Fit equation not correct for x vector

Hi everyone,

I am trying to fit a serie (x,y) of data with a polynomial equation. I have found the equation that fit my data and well reproduce the trend. Nevertheless, the equation solution for the x vector does not give the initial y values. I post the code to be clearer.

Thank you in advance for your help!

x2= [8;8.5;9;9.3;9.5;9.8]; y2=[0.280;0.283;0.287;0.293;0.309;0.350]; curvefit = fit(x2,y2,'poly4','normalize','off'); figure,plot(curvefit,x2,y2) %the fitting weel reproduce data

% equation given by the fitting p1 = 0.005065 ; p2 = 0.0148 ; p3 = 0.01398 ; p4 = 0.008919 ; p5 = 0.2871 ; y2f=p1*x2.^4 + p2*x2.^3 + p3*x2.^2 + p4*x2 + p5; figure,scatter(x2,y2) hold on,plot(x2,y2f) %the y values given by the fitting equation are far from the initial y values

Answer 1

[John D'Errico]

NEVER use short digit approximations to double precision numbers.

x2= [8;8.5;9;9.3;9.5;9.8];
y2=[0.280;0.283;0.287;0.293;0.309;0.350];
curvefit = fit(x2,y2,'poly4','normalize','off'); 

curvefit
curvefit = 
     Linear model Poly4:
     curvefit(x) = p1*x^4 + p2*x^3 + p3*x^2 + p4*x + p5
     Coefficients (with 95% confidence bounds):
       p1 =     0.02196  (-0.1672, 0.2111)
       p2 =     -0.7435  (-7.493, 6.006)
       p3 =       9.434  (-80.72, 99.59)
       p4 =      -53.16  (-587.4, 481.1)
       p5 =       112.5  (-1073, 1298)

So I don't know where you got those coefficients from, but they don't even correspond to the fit! This is the biggest reason for your problem.

The data that you gave us has a completely different set of fitted coefficients. But even there, DON'T ever use short approximations to those coefficients!!!!!!!!!!!!!!

Better approximations are:

coeffvalues(curvefit)
ans =
    0.0219598636195908        -0.743494138038658           9.4337274043818         -53.1599385897607          112.522326044507

But even they are not exact.

Don't forget that if x is on the order of 10, then when you raise x to the 4th power, x^4 is on the order of 10000. So an error of 1 part in 10000 in the coefficient of x^4 may well produce an error on the order of +/- 1 in p(1)*x^4.

Is that significant? It sure as hell is! The data itself has y on the order of 0.3. If you add some random component that is +/- 1 to 0.3, what do you expect? Complete garbage.

(Actually, it won't be quite that large an error, since p(1) was~0.02, but the trash so created by use of poor approximations to the coefficients will be significant. Had the model been a slightly higher order, you would have gotten complete garbage out.)

[claral]

Thank you!