[Patch] Polyfit with scaling
Ben Abbott
bpabbott at mac.com
Thu Feb 14 20:01:57 CST 2008
Both polyfit.m and polyval.m have been modified.
2008-02-14 Ben Abbott <bpabbott at mac.com>
* polynomial/polyfit.m: Modified algorithm to use QR decomposition,
and added normalizaton option for Matlab compatibility.
* polynomial/polyval.m: Modified to respect normalization of the
dependent variable, and added optional 50% prediction intervals.
Patches for each are both below and attached.
Ben
diff -r 78f3811155f7 scripts/polynomial/polyfit.m
--- a/scripts/polynomial/polyfit.m Thu Feb 14 17:14:23 2008 -0500
+++ b/scripts/polynomial/polyfit.m Thu Feb 14 20:59:43 2008 -0500
@@ -18,39 +18,38 @@
## <http://www.gnu.org/licenses/>.
## -*- texinfo -*-
-## @deftypefn {Function File} {[@var{p}, @var{s}] =} polyfit
(@var{x}, @var{y}, @var{n})
+## @deftypefn {Function File} {[@var{p}, @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 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} = (@var{x}- at var{mu}(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 <Kurt.Hornik at wu-wien.ac.at>
@@ -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 78f3811155f7 scripts/polynomial/polyval.m
--- a/scripts/polynomial/polyval.m Thu Feb 14 17:14:23 2008 -0500
+++ b/scripts/polynomial/polyval.m Thu Feb 14 20:59:43 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} {@var{y}=} polyval (@var{p}, @var{x})
+## @deftypefnx {Function File} {@var{y}=} 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 (@var{x}- at var{mu}(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} {[@var{y}, @var{dy}] =} polyval
(@var{p}, @var{x}, @var{S})
+## @deftypefnx {Function File} {[@var{y}, @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])));
-
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