Least Squares Method with Adaptive Weights for Nonlinear Function Approximation in Arbitrary Bases

The least squares method is widely used for solving data approximation problems in various domains such as signal processing, statistical analysis, and complex system modeling. This study investigates the application of the least squares method and its modifications for approximating nonlinear functions with normally distributed noise. The experiment employs the least squares method with adaptive weights and a set of basis functions that allow for tailoring the approximating function to the data. The results demonstrate that approximating functions across the entire data range, even when using adaptive weights, can overlook significant data points or require expanding the basis, failing to achieve the desired accuracy. A local approach is proposed, involving partitioning the function into individual segments and applying LSM independently to each, to enhance approximation accuracy. The obtained results support the advantages of the local approach.