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annotate scripts/polynomial/polyfit.m @ 19790:446c46af4b42 stable

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strip trailing whitespace from most source files * Makefile.am, NEWS, build-aux/common.mk, configure.ac, doc/Makefile.am, doc/doxyhtml/Makefile.am, doc/interpreter/Makefile.am, doc/interpreter/arith.txi, doc/interpreter/audio.txi, doc/interpreter/basics.txi, doc/interpreter/bugs.txi, doc/interpreter/container.txi, doc/interpreter/cp-idx.txi, doc/interpreter/data.txi, doc/interpreter/debug.txi, doc/interpreter/diagperm.txi, doc/interpreter/diffeq.txi, doc/interpreter/doccheck/README, doc/interpreter/doccheck/spellcheck, doc/interpreter/emacs.txi, doc/interpreter/errors.txi, doc/interpreter/eval.txi, doc/interpreter/expr.txi, doc/interpreter/external.txi, doc/interpreter/fn-idx.txi, doc/interpreter/func.txi, doc/interpreter/geometry.txi, doc/interpreter/geometryimages.m, doc/interpreter/gpl.txi, doc/interpreter/grammar.txi, doc/interpreter/gui.txi, doc/interpreter/image.txi, doc/interpreter/install.txi, doc/interpreter/interp.txi, doc/interpreter/interpimages.m, 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scripts/statistics/distributions/frnd.m, scripts/statistics/distributions/gamrnd.m, scripts/statistics/distributions/geornd.m, scripts/statistics/distributions/hygernd.m, scripts/statistics/distributions/kolmogorov_smirnov_cdf.m, scripts/statistics/distributions/laplace_cdf.m, scripts/statistics/distributions/laplace_pdf.m, scripts/statistics/distributions/logistic_cdf.m, scripts/statistics/distributions/logistic_pdf.m, scripts/statistics/distributions/lognrnd.m, scripts/statistics/distributions/nbincdf.m, scripts/statistics/distributions/nbininv.m, scripts/statistics/distributions/nbinpdf.m, scripts/statistics/distributions/nbinrnd.m, scripts/statistics/distributions/normrnd.m, scripts/statistics/distributions/poissinv.m, scripts/statistics/distributions/poissrnd.m, scripts/statistics/distributions/tinv.m, scripts/statistics/distributions/trnd.m, scripts/statistics/distributions/unidcdf.m, scripts/statistics/distributions/unidpdf.m, scripts/statistics/distributions/unidrnd.m, scripts/statistics/distributions/unifrnd.m, scripts/statistics/distributions/wblrnd.m, scripts/statistics/models/module.mk, scripts/statistics/tests/kruskal_wallis_test.m, scripts/strings/base2dec.m, scripts/strings/deblank.m, scripts/strings/dec2base.m, scripts/strings/dec2bin.m, scripts/strings/dec2hex.m, scripts/strings/mat2str.m, scripts/strings/ostrsplit.m, scripts/strings/regexptranslate.m, scripts/strings/str2num.m, scripts/strings/strcat.m, scripts/strings/strjoin.m, scripts/strings/strsplit.m, scripts/strings/strtok.m, scripts/strings/strtrim.m, scripts/strings/strtrunc.m, scripts/strings/substr.m, scripts/testfun/__run_test_suite__.m, scripts/testfun/speed.m, scripts/testfun/test.m, scripts/time/asctime.m, scripts/time/datenum.m, scripts/time/datevec.m, scripts/time/weekday.m, src/Makefile.am, test/Makefile.am, test/build-bc-overload-tests.sh, test/build-sparse-tests.sh, test/jit.tst, test/line-continue.tst: Strip trailing whitespace.
author John W. Eaton <jwe@octave.org>
date Tue, 20 Jan 2015 08:26:57 -0500
parents d63878346099
children e5facc6eec13
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1 ## Copyright (C) 1996-2013 John W. Eaton
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2 ##
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3 ## This file is part of Octave.
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4 ##
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5 ## Octave is free software; you can redistribute it and/or modify it
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6 ## under the terms of the GNU General Public License as published by
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7 ## the Free Software Foundation; either version 3 of the License, or (at
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8 ## your option) any later version.
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9 ##
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10 ## Octave is distributed in the hope that it will be useful, but
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe]
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11 ## WITHOUT ANY WARRANTY; without even the implied warranty of
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe]
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12 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
5ca126254d15 [project @ 1996-07-11 21:25:22 by jwe]
jwe
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13 ## General Public License for more details.
