24.4 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.

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.