# Generalized Linear Least-Squares Algorithm

A Generalized Linear Least-Squares Algorithm is a linear least-squares algorithm that estimates the unknown function parameters using linear regression.

**AKA:**GLS, Generalized Least Squares.**Example(s):****Counter-Example(s):****See:**Lasso Algorithm.

## References

### 2013

- http://en.wikipedia.org/wiki/Generalized_least_squares
- In statistics,
**generalized least squares (GLS)**is a technique for estimating the unknown parameters in a linear regression model. The GLS is applied when the variances of the observations are unequal (heteroscedasticity), or when there is a certain degree of correlation between the observations. In these cases ordinary least squares can be statistically inefficient, or even give misleading inferences.

- In statistics,

- http://en.wikipedia.org/wiki/Generalized_least_squares#Method_outline
- In a typical linear regression model we observe data [math]\{y_i,x_{ij}\}_{i=1..n,j=1..p}[/math] on
*n*statistical units. The response values are placed in a vector Y*= (*yy_{1}, ...,n)′, and the predictor values are placed in the design matrix_{}*X*= [[*x*_{ij}*]], where*x_{ij}is the value of the*j*th predictor variable for the*i*th unit. The model assumes that the conditional mean of*Y*given*X*is a linear function of X*, whereas the conditional variance of the error term given*X is a*known*matrix Ω. This is usually written as : [math] Y = X\beta + \varepsilon, \qquad \mathrm{E}[\varepsilon|X]=0,\ \operatorname{Var}[\varepsilon|X]=\Omega. [/math] Here β*is a vector of unknown “regression coefficients” that must be estimated from the data.**Suppose*b*is a candidate estimate for*β*. Then the residual vector for*b*will be*Y*−*Xb*. Generalized least squares method estimates*β*by minimizing the squared Mahalanobis length of this residual vector: : [math] \hat\beta = \underset{b}{\rm arg\,min}\,(Y-Xb)'\,\Omega^{-1}(Y-Xb), [/math] Since the objective is a quadratic form in*b*, the estimator has an explicit formula: : [math] \hat\beta = (X'\Omega^{-1}X)^{-1} X'\Omega^{-1}Y. [/math]*

- In a typical linear regression model we observe data [math]\{y_i,x_{ij}\}_{i=1..n,j=1..p}[/math] on