I don't know the answer to your question about fitting, but your description of the data rang bells from years ago when I worked with signal processing - as though your data is in the frequency spectrum and has a main signal frequency and a number of harmonics, as you might get by taking the Laplace/Fourier transform of a square wave or sawtooth. Not sure if this helps at all! Perhaps if you took the inverse Fourier transform of the data, it might look more like something that you can fit a more simple function to, which you could then take the Laplace transform of.

Thank You very much Jonathan ! As a matter of fact this is exactly what the distribution looks like, the point is that the distribution deals with genetic distances computed over subsequent patches of genome by means of SNPs data and I was wandering if this could be an artifact of discretization ... but I'll try with inverse harmonics, is in amy case first step, if this bizzare behavior is eliminated I have an indication of some discrete character anyway....

I'd be interested to hear if it leads anywhere interesting!

I'll sure let you know !

Dear Jonathan,

in the attached file you will see the original distribution I talked you about. It refers to principal component scored relative to pc3 (the same happens with pc2) of the below linked paper:

http://www.la-press.com/a-novel-multi-scale-modeling-approach-to-infer-whole-genome-divergence-article-a3417-abstract

as you will discover reading the paper pc2 and pc3 are able to discriminate between different musculus strains while pc1 (the by far most important component) is identical among the different strains and represents the 'macroevolution signal', while pc2 and pc3 are 'microevolution'. Interesting enough pc1 has a significant score in very few genes while pc2 and pc3 demonstate a much more 'peaked' and 'gene centered' distribution with higher numebr of genes significantly above the mean score value (334 and 169 respectively for pc3 and pc2). This suggests macroevolutio is a global process jointly encompassing the entire genome while micro is local.

Now pc1 scores have a classical gaussina shape, while pc2 and pc3 have the strange form I told you, when we shuffled 25% of distance locations and recomputed PCA the strange distribution disappeared and the scores of pc2 and pc3 present a very clear single peak symmetical distribution, thus what we observe is not an artifact of discretization but a real phenomenon due to the internal genome correlations as for possible genetic distances (allowed mutations ? recombination ?), when we made the inverse FFT as you kindly suggested me, we totally loose the discretization and we obtain a continuous function....

here is the attachment I told you in the previous message, ...

I was looking a little more into this distribution, and it is similar in some ways to the frequency spectrum of an AM (amplitude modulated) radio signal. In AM, the 'carrier frequency' is the main frequency of the broadcast (the large peak at the centre), and the music signal is carried by modulating the amplitude of this sin wave, and the music shows up in the frequency spectrum as symmetrical 'side bands'. Here it looks as though each side band also has its own harmonic side bands (the self-similar nature).

If you looked at the inverse FFT of your function I guess you would see roughly a sin wave whose amplitude varies with time? If the function is self-similar, then I am guessing that the 'music' signal here is also an AM sin wave.

This leads me to the idea that a good fit of the inverse FFT of the data might be a product of sin waves of decreasing amplitude (each sin wave amplitude-modulated by the next in the series).

Well, I must confess you I have some problems to interpretate the results of inverse FFT for the simple fact we have nothing that can be quated to a 'time' dimension, as you probably seen by the links I attached the distribution refers to the values of second component scores relative to the genetic distances distribution along the DNA of the entire genome of a set of mus musculus strains. Thus the 'strange spectrum' has as X axis the value of component scores (this is why is centered to zero) and as Y axis the frequency of the corresponding values over 8 million of paches (each patch is 200 bases long). The first idea that came in our minds is that we had a pure artifactual result coming from a truncation (Pc2 explains a very low portion of total variability, the possible distance values are very few..) but this was not the case, shuffling along the DNA the distance values (keeping identical their values) provokes the disapperaing of the strange spectrum and the distribution becomes gaussian, thus we are looking to something that comes from the mutual correlation between distances and not from their specific values. Now, what plays the role of time in this metaphor ? Attached you will find the output of R for inverse FFT, we can only say that the structure is destroyed but we remain with the problem of interpretation...

I was wondering whether distance along the chromosome was the analog of time, and that there are some repetitive objects which have a consistent size, or consistent spacing along the chromosome, which account for the main peak, like maybe genes, or exons or maybe histones... It is certainly an interesting problem

In addition to what has already been suggested, you may find the Manipulate function from Mathematica helpful.    For a detailed explanation and a number of examples, see

http://reference.wolfram.com/language/tutorial/IntroductionToManipulate.html

Thank you James, I will go and see....