Re: [changeset] Polyfit with scaling

[Ben Abbott]

Replacing the patch with a mercurial export.

# HG changeset patch
# User Ben Abbott <address@hidden>
# Date 1203424120 18000
# Node ID 37bf6aca402506dd0cb12bbed25b8dfbb32b59eb
# Parent  c1279a36706dbd1d0cbc99c1f679f01c0e7f3714
Modified polyfit to use QR decompsition (more stable), added  
normalization (DWMD) of the dependent variable. Modified polyval to  
respect normalization and added optional output of 50% prediction  
intervals (DWMD).

diff -r c1279a36706d -r 37bf6aca4025 scripts/ChangeLog
--- a/scripts/ChangeLog Tue Feb 19 07:24:49 2008 -0500
+++ b/scripts/ChangeLog Tue Feb 19 07:28:40 2008 -0500
@@ -1,3 +1,10 @@ 2008-02-18  Ben Abbott <address@hidden
+2008-02-19  Ben Abbott <address@hidden>
+
+       * polynomial/polyfit.m: Modified algorithm to use QR decomposition,
+         and added Matlab's normalizaton option.
+       * polynomial/polyval.m: Modified to respect normalization of the
+         dependent variable, and added optional 50% prediction intervals.
+
 2008-02-18  Ben Abbott <address@hidden>

        * miscellaneous/ver: Added package version information, and
diff -r c1279a36706d -r 37bf6aca4025 scripts/polynomial/polyfit.m
--- a/scripts/polynomial/polyfit.m      Tue Feb 19 07:24:49 2008 -0500
+++ b/scripts/polynomial/polyfit.m      Tue Feb 19 07:28:40 2008 -0500
@@ -18,39 +18,38 @@
 ## <http://www.gnu.org/licenses/>.

 ## -*- texinfo -*-
-## @deftypefn {Function File} address@hidden, @var{s}] =} polyfit  
(@var{x}, @var{y}, @var{n})
+## @deftypefn {Function File} address@hidden, @var{s}, @var{mu}] =}  
polyfit (@var{x}, @var{y}, @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.
+## @var{n} that minimizes the least-squares-error of the fit.
 ##
 ## The polynomial coefficients are returned in a row vector.
 ##
-## If two output arguments are requested, the second is a structure
+## The second output is a structured variable, @var{s},
 ## containing the following fields:
 ##
-## @table @code
+## @table @samp
 ## @item R
-## The Cholesky factor of the Vandermonde matrix used to compute the
-## polynomial coefficients.
+##  Triangular factor R from the QR decomposition.
 ## @item X
-## The Vandermonde matrix used to compute the polynomial coefficients.
+##  The Vandermonde matrix used to compute the polynomial coefficients.
 ## @item df
-## The degrees of freedom.
+##  The degrees of freedom.
 ## @item normr
-## The norm of the residuals.
+##  The norm of the residuals.
 ## @item yf
-## The values of the polynomial for each value of @var{x}.
+##  The values of the polynomial for each value of @var{x}.
 ## @end table
+##
+## The second output may be used by @code{polyval} to calculate the
+## statistical error limits of the predicted values.
+##
+## When the third output, @var{mu}, is present the
+## coefficients, @var{p}, are associated with a polynomial in
+## @var{xhat} = (@address@hidden(1))/@var{mu}(2).
+## Where @var{mu}(1) = mean (@var{x}), and @var{mu}(2) = std (@var{x}).
+## This linear transformation of @var{x} improves the numerical
+## stability of the fit.
+## @seealso{polyval, polyconf, residue}
 ## @end deftypefn

 ## Author: KH <address@hidden>
@@ -59,12 +58,17 @@

 function [p, s, mu] = polyfit (x, y, n)

-
-  if (nargin != 3)
+  if (nargin < 3 || nargin > 4)
     print_usage ();
   endif

-  if (! (isvector (x) && isvector (y) && size_equal (x, y)))
+  if (nargout > 2)
+    ## Normalized the x values.
+    mu = [mean(x), std(x)];
+    x = (x - mu(1)) / mu(2);
+  endif
+
+  if (! size_equal (x, y))
     error ("polyfit: x and y must be vectors of the same size");
   endif

@@ -74,17 +78,21 @@ function [p, s, mu] = polyfit (x, y, n)

   y_is_row_vector = (rows (y) == 1);

-  l = length (x);
+  ## Reshape x & y into column vectors.
+  l = numel (x);
   x = reshape (x, l, 1);
   y = reshape (y, l, 1);

-  X = (x * ones (1, n+1)) .^ (ones (l, 1) * (n : -1 : 0));
+  ## Construct the Vandermonde matrix.
+  v = (x * ones (1, n+1)) .^ (ones (l, 1) * (n : -1 : 0));

-  p = X \ y;
+  ## Solve by QR decomposition.
+  [q, r, k] = qr (v, 0);
+  p = r \ (y' * q)';
+  p(k) = p;

   if (nargout > 1)
-
-    yf = X*p;
+    yf = v*p;

     if (y_is_row_vector)
       s.yf = yf.';
@@ -92,15 +100,13 @@ function [p, s, mu] = polyfit (x, y, n)
       s.yf = yf;
     endif

