Polynomial fit with centering & scaling

Hello,

When I plotted the following data and applied polyfit with second order, it showed the warning like: Warning: Polynomial is badly conditioned. Add points with distinct X values, reduce the degree of the polynomial, or try centering and scaling as described in HELP POLYFIT. This fitting can also be done using [p, S, mu] = polyfit(x,y,2). However, the fitting coefficients from [p, S, mu] = polyfit(x, y, 2) are totally different from p=polyfit(x,y,2). How can I use the [p, S, mu] = polyfit(x,y,2) and obtain coefficients that correspond to my unscaled system p=polyfit(x,y,2)?

x y

xy = [
1508.936            19.2189 
1509.159            54.27226 
1509.472            102.8022 
1509.852           154.8707 
1510.224            199.6335 
1510.809333      258.6648 
1511.443429      316.4253 
1512.112            370.7319 
1512.4112          391.8055 
1513.118545      441.4118 
1514.002667      497.8672 
1514.777778      547.8429 
1515.6144           598.6195 
1516.467             650.4304 
1517.3032          701.3778 ]

Answer 1

[Torsten]

Here is an example that shows how to proceed:

x = [1,2,3];
y = [4,9,25];
order = 1;
p = polyfit(x,y,order);

[p1,S,mu] = polyfit(x,y,order);

% Transform coefficients to those of the polynomial p centered at 0
p1 = flip(p1);  % Flip order to [p0, ..., pn]

p2 = zeros(1,order+1); 
for i = 0:order
    for k = i:order
        p2(i+1) = p2(i+1) + nchoosek(k, k-i) * p1(k+1)/mu(2)^k * (-mu(1))^(k-i);
    end
end
p2 = flip(p2);  % Back to original order [pn, ..., p0] 

% Compare
p
p2

and here is the link:

https://de.mathworks.com/matlabcentral/answers/289815-how-do-i-rescale-the-polyfit-coefficients-back-to-normal-coordinate-system

Answer 2

[Walter Roberson]

xy = [
1508.936            19.2189 
1509.159            54.27226 
1509.472            102.8022 
1509.852           154.8707 
1510.224            199.6335 
1510.809333      258.6648 
1511.443429      316.4253 
1512.112            370.7319 
1512.4112          391.8055 
1513.118545      441.4118 
1514.002667      497.8672 
1514.777778      547.8429 
1515.6144           598.6195 
1516.467             650.4304 
1517.3032          701.3778 ];

x = xy(:,1); y = xy(:,2);

format long g
[p, S, mu] = polyfit(x,y,2)

p = 1×3

-36.6930527200109 231.045137444742 387.978493205332

S = struct with fields:
        R: [3×3 double]
       df: 12
    normr: 46.6308023944645

mu = 2×1

1.0e+00 * 1512.38017013333 2.74485447071878

 xscaled = (x-mu(1))/mu(2);
 pscaled = polyfit(xscaled, y, 2)

pscaled = 1×3

-36.6930527200109 231.045137444742 387.978493205332

 syms X t
 Xs = (X-mu(1))/mu(2)

Xs = 

punscaled = collect(subs(poly2sym(p, t), t, Xs), X)

punscaled = 

vpa(punscaled, 16)

ans = 

polyfit(x, y, 2)

ans = 1×3

1.0e+00 * -4.8701820734741 14815.3074972764 -11266452.1354997

Answer 3

[Torsten]

Another option:

xy = [
    1508.936            19.2189 
    1509.159            54.27226 
    1509.472            102.8022 
    1509.852           154.8707 
    1510.224            199.6335 
    1510.809333      258.6648 
    1511.443429      316.4253 
    1512.112            370.7319 
    1512.4112          391.8055 
    1513.118545      441.4118 
    1514.002667      497.8672 
    1514.777778      547.8429 
    1515.6144           598.6195 
    1516.467             650.4304 
    1517.3032          701.3778 ];

x = xy(:,1); y = xy(:,2);

order = 2;

p = polyfit(x,y,order);

[p1,S,mu] = polyfit(x,y,order);

syms t
p1s = flip(p1)*((t-mu(1))/mu(2)).^(0:order).';  % create polynomial p1
cp1s = coeffs(taylor(p1s,t,0));                 % compute Taylor coefficients in t=0
cp1 = double(cp1s);                             % convert to double
pp = flip(cp1);                                 % flip to original order
p
pp