.
* L-curve: Plot the sum of the loss function and regularization term as `lambda` varies, and select the `lambda` at the inflection point.

### 2.3 Ridge Regression Implementation in MATLAB

#### 2.3.1 Data Preparation and Model Training

% Load data
data = load('data.mat');

% Split data
X = data.X;
y = data.y;

% Standardize data
X = (X - mean(X)) ./ std(X);
y = (y - mean(y)) ./ std(y);

% Create a ridge regression model
lambda = 0.1; % Regularization parameter
model = ridge(X, y, lambda);

#### 2.3.2 Model Evaluation and Results Interpretation

% Predict
y_pred = predict(model, X);

% Evaluate model
mse = mean((y_pred - y).^2); % Mean Squared Error
rmse = sqrt(mse); % Root Mean Squared Error
r2 = 1 - sum((y_pred - y).^2) / sum((y - mean(y)).^2); % R^2 Score

% Print results
fprintf('Mean Squared Error: %.4f\n', mse);
fprintf('Root Mean Squared Error: %.4f\n', rmse);
fprintf('R^2 Score: %.4f\n', r2);

# 3. Lasso Regression in MATLAB

### 3.1 Principles and Mathematical Derivation of Lasso Regression

Lasso regression, or Least Absolute Shrinkage and Selection Operator Regression, is a linear regression model that incorporates an L1 regularization term into the objective function to penalize the regression coefficients. Unlike ridge regression, Lasso regression's regularization term is the absolute value, which can lead to some regression coefficients being exactly zero, thus achieving feature selection.

The objective function of Lasso regression is:

min_β (1/2n) ||y - Xβ||^2 + λ||β||_1

Where:

* y is the target variable
* X is the independent variable
* β is the regression coefficients
* n is the number of samples
* λ is the regularization parameter

### 3.2 Lasso Regression Parameter Estimation and Regularization Parameter Selection

#### 3.2.1 Ordinary Least Squares Estimation

Like ridge regression, Lasso regression parameters can be estimated by minimizing the objective function. However, due to the non-smoothness of the L1 regularization term, Lasso regression does not have an analytical solution for the parameter estimation, requiring numerical optimization methods.

#### 3.2.2 Regularization Parameter Selection Methods

The choice of the regularization parameter λ i