InverseFourierTransform very slow

Question

I determined a distribution dist = SmoothKernelDistribution[ListOfValues];. Then I determined the FT of the PDF, which is the CharacteristicFunction. For testing purposes I want to show, that the InverseFourierTransform will give me back the PDF. But the InverseFourierTransform is very slow.

Convince yourself. ListOfValues ist just a bunch of zeros:

    {0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0., 0.,
0., 0., 0.}

For some reason the execution

cf1[w_] = CharacteristicFunction[dist, w];
Plot[{10*Norm[cf1[w]], Arg[cf1[w]]}, {w, 0.001, 2}]

gives me the following CF:

That thing should be constant. Maybe it's just an issue with SmoothKernelDist. Anyway, the expression InverseFourierTransform[cf1[w], w, t] takes so long, that I cant even tell if it would ever give a useful result.

How can I perform the InverseFT? My goal is to find the inverse of a convolution, for example: 'a' convolved with 'b' gives 'c'. Given 'a' and 'c', I want to find 'b'. Doing this in Fourierspace should theoretically be , at least symbolically easier?

EDIT:

As a compromise, instead of InverseFourierTransform I could used something like

tf[t_] := 
  Sqrt[1/(2 \[Pi])]
    NIntegrate[cf1[\[Omega]] *Exp[-I*\[Omega]*t], {\[Omega], 0, 2}];

and then do a pointwise evaluation. That would lead to:

ListPlot[Table[Norm[tf[t]], {t, 10, 11, 0.1}]]

and the actual pdf should look like this:

Plot[PDF[dist[1], x], {x, 10, 11}]

Not only that they don't match. I also had to take the Norm because otherwise the numerical one would be complexvalued.

Answer 1

Your question seems to raise more questions than it could raise answers.

• If you want to show that the InverseFourierTransform will give you back the original PDF, why not do this for a simple PDF? Why are you doing it with a Smoothed kernel density estimate (which is, in in-built Mma, a numerical interpolated routine). Have you actually checked what you are actually working with?

For example:

  lis = {0, 0, 0};
  dist = SmoothKernelDistribution[lis];
  PDF[dist, z]

returns an InterpolatingFunction object. How do you expect Mma to find the Characteristic Function of this, or even worse, to then find the InverseFourierTransform of the latter??

• Why do you want the Characteristic Function anyway? How will it help you? What will you do with it, in the context of a smoothed kernel density estimate?

• It is possible to derive an exact symbolic construct for a smoothed kernel density estimator, though I am not sure that the in-built mma funcs do it. The NPKDE (non-parametric kernel density estimator) function in the mathStatica add-on package does this.

For example, given:

 data = {a,b,c};

... and for a given kernel g, say an Epanechnikov kernel:

 g = (3/4)*(1 - x^2);     domain[g] = {x, -1, 1}; 

... and any arbitrary bandwidth bin, the symbolic non-parametric kernel density estimator, calculated at an arbitrary point x is:

   NPKDE[x, data, bin, g]

returns:

_{(source: tri.org.au)}

As you can see, the structures are not trivial, and deriving characteristic functions, or even worse, then trying to invert them, is frankly unrealistic.