How do I restrict an interpolated function to only take values > 0?

Question

I have a list of points which looks like this when plotted with Joined -> True:

I'm interested in the area under the curve with x-axis > 0 as a ratio to the area under the curve as a whole. The obvious way to do this is to define function = Interpolation[list], then use Integrate[function[x],{x,0,Infinity}/Integrate[function[x]],{x,-Infinity,Infinity}].

However, for some of the parameters used to generate this plot, this ratio goes above one. My reading is that the interpolated function goes negative at some point. I know for physical reasons that the function is strictly positive (it is an unnormalized probability). Is there a way to feed this restriction to Interpolation, or perhaps to simply zero out the interpolated function past a certain point?

Edit: some explicit code that shows the problem I'm facing. For this list:

{{-2.,-1.5,-1.,-0.5,0.,0.5,1.,1.5,2.,2.5,3.,3.5,4.,4.5,5.},{2.447482917*10^-26,7.166525422*10^-23,6.828228711*10^-20,2.221078019*10^-17,2.603521623*10^-15,1.167198034*10^-13,2.134212354*10^-12,1.702246816*10^-11,6.341646627*10^-11,1.183984093*10^-10,1.195443416*10^-10,7.082172125*10^-11,2.655421922*10^-11,6.681130557*10^-12,1.175390799*10^-12}}

(first list is x-axis values, second list is y-axis values). Define interpolating function:

Testfunc = Interpolation[{2.447482916954607`*^-26, 7.166525421661271`*^-23, 6.82822871054717`*^-20, 2.2210780189277698`*^-17, 2.6035216228330743`*^-15, 1.1671980340015243`*^-13, 2.134212354193162`*^-12, 1.7022468159138925`*^-11, 6.341646627292107`*^-11, 1.1839840933620157`*^-10, 1.195443415545412`*^-10, 7.082172125430078`*^-11, 2.655421922388246`*^-11, 6.681130557379854`*^-12, 1.1753907990705456`*^-12}]

Take the numerical integral: NIntegrate[Testfunc[x], {x, 5, 15}]/NIntegrate[Testfunc[x], {x, 1, 15}]

Result is: 1.000008312

Answer 1

Since the second coordinates are positive you can apply Log and then apply Exp to the function approximating the data before integrating it numerically:

data = {{-2., -1.5, -1., -0.5, 0., 0.5, 1., 1.5, 2., 2.5, 3., 3.5, 4., 4.5, 5.}, 
        {2.447482917*10^-26, 7.166525422*10^-23, 6.828228711*10^-20, 2.221078019*10^-17,
         2.603521623*10^-15, 1.167198034*10^-13, 2.134212354*10^-12, 1.702246816*10^-11,
         6.341646627*10^-11, 1.183984093*10^-10, 1.195443416*10^-10, 7.082172125*10^-11,
         2.655421922*10^-11, 6.681130557*10^-12, 1.175390799*10^-12}};
dataL = Transpose[{data[[1]], data[[2]] // Log}];

f[x_] = Exp[Fit[dataL, x^Range[0,4], x]];

#/(# + #2) &[
  NIntegrate[f[x],{x, 0, Max[dataL[[All,1]]]}],
  NIntegrate[f[x],{x, Min[dataL[[All,1]]], 0}]]

0.999999

Plotting the non-transformed data looks like a normal distribution density, so you could also try:

normal=(E^(-((-m + x)^2/(2 s^2))) n)/(Sqrt[2 Pi] s);
f[x_] = normal /. FindFit[Transpose[data], normal, {m, n, s}, x];

#/(# + #2) &[
  NIntegrate[f[x],{x, 0, Max[dataL[[All,1]]]}],
  NIntegrate[f[x],{x, Min[dataL[[All,1]]], 0}]]

0.999986

Answer 2

No, you can not specify a limit on the interpolating function. Two options that work for this data set are the use of:

t2=Interpolation[Transpose[data],InterpolationOrder\[Rule]1];

or

t2 = Interpolation[Transpose[data], Method -> "Spline"];
Plot[t2[x], {x, -2, 5}]
NIntegrate[t2[x], {x, 0, 5}]/NIntegrate[t2[x], {x, -2, 5}]
0.9999993190296589`