This file is indexed.

/usr/share/octave/packages/optim-1.3.0/wpolyfit.m is in octave-optim 1.3.0-1.

This file is owned by root:root, with mode 0o644.

The actual contents of the file can be viewed below.

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
## Author: Paul Kienzle <pkienzle@gmail.com>
## This program is granted to the public domain.

## -*- texinfo -*-
## @deftypefn {Function File} {[@var{p}, @var{s}] =} wpolyfit (@var{x}, @var{y}, @var{dy}, @var{n})
## Return the coefficients of a polynomial @var{p}(@var{x}) of degree
## @var{n} that minimizes
## @iftex
## @tex
## $$
## \sum_{i=1}^N (p(x_i) - y_i)^2
## $$
## @end tex
## @end iftex
## @ifinfo
## @code{sumsq (p(x(i)) - y(i))},
## @end ifinfo
## to best fit the data in the least squares sense.  The standard error
## on the observations @var{y} if present are given in @var{dy}.
##
## The returned value @var{p} contains the polynomial coefficients 
## suitable for use in the function polyval.  The structure @var{s} returns
## information necessary to compute uncertainty in the model.
##
## To compute the predicted values of y with uncertainty use
## @example
## [y,dy] = polyconf(p,x,s,'ci');
## @end example
## You can see the effects of different confidence intervals and
## prediction intervals by calling the wpolyfit internal plot
## function with your fit:
## @example
## feval('wpolyfit:plt',x,y,dy,p,s,0.05,'pi')
## @end example
## Use @var{dy}=[] if uncertainty is unknown.
##
## You can use a chi^2 test to reject the polynomial fit:
## @example
## p = 1-chi2cdf(s.normr^2,s.df);
## @end example
## p is the probability of seeing a chi^2 value higher than that which 
## was observed assuming the data are normally distributed around the fit.
## If p < 0.01, you can reject the fit at the 1% level.
##
## You can use an F test to determine if a higher order polynomial 
## improves the fit:
## @example
## [poly1,S1] = wpolyfit(x,y,dy,n);
## [poly2,S2] = wpolyfit(x,y,dy,n+1);
## F = (S1.normr^2 - S2.normr^2)/(S1.df-S2.df)/(S2.normr^2/S2.df);
## p = 1-f_cdf(F,S1.df-S2.df,S2.df);
## @end example
## p is the probability of observing the improvement in chi^2 obtained
## by adding the extra parameter to the fit.  If p < 0.01, you can reject 
## the lower order polynomial at the 1% level.
##
## You can estimate the uncertainty in the polynomial coefficients 
## themselves using
## @example
## dp = sqrt(sumsq(inv(s.R'))'/s.df)*s.normr;
## @end example
## but the high degree of covariance amongst them makes this a questionable
## operation.
##
## @deftypefnx {Function File} {[@var{p}, @var{s}, @var{mu}] =} wpolyfit (...)
##
## If an additional output @code{mu = [mean(x),std(x)]} is requested then 
## the @var{x} values are centered and normalized prior to computing the fit.
## This will give more stable numerical results.  To compute a predicted 
## @var{y} from the returned model use
## @code{y = polyval(p, (x-mu(1))/mu(2)}
##
## @deftypefnx {Function File} wpolyfit (...)
##
## If no output arguments are requested, then wpolyfit plots the data,
## the fitted line and polynomials defining the standard error range.
##
## Example
## @example
## x = linspace(0,4,20);
## dy = (1+rand(size(x)))/2;
## y = polyval([2,3,1],x) + dy.*randn(size(x));
## wpolyfit(x,y,dy,2);
## @end example
##
## @deftypefnx {Function File} wpolyfit (..., 'origin')
##
## If 'origin' is specified, then the fitted polynomial will go through
## the origin.  This is generally ill-advised.  Use with caution.
##
## Hocking, RR (2003). Methods and Applications of Linear Models.
## New Jersey: John Wiley and Sons, Inc.
##
## @end deftypefn
## @seealso{polyfit,polyconf}

function [p_out, s, mu] = wpolyfit (varargin)

