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maint: Remove "Created:" lines from code base. * lin2mu.m, mu2lin.m, bincoeff.m, common_size.m, deal.m, nextpow2.m, postpad.m, prepad.m, repmat.m, shift.m, xor.m, inpolygon.m, rotx.m, roty.m, rotz.m, getappdata.m, isappdata.m, rmappdata.m, setappdata.m, colormap.m, gray.m, gray2ind.m, image.m, imagesc.m, ind2gray.m, ind2rgb.m, ocean.m, rgb2ind.m, javachk.m, javamem.m, commutation_matrix.m, cross.m, duplication_matrix.m, gls.m, housh.m, isdefinite.m, ishermitian.m, issymmetric.m, null.m, ols.m, orth.m, qzhess.m, vech.m, __w2mpth__.m, expand_rel_paths.m, make_rel_paths.m, daspect.m, pbaspect.m, rticks.m, thetaticks.m, xticklabels.m, xticks.m, yticklabels.m, yticks.m, zticklabels.m, zticks.m, comet.m, errorbar.m, loglogerr.m, __errplot__.m, semilogxerr.m, semilogyerr.m, cla.m, copyobj.m, hdl2struct.m, linkaxes.m, __ghostscript__.m, __gnuplot_get_var__.m, __gnuplot_has_feature__.m, __gnuplot_has_terminal__.m, __gnuplot_open_stream__.m, struct2hdl.m, compan.m, conv.m, deconv.m, mpoles.m, poly.m, polyder.m, polyfit.m, polyint.m, polyout.m, polyreduce.m, polyval.m, polyvalm.m, residue.m, roots.m, __parse_movargs__.m, detrend.m, fftconv.m, fftfilt.m, fftshift.m, ifftshift.m, movfun.m, movslice.m, beta.m, betaln.m, lcm.m, pow2.m, corr.m, kurtosis.m, skewness.m, untabify.m, calendar.m, datestr.m, datevec.m, eomday.m, now.m, weekday.m: Remove "Created:" lines from code base.
author Rik <rik@octave.org>
date Tue, 21 Jan 2020 13:06:02 -0800
parents bd51beb6205e
children 7f1cec12c240
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
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1 ########################################################################
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2 ##
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3 ## Copyright (C) 1996-2020 The Octave Project Developers
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4 ##
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5 ## See the file COPYRIGHT.md in the top-level directory of this
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6 ## distribution or <https://octave.org/copyright/>.
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7 ##
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8 ## This file is part of Octave.
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9 ##
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10 ## Octave is free software: you can redistribute it and/or modify it
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11 ## under the terms of the GNU General Public License as published by
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12 ## the Free Software Foundation, either version 3 of the License, or
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13 ## (at your option) any later version.
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14 ##
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15 ## Octave is distributed in the hope that it will be useful, but
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16 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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18 ## GNU General Public License for more details.
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19 ##
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20 ## You should have received a copy of the GNU General Public License
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21 ## along with Octave; see the file COPYING. If not, see
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22 ## <https://www.gnu.org/licenses/>.
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23 ##
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24 ########################################################################
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25
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26 ## -*- texinfo -*-
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27 ## @deftypefn {} {[@var{beta}, @var{sigma}, @var{r}] =} ols (@var{y}, @var{x})
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28 ## Ordinary least squares (OLS) estimation.
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29 ##
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30 ## OLS applies to the multivariate model
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31 ## @tex
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32 ## $@var{y} = @var{x}\,@var{b} + @var{e}$
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33 ## @end tex
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34 ## @ifnottex
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35 ## @w{@math{@var{y} = @var{x}*@var{b} + @var{e}}}
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36 ## @end ifnottex
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37 ## where
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38 ## @tex
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39 ## $@var{y}$ is a $t \times p$ matrix, $@var{x}$ is a $t \times k$ matrix,
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40 ## $@var{b}$ is a $k \times p$ matrix, and $@var{e}$ is a $t \times p$ matrix.
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41 ## @end tex
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42 ## @ifnottex
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43 ## @math{@var{y}} is a @math{t}-by-@math{p} matrix, @math{@var{x}} is a
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44 ## @math{t}-by-@math{k} matrix, @var{b} is a @math{k}-by-@math{p} matrix, and
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45 ## @var{e} is a @math{t}-by-@math{p} matrix.
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46 ## @end ifnottex
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47 ##
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48 ## Each row of @var{y} is a @math{p}-variate observation in which each column
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49 ## represents a variable. Likewise, the rows of @var{x} represent
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50 ## @math{k}-variate observations or possibly designed values. Furthermore,
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51 ## the collection of observations @var{x} must be of adequate rank, @math{k},
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52 ## otherwise @var{b} cannot be uniquely estimated.
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53 ##
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54 ## The observation errors, @var{e}, are assumed to originate from an
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55 ## underlying @math{p}-variate distribution with zero mean and
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56 ## @math{p}-by-@math{p} covariance matrix @var{S}, both constant conditioned
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57 ## on @var{x}. Furthermore, the matrix @var{S} is constant with respect to
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58 ## each observation such that
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59 ## @tex
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60 ## $\bar{@var{e}} = 0$ and cov(vec(@var{e})) = kron(@var{s},@var{I}).