-    [s.R, dummy] = chol (X'*X);
-    s.X = X;
+    s.R = r;
+    s.X = v;
     s.df = l - n - 1;
     s.normr = norm (yf - y);
-
   endif

-  ## Return value should be a row vector.
-
+  ## Return a row vector.
   p = p.';

 endfunction
@@ -109,11 +115,11 @@ endfunction
 %! x = [-2, -1, 0, 1, 2];
 %! assert(all (all (abs (polyfit (x, x.^2+x+1, 2) - [1, 1, 1]) <  
sqrt (eps))));

+%!error(polyfit ([1, 2; 3, 4], [1, 2, 3, 4], 2))
+
 %!test
 %! x = [-2, -1, 0, 1, 2];
 %! assert(all (all (abs (polyfit (x, x.^2+x+1, 3) - [0, 1, 1, 1]) <  
sqrt (eps))));
-
-%!error polyfit ([1, 2; 3, 4], [1, 2; 3, 4], 4);

 %!test
 %! x = [-2, -1, 0, 1, 2];
@@ -123,3 +129,45 @@ endfunction
 %! x = [-2, -1, 0, 1, 2];
 %! fail("polyfit (x, x.^2+x+1, [])");

+## Test difficult case where scaling is really needed. This example
+## demonstrates the rather poor result which occurs when the dependent
+## variable is not normalized properly.
+## Also check the usage of 2nd & 3rd output arguments.
+%!test
+%! x = [ -1196.4, -1195.2, -1194, -1192.8, -1191.6, -1190.4, -1189.2,  
-1188, \
+%!       -1186.8, -1185.6, -1184.4, -1183.2, -1182];
+%! y = [ 315571.7086, 315575.9618, 315579.4195, 315582.6206,  
315585.4966,    \
+%!       315588.3172, 315590.9326, 315593.5934, 315596.0455,  
315598.4201,    \
+%!       315600.7143, 315602.9508, 315605.1765 ];
+%! [p1, s1] = polyfit (x, y, 10);
+%! [p2, s2, mu] = polyfit (x, y, 10);
+%! assert (s1.normr, 0.11264, 0.1)
+%! assert (s2.normr < s1.normr)
+
+%!test
+%! x = 1:4;
+%! p0 = [1i, 0, 2i, 4];
+%! y0 = polyval (p0, x);
+%! p = polyfit (x, y0, numel(p0)-1);
+%! assert (p, p0, 1000*eps)
+
+%!test
+%! x = 1000 + (-5:5);
+%! xn = (x - mean (x)) / std (x);
+%! pn = ones (1,5);
+%! y = polyval (pn, xn);
+%! [p, s, mu] = polyfit (x, y, numel(pn)-1);
+%! [p2, s2] = polyfit (x, y, numel(pn)-1);
+%! assert (p, pn, s.normr)
+%! assert (s.yf, y, s.normr)
+%! assert (mu, [mean(x), std(x)])
+%! assert (s.normr/s2.normr < 1e-9)
+
+%!test
+%! x = [1, 2, 3; 4, 5, 6];
+%! y = [0, 0, 1; 1, 0, 0];
+%! p = polyfit (x, y, 5);
+%! expected = [0, 1, -14, 65, -112, 60]/12;
+%! assert (p, expected, sqrt(eps))
+
+
diff -r c1279a36706d -r 37bf6aca4025 scripts/polynomial/polyval.m
--- a/scripts/polynomial/polyval.m      Tue Feb 19 07:24:49 2008 -0500
+++ b/scripts/polynomial/polyval.m      Tue Feb 19 07:28:40 2008 -0500
@@ -18,15 +18,20 @@
 ## <http://www.gnu.org/licenses/>.

 ## -*- texinfo -*-
-## @deftypefn {Function File} {} polyval (@var{c}, @var{x})
-## Evaluate a polynomial.
-##
-## @code{polyval (@var{c}, @var{x})} will evaluate the polynomial at  
the
-## specified value of @var{x}.
-##
-## If @var{x} is a vector or matrix, the polynomial is evaluated at  
each of
+## @deftypefn {Function File} address@hidden polyval (@var{p}, @var{x})
+## @deftypefnx {Function File} address@hidden polyval (@var{p}, @var{x},  
[], @var{mu})
+## Evaluate the polynomial at of the specified values for @var{x}.  
When @var{mu}
+## is present evaluate the polynomial for (@address@hidden(1))/ 
@var{mu}(2).
+## If @var{x} is a vector or matrix, the polynomial is evaluated for  
each of
 ## the elements of @var{x}.
-## @seealso{polyvalm, poly, roots, conv, deconv, residue, filter,
+## @deftypefnx {Function File} address@hidden, @var{dy}] =} polyval  
(@var{p}, @var{x}, @var{S})
+## @deftypefnx {Function File} address@hidden, @var{dy}] =} polyval  
(@var{p}, @var{x}, @var{S}, @var{mu})
+## In addition to evaluating the polynomial, the second output
+## represents the prediction interval, @var{y} +/- @var{dy}, which
+## contains at least 50% of the future predictions. To calculate the
+## prediction interval, the structured variable @var{s}, originating
+## form `polyfit', must be present.
+## @seealso{polyfit, polyvalm, poly, roots, conv, deconv, residue,  
filter,
 ## polyderiv, polyinteg}
 ## @end deftypefn