  ## strip 'origin' of the end
  args = length(varargin);
  if args>0 && ischar(varargin{args})
    origin = varargin{args};
    args--;
  else
    origin='';
  endif
  ## strip polynomial order off the end
  if args>0
    n = varargin{args};
    args--;
  else
    n = [];
  end
  ## interpret the remainder as x,y or x,y,dy or [x,y] or [x,y,dy]
  if args == 3
    x = varargin{1};
    y = varargin{2};
    dy = varargin{3};
  elseif args == 2
    x = varargin{1};
    y = varargin{2};
    dy = [];
  elseif args == 1
    A = varargin{1};
    [nr,nc]=size(A);
    if all(nc!=[2,3])
      error("wpolyfit expects vectors x,y,dy or matrix [x,y,dy]");
    endif
    dy = [];
    if nc == 3, dy = A(:,3); endif
    y = A(:,2);
    x = A(:,1);
  else
    usage ("wpolyfit (x, y [, dy], n [, 'origin'])");
  end

  if (length(origin) == 0)
    through_origin = 0;
  elseif strcmp(origin,'origin')
    through_origin = 1;
  else
    error ("wpolyfit: expected 'origin' but found '%s'", origin)
  endif

  if any(size (x) != size (y))
    error ("wpolyfit: x and y must be vectors of the same size");
  endif
  if length(dy)>1 && length(y) != length(dy)
    error ("wpolyfit: dy must be a vector the same length as y");
  endif

  if (! (isscalar (n) && n >= 0 && ! isinf (n) && n == round (n)))
    error ("wpolyfit: n must be a nonnegative integer");
  endif

  if nargout == 3
    mu = [mean(x), std(x)];
    x = (x - mu(1))/mu(2);
  endif

  k = length (x);

  ## observation matrix
  if through_origin
    ## polynomial through the origin y = ax + bx^2 + cx^3 + ...
    A = (x(:) * ones(1,n)) .^ (ones(k,1) * (n:-1:1));
  else
    ## polynomial least squares y = a + bx + cx^2 + dx^3 + ...
    A = (x(:) * ones (1, n+1)) .^ (ones (k, 1) * (n:-1:0));
  endif

  [p,s] = wsolve(A,y(:),dy(:));

  if through_origin
    p(n+1) = 0;
  endif

  if nargout == 0
    good_fit = 1-chi2cdf(s.normr^2,s.df);
    printf("Polynomial: %s  [ p(chi^2>observed)=%.2f%% ]\n", polyout(p,'x'), good_fit*100);
    plt(x,y,dy,p,s,'ci');
  else
    p_out = p';
  endif

function plt(x,y,dy,p,s,varargin)

  if iscomplex(p)
    # XXX FIXME XXX how to plot complex valued functions?
    # Maybe using hue for phase and saturation for magnitude
    # e.g., Frank Farris (Santa Cruz University) has this:
    # http://www.maa.org/pubs/amm_complements/complex.html
    # Could also look at the book
    #   Visual Complex Analysis by Tristan Needham, Oxford Univ. Press
    # but for now we punt
    return
  end

  ## decorate the graph
  grid('on');
  xlabel('abscissa X'); ylabel('data Y');
  title('Least-squares Polynomial Fit with Error Bounds');

  ## draw fit with estimated error bounds
  xf = linspace(min(x),max(x),150)';
  [yf,dyf] = polyconf(p,xf,s,varargin{:});
  plot(xf,yf+dyf,"g.;;", xf,yf-dyf,"g.;;", xf,yf,"g-;fit;");

  ## plot the data
  hold on;
  if (isempty(dy))
    plot(x,y,"x;data;");
  else
    if isscalar(dy), dy = ones(size(y))*dy; end
    errorbar (x, y, dy, "~;data;");
  endif
  hold off;

  if strcmp(deblank(input('See residuals? [y,n] ','s')),'y')
    clf;
    if (isempty(dy))
      plot(x,y-polyval(p,x),"x;data;");
    else
      errorbar(x,y-polyval(p,x),dy, '~;data;');
    endif
    hold on;
    grid on;
    ylabel('Residuals');
    xlabel('abscissa X'); 
    plot(xf,dyf,'g.;;',xf,-dyf,'g.;;');
    hold off;
  endif

%!demo % #1  
%! x = linspace(0,4,20);
%! dy = (1+rand(size(x)))/2;
%! y = polyval([2,3,1],x) + dy.*randn(size(x));
%! wpolyfit(x,y,dy,2);
  
%!demo % #2
%! x = linspace(-i,+2i,20);
%! noise = ( randn(size(x)) + i*randn(size(x)) )/10;
%! P = [2-i,3,1+i];
%! y = polyval(P,x) + noise;
%! wpolyfit(x,y,2)

%!demo
%! pin = [3; -1; 2];
%! x = -3:0.1:3;
%! y = polyval (pin, x);
%!
%! ## Poisson weights
%! # dy = sqrt (abs (y));
%! ## Uniform weights in [0.5,1]
%! dy = 0.5 + 0.5 * rand (size (y));
%!
%! y = y + randn (size (y)) .* dy;
%! printf ("Original polynomial: %s\n", polyout (pin, 'x'));
%! wpolyfit (x, y, dy, length (pin)-1);