Previous: Nonlinear Programming, Up: Optimization [Contents][Index]
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.
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.
The OLS estimator for b.
beta is calculated directly via inv (x'*x) * x' * y
if the
matrix x'*x
is of full rank.
Otherwise, beta = pinv (x) * y
where
pinv (x)
denotes the pseudoinverse of x.
The OLS estimator for the matrix s,
sigma = (y-x*beta)' * (y-x*beta) / (t-rank(x))
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.
The GLS estimator for b.
The GLS estimator for s^2.
The matrix of GLS residuals, r = y - x*beta.
See also: ols.
Minimize norm (c*x - d)
subject to
x >= 0
. c and d must be real. x0 is an
optional initial guess for x.
Outputs:
The squared 2-norm of the residual: norm(c*x-d)^2
The residual: d-c*x
An indicator of convergence. 0 indicates that the iteration count was exceeded, and therefore convergence was not reached; >0 indicates that the algorithm converged. (The algorithm is stable and will converge given enough iterations.)
A structure with two fields:
Not implemented.
Create options struct for optimization functions.
Valid parameters are:
TolFun
the optimization stops. Must be a positive
scalar.
TolX
the optimization stops. Must be a positive scalar.
Return a specific option from a structure created by
optimset
. If parname is not a field of the options
structure, return default if supplied, otherwise return an
empty matrix.
Previous: Nonlinear Programming, Up: Optimization [Contents][Index]