Binary Classification Implementation Using Logistic Regression

Logistic Regression for Binary Classification Logistic regression is a widely used machine learning algorithm for binary classification problems. Despite its name containing "regression," it is fundamentally a classification model. This article demonstrates how to implement logistic regression in MATLAB using gradient descent for training and visualizes the classification boundary. Fundamental Principles of Logistic Regression Logistic regression maps the output of linear regression to a probability between 0 and 1 using the sigmoid function, representing the probability that a sample belongs to a particular class. Given input features X and weight parameters θ, the prediction probability is calculated as: [ P(y=1 | X) = rac{1}{1 + e^{-θ^T X}} ] In MATLAB implementation, the sigmoid function can be coded as: `sigmoid = @(z) 1./(1+exp(-z));` Model Training: Gradient Descent Method Gradient descent is an optimization algorithm used to adjust model parameters θ to minimize the loss function (typically cross-entropy loss). The core concept involves iteratively updating parameters by moving in the negative gradient direction of the loss function: [ θ := θ - α · rac{∂J(θ)}{∂θ} ] where α is the learning rate controlling the step size of parameter updates. The gradient calculation can be efficiently implemented using MATLAB's matrix operations: `gradient = (1/m)*X'*(sigmoid(X*theta)-y);` Implementation Steps Data Generation: Randomly generate two-class sample data ensuring linear or approximate separability. Use MATLAB's `randn` function for Gaussian distributed data. Feature Normalization: Standardize the data using z-score normalization to improve gradient descent convergence: `X_normalized = (X - mean(X))./std(X);` Parameter Initialization: Set initial weights θ and bias term, typically initialized to zeros or small random values: `theta = zeros(n_features, 1);` Iterative Training: Repeatedly compute gradients and update parameters until loss convergence or maximum iterations reached. Implement learning rate scheduling for better convergence. Classification Boundary Plotting: Calculate the decision boundary using trained parameters θ and plot it on the graph using MATLAB's plotting functions. Classification Boundary Visualization The decision boundary in logistic regression corresponds to the line where probability equals 0.5: [ θ^T X = 0 ] By solving this equation, the separation line can be drawn in the feature space, providing intuitive visualization of the model's classification performance. In MATLAB, use `contour` or `plot` functions to visualize the boundary alongside scatter plots of the classified data points. Conclusion Logistic regression combined with gradient descent provides an efficient solution for binary classification problems. MATLAB's matrix computation capabilities simplify the implementation of gradient calculations and parameter updates, while its data visualization functions facilitate intuitive evaluation of model performance. The implementation demonstrates key machine learning concepts including probability estimation, optimization techniques, and model evaluation through boundary visualization.