Mercurial > octave
view scripts/linear-algebra/ols.m @ 30564:796f54d4ddbf stable
update Octave Project Developers copyright for the new year
In files that have the "Octave Project Developers" copyright notice,
update for 2021.
In all .txi and .texi files except gpl.txi and gpl.texi in the
doc/liboctave and doc/interpreter directories, change the copyright
to "Octave Project Developers", the same as used for other source
files. Update copyright notices for 2022 (not done since 2019). For
gpl.txi and gpl.texi, change the copyright notice to be "Free Software
Foundation, Inc." and leave the date at 2007 only because this file
only contains the text of the GPL, not anything created by the Octave
Project Developers.
Add Paul Thomas to contributors.in.
author | John W. Eaton <jwe@octave.org> |
---|---|
date | Tue, 28 Dec 2021 18:22:40 -0500 |
parents | 7854d5752dd2 |
children | 597f3ee61a48 |
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######################################################################## ## ## Copyright (C) 1996-2022 The Octave Project Developers ## ## See the file COPYRIGHT.md in the top-level directory of this ## distribution or <https://octave.org/copyright/>. ## ## This file is part of Octave. ## ## Octave is free software: you can redistribute it and/or modify it ## under the terms of the GNU General Public License as published by ## the Free Software Foundation, either version 3 of the License, or ## (at your option) any later version. ## ## Octave is distributed in the hope that it will be useful, but ## WITHOUT ANY WARRANTY; without even the implied warranty of ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ## GNU General Public License for more details. ## ## You should have received a copy of the GNU General Public License ## along with Octave; see the file COPYING. If not, see ## <https://www.gnu.org/licenses/>. ## ######################################################################## ## -*- texinfo -*- ## @deftypefn {} {[@var{beta}, @var{sigma}, @var{r}] =} ols (@var{y}, @var{x}) ## Ordinary least squares (OLS) estimation. ## ## OLS applies to the multivariate model ## @tex ## $@var{y} = @var{x}\,@var{b} + @var{e}$ ## @end tex ## @ifnottex ## @w{@math{@var{y} = @var{x}*@var{b} + @var{e}}} ## @end ifnottex ## where ## @tex ## $@var{y}$ is a $t \times p$ matrix, $@var{x}$ is a $t \times k$ matrix, ## $@var{b}$ is a $k \times p$ matrix, and $@var{e}$ is a $t \times p$ matrix. ## @end tex ## @ifnottex ## @math{@var{y}} is a @math{t}-by-@math{p} matrix, @math{@var{x}} is a ## @math{t}-by-@math{k} matrix, @var{b} is a @math{k}-by-@math{p} matrix, and ## @var{e} is a @math{t}-by-@math{p} matrix. ## @end ifnottex ## ## Each row of @var{y} is a @math{p}-variate observation in which each column ## represents a variable. Likewise, the rows of @var{x} represent ## @math{k}-variate observations or possibly designed values. Furthermore, ## the collection of observations @var{x} must be of adequate rank, @math{k}, ## otherwise @var{b} cannot be uniquely estimated. ## ## The observation errors, @var{e}, are assumed to originate from an ## underlying @math{p}-variate distribution with zero mean and ## @math{p}-by-@math{p} covariance matrix @var{S}, both constant conditioned ## on @var{x}. Furthermore, the matrix @var{S} is constant with respect to ## each observation such that ## @tex ## $\bar{@var{e}} = 0$ and cov(vec(@var{e})) = kron(@var{s},@var{I}). ## @end tex ## @ifnottex ## @code{mean (@var{e}) = 0} and ## @code{cov (vec (@var{e})) = kron (@var{s}, @var{I})}. ## @end ifnottex ## (For cases ## that don't meet this criteria, such as autocorrelated errors, see ## generalized least squares, gls, for more efficient estimations.) ## ## The return values @var{beta}, @var{sigma}, and @var{r} are defined as ## follows. ## ## @table @var ## @item beta ## The OLS estimator for matrix @var{b}. ## @tex ## @var{beta} is calculated directly via $(@var{x}^T@var{x})^{-1} @var{x}^T ## @var{y}$ if the matrix $@var{x}^T@var{x}$ is of full rank. ## @end tex ## @ifnottex ## @var{beta} is calculated directly via ## @code{inv (@var{x}'*@var{x}) * @var{x}' * @var{y}} if the matrix ## @code{@var{x}'*@var{x}} is of full rank. ## @end ifnottex ## Otherwise, @code{@var{beta} = pinv (@var{x}) * @var{y}} where ## @code{pinv (@var{x})} denotes the pseudoinverse of @var{x}. ## ## @item sigma ## The OLS estimator for the matrix @var{s}, ## ## @example ## @group ## @var{sigma} = (@var{y}-@var{x}*@var{beta})' * (@var{y}-@var{x}*@var{beta}) / (@math{t}-rank(@var{x})) ## @end group ## @end example ## ## @item r ## The matrix of OLS residuals, @code{@var{r} = @var{y} - @var{x}*@var{beta}}. ## @end table ## @seealso{gls, pinv} ## @end deftypefn function [beta, sigma, r] = ols (y, x) if (nargin != 2) print_usage (); endif if (! (isnumeric (x) && isnumeric (y))) error ("ols: X and Y must be numeric matrices or vectors"); endif if (ndims (x) != 2 || ndims (y) != 2) error ("ols: X and Y must be 2-D matrices or vectors"); endif [nr, nc] = size (x); [ry, cy] = size (y); if (nr != ry) error ("ols: number of rows of X and Y must be equal"); endif if (isinteger (x)) x = double (x); endif if (isinteger (y)) y = double (y); endif ## Start of algorithm z = x' * x; [u, p] = chol (z); if (p) beta = pinv (x) * y; else beta = u \ (u' \ (x' * y)); endif if (isargout (2) || isargout (3)) r = y - x * beta; endif if (isargout (2)) ## z is of full rank, avoid the SVD in rnk if (p == 0) rnk = columns (z); else rnk = rank (z); endif sigma = r' * r / (nr - rnk); endif endfunction %!test %! x = [1:5]'; %! y = 3*x + 2; %! x = [x, ones(5,1)]; %! assert (ols (y,x), [3; 2], 50*eps); %!test %! x = [1, 2; 3, 4]; %! y = [1; 2]; %! [b, s, r] = ols (x, y); %! assert (b, [1.4, 2], 2*eps); %! assert (s, [0.2, 0; 0, 0], 2*eps); %! assert (r, [-0.4, 0; 0.2, 0], 2*eps); %!test %! x = [1, 2; 3, 4]; %! y = [1; 2]; %! [b, s] = ols (x, y); %! assert (b, [1.4, 2], 2*eps); %! assert (s, [0.2, 0; 0, 0], 2*eps); %!test %! x = [1, 2; 3, 4]; %! y = [1; 2]; %! b = ols (x, y); %! assert (b, [1.4, 2], 2*eps); ## Test input validation %!error <Invalid call> ols () %!error <Invalid call> ols (1) %!error ols ([true, true], [1, 2]) %!error ols ([1, 2], [true, true]) %!error ols (ones (2,2,2), ones (2,2)) %!error ols (ones (2,2), ones (2,2,2)) %!error ols (ones (1,2), ones (2,2))