GNU Octave Manual Version 3by John W. Eaton, David Bateman, Søren Hauberg Paperback (6"x9"), 568 pages ISBN 095461206X RRP £24.95 ($39.95) |

## 23.4 Linear Least Squares

Octave also supports linear least squares minimization. That is, Octave can find the parameter b such that the model y = x*b

fits data (x,y) as well as possible, assuming zero-mean
Gaussian noise. If the noise is assumed to be isotropic the problem
can be solved using the `‘\’` or `‘/’` operators, or the `ols`

function. In the general case where the noise is assumed to be anisotropic
the `gls`

is needed.

__Function File:__[`beta`,`sigma`,`r`] =**ols***(*`y`,`x`)- Ordinary least squares estimation for the multivariate model
y = x b + e with
mean (e) = 0 and cov (vec (e)) = kron (s, I).
where y is a t by p matrix, x is a t by k matrix, b is a k by p matrix, and e is a t by p matrix.

Each row of

`y`and`x`is an observation and each column a variable.The return values

`beta`,`sigma`, and`r`are defined as follows.`beta`-
The OLS estimator for
`b`,

, where`beta`= pinv (`x`) *`y``pinv (`

denotes the pseudoinverse of`x`)`x`. `sigma`-
The OLS estimator for the matrix
`s`,`sigma`= (`y`-`x`*`beta`)' * (`y`-`x`*`beta`) / (`t`-rank(`x`)) `r`-
The matrix of OLS residuals,

.`r`=`y`-`x`*`beta`

__Function File:__[`beta`,`v`,`r`] =**gls***(*`y`,`x`,`o`)- Generalized least squares estimation for the multivariate model
y = x b + e with mean (e) = 0 and
cov (vec (e)) = (s^2) o,
where y is a t by p matrix, x is a t by k matrix, b is a k by p matrix, e is a t by p matrix, and o is a t p by t p matrix.

Each row of

`y`and`x`is an observation and each column a variable. The return values`beta`,`v`, and`r`are defined as follows.`beta`- The GLS estimator for b.
`v`- The GLS estimator for s^2.
`r`- The matrix of GLS residuals, r = y - x beta.

ISBN 095461206X | GNU Octave Manual Version 3 | See the print edition |