Least squares


In regression analysis, least squares is a method to determine the best-fit model by minimizing the sum of the squared residuals—the differences between observed values and the values predicted by the model.
Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the model functions are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution. The nonlinear problem is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, and thus the core calculation is similar in both cases.
Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations from the fitted curve.
When the observations come from an exponential family with identity as its natural sufficient statistics and mild-conditions are satisfied, standardized least-squares estimates and maximum-likelihood estimates are identical. The method of least squares can also be derived as a method of moments estimator.

History

The method was the culmination of several advances that took place during the course of the eighteenth century:
  • The combination of different observations as being the best estimate of the true value; errors decrease with aggregation rather than increase, first appeared in Isaac Newton's work in 1671, though it went unpublished, and again in 1700. It was perhaps first expressed formally by Roger Cotes in 1722.
  • The combination of different observations taken under the same conditions contrary to simply trying one's best to observe and record a single observation accurately. The approach was known as the method of averages. This approach was notably used by Newton while studying equinoxes in 1700, also writing down the first of the 'normal equations' known from ordinary least squares, Tobias Mayer while studying the librations of the Moon in 1750, and by Pierre-Simon Laplace in his work in explaining the differences in motion of Jupiter and Saturn in 1788.
  • The combination of different observations taken under conditions. The method came to be known as the method of least absolute deviation. It was notably performed by Roger Joseph Boscovich in his work on the shape of the Earth in 1757 and by Pierre-Simon Laplace for the same problem in 1789 and 1799.
  • The development of a criterion that can be evaluated to determine when the solution with the minimum error has been achieved. Laplace tried to specify a mathematical form of the probability density for the errors and define a method of estimation that minimizes the error of estimation. For this purpose, Laplace used a symmetric two-sided exponential distribution we now call Laplace distribution to model the error distribution, and used the sum of absolute deviation as error of estimation. 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 method

The first clear and concise exposition of the method of least squares was published by Legendre in 1805. 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. Within ten years after Legendre's publication, the method of least squares had been adopted as a standard tool in astronomy and geodesy in France, Italy, and Prussia, which constitutes an extraordinarily rapid acceptance of a scientific technique.
In 1809 Carl Friedrich Gauss published his method of calculating the orbits of celestial bodies. 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. However, to Gauss's credit, he went beyond Legendre and succeeded in connecting the method of least squares with the principles of probability and to the normal distribution. 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. Gauss showed that the arithmetic mean is indeed the best estimate of the location parameter by changing both the probability density and the method 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.
An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid Ceres. On 1 January 1801, the Italian astronomer Giuseppe Piazzi discovered Ceres and was able to track its path for 40 days before it was lost in the glare of the Sun. Based on these data, astronomers desired to determine the location of Ceres after it emerged from behind the Sun without solving Kepler's complicated nonlinear equations of planetary motion. The only predictions that successfully allowed Hungarian astronomer Franz Xaver von Zach to relocate Ceres were those performed by the 24-year-old Gauss using least-squares analysis.
In 1810, after reading Gauss's work, Laplace, after proving the central limit theorem, used it to give a large sample justification for the method of least squares and the normal distribution. 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, normally distributed, and have equal variances, the best linear unbiased estimator of the coefficients is the least-squares estimator. An extended version of this result is known as the Gauss–Markov theorem.
The idea of least-squares analysis was also independently formulated by the American Robert Adrain in 1808. In the next two centuries workers in the theory of errors and in statistics found many different ways of implementing least squares.

Problem statement

The objective consists of adjusting the parameters of a model function to best fit a data set. A simple data set consists of n points , i = 1, …, n, where is an independent variable and is a dependent variable whose value is found by observation. The model function has the form, where m adjustable parameters are held in the vector. The goal is to find the parameter values for the model that "best" fits the data. The fit of a model to a data point is measured by its residual, defined as the difference between the observed value of the dependent variable and the value predicted by the model:
The least-squares method finds the optimal parameter values by minimizing the sum of squared residuals, :
In the simplest case, and the result of the least-squares method is the arithmetic mean of the input data.
An example of a model in two dimensions is that of the straight line. Denoting the y-intercept as and the slope as, the model function is given by. See linear least squares for a fully worked out example of this model.
A data point may consist of more than one independent variable. For example, when fitting a plane to a set of height measurements, the plane is a function of two independent variables, x and z, say. In the most general case there may be one or more independent variables and one or more dependent variables at each data point.
To the right is a residual plot illustrating random fluctuations about, indicating that a linear model is appropriate. is an independent, random variable.  

If the residual points had some sort of a shape and were not randomly fluctuating, a linear model would not be appropriate. For example, if the residual plot had a parabolic shape as seen to the right, a parabolic model would be appropriate for the data. The residuals for a parabolic model can be calculated via.

Limitations

This regression formulation considers only observational errors in the dependent variable. There are two rather different contexts with different implications:
  • Regression for prediction. 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. It is therefore logically consistent to use the least-squares prediction rule for such data.
  • Regression for fitting a "true relationship". In standard regression analysis that leads to fitting by least squares there is an implicit assumption that errors in the independent variable are zero or strictly controlled so as to be negligible. When errors in the independent variable are non-negligible, models of measurement error can be used; such methods can lead to parameter estimates, hypothesis testing and confidence intervals that take into account the presence of observation errors in the independent variables. An alternative approach is to fit a model by total least squares; this can be viewed as taking a pragmatic approach to balancing the effects of the different sources of error in formulating an objective function for use in model-fitting.

    Solving the least squares problem

The minimum of the sum of squares is found by setting the gradient to zero. Since the model contains m parameters, there are m gradient equations:
and since, the gradient equations become
The gradient equations apply to all least squares problems. Each particular problem requires particular expressions for the model and its partial derivatives.