Polynomial Curve Fitting for Sine Wave Approximation

Polynomial curve fitting serves as a fundamental task in data analysis and machine learning, particularly when approximating nonlinear patterns. The sine wave, with its inherent periodic characteristics, provides an excellent benchmark for testing polynomial model flexibility. This article demonstrates MATLAB implementation of polynomial fitting for sinusoidal data, incorporating k-fold cross-validation for robust model evaluation.

### 1. Fundamental Principles of Polynomial Fitting The core concept involves using polynomial functions to approximate given data points. For periodic data like sine waves, higher-degree polynomials can achieve satisfactory fitting within specific intervals, though overfitting risks must be carefully managed. The mathematical foundation relies on minimizing residual errors through least squares optimization.

### 2. MATLAB Implementation with Code Insights MATLAB's built-in `polyfit(x,y,n)` function efficiently performs polynomial regression, where n represents the polynomial degree. For sine wave fitting, typical implementations involve: - Generating sample data: `x = 0:0.1:2*pi; y = sin(x);` - Fitting with cubic polynomial: `p = polyfit(x,y,3);` - Evaluating results: `y_fit = polyval(p,x);` Higher degrees (e.g., 5th-order) may improve accuracy but require validation against overfitting through cross-validation techniques.

### 3. K-Fold Cross-Validation Implementation To assess model generalization, 10-fold cross-validation partitions data randomly into 10 subsets, iteratively using 9 subsets for training and 1 for testing. MATLAB offers multiple approaches: - Using `crossval` function with custom loss functions - Manual implementation with for-loops and data indexing - Integration with `cvpartition` for structured data splitting The final model performance is averaged across all iterations to ensure reliability.

### 4. Model Evaluation and Optimization Metrics Key evaluation metrics include: - Mean Squared Error (MSE): `mse = mean((y_true - y_pred).^2)` - R-squared (R²): Measures proportion of variance explained Comparative analysis across different polynomial degrees helps identify optimal complexity, balancing bias-variance tradeoffs through visualization and statistical testing.

### 5. Critical Considerations and Best Practices - Polynomial degree selection: Overly high degrees may capture noise instead of signal - Cross-validation reliability: Requires representative data distribution - Alternative methods: Fourier transforms may better handle pure periodic data - Code implementation should include residual analysis and confidence interval estimation

Through systematic implementation of polynomial fitting with cross-validation in MATLAB, practitioners can develop robust models for sinusoidal pattern recognition while maintaining generalization capabilities for predictive accuracy.