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.

— Function File: x = lsqnonneg (c, d)
— Function File: x = lsqnonneg (c, d, x0)
— Function File: [x, resnorm] = lsqnonneg (...)
— Function File: [x, resnorm, residual] = lsqnonneg (...)
— Function File: [x, resnorm, residual, exitflag] = lsqnonneg (...)
— Function File: [x, resnorm, residual, exitflag, output] = lsqnonneg (...)
— Function File: [x, resnorm, residual, exitflag, output, lambda] = lsqnonneg (...)

Minimize norm (c*x-d) subject to x>= 0. c and d must be real. x0 is an optional initial guess for x.
Outputs:

resnorm
The squared 2-norm of the residual: norm(c*x-d)^2
residual
The residual: d-c*x
exitflag
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.)
output
A structure with two fields:

"algorithm": The algorithm used ("nnls")
"iterations": The number of iterations taken.

lambda
Not implemented.

See also: optimset.

— Function File: optimset ()
— Function File: optimset (par, val, ...)
— Function File: optimset (old, par, val, ...)
— Function File: optimset (old, new)

Create options struct for optimization functions.

— Function File: optimget (options, parname)
— Function File: optimget (options, parname, default)

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.

24.4 Linear Least Squares