Neliöintimenetelmä

Neliöintimenetelmä, known in English as the method of least squares, is a standard approach in regression analysis to approximate the solution of overdetermined systems, i.e., systems of equations in which there are more equations than unknowns. The method aims to find the "best fit" for a set of data points by minimizing the sum of the squares of the differences between the observed values and the values predicted by a mathematical model.

In simpler terms, imagine you have a scatter plot of data points. The neliöintimenetelmä tries to draw a line or curve through these points such that the total squared vertical distances from each point to the line or curve are as small as possible. This minimizes the overall error of the model in representing the data.

The core idea is to define an error function, typically the sum of squared residuals (the differences between observed and predicted values), and then find the parameters of the model that minimize this function. This minimization is often achieved using calculus, by taking the partial derivatives of the error function with respect to each parameter and setting them to zero. Solving these resulting equations yields the optimal parameter values.

The neliöintimenetelmä is widely used in various fields, including statistics, physics, engineering, economics, and computer science, for tasks such as fitting curves to experimental data, estimating parameters in statistical models, and solving systems of linear equations that do not have an exact solution. Its robustness and theoretical underpinnings make it a fundamental tool in data analysis.