Linear least squares

Linear least squares (LLS) is the least squares approximation of linear functions to data.
It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals.
Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.
Main formulations.
The three main linear least squares formulations are:
Alternative formulations.
Other formulations include:
In addition, percentage least squares focuses on reducing percentage errors, which is useful in the field of forecasting or time series analysis. It is also useful in situations where the dependent variable has a wide range without constant variance, as here the larger residuals at the upper end of the range would dominate if OLS were used. When the percentage or relative error is normally distributed, least squares percentage regression provides maximum likelihood estimates. Percentage regression is linked to a multiplicative error model, whereas OLS is linked to models containing an additive error term.
In constrained least squares, one is interested in solving a linear least squares problem with an additional constraint on the solution.
Objective function.
In OLS (i.e., assuming unweighted observations), the optimal value of the objective function is found by substituting the optimal expression for the coefficient vector:
where formula_2, the latter equality holding since formula_3 is symmetric and idempotent. It can be shown from this that under an appropriate assignment of weights the expected value of "S" is "m" − "n". If instead unit weights are assumed, the expected value of "S" is formula_4, where formula_5 is the variance of each observation.
If it is assumed that the residuals belong to a normal distribution, the objective function, being a sum of weighted squared residuals, will belong to a chi-squared (formula_6) distribution with "m" − "n" degrees of freedom. Some illustrative percentile values of formula_6 are given in the following table.
These values can be used for a statistical criterion as to the goodness of fit. When unit weights are used, the numbers should be divided by the variance of an observation.
For WLS, the ordinary objective function above is replaced for a weighted average of residuals.
Discussion.
In statistics and mathematics, linear least squares is an approach to fitting a mathematical or statistical model to data in cases where the idealized value provided by the model for any data point is expressed linearly in terms of the unknown parameters of the model. The resulting fitted model can be used to summarize the data, to predict unobserved values from the same system, and to understand the mechanisms that may underlie the system.
Mathematically, linear least squares is the problem of approximately solving an overdetermined system of linear equations A x = b, where b is not an element of the column space of the matrix A. The approximate solution is realized as an exact solution to A x = b', where b' is the projection of b onto the column space of A. The best approximation is then that which minimizes the sum of squared differences between the data values and their corresponding modeled values. The approach is called "linear" least squares since the assumed function is linear in the parameters to be estimated. Linear least squares problems are convex and have a closed-form solution that is unique, provided that the number of data points used for fitting equals or exceeds the number of unknown parameters, except in special degenerate situations. In contrast, non-linear least squares problems generally must be solved by an iterative procedure, and the problems can be non-convex with multiple optima for the objective function. If prior distributions are available, then even an underdetermined system can be solved using the Bayesian MMSE estimator.
In statistics, linear least squares problems correspond to a particularly important type of statistical model called linear regression which arises as a particular form of regression analysis. One basic form of such a model is an ordinary least squares model. The present article concentrates on the mathematical aspects of linear least squares problems, with discussion of the formulation and interpretation of statistical regression models and statistical inferences related to these being dealt with in the articles just mentioned. See outline of regression analysis for an outline of the topic.
Properties.
If the experimental errors, formula_9, are uncorrelated, have a mean of zero and a constant variance, formula_10, the Gauss–Markov theorem states that the least-squares estimator, formula_11, has the minimum variance of all estimators that are linear combinations of the observations. In this sense it is the best, or optimal, estimator of the parameters. Note particularly that this property is independent of the statistical distribution function of the errors. In other words, "the distribution function of the errors need not be a normal distribution". However, for some probability distributions, there is no guarantee that the least-squares solution is even possible given the observations; still, in such cases it is the best estimator that is both linear and unbiased.
For example, it is easy to show that the arithmetic mean of a set of measurements of a quantity is the least-squares estimator of the value of that quantity. If the conditions of the Gauss–Markov theorem apply, the arithmetic mean is optimal, whatever the distribution of errors of the measurements might be.
However, in the case that the experimental errors do belong to a normal distribution, the least-squares estimator is also a maximum likelihood estimator.
These properties underpin the use of the method of least squares for all types of data fitting, even when the assumptions are not strictly valid.
