Polynomial Curve Fitting (Sine Wave) with Cross-Validation (10-Fold Cross-Validation)

Polynomial curve fitting is a method that approximates given data points using polynomial functions, widely applied in fields like signal processing. Using a sine wave as an example, we can implement polynomial fitting of different degrees to sinusoidal data in MATLAB and evaluate model generalization capability through cross-validation. First, generate a set of noisy sine wave data as the training set. The core of polynomial fitting lies in finding optimal polynomial coefficients that minimize the error between the fitted curve and original data. In MATLAB implementation, this typically involves using the polyfit function for coefficient calculation and polyval for curve evaluation. As polynomial degree increases, the fitted curve becomes more closely aligned with training data, but may exhibit overfitting—excellent performance on training data but poor generalization to new data. To assess true model performance, we employ 10-fold cross-validation: randomly partition original data into ten subsets, repeatedly use nine subsets for training and the remaining one for validation, cycling through all subsets. The average error across all ten iterations serves as the evaluation metric. This approach efficiently utilizes limited data while preventing evaluation bias from improper data partitioning. MATLAB implementation can use crossval function or custom looping structures with data indexing. Key observation involves monitoring the trend of training error versus validation error across different polynomial degrees. The optimal degree should be selected just before validation error begins to increase, balancing fitting accuracy with generalization capability. Through the sine wave fitting case study, we can clearly demonstrate three states: underfitting, appropriate fitting, and overfitting, while cross-validation provides quantitative basis for optimal model selection. This methodology, implementable through MATLAB's built-in functions and custom scripts, equally applies to other complex curve fitting scenarios. Critical implementation considerations include proper data normalization, efficient computation of polynomial coefficients using matrix operations, and systematic error calculation using metrics like mean squared error (MSE).