Weighted least squares

This article or section may need to be cleaned up or summarized because it has been split from/to Linear least squares (mathematics) .

Part of a series on
Regression analysis
Models
Linear regression Simple regression Polynomial regression General linear model
Generalized linear model Vector generalized linear model Discrete choice Binomial regression Binary regression Logistic regression Multinomial logistic regression Mixed logit Probit Multinomial probit Ordered logit Ordered probit Poisson
Multilevel model Fixed effects Random effects Linear mixed-effects model Nonlinear mixed-effects model
Nonlinear regression Nonparametric Semiparametric Robust Quantile Isotonic Principal components Least angle Local Segmented
Errors-in-variables
Estimation
Least squares Linear Non-linear
Ordinary Weighted Generalized Generalized estimating equation
Partial Total Non-negative Ridge regression Regularized
Least absolute deviations Iteratively reweighted Bayesian Bayesian multivariate Least-squares spectral analysis
Background
Regression validation Mean and predicted response Errors and residuals Goodness of fit Studentized residual Gauss–Markov theorem
Mathematics portal
v t e

Weighted least squares (WLS), also known as weighted linear regression,

) is incorporated into the regression. WLS is also a specialization of generalized least squares, when all the off-diagonal entries of the covariance matrix of the errors, are null.

Formulation

The fit of a model to a data point is measured by its

, ${\displaystyle r_{i}}$, defined as the difference between a measured value of the dependent variable, ${\displaystyle y_{i}}$ and the value predicted by the model, ${\displaystyle f(x_{i},{\boldsymbol {\beta }})}$:

If the errors are uncorrelated and have equal variance, then the function

is minimised at ${\displaystyle {\boldsymbol {\hat {\beta }}}}$, such that ${\displaystyle {\frac {\partial S}{\partial \beta _{j}}}({\hat {\boldsymbol {\beta }}})=0}$.

The Gauss–Markov theorem shows that, when this is so, ${\displaystyle {\hat {\boldsymbol {\beta }}}}$ is a

showed that when a weighted sum of squared residuals is minimized, ${\displaystyle {\hat {\boldsymbol {\beta }}}}$ is the

BLUE

if each weight is equal to the reciprocal of the variance of the measurement

$${\displaystyle {\begin{aligned}S&=\sum _{i=1}^{n}W_{ii}{r_{i}}^{2},&W_{ii}&={\frac {1}{{\sigma _{i}}^{2}}}\end{aligned}}}$$

The gradient equations for this sum of squares are

$${\displaystyle -2\sum _{i}W_{ii}{\frac {\partial f(x_{i},{\boldsymbol {\beta }})}{\partial \beta _{j}}}r_{i}=0,\quad j=1,\ldots ,m}$$

which, in a linear least squares system give the modified normal equations,

$${\displaystyle \sum _{i=1}^{n}\sum _{k=1}^{m}X_{ij}W_{ii}X_{ik}{\hat {\beta }}_{k}=\sum _{i=1}^{n}X_{ij}W_{ii}y_{i},\quad j=1,\ldots ,m\,.}$$

When the observational errors are uncorrelated and the weight matrix, W=Ω⁻¹, is diagonal, these may be written as

$${\displaystyle \mathbf {\left(X^{\textsf {T}}WX\right){\hat {\boldsymbol {\beta }}}=X^{\textsf {T}}Wy} .}$$

If the errors are correlated, the resulting estimator is the

variance-covariance matrix

of the observations.

When the errors are uncorrelated, it is convenient to simplify the calculations to factor the weight matrix as ${\displaystyle w_{ii}={\sqrt {W_{ii}}}}$. The normal equations can then be written in the same form as ordinary least squares:

$${\displaystyle \mathbf {\left(X'^{\textsf {T}}X'\right){\hat {\boldsymbol {\beta }}}=X'^{\textsf {T}}y'} \,}$$

where we define the following scaled matrix and vector:

$${\displaystyle {\begin{aligned}\mathbf {X'} &=\operatorname {diag} \left(\mathbf {w} \right)\mathbf {X} ,\\\mathbf {y'} &=\operatorname {diag} \left(\mathbf {w} \right)\mathbf {y} =\mathbf {y} \oslash \mathbf {\sigma } .\end{aligned}}}$$

This is a type of

entrywise division

.

For non-linear least squares systems a similar argument shows that the normal equations should be modified as follows.

Note that for empirical tests, the appropriate W is not known for sure and must be estimated. For this

(FGLS) techniques may be used; in this case it is specialized for a diagonal covariance matrix, thus yielding a feasible weighted least squares solution.

