Curve fitting with loglog data

I am having some issues fitting a curve using polyfit to log data. I think I am making a silly mistake during the fit plotting as the fitting is really bad during the beggining but can't seem to figure it out. Any help would be greatly appreciated! I have attached the data as: msd_help

load('msd_help')

figure()
loglog(s,m)
hold on

linearCoefficients = polyfit((s),(m), 1);
yFit = polyval(linearCoefficients, s);
loglog(s, yFit, 'r-', 'LineWidth', 2)

[Yazan]

You are not doing anything wrong, and the fitting is working. However, you are plotting the results using a base 10 logarithmic scale. See below the same result plotted using a linear scale.

Answer 1

[Star Strider]

Fit it to a power function:

LD = load('msd_help.mat');
m = LD.m;
s = LD.s;

fcn = @(b,x) x.^b(1).*exp(b(2));
B = fminsearch(@(b) norm(m - fcn(b,s)), rand(2,1))
yFit = fcn(B,s);

figure
loglog(s,m,'.')
hold on
plot(s, yFit,'-r')
hold off
grid

producing:

B =
   0.867368600071621
 -27.933653082691194

and:

.

[Yazan]

A 5th order polynomial also gives reasonable results.

[Alex Sha]

If the fitting function types are free, the follow is good enough:

y = p1*x/(1+p2*x^2)^p3+p4

Root of Mean Square Error (RMSE): 5.64364594544832E-13
Sum of Squared Residual: 1.91104437345451E-21
Correlation Coef. (R): 0.999945510161687
R-Square: 0.999891023292516

Parameter	Best Estimate
----------	-------------
p1	3.82443123959562E-13
p2	2.48820059997632E-6
p3	0.334013945130774
p4	-8.79374499358471E-13

[Manny Kins]

Thanks all for your help. In reference to Star Strider's answer, it is good for me because I wanted to fit to a straight line (apologies for not mentioning this in the initial question). Is there a reason why the straight line fit on the log plot only passes the later stages of the data (after s=100) and misses the rest?

[Star Strider]

‘Is there a reason why the straight line fit on the log plot only passes the later stages of the data (after s=100) and misses the rest?’

Yes.

The first-order unweighted power fit may not be the ideal model. (However the actual error is exceedingly small, and close to the limit of IEEE 754 floating-point representation.)

The best model is a mathematical expression of the process that created those data.

.

[Manny Kins]

Thanks for your help and insight

[Star Strider]

As always, my pleasure!

.