### 25.4 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.

: [beta, sigma, r] = ols (y, x)

Ordinary least squares (OLS) estimation.

OLS applies to the multivariate model y = x*b + e 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 is a p-variate observation in which each column represents a variable. Likewise, the rows of x represent k-variate observations or possibly designed values. Furthermore, the collection of observations x must be of adequate rank, k, otherwise b cannot be uniquely estimated.

The observation errors, e, are assumed to originate from an underlying p-variate distribution with zero mean and p-by-p covariance matrix S, both constant conditioned on x. Furthermore, the matrix S is constant with respect to each observation such that mean (e) = 0 and cov (vec (e)) = kron (s, I). (For cases that don’t meet this criteria, such as autocorrelated errors, see generalized least squares, gls, for more efficient estimations.)

The return values beta, sigma, and r are defined as follows.

beta

The OLS estimator for matrix 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.

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.

: [beta, v, r] = gls (y, x, o)

Generalized least squares (GLS) model.

Perform a generalized least squares estimation for the multivariate model y = x*B + E 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 is a p-variate observation in which each column represents a variable. Likewise, the rows of x represent k-variate observations or possibly designed values. Furthermore, the collection of observations x must be of adequate rank, k, otherwise b cannot be uniquely estimated.

The observation errors, e, are assumed to originate from an underlying p-variate distribution with zero mean but possibly heteroscedastic observations. That is, in general, mean (e) = 0 and cov (vec (e)) = (s^2)*o in which s is a scalar and o is a t*p-by-t*p matrix.

The return values beta, v, and r are defined as follows.

beta

The GLS estimator for matrix b.

v

The GLS estimator for scalar s^2.

r

The matrix of GLS residuals, r = y - x*beta.

: x = lsqnonneg (c, d)
: x = lsqnonneg (c, d, x0)
: x = lsqnonneg (c, d, x0, options)
: [x, resnorm] = lsqnonneg (…)
: [x, resnorm, residual] = lsqnonneg (…)
: [x, resnorm, residual, exitflag] = lsqnonneg (…)
: [x, resnorm, residual, exitflag, output] = lsqnonneg (…)
: [x, resnorm, residual, exitflag, output, lambda] = lsqnonneg (…)

Minimize norm (c*x - d) subject to x >= 0.

c and d must be real matrices.

x0 is an optional initial guess for the solution x.

options is an options structure to change the behavior of the algorithm (see optimset). lsqnonneg recognizes these options: "MaxIter", "TolX".

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

Lagrange multipliers. If these are nonzero, the corresponding x values should be zero, indicating the solution is pressed up against a coordinate plane. The magnitude indicates how much the residual would improve if the x >= 0 constraints were relaxed in that direction.

: x = lscov (A, b)
: x = lscov (A, b, V)
: x = lscov (A, b, V, alg)
: [x, stdx, mse, S] = lscov (…)

Compute a generalized linear least squares fit.

Estimate x under the model b = Ax + w, where the noise w is assumed to follow a normal distribution with covariance matrix {\sigma^2} V.

If the size of the coefficient matrix A is n-by-p, the size of the vector/array of constant terms b must be n-by-k.

The optional input argument V may be an n-element vector of positive weights (inverse variances), or an n-by-n symmetric positive semi-definite matrix representing the covariance of b. If V is not supplied, the ordinary least squares solution is returned.

The alg input argument, a guidance on solution method to use, is currently ignored.

Besides the least-squares estimate matrix x (p-by-k), the function also returns stdx (p-by-k), the error standard deviation of estimated x; mse (k-by-1), the estimated data error covariance scale factors (\sigma^2); and S (p-by-p, or p-by-p-by-k if k > 1), the error covariance of x.

Reference: Golub and Van Loan (1996), Matrix Computations (3rd Ed.), Johns Hopkins, Section 5.6.3

: optimset ()
: options = optimset ()
: options = optimset (par, val, …)
: options = optimset (old, par, val, …)
: options = optimset (old, new)

Create options structure for optimization functions.

When called without any input or output arguments, optimset prints a list of all valid optimization parameters.

When called with one output and no inputs, return an options structure with all valid option parameters initialized to [].

When called with a list of parameter/value pairs, return an options structure with only the named parameters initialized.

When the first input is an existing options structure old, the values are updated from either the par/val list or from the options structure new.

If par does not exactly match the name of a standard parameter, optimset will attempt to match par to a standard parameter and will set the value of that parameter if a match is found. Matching is case insensitive and is based on character matching at the start of the parameter name. optimset produces an error if it finds multiple ambiguous matches. If no standard parameter matches are found a warning is issued and the non-standard parameter is created.

Standard list of valid parameters:

AutoScaling
ComplexEqn
Display

Request verbose display of results from optimizations. Values are:

"off" [default]

No display.

"iter"

Display intermediate results for every loop iteration.

"final"

Display the result of the final loop iteration.

"notify"

Display the result of the final loop iteration if the function has failed to converge.

FinDiffType
FunValCheck

When enabled, display an error if the objective function returns an invalid value (a complex number, NaN, or Inf). Must be set to "on" or "off" [default]. Note: the functions fzero and fminbnd correctly handle Inf values and only complex values or NaN will cause an error in this case.

When set to "on", the function to be minimized must return a second argument which is the gradient, or first derivative, of the function at the point x. If set to "off" [default], the gradient is computed via finite differences.

Jacobian

When set to "on", the function to be minimized must return a second argument which is the Jacobian, or first derivative, of the function at the point x. If set to "off" [default], the Jacobian is computed via finite differences.

MaxFunEvals

Maximum number of function evaluations before optimization stops. Must be a positive integer.

MaxIter

Maximum number of algorithm iterations before optimization stops. Must be a positive integer.

OutputFcn

A user-defined function executed once per algorithm iteration.

TolFun

Termination criterion for the function output. If the difference in the calculated objective function between one algorithm iteration and the next is less than TolFun the optimization stops. Must be a positive scalar.

TolX

Termination criterion for the function input. If the difference in x, the current search point, between one algorithm iteration and the next is less than TolX the optimization stops. Must be a positive scalar.

TypicalX
Updating

This list can be extended by the user or other loaded Octave packages. An updated valid parameters list can be queried using the no-argument form of optimset.

Note 1: Only parameter names from the standard list are considered when matching short parameter names, and par will always be expanded to match a standard parameter even if an exact non-standard match exists. The value of a non-standard parameter that is ambiguous with one or more standard parameters cannot be set by optimset and can only be set using setfield or dot notation for structs.

Note 2: The optimization options structure is primarily intended for manipulation of known parameters by optimset and optimget. Unpredictable behavior on future calls to optimset or optimget can result from creating non-standard or ambiguous parameters or from loading/unloading packages that change the known parameter list after creation of an optimization options structure.

: val = optimget (options, par)
: val = optimget (options, par, default)

Return the value of the specific parameter par from the optimization options structure options created by optimset.

If par is not defined then return the default value if supplied, otherwise return an empty matrix.

If par does not exactly match the name of a standard parameter, optimget will attempt to match par to a standard parameter and will return that parameter’s value if a match is found. Matching is case insensitive and is based on character matching at the start of the parameter name. optimget produces an error if it finds multiple ambiguous matches. If no standard parameter matches are found a warning is issued. See optimset for information about the standard options list.

Note: Only parameter names from the standard list are considered when matching short parameter names, and par will always be expanded to match a standard parameter even if an exact non-standard match exists. The value of a non-standard parameter that is ambiguous with one or more standard parameters cannot be returned by optimget and can only be accessed using getfield or dot notation for structs.