@@ -34,14 +39,22 @@
 ## Created: June 1994
 ## Adapted-By: jwe

-function y = polyval (c, x)
+function [y, dy] = polyval (p, x, s, mu)

-  if (nargin != 2)
+  if (nargin < 2 || nargin > 4 || (nargout == 2 && nargin < 3))
     print_usage ();
   endif

-  if (! (isvector (c) || isempty (c)))
+  if (nargin < 3)
+    s = [];
+  endif
+
+  if (! (isvector (p) || isempty (p)))
     error ("polyval: first argument must be a vector");
+  endif
+
+  if (nargin < 4)
+    mu = [0, 1];
   endif

   if (isempty (x))
@@ -49,28 +62,70 @@ function y = polyval (c, x)
     return;
   endif

-  if (length (c) == 0)
-    y = c;
+  if (length (p) == 0)
+    y = p;
     return;
   endif

-  n = length (c);
-  y = c (1) * ones (rows (x), columns (x));
-  for index = 2:n
-    y = c (index) + x .* y;
-  endfor
+  n = length (p) - 1;
+  k = numel (x);
+  x = (x - mu(1)) / mu(2);
+  A = (x(:) * ones (1, n+1)) .^ (ones (k, 1) * (n:-1:0));
+  y(:) = A * p(:);
+  y = reshape (y, size (x));
+
+  if (nargout == 2)
+    ## The line below is *not* the result of a conceptual grasp of  
statistics.
+    ## Instead, after reading the links below and comparing to the  
output of Matlab's polyval.m,
+    ## 
http://www.mathworks.com/access/helpdesk/help/toolbox/stats/index.html?/access/helpdesk/help/toolbox/stats/finv.html
+    ## 
http://www.mathworks.com/access/helpdesk/help/toolbox/curvefit/index.html?/access/helpdesk/help/toolbox/curvefit/bq_5ka6-1_1.html
+    ## Note: the F-Distribution is generally considered to be single- 
sided.
+    ## http://www.itl.nist.gov/div898/handbook/eda/section3/eda3673.htm
+    ##   t = finv (1-alpha, s.df, s.df);
+    ##   dy = t * sqrt (1 + sumsq (A/s.R, 2)) * s.normr / sqrt (s.df)
+    ## If my inference is correct, then t must equal 1 for polyval.
+    ## This is because finv (0.5, n, n) = 1.0 for any n.
+    dy = sqrt (1 + sumsq (A/s.R, 2)) * s.normr / sqrt (s.df);
+    dy = reshape (dy, size (x));
+  endif

 endfunction

-%!assert(polyval ([1, 1, 1], 2) == 7);
+%!test
+%! fail("polyval([1,0;0,1],0:10)");

-%!assert(all (all (polyval ([1, 1, 1], [0; 1; 2]) == [1; 3; 7])));
+%!test
+%! r = 0:10:50;
+%! p = poly (r);
+%! p = p / max(abs(p));
+%! x = linspace(0,50,11);
+%! y = polyval(p,x) + 0.25*sin(100*x);
+%! [pf, s] = polyfit (x, y, numel(r));
+%! [y1, delta] = polyval (pf, x, s);
+%! expected = [0.37235, 0.35854, 0.32231, 0.32448, 0.31328, ...
+%!    0.32036, 0.31328, 0.32448, 0.32231, 0.35854, 0.37235];
+%! assert (delta, expected, 0.00001)

-%!assert(isempty (polyval ([1, 1, 1], [])));
+%!test
+%! x = 10 + (-2:2);
+%! y = [0, 0, 1, 0, 2];
+%! p = polyfit (x, y, numel (x) - 1);
+%! [pn, s, mu] = polyfit (x, y, numel (x) - 1);
+%! y1 = polyval (p, x);
+%! yn = polyval (pn, x, [], mu);
+%! assert (y1, y, sqrt(eps))
+%! assert (yn, y, sqrt(eps))

-%!assert(all (all (polyval ([1, 1, 1], [-1, 0; 1, 2]) == [1, 1; 3,  
7])));
+%!test
+%! p = [0, 1, 0];
+%! x = 1:10;
+%! assert (x, polyval(p,x), eps)
+%! x = x(:);
+%! assert (x, polyval(p,x), eps)
+%! x = reshape(x, [2, 5]);
+%! assert (x, polyval(p,x), eps)
+%! x = reshape(x, [5, 2]);
+%! assert (x, polyval(p,x), eps)
+%! x = reshape(x, [1, 1, 5, 2]);
+%! assert (x, polyval(p,x), eps)

-%!error polyval ([1, 2; 3, 4], [-1, 0; 1, 2]);
-
-%!assert(isempty (polyval ([], [-1, 0; 1, 2])));
-