Limitations.
An assumption underlying the treatment given above is that the independent variable, "x", is free of error. In practice, the errors on the measurements of the independent variable are usually much smaller than the errors on the dependent variable and can therefore be ignored. When this is not the case, total least squares or more generally errors-in-variables models, or "rigorous least squares", should be used. This can be done by adjusting the weighting scheme to take into account errors on both the dependent and independent variables and then following the standard procedure.
In some cases the (weighted) normal equations matrix "X"T"X" is ill-conditioned. When fitting polynomials the normal equations matrix is a Vandermonde matrix. Vandermonde matrices become increasingly ill-conditioned as the order of the matrix increases. In these cases, the least squares estimate amplifies the measurement noise and may be grossly inaccurate. Various regularization techniques can be applied in such cases, the most common of which is called ridge regression. If further information about the parameters is known, for example, a range of possible values of formula_12, then various techniques can be used to increase the stability of the solution. For example, see constrained least squares.
Another drawback of the least squares estimator is the fact that the norm of the residuals, formula_13 is minimized, whereas in some cases one is truly interested in obtaining small error in the parameter formula_12, e.g., a small value of formula_15. However, since the true parameter formula_16 is necessarily unknown, this quantity cannot be directly minimized. If a prior probability on formula_11 is known, then a Bayes estimator can be used to minimize the mean squared error, formula_18. The least squares method is often applied when no prior is known. Surprisingly, when several parameters are being estimated jointly, better estimators can be constructed, an effect known as Stein's phenomenon. For example, if the measurement error is Gaussian, several estimators are known which dominate, or outperform, the least squares technique; the best known of these is the James–Stein estimator. This is an example of more general shrinkage estimators that have been applied to regression problems.
Applications.
Uses in data fitting.
The primary application of linear least squares is in data fitting. Given a set of "m" data points formula_21 consisting of experimentally measured values taken at "m" values formula_22 of an independent variable (formula_23 may be scalar or vector quantities), and given a model function formula_24 with formula_25 it is desired to find the parameters formula_26 such that the model function "best" fits the data. In linear least squares, linearity is meant to be with respect to parameters formula_27 so
Here, the functions formula_29 may be nonlinear with respect to the variable x.
Ideally, the model function fits the data exactly, so
for all formula_31 This is usually not possible in practice, as there are more data points than there are parameters to be determined. The approach chosen then is to find the minimal possible value of the sum of squares of the residuals
so to minimize the function
After substituting for formula_34 and then for formula_35, this minimization problem becomes the quadratic minimization problem above with
and the best fit can be found by solving the normal equations.
Example.
As a result of an experiment, four formula_37 data points were obtained, formula_38 formula_39 formula_40 and formula_41 (shown in red in the diagram on the right). We hope to find a line formula_42 that best fits these four points. In other words, we would like to find the numbers formula_43 and formula_44 that approximately solve the overdetermined linear system:
of four equations in two unknowns in some "best" sense.
formula_46 represents the residual, at each point, between the curve fit and the data:
The least squares approach to solving this problem is to try to make the sum of the squares of these residuals as small as possible; that is, to find the minimum of the function:
The minimum is determined by calculating the partial derivatives of formula_49 with respect to formula_43 and formula_44 and setting them to zero:
This results in a system of two equations in two unknowns, called the normal equations, which when solved give:
and the equation formula_56 is the line of best fit. The residuals, that is, the differences between the formula_57 values from the observations and the formula_57 predicated variables by using the line of best fit, are then found to be formula_59 formula_60 formula_61 and formula_62 (see the diagram on the right). The minimum value of the sum of squares of the residuals is formula_63
More generally, one can have formula_64 regressors formula_65, and a linear model
Using a quadratic model.
Importantly, in "linear least squares", we are not restricted to using a line as the model as in the above example. For instance, we could have chosen the restricted quadratic model formula_67. This model is still linear in the formula_43 parameter, so we can still perform the same analysis, constructing a system of equations from the data points:
The partial derivatives with respect to the parameters (this time there is only one) are again computed and set to 0:
formula_70
and solved
formula_71
leading to the resulting best fit model formula_72