Principles of Weighted Least Squares Method

This article provides an in-depth exploration of the principles, applications, and computational methods of the Weighted Least Squares (WLS) algorithm.

First, let's revisit ordinary Least Squares (LS). LS is a widely used regression analysis method that aims to establish regression models by finding estimates that minimize the sum of squared residuals. However, when different observations carry varying levels of importance, Weighted Least Squares becomes necessary.

The key distinction between WLS and standard LS lies in the introduction of weight coefficients that account for the reliability of each observation. This approach enhances model fitting accuracy and better reflects real-world scenarios. In implementation, weights are typically assigned through a weight matrix where diagonal elements represent the relative importance of each data point.

WLS finds extensive applications across various regression analysis scenarios, including multiple linear regression, nonlinear regression, and generalized linear models. In time series analysis, WLS proves particularly valuable for modeling and forecasting heteroscedastic data. The algorithm can be implemented using statistical software packages like R (using lm() with weights parameter) or Python's statsmodels (WLS class).

To perform WLS computations, understanding its mathematical formulation is essential. The solution typically involves matrix operations, where the weighted normal equations are solved to obtain optimal parameters. The fundamental equation takes the form: (X'WX)β = X'WY, where W is the weight matrix, X is the design matrix, and Y is the response vector. Computational methods often employ QR decomposition or singular value decomposition for numerical stability.

In summary, Weighted Least Squares is a highly practical regression analysis method that enables more accurate model building and prediction. The implementation typically involves constructing appropriate weight matrices based on residual analysis or domain knowledge, followed by solving the weighted normal equations using linear algebra techniques.