polyfitweighted2.m and polyval2.m

Contents

Usage polyfitweighted2

P = polyfitweighted2(X,Y,Z,N,W) finds the coefficients of a polynomial P(X,Y) of degree N that fits the data Z best in a least-squares sense. P is a row vector of length (N+1)*(N+2)/2 containing the polynomial coefficients in ascending powers, 0th order first.

 P = [p00 p10 p01 p20 p11 p02 p30 p21 p12 p03...]

e.g. For a 3rd order fit, the regression problem is formulated in matrix format as:

 wZ = V*P    or

                    2       2   3   2     2      3
 wZ = [w  wx  wy  wx  xy  wy  wx  wx y  wx y   wy ]  [p00
                                                      p10
                                                      p01
                                                      p20
                                                      p11
                                                      p02
                                                      p30
                                                      p21
                                                      p12
                                                      p03]

X,Y must be vectors Z,W must be 2D arrays of size [length(X) length(Y)]

based on polyfit.m by The Mathworks Inc. - see doc polyfit for more details

Class support for inputs X,Y,Z,W: float: double, single

Note: P is not in the format of standard Matlab 1D polynomials. Use polval2.m to evaluate the polynomial in this format, at given values of x,y.

Usage polyval2

Z = POLYVAL2(P,X,Y) returns the value of a 2D polynomial P evaluated at (X,Y). P is a vector of length (N+1)*(N+2)/2 containing the polynomial coefficients in ascending powers:

 P = [p00 p10 p01 p20 p11 p02 p30 p21 p12 p03...]

e.g. For a 3rd order fit, polyval2.m evaluates the matrix equation:

  Z = V*P    or

                 2      2  3  2      2   3
  Z = [1  x  y  x  xy  y  x  x y  x y   y ]  [p00
                                              p10
                                              p01
                                              p20
                                              p11
                                              p02
                                              p30
                                              p21
                                              p12
                                              p03]

Note: P is not in the format of standard Matlab 1D polynomials.

X and Y should be vectors; the polynomial is evaluated at all points (X,Y).

Class support for inputs P,X,Y: float: double, single

Example

Define some data z: Calculate best fit cubic with various weights:

- equal weight on all data points

- weight by the data itself

use polyval2.m to evaluate and plot best fit polynomial in each case. See help polyval2.m

x=[-50:50]';
y=[-50:50];
z=exp(-(x.^2*y.^0/100+x*y/140+x.^0*y.^2/50));

P_equalweight=polyfitweighted2(x,y,z,4,ones(101));
P_weighted=polyfitweighted2(x,y,z,4,z);


figure
subplot(2,2,1)
contour(x,y,z,[0:0.02:1])
title('data')
subplot(2,2,2)
contour(x,y,polyval2(P_equalweight,x,y),[0:0.02:1])
title('equal weight')
subplot(2,2,3)
contour(x,y,polyval2(P_weighted,x,y),[0:0.02:1])
title('weighted by z')

Published with MATLAB®