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 GNU Octave Manual Version 3 by John W. Eaton, David Bateman, Søren HaubergPaperback (6"x9"), 568 pagesISBN 095461206XRRP £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, ```beta = pinv (x) * y```, where `pinv (x)` denotes the pseudoinverse of 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