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61 ## @end tex
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62 ## @ifnottex
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63 ## @code{mean (@var{e}) = 0} and
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64 ## @code{cov (vec (@var{e})) = kron (@var{s}, @var{I})}.
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65 ## @end ifnottex
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66 ## (For cases
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67 ## that don't meet this criteria, such as autocorrelated errors, see
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68 ## generalized least squares, gls, for more efficient estimations.)
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69 ##
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70 ## The return values @var{beta}, @var{sigma}, and @var{r} are defined as
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71 ## follows.
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72 ##
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73 ## @table @var
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74 ## @item beta
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75 ## The OLS estimator for matrix @var{b}.
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76 ## @tex
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77 ## @var{beta} is calculated directly via $(@var{x}^T@var{x})^{-1} @var{x}^T
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78 ## @var{y}$ if the matrix $@var{x}^T@var{x}$ is of full rank.
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79 ## @end tex
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80 ## @ifnottex
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81 ## @var{beta} is calculated directly via
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82 ## @code{inv (@var{x}'*@var{x}) * @var{x}' * @var{y}} if the matrix
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83 ## @code{@var{x}'*@var{x}} is of full rank.
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84 ## @end ifnottex
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85 ## Otherwise, @code{@var{beta} = pinv (@var{x}) * @var{y}} where
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86 ## @code{pinv (@var{x})} denotes the pseudoinverse of @var{x}.
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87 ##
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88 ## @item sigma
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89 ## The OLS estimator for the matrix @var{s},
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90 ##
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91 ## @example
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92 ## @group
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93 ## @var{sigma} = (@var{y}-@var{x}*@var{beta})' * (@var{y}-@var{x}*@var{beta}) / (@math{t}-rank(@var{x}))
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94 ## @end group
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95 ## @end example
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96 ##
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97 ## @item r
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98 ## The matrix of OLS residuals, @code{@var{r} = @var{y} - @var{x}*@var{beta}}.
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99 ## @end table
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100 ## @seealso{gls, pinv}
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101 ## @end deftypefn
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102
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103 ## Author: Teresa Twaroch <twaroch@ci.tuwien.ac.at>
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104 ## Adapted-By: jwe
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105
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106 function [beta, sigma, r] = ols (y, x)
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107
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108 if (nargin != 2)
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109 print_usage ();
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110 endif
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111
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112 if (! (isnumeric (x) && isnumeric (y)))
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113 error ("ols: X and Y must be numeric matrices or vectors");
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114 endif
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115
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116 if (ndims (x) != 2 || ndims (y) != 2)
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117 error ("ols: X and Y must be 2-D matrices or vectors");
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118 endif
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119
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120 [nr, nc] = size (x);
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121 [ry, cy] = size (y);
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122 if (nr != ry)
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123 error ("ols: number of rows of X and Y must be equal");
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124 endif
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125
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126 if (isinteger (x))
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127 x = double (x);
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128 endif
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129 if (isinteger (y))
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130 y = double (y);
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131 endif
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132
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133 ## Start of algorithm
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134 z = x' * x;
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135 [u, p] = chol (z);
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136
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137 if (p)
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138 beta = pinv (x) * y;
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139 else
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140 beta = u \ (u' \ (x' * y));
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141 endif
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142
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143 if (isargout (2) || isargout (3))
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144 r = y - x * beta;
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145 endif
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146 if (isargout (2))
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147
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148 ## z is of full rank, avoid the SVD in rnk
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149 if (p == 0)
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150 rnk = columns (z);
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151 else
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152 rnk = rank (z);
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153 endif
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154
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155 sigma = r' * r / (nr - rnk);
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156 endif
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157
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158 endfunction
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159
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160
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161 %!test
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162 %! x = [1:5]';
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163 %! y = 3*x + 2;
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164 %! x = [x, ones(5,1)];
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165 %! assert (ols (y,x), [3; 2], 50*eps);
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166
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167 %!test
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168 %! x = [1, 2; 3, 4];
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169 %! y = [1; 2];
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170 %! [b, s, r] = ols (x, y);
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171 %! assert (b, [1.4, 2], 2*eps);
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172 %! assert (s, [0.2, 0; 0, 0], 2*eps);
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173 %! assert (r, [-0.4, 0; 0.2, 0], 2*eps);
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174
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175 %!test
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176 %! x = [1, 2; 3, 4];
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177 %! y = [1; 2];
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178 %! [b, s] = ols (x, y);
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179 %! assert (b, [1.4, 2], 2*eps);
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180 %! assert (s, [0.2, 0; 0, 0], 2*eps);
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181
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182 %!test
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183 %! x = [1, 2; 3, 4];
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184 %! y = [1; 2];
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185 %! b = ols (x, y);
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186 %! assert (b, [1.4, 2], 2*eps);
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187
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188 ## Test input validation
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189 %!error ols ()
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190 %!error ols (1)
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191 %!error ols (1, 2, 3)
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192 %!error ols ([true, true], [1, 2])
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193 %!error ols ([1, 2], [true, true])
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194 %!error ols (ones (2,2,2), ones (2,2))
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195 %!error ols (ones (2,2), ones (2,2,2))
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196 %!error ols (ones (1,2), ones (2,2))