This article is about regression analysis in statistics. Least squares” residual analysis in regression pdf that the overall solution minimizes the sum of the squares of the residuals made in the results of every single equation.
The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on land sightings for navigation. The approach was known as the method of averages. The method came to be known as the method of least absolute deviation. The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved. He felt these to be the simplest assumptions he could make, and he had hoped to obtain the arithmetic mean as the best estimate.
Instead, his estimator was the posterior median. The technique is described as an algebraic procedure for fitting linear equations to data and Legendre demonstrates the new method by analyzing the same data as Laplace for the shape of the earth. The value of Legendre’s method of least squares was immediately recognized by leading astronomers and geodesists of the time. In that work he claimed to have been in possession of the method of least squares since 1795. This naturally led to a priority dispute with Legendre. He had managed to complete Laplace’s program of specifying a mathematical form of the probability density for the observations, depending on a finite number of unknown parameters, and define a method of estimation that minimizes the error of estimation. He then turned the problem around by asking what form the density should have and what method of estimation should be used to get the arithmetic mean as estimate of the location parameter.
In this attempt, he invented the normal distribution. Ceres and was able to track its path for 40 days before it was lost in the glare of the sun. Ceres were those performed by the 24-year-old Gauss using least-squares analysis. In 1822, Gauss was able to state that the least-squares approach to regression analysis is optimal in the sense that in a linear model where the errors have a mean of zero, are uncorrelated, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. In the next two centuries workers in the theory of errors and in statistics found many different ways of implementing least squares. The objective consists of adjusting the parameters of a model function to best fit a data set.
The goal is to find the parameter values for the model that “best” fits the data. An example of a model in two dimensions is that of the straight line. A data point may consist of more than one independent variable. In the most general case there may be one or more independent variables and one or more dependent variables at each data point. This regression formulation considers only residuals in the dependent variable. Here a model is fitted to provide a prediction rule for application in a similar situation to which the data used for fitting apply. Here the dependent variables corresponding to such future application would be subject to the same types of observation error as those in the data used for fitting.