Curve Fitting Using Least Squares Method

In statistics and mathematics, the least squares method is a fundamental mathematical optimization technique used to find the best-fitting function curve for a given set of data points. This method determines the optimal fitting curve by minimizing the sum of squared errors. It finds extensive applications across various fields including economics, physics, and engineering. The core principle involves minimizing the total sum of squared differences between each data point and the corresponding value on the fitted curve, ensuring optimal data alignment.

The fundamental concept of least squares fitting involves approximating data with a mathematical function that minimizes the distances between all data points and the curve. By minimizing the residual sum of squares (RSS), the method identifies the optimal curve parameters that can be utilized for future predictions. In computational implementations, this typically involves solving a system of linear equations or using matrix operations like the normal equation: θ = (XᵀX)⁻¹Xᵀy, where X represents the design matrix and y contains the observed values.

While least squares is a widely adopted fitting technique, it possesses certain limitations. For instance, some datasets may not be appropriately represented by functional curves, necessitating alternative fitting approaches like robust regression or nonparametric methods. Additionally, the method's performance can be influenced by factors such as dataset size and noise levels. Key implementation considerations include checking for multicollinearity in independent variables and assessing residual patterns. Therefore, careful evaluation of both data characteristics and fitting function suitability is essential before applying least squares, including diagnostic measures like R-squared values and residual analysis to verify method appropriateness.