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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 good 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.
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, wherepinv (
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.
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.