If the uncertainty of the observations is not known from external sources, then the weights could be estimated from the given observations. This can be useful, for example, to identify outliers. After the outliers have been removed from the data set, the weights should be reset to one.^[3]

Motivation

In some cases the observations may be weighted—for example, they may not be equally reliable. In this case, one can minimize the weighted sum of squares:

where w_i > 0 is the weight of the ith observation, and W is the diagonal matrix of such weights.

The weights should, ideally, be equal to the

, the expression ${\textstyle S=\sum _{k}\sum _{j}r_{k}W_{kj}r_{j}\,}$ applies. In this case the weight matrix should ideally be equal to the inverse of the The normal equations are then:

$${\displaystyle \left(X^{\textsf {T}}WX\right){\hat {\boldsymbol {\beta }}}=X^{\textsf {T}}W\mathbf {y} .}$$

This method is used in iteratively reweighted least squares.

Solution

Parameter errors and correlation

The estimated parameter values are linear combinations of the observed values

Therefore, an expression for the estimated

from the errors in the observations. Let the variance-covariance matrix for the observations be denoted by M and that of the estimated parameters by M^β. Then

When W = M⁻¹, this simplifies to

When unit weights are used (W = I, the

${\displaystyle \chi _{\nu }^{2}}$:

where S is the minimum value of the weighted objective function:

The denominator, ${\displaystyle \nu =n-m}$, is the number of degrees of freedom; see effective degrees of freedom for generalizations for the case of correlated observations.

In all cases, the variance of the parameter estimate ${\displaystyle {\hat {\beta }}_{i}}$ is given by ${\displaystyle M_{ii}^{\beta }}$ and the covariance between the parameter estimates ${\displaystyle {\hat {\beta }}_{i}}$ and ${\displaystyle {\hat {\beta }}_{j}}$ is given by ${\displaystyle M_{ij}^{\beta }}$. The standard deviation is the square root of variance, ${\displaystyle \sigma _{i}={\sqrt {M_{ii}^{\beta }}}}$, and the correlation coefficient is given by ${\displaystyle \rho _{ij}=M_{ij}^{\beta }/(\sigma _{i}\sigma _{j})}$. These error estimates reflect only

, which, by definition, cannot be quantified. Note that even though the observations may be uncorrelated, the parameters are typically .

Parameter confidence limits

It is often assumed, for want of any concrete evidence but often appealing to the central limit theorem—see Normal distribution#Occurrence and applications—that the error on each observation belongs to a normal distribution with a mean of zero and standard deviation ${\displaystyle \sigma }$. Under that assumption the following probabilities can be derived for a single scalar parameter estimate in terms of its estimated standard error ${\displaystyle se_{\beta }}$ (given here):

• 68% that the interval ${\displaystyle {\hat {\beta }}\pm se_{\beta }}$ encompasses the true coefficient value
• 95% that the interval ${\displaystyle {\hat {\beta }}\pm 2se_{\beta }}$ encompasses the true coefficient value
• 99% that the interval ${\displaystyle {\hat {\beta }}\pm 2.5se_{\beta }}$ encompasses the true coefficient value

The assumption is not unreasonable when n >> m. If the experimental errors are normally distributed the parameters will belong to a Student's t-distribution with n − m degrees of freedom. When n ≫ m Student's t-distribution approximates a normal distribution. Note, however, that these confidence limits cannot take systematic error into account. Also, parameter errors should be quoted to one significant figure only, as they are subject to sampling error.^[4]

When the number of observations is relatively small,

can be used for an upper bound on probabilities, regardless of any assumptions about the distribution of experimental errors: the maximum probabilities that a parameter will be more than 1, 2, or 3 standard deviations away from its expectation value are 100%, 25% and 11% respectively.

Residual values and correlation

The

are related to the observations by

where H is the

:

and I is the identity matrix. The variance-covariance matrix of the residuals, M ^r is given by

Thus the residuals are correlated, even if the observations are not.

When ${\displaystyle W=M^{-1}}$,

The sum of weighted residual values is equal to zero whenever the model function contains a constant term. Left-multiply the expression for the residuals by X^T W^T:

Say, for example, that the first term of the model is a constant, so that ${\displaystyle X_{i1}=1}$ for all i. In that case it follows that

Thus, in the motivational example, above, the fact that the sum of residual values is equal to zero is not accidental, but is a consequence of the presence of the constant term, α, in the model.

If experimental error follows a normal distribution, then, because of the linear relationship between residuals and observations, so should residuals,^[5] but since the observations are only a sample of the population of all possible observations, the residuals should belong to a Student's t-distribution. Studentized residuals are useful in making a statistical test for an outlier when a particular residual appears to be excessively large.

See also

• Iteratively reweighted least squares
• Heteroscedasticity-consistent standard errors
• Weighted mean

References