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14 ##
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15 ## You should have received a copy of the GNU General Public License
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16 ## along with Octave; see the file COPYING. If not, see
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17 ## <http://www.gnu.org/licenses/>.
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19 ## -*- texinfo -*-
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20 ## @deftypefn {Function File} {@var{p} =} polyfit (@var{x}, @var{y}, @var{n})
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21 ## @deftypefnx {Function File} {[@var{p}, @var{s}] =} polyfit (@var{x}, @var{y}, @var{n})
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22 ## @deftypefnx {Function File} {[@var{p}, @var{s}, @var{mu}] =} polyfit (@var{x}, @var{y}, @var{n})
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23 ## Return the coefficients of a polynomial @var{p}(@var{x}) of degree
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24 ## @var{n} that minimizes the least-squares-error of the fit to the points
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25 ## @code{[@var{x}, @var{y}]}. If @var{n} is a logical vector, it is used
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26 ## as a mask to selectively force the corresponding polynomial
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27 ## coefficients to be used or ignored.
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28 ##
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29 ## The polynomial coefficients are returned in a row vector.
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30 ##
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31 ## The optional output @var{s} is a structure containing the following fields:
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32 ##
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33 ## @table @samp
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34 ## @item R
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35 ## Triangular factor R from the QR@tie{}decomposition.
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36 ##
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37 ## @item X
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38 ## The Vandermonde matrix used to compute the polynomial coefficients.
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39 ##
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40 ## @item C
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41 ## The unscaled covariance matrix, formally equal to the inverse of
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42 ## @var{x'}*@var{x}, but computed in a way minimizing roundoff error
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43 ## propagation.
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44 ##
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45 ## @item df
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46 ## The degrees of freedom.
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47 ##
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48 ## @item normr
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49 ## The norm of the residuals.
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50 ##
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51 ## @item yf
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52 ## The values of the polynomial for each value of @var{x}.
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53 ## @end table
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54 ##
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55 ## The second output may be used by @code{polyval} to calculate the
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56 ## statistical error limits of the predicted values. In particular, the
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57 ## standard deviation of @var{p} coefficients is given by @*
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58 ## @code{sqrt (diag (s.C)/s.df)*s.normr}.
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59 ##
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60 ## When the third output, @var{mu}, is present the
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61 ## coefficients, @var{p}, are associated with a polynomial in
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62 ## @var{xhat} = (@var{x}-@var{mu}(1))/@var{mu}(2).
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63 ## Where @var{mu}(1) = mean (@var{x}), and @var{mu}(2) = std (@var{x}).
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64 ## This linear transformation of @var{x} improves the numerical
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65 ## stability of the fit.
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66 ## @seealso{polyval, polyaffine, roots, vander, zscore}
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67 ## @end deftypefn
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68
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69 ## Author: KH <Kurt.Hornik@wu-wien.ac.at>
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70 ## Created: 13 December 1994
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71 ## Adapted-By: jwe
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72 ## Modified on 20120204 by P. Dupuis; added the ability to specify a
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73 ## polynomial mask instead of a polynomial degree.
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74
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75 function [p, s, mu] = polyfit (x, y, n)
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76
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77 if (nargin < 3 || nargin > 4)
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78 print_usage ();
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79 endif
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80
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81 if (nargout > 2)
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82 ## Normalized the x values.
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83 mu = [mean(x), std(x)];
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84 x = (x - mu(1)) / mu(2);
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85 endif
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86
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87 if (! size_equal (x, y))
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88 error ("polyfit: X and Y must be vectors of the same size");
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89 endif
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90
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91 if (islogical (n))
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92 polymask = n;
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93 ## n is the polynomial degree as given the polymask size; m is the
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94 ## effective number of used coefficients.
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95 n = length (polymask) - 1; m = sum (polymask) - 1;
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96 else
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97 if (! (isscalar (n) && n >= 0 && ! isinf (n) && n == fix (n)))
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98 error ("polyfit: N must be a non-negative integer");
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99 endif
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100 polymask = logical (ones (1, n+1)); m = n;
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101 endif
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102
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103 y_is_row_vector = (rows (y) == 1);
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104
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105 ## Reshape x & y into column vectors.
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106 l = numel (x);
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107 x = x(:);
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108 y = y(:);
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109
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110 ## Construct the Vandermonde matrix.
