Two ways to fit a function on data points

Question

I have the following data:

eR = {1.1*10^-4 , 3.3*10^-4 , 1.1*10^-3 , 2*10^-3 , 
   3.3*10^-3 , 5.8*10^-3 , 1.1*10^-2 , 1.9*10^-2 , 3.3*10^-2 , 
   5.8*10^-2 , 1.1*10^-1} ;
NoDF = {2 , 2 , 2 , 3 , 4 , 4 , 7 , 11 , 18 , 21 , 35};
DSF = Transpose@{eR , NoDF};

Now, what I would like to do is to linearly fit a curve on the above set of points. Thus, I use the following

fitF = FindFit[DSF, a*n^b, {a, b}, n];

So, when I plot the data and the curve together they look ok. First I plot the data point

plot1F = {DSF } // 
  ListLogLogPlot[#, Joined -> False, FrameTicks -> All ,  
    Frame -> True] &

Secondly, I plot the fitting curve

a = Last[First[fitF]];
b = Last[Last[fitF]];
func[x_] := a x^b;
Data = Table[func[x] , {x , 1.1*10^-4 , 1.1*10^-1 , 0.00001}];
XF = Table[x , {x , 1.1*10^-4 , 1.1*10^-1 , 0.00001}];
DATAfit  = Transpose@{XF , Data};
plot2F = {DATAfit} // 
  ListLogLogPlot[# , PlotRange -> All , FrameTicks -> All,  
    Frame -> True] &

Then, I put them together

Show[plot1F , plot2F , PlotRange -> All]

The result looks ok for me. But then I was asked to change the function and first, take the logarithm of the points and then fit them with a + b*x function. So first I took the logarithm of the data point

DSFL = Transpose@{Log[eR] , Log[NoDF]};

and then fit the data

fitF2 = FindFit[DSFL, c + d n, {c, d}, n];
 c = Last[First[fitF2]];
 d = Last[Last[fitF2]];
funcL[x_] := c + d x;

But when I plot the data, it seems that there is something wrong with my plots. Because the curve does not fit the data

DataL = Table[funcL[x]  , {x , 1.1*10^-4 , 1.1*10^-1 , 0.00001}]; 
XF = Table[x , {x , 1.1*10^-4 , 1.1*10^-1 , 0.00001}];
DATAL = Transpose@{XF , DataL};
plot3F = {DATAL} // 
  ListPlot[# , PlotRange -> All , FrameTicks -> All,  Frame -> True] &
Show[plot1F , plot3F , PlotRange -> All]

Now, I want to know if there is something wrong with my function or code. Because I know that the method is correct and I should be able to fit the data with a c + d*x function.

Answer 1

I think not all your data adheres to your model, or perhaps your model is too simple to fit here.

Here is a somewhat simplified approach to your problem. Let's start with fitting the data directly to the non-linear model (in general a preferable approach, since no distortion is introduced in error statistics):

NonlinearModelFit[DSF, a n^b, {a, b}, n]
Plot[
  %[n], Evaluate@Flatten@{n, MinMax@eR},
  Epilog -> {Red, PointSize[0.015], Point@DSF}
]

That's not bad.

---

Let's transform the data and try the linear fit then:

logData = Log10@DSF;

LinearModelFit[logData, x, x]
Plot[%[x], 
  Evaluate@Flatten@{x, MinMax@Log10@eR},
  Epilog -> {Red, PointSize[0.015], Point@logData}
]

This is also not bad, but definitely the first two points don't seem to fit in the linear model.

---

We can try to redo the fit by excluding those points:

LinearModelFit[logData[[3 ;;]], x, x]
Plot[%[x], 
  Evaluate@Flatten@{x, MinMax@Log10@eR},
  Epilog -> {Red, PointSize[0.015], Point@logData}
]

This is a lot better; notice also that the slope parameter here is numerically very close to the fitted exponent from the non-linear model fit, as it should be.