Fitting a three-dimensional sphere by the least squares method

Abstract

As a traditional parameter estimation method, the least squares method is an important tool for studying the optimization of functions in an interval and for applying the optimal function to an unknown function, as well as a mathematical optimization method using the sum of squares of the errors to find the best-fitting function. When the error value between the estimated data and the actual data reaches the minimum, it means that the estimation model has a better fitting effect. The article derives the principle of the least squares method from a differential perspective and explains the origin of multivariate linear fitting. Curve fitting is described in detail, and the nonlinear fitting process is deduced on the basis of the polynomial fitting principle. In the article, in order to fit a three-dimensional sphere with the smallest possible error under the mean-square significance, the data of the three-dimensional sphere are nonlinearly fitted on the basis of utilizing the principle of the method of least squares, and Gaussian-distributed noises with zero mean are also added to compare with the original fitted sphere. On this basis, a weighted least squares is proposed using weight analysis thereby minimizing the error and finally fitting the optimal sphere.