MATLAB Code Implementation for Surface Fitting

Surface fitting in MATLAB typically involves three key steps: data preparation, model fitting, and prediction computation. The core concept of surface fitting is to approximate known discrete data points using mathematical functions, enabling estimation of values at unknown locations. First, you need to prepare known discrete data points including X and Y coordinates along with corresponding Z values. These datasets may originate from experimental measurements or numerical simulations, usually stored as matrices or vectors in the MATLAB workspace. To handle this efficiently, you can organize data using column vectors or meshgrid arrays, with missing values handled through interpolation or removal techniques. Next, select an appropriate fitting method. MATLAB provides various fitting functions such as `polyfitn` (for polynomial fitting) or the `fit` function combined with surface models (like quadratic polynomials, Gaussian models, etc.). The least squares method remains the most common fitting approach, determining optimal parameters by minimizing the sum of squared errors. For polynomial surface fitting, you can construct a model containing higher-order terms of X and Y (such as X², XY, Y²) and solve for coefficients using linear regression. Implementation involves creating a design matrix using terms like [ones(size(X)) X Y X.*Y X.^2 Y.^2] for quadratic fitting, then applying the backslash operator (\) for coefficient calculation: coefficients = design_matrix \ Z. After fitting completes, you obtain a set of coefficients representing the surface shape. These coefficients define the mathematical expression of the surface. For predicting unknown points, simply substitute their X and Y coordinates into the fitted equation to compute corresponding Z values. MATLAB's `polyvaln` function or custom expressions can conveniently perform this calculation. For custom implementations, you might evaluate using dot operations: Z_pred = coefficients(1) + coefficients(2)*X_new + coefficients(3)*Y_new + coefficients(4)*X_new.*Y_new + ... Surface fitting finds extensive applications in engineering and scientific fields such as terrain modeling, temperature distribution analysis, and mechanical stress prediction. By adjusting model complexity (e.g., polynomial degree), you can balance between overfitting and underfitting, ensuring the model maintains both strong approximation capability and generalization performance. Cross-validation techniques like k-fold validation can help determine optimal polynomial orders programmatically.