Mathematica: Interpolation Functions

[Niles]

Hi

I have a data set of the form:

data = {{0, 0}, {1, 1}, {2, 2}, {3, 20}, {4, 1}, {20, 1}, {21, 1}, {22, 0}};

This data set is a probability density function (PDF), and I need to be able to integrate it from 0 to some x<22. I thought that I would use an interpolating function for this. However, there is a "gap" between 4 and 20, where the PDF is zero, but an interpolating function doesn't take that into account by itself (I think). Is there a smarter way of accounting for this other than manually inserting data points?

The second problem is that ListInterpolation only takes a 1D-list. So I can only feed it the 2nd coordinate of the data points. However, I need somehow for the interpolation function to run from 0 to 22. I don't know how to do that.

I appreciate any help and suggestions.


Niles.

 

[Bill Simpson]

As
http://reference.wolfram.com/mathematica/ref/Interpolation.html
shows, Interpolation happily accepts a list of {x,f[x]}.

For example
In[1]:= f=Interpolation[{{0,0},{1,1},{2,2},{3,20},{4,1},{20,1},{21,1},{22,0}}]
Out[1]= InterpolatingFunction[{{0,22}},<>]

But the question becomes whether this is an adequate PDF.

In[2]:= Table[{t,f[t]},{t,0,22}]
Out[2]= {{0, 0}, {1, 1}, {2, 2}, {3, 20}, {4, 1}, {5, -709/51}, {6, -1279/51}, {7, -1678/51}, {8, -1925/51}, {9, -2039/51}, {10, -2039/51}, {11, -648/17}, {12, -591/17}, {13, -515/17}, {14, -1279/51}, {15, -994/51}, {16, -709/51}, {17, -443/51}, {18, -215/51}, {19, -44/51}, {20, 1}, {21, 1}, {22, 0}}

That clearly shows that the interpolation fits your given points exactly, but is not PDF-aware and you need to find a radically different approach to your problem to come up with something that will be non negative and with an area=1.

 

[Dickfore]

Yes, split your list into two sublists:

data1 = {{0, 0}, {1, 1}, {2, 2}, {3, 20}, {4, 1}}

and

data2 = {{20, 1}, {21, 1}, {22, 0}}

An interpolating function is given only over the range where the data set is given. That is why you introduce two interpolating functions:

inter1= Interpolation[data1]

and

inter2 = Interpolation[data2]

Then, define your function conditionally:

f[x_] := 0 \; x < 0 || (x > 4 && x < 20) || x > 22;
f[x_] := inter1[x] \; x >= 0 && x <= 4;
f[x_] := inter2[x] \; x >= 20 && x <= 22;

Now f[x] is a good function.

 

[Niles]

Thanks for both your suggestions, that is very kind. You are right Bill Simpson, I need some other way to do this. I'll have to think about this for some time.

Thanks for the help.Niles.

 

[alphy]

Hello Niles,

Thank you for sharing your question about interpolation functions in Mathematica. Interpolation functions are a powerful tool for approximating a function from a set of discrete data points. In your case, it sounds like you have a PDF with a gap and you want to find the integral of the function from 0 to a specific value.

To account for the gap in your data, you can use the option "ExtrapolationHandler" when creating your interpolation function. This allows you to specify how the function should behave outside of the given data range. In your case, you can use the option "Constant" to specify that the function should remain constant outside of the data range.

For example, you can create an interpolation function with this option as follows:

Interpolation[data, InterpolationOrder -> 1, ExtrapolationHandler -> {0 &}]

This will create an interpolation function that is 0 for all values outside of the data range.

As for your second problem, you can use the option "Domain" to specify the desired range of your interpolation function. For example, you can create an interpolation function that runs from 0 to 22 as follows:

Interpolation[data, InterpolationOrder -> 1, Domain -> {0, 22}]

I hope this helps! Let me know if you have any other questions or need further clarification. Best of luck with your research.

Sincerely,