Before you EVER make any fit to data, PLOT IT!!!!!!!! Plot your data. Then plot it in another way, until you have drained all the information you can learn from that fit.
x1=[6.71214E-05 0.00112676 0.047319082 0.142706219 0.273947664 0.431651447 0.548219473 0.615322614 0.707701444 0.80169938 0.859014989 0.91129514 0.942910695 0.960171921 0.983861934 0.99297485 1];
y1=[4.78684E-06 0.008166282 0.22229314 0.456984391 0.648856946 0.795065728 0.849101429 0.861065276 0.910476057 0.947345705 0.96719942 0.982912047 0.990964548 0.994569633 0.998431303 0.999498993 1];
title 'Direct plot of data'
And that does look vaguely like a power law curve might look, with a fractional exponent. So something like sqrt(x) might be not unreasonable, just as a wild guess.
But you have decided to fit it using a transformation of your data. Thus, if the model
y = a*x^b + noise
is what you think is the model, then by logging both sides of the equation, we will see:
log(y) = log(a) + b*log(x)
And that SHOULD be a straight line, if we plot log(y) vs log(x), IF the power law model is truly reasonable. Note that this transformation implies that the additive noise we saw in the first model is not appropriate. In fact, we would presume multiplicative noise. That can often be a reasonable presumptino when your data spans many orders of magnitide.
But the assumptions implicit in this transformation are sometimes a big IF. So what does the log log plot look like?
title 'Log-log plot of data'
I don't really like it. That is NOT a straight line. In fact, while things are clearly dominated by that point at the bottom end, even if we look at the rest of the curve, it seems to be clearly curved over the entire domain. And that suggests why things failed you. I'll try to explain a little more deeply.
Consider the two fits you will get:
mdl = fittype('power1')
mdl =
General model Power1:
mdl(a,b,x) = a*x^b
PLmdl = fit(x1',y1',mdl)
PLmdl =
General model Power1:
PLmdl(x) = a*x^b
Coefficients (with 95% confidence bounds):
a = 1.028 (1, 1.056)
b = 0.4066 (0.3574, 0.4559)
Does that fit reasonably?
title 'Direct fit of power law model, in the untransformed doamin'
The fit is not great, with what appears to me to be some clear lack of fit. But it looks sort of like what you may expect.
Now, contrast that to what you get from a fit in the log domain:
linmdl = fit(log(x1'),log(y1'),'poly1')
linmdl =
Linear model Poly1:
linmdl(x) = p1*x + p2
Coefficients (with 95% confidence bounds):
p1 = 1.088 (0.8988, 1.278)
p2 = 0.4279 (-0.1438, 0.9996)
Just to be clear, look carefully at that bottom point. I'll overlay the linear fit on top of the plot.
plot(log(x1),log(y1),'o')
What happens is that bottom left point now introduces a huge bias into the linear fit. We should see that the log transformation is not helping you. But the real problem is that power law model is not a very good approximation to your data. And the assumption that logging the curve will allow you to use a linear fit is invalid, if a power law model is not appropriate.
Are there better models? Probably, yes. If we look at the original data, a negative power law does imply the curve has an infinite slope at x==0. Thus remember what the sqrt curve looks like. a*x^0.4 would look very much the same.
Even a simple polynomial model would seem to fit your data much better, but perhaps a simple sum of exponentials might be a better choice.
exp2mdl = fit(x1',y1','exp2')
polymdl =
General model Exp2:
polymdl(x) = a*exp(b*x) + c*exp(d*x)
Coefficients (with 95% confidence bounds):
a = 0.7275 (0.69, 0.7651)
b = 0.327 (0.2709, 0.3832)
c = -0.7201 (-0.7575, -0.6828)
d = -6.143 (-6.841, -5.446)
Now, you are the only one who knows why you chose a power law model. My guess is, it looked like it would fit. That would have been a possible choice given a cursory glance at your data. But a deeper glance tells me that power law model is not correct here.
