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The Origin of the Least Squares Method: How was this mathematical magic born?
In data analysis and regression models, the method of least squares is one of the most popular parameter estimation methods. The core of this method is to minimize the sum of squared errors between observed values and model predicted values. The birth of the least squares method is deeply rooted in scientific developments in the 18th century, especially in the fields of astronomy and geodesy. Scientists at the time needed precise data for navigation, which led to the gradual maturity of the method of least squares.
The method of least squares was born in the quest to solve the challenges of navigating the Earth's oceans.
The Development of Least Squares Method
The origins of the least squares method can be traced back to Adrien-Marie Legendre who first publicly proposed this method in 1805. The essence of this technique is to fit a linear equation to the data through an algebraic procedure. In his published article, Legendre used data previously used by Pierre-Simon Laplace to analyze the shape of the Earth.
Before Legendre, as early as 1671, Ivy Newton had begun exploring the combination of different observations, suggesting the existence of best estimates, where the errors of these observations would gradually decrease rather than increase after aggregation. The concept was further developed in 1700 and 1722. Many methods around these principles were embodied in later discoveries, including the "method of averages" and the "method of least absolute deviations." These methods all emphasize combining observational data under different conditions.
The development of the least squares method was a response to many challenges in astronomy at the time, especially in the prediction of celestial motions.
Important milestones in history
In 1810, Carl Friedrich Gauss further refined the method of least squares, relating it to probability theory and normal distribution. Gauss claimed in his works that he had acquired this method since 1795 and had used it extensively in his research. Although there was a dispute over priority between him and Legendre, Gauss deserves recognition for his successful combination of the method of least squares with the theory of errors into a broader mathematical framework.
Gauss's advantage lies in that he combined the arithmetic mean with the optimal estimation regression model of the location parameter, transformed the basis of the least squares method, and clarified its superiority in regression analysis. He further improved this method by discovering the normal distribution. After Gauss, Laplace also verified the method of least squares in 1810, further establishing its position in statistics.
Gauss's work demonstrated the powerful potential of the method of least squares in predicting future events, especially in the accuracy of astronomical observations.
Applications and Challenges of Least Squares Method
As the term least squares-based model implies, the goal is to adjust the model parameters to best fit a set of observed data. In the most common scenarios, these data points may come from single or multivariate analyses. Although the least squares method is widely used in many practical situations, it also suffers from algorithmic limitations, especially in the face of observation errors. If the errors of the independent variables cannot be ignored, the total least squares method can be considered to seek more robust estimates.
The method of least squares remains the cornerstone of many modern simulations and data analyses today. Nevertheless, the approach is not completely immune to the difficulties that arise with the increase of complex variables. For example, nonlinear least squares methods often require iterative approximations, which can be computationally expensive.
ConclusionThe success of the least squares method lies not only in its wide application in data fitting, but also in its unlimited possibilities for future data exploration.
The method of least squares is not only a mathematical technique, its birth and development represent the journey of scientific progress. Over the centuries, this method has evolved from simple observations to complex mathematical models and remains an indispensable tool in data science today. This makes us wonder how future mathematical technology will change our understanding and use of data?
