“Linear least squares”的意思、由来-开放百科全书

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

Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.

Main formulations

The OLS method minimizes the sum of squared residuals, and leads to a closed-form expression for the estimated value of the unknown parameter vector β:

where

is a vector whose ith element is the ith observation of the dependent variable, and

is a matrix whose ij element is the ith observation of the jth independent variable. The estimator is unbiased and consistent if the errors have finite variance and are uncorrelated with the regressors:^[1]

where

is the transpose of row i of the matrix

It is also efficient under the assumption that the errors have finite variance and are homoscedastic, meaning that E[ε_i²{{!}}x_i] does not depend on i. The condition that the errors are uncorrelated with the regressors will generally be satisfied in an experiment, but in the case of observational data, it is difficult to exclude the possibility of an omitted covariate z that is related to both the observed covariates and the response variable. The existence of such a covariate will generally lead to a correlation between the regressors and the response variable, and hence to an inconsistent estimator of β. The condition of homoscedasticity can fail with either experimental or observational data. If the goal is either inference or predictive modeling, the performance of OLS estimates can be poor if multicollinearity is present, unless the sample size is large.

where Ω is the covariance matrix of the errors. GLS can be viewed as applying a linear transformation to the data so that the assumptions of OLS are met for the transformed data. For GLS to be applied, the covariance structure of the errors must be known up to a multiplicative constant.

Alternative formulations

Optimal instruments regression is an extension of classical IV regression to the situation where E[ε_i {{!}} z_i] = 0.

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.^[6]

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 in the optimal expression for the coefficient vector, can be written as:

where

, the latter equality holding since

is symmetric and idempotent. It can be shown from this^[7] 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

, where

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 (

) distribution with m − n degrees of freedom. Some illustrative percentile values of

are given in the following table.^[8]

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, where the best approximation is defined as 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,

, are uncorrelated, have a mean of zero and a constant variance,

, the Gauss–Markov theorem states that the least-squares estimator,

, 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.^[9]

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.^[10]^[11]

In some cases the (weighted) normal equations matrix X^TX 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.{{citation needed|date=December 2010}} In these cases, the least squares estimate amplifies the measurement noise and may be grossly inaccurate.{{citation needed|date=December 2010}} 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

, 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,

is minimized, whereas in some cases one is truly interested in obtaining small error in the parameter

, e.g., a small value of

.{{citation needed|date=December 2010}} However, since the true parameter

is necessarily unknown, this quantity cannot be directly minimized. If a prior probability on

is known, then a Bayes estimator can be used to minimize the mean squared error,

. 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

consisting of experimentally measured values taken at m values

of an independent variable (

may be scalar or vector quantities), and given a model function

with

it is desired to find the parameters

such that the model function "best" fits the data. In linear least squares, linearity is meant to be with respect to parameters

for all

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

After substituting for

and then for

, this minimization problem becomes the quadratic minimization problem above with

Example

As a result of an experiment, four

data points were obtained,

and

(shown in red in the diagram on the right). We hope to find a line

that best fits these four points. In other words, we would like to find the numbers

and

that approximately solve the overdetermined linear system

The residual, at each point, between the curve fit and the data is the difference between the right- and left-hand sides of the equations above. 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

with respect to

and

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

of the line of best fit. The residuals, that is, the differences between the

values from the observations and the

predicated variables by using the line of best fit, are then found to be

and

(see the diagram on the right). The minimum value of the sum of squares of the residuals is

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

. This model is still linear in the

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:

Main formulations

Alternative formulations

Objective function

Discussion

Properties

Limitations

Applications

Uses in data fitting

Example

Using a quadratic model

See also

References

Further reading

External links