One-Dimensional Moving Least Squares (MLS) MATLAB Implementation

Moving Least Squares (MLS) is a numerical method commonly used for curve fitting and surface reconstruction. Compared to traditional least squares methods, MLS performs weighted least squares fitting within local regions, providing greater flexibility to adapt to local data characteristics. This makes it particularly suitable for non-uniformly distributed or noisy data. Implementing one-dimensional MLS computation in MATLAB typically involves the following core steps: Data Preparation First, prepare one-dimensional data point sets including independent variable X and dependent variable Y. This data can originate from experimental measurements, simulation results, or other sources. In MATLAB implementation, data can be organized as column vectors using syntax like: X = [x1; x2; ...; xn]; Y = [y1; y2; ...; yn]; Weight Function Selection The core concept of MLS involves assigning different weights to each fitting point. Common weight functions include Gaussian functions, cubic spline functions, or other smooth functions. The weight magnitude depends on the distance between data points and the fitting point - closer points receive higher weights. For code implementation, a Gaussian weight function can be defined as: w = exp(-(distance.^2)/(h^2)), where h controls the bandwidth parameter. Local Fitting For each target point, select neighboring data points for local weighted least squares fitting. Calculate weights for points within the neighborhood using the weight function, then solve the weighted least squares problem to obtain the fitted value at that point. The MATLAB implementation typically involves constructing a design matrix A = [ones(size(X_local)), X_local] and solving (A' * W * A) \ (A' * W * Y_local) where W is the diagonal weight matrix. Global Fitting Iterate through all target points, repeating the local fitting process to ultimately obtain the complete fitted curve. This can be implemented using for-loops or optimized through vectorization techniques in MATLAB for better computational efficiency. The advantage of MLS lies in its local adaptability, effectively handling noise and non-uniform distributions in data, though it requires relatively larger computational resources. In MATLAB, efficient MLS computation can be achieved through loop structures or vectorized operations using built-in functions like meshgrid and array operations. For more complex applications, the MLS method can be extended by incorporating higher-order polynomial basis functions for advanced fitting, or by adjusting weight function parameters to optimize fitting performance. The polyval and polyfit functions in MATLAB can facilitate polynomial basis implementations, while parameter tuning can be automated using optimization techniques like fminsearch.