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111 v = vander (x, n+1);
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112
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113 ## Solve by QR decomposition.
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114 [q, r, k] = qr (v(:, polymask), 0);
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115 p = r \ (q' * y);
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116 p(k) = p;
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117
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118 if (n != m)
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119 q = p; p = zeros (n+1, 1);
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120 p(polymask) = q;
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121 endif
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122
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123 if (nargout > 1)
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124 yf = v*p;
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125
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126 if (y_is_row_vector)
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127 s.yf = yf.';
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128 else
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129 s.yf = yf;
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130 endif
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131 s.X = v;
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132
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133 ## r.'*r is positive definite if X(:, polymask) is of full rank.
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134 ## Invert it by cholinv to avoid taking the square root of squared
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135 ## quantities. If cholinv fails, then X(:, polymask) is rank
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136 ## deficient and not invertible.
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137 try
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138 C = cholinv (r.'*r)(k, k);
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139 catch
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140 C = NaN (m+1, m+1);
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141 end_try_catch
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142
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143 if (n != m)
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144 ## fill matrices if required
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145 s.X(:, !polymask) = 0;
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146 s.R = zeros (n+1, n+1); s.R(polymask, polymask) = r;
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147 s.C = zeros (n+1, n+1); s.C(polymask, polymask) = C;
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148 else
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149 s.R = r;
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150 s.C = C;
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151 endif
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152 s.df = l - m - 1;
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153 s.normr = norm (yf - y);
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154 endif
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155
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156 ## Return a row vector.
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157 p = p.';
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158
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159 endfunction
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160
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161
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162 %!shared x
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163 %! x = [-2, -1, 0, 1, 2];
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164 %!assert (polyfit (x, x.^2+x+1, 2), [1, 1, 1], sqrt (eps))
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165 %!assert (polyfit (x, x.^2+x+1, 3), [0, 1, 1, 1], sqrt (eps))
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166 %!fail ("polyfit (x, x.^2+x+1)")
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167 %!fail ("polyfit (x, x.^2+x+1, [])")
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168
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169 ## Test difficult case where scaling is really needed. This example
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170 ## demonstrates the rather poor result which occurs when the dependent
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171 ## variable is not normalized properly.
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172 ## Also check the usage of 2nd & 3rd output arguments.
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173 %!test
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174 %! x = [ -1196.4, -1195.2, -1194, -1192.8, -1191.6, -1190.4, -1189.2, -1188, ...
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175 %! -1186.8, -1185.6, -1184.4, -1183.2, -1182];
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176 %! y = [ 315571.7086, 315575.9618, 315579.4195, 315582.6206, 315585.4966, ...
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177 %! 315588.3172, 315590.9326, 315593.5934, 315596.0455, 315598.4201, ...
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178 %! 315600.7143, 315602.9508, 315605.1765 ];
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179 %! [p1, s1] = polyfit (x, y, 10);
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180 %! [p2, s2, mu] = polyfit (x, y, 10);
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181 %! assert (s2.normr < s1.normr);
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182
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183 %!test
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184 %! x = 1:4;
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185 %! p0 = [1i, 0, 2i, 4];
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186 %! y0 = polyval (p0, x);
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187 %! p = polyfit (x, y0, numel (p0) - 1);
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188 %! assert (p, p0, 1000*eps);
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189
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190 %!test
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191 %! x = 1000 + (-5:5);
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192 %! xn = (x - mean (x)) / std (x);
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193 %! pn = ones (1,5);
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194 %! y = polyval (pn, xn);
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195 %! [p, s, mu] = polyfit (x, y, numel (pn) - 1);
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196 %! [p2, s2] = polyfit (x, y, numel (pn) - 1);
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197 %! assert (p, pn, s.normr);
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198 %! assert (s.yf, y, s.normr);
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199 %! assert (mu, [mean(x), std(x)]);
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200 %! assert (s.normr/s2.normr < sqrt (eps));
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201
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202 %!test
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203 %! x = [1, 2, 3; 4, 5, 6];
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204 %! y = [0, 0, 1; 1, 0, 0];
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205 %! p = polyfit (x, y, 5);
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206 %! expected = [0, 1, -14, 65, -112, 60] / 12;
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207 %! assert (p, expected, sqrt (eps));
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208
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209 %!error <vectors of the same size> polyfit ([1, 2; 3, 4], [1, 2, 3, 4], 2)
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210