How to fit a specific range of data?

Question

Let me explain my question by a toy example: Suppose I have a simple Gaussian function as follows:

test[x_] = 4*Exp[-(x)^2];

If I its data to a new Gaussian function I do:

data = Table[{x, test[x]}, {x, 0, 10, 10/100}];

fit = NonlinearModelFit[data, {a1*Exp[-b1*(x)^2]}, {a1, b1}, x, 
   Method -> NMinimize];
fit[x]
(*3.99999994699 E^(-0.999999974954 x^2)*)

where I see it recovers the original function very good. Now I want to fit just a part of data, for example instead of {0 to 10} I want to pick data of {5 to 10} from original function and then fit them (so I think the center of gaussian is not 0 any more and it shifts +5 namely in exponent I have to write $(x+5)^2$ instead $x^2$, however I don't know what is the best way to implement it). How can I do this? Does Mathematica has a built-in function or option to do this?

Answer 1

---

Edit: in this edited version I will try to be more concrete after the comments of the author of the OP. All detailed steps remain as they were presented in the first version at the end

---

Breaking down the main point of the OP assuming that there is no miscommunication.

In the OP the author provided us with the following data

test[x_] = 4*Exp[-(x)^2];
data = Table[{x, test[x]}, {x, 0, 10, 1/10}];

The task at hand is to choose the subset of the above data -already generated- in the region {x,5,10} and their corresponding test[x] values. This can be done programmatically in the following manner

data[[5*10 + 1 ;; All]]

So, as we see we sorted out the subset of the original data for which we have $x \geq 5$ and their corresponding test[x] values. Since we are done with this, we can this new set of data like so:

fit = NonlinearModelFit[
   data[[5*10 + 1 ;; All]], {a1*Exp[-b1*(x)^2]}, {a1, b1}, x, 
   Method -> NMinimize];
fit[x]

We plot the specific set of data used for the fit and the fitted function we obtained for comparion

p1 = ListPlot[data[[5*10 + 1 ;; All]], 
   PlotStyle -> PointSize /@ {Large}, PlotRange -> {All, All}];
p2 = Plot[
   2.6960551596217814` E^(-0.9843743450389705` x^2), {x, 5, 10}, 
   PlotStyle -> {Thick, Dashed, Red}, PlotRange -> {All, All}];
Show[p1, p2, ImageSize -> Large]

---

Original answer: step-by-step explanation

---

Comment: I think that the main point of this question is how to single out specific sublists from a mother list.

Also, I do not understand the reason of writing 10/100 and not 1/10, so I am changing this to 1/10

So, from the OP

test[x_] = 4*Exp[-(x)^2];

Now we generate the data

data = Table[{x, test[x]}, {x, 0, 10, 1/10}];

How to single out the ones that {x,5,10} from the above list? I think that this is the question.

The following

data[[5*10 + 1 ;; All]]

takes care of that. You get

Another check for the validity of the above. If you want to single out data with x=5 or higher you could have started by generating only those. I understand that this might not be useful for practical purposes, but serves well for a check.

lst = Table[{x, test[x]}, {x, 5, 10, 1/10}];

and then

data[[5*10 + 1 ;; All]] - lst

Finally, doing the fit is easy now

fit = NonlinearModelFit[
   data[[5*10 + 1 ;; All]], {a1*Exp[-b1*(x)^2]}, {a1, b1}, x, 
   Method -> NMinimize];
fit[x]

Checking what we did

p1 = ListPlot[data[[5*10 + 1 ;; All]], 
   PlotStyle -> PointSize /@ {Large}, PlotRange -> {All, All}];
p2 = Plot[
   2.6960551596217814` E^(-0.9843743450389705` x^2), {x, 5, 10}, 
   PlotStyle -> {Thick, Dashed, Red}, PlotRange -> {All, All}];
Show[p1, p2, ImageSize -> Large]