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If I have a spectral data set showing measured energy on the y-axis and frequency on the x-axis, how would I go about fitting this to a beta distribution?

For example, lets say I had the data for the following curves showing spectrum for this ship:

enter image description here

How do I determine the beta curve fit parameters to say the blue line?

Taking it one step further. If I have a wav sound file of the sound of an engine, how would I do the same thing?


I attempted to use the supplied answer and got an error as follows:

enter image description here

The data file can be downloaded from the following URL: acoustic power spectrum data to 3k hz

The overall goal here is to generate a curve approximating the shape like this:

enter image description here

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Are you sure about fitting a Beta Distribution to this data? A Beta distribution has a finite support (0..1) which your data doesn't have and an interpretation related to probability, which I find hard to make in the context of frequency spectra of sounds. – Sjoerd C. de Vries Jan 15 '14 at 12:55
What do you mean by finite support? – Tyler Durden Jan 15 '14 at 15:41
A beta distribution is defined only between 0 and 1. – Sjoerd C. de Vries Jan 15 '14 at 16:00
I get the same answer if I scale the values so they are less than 1. – Tyler Durden Jan 16 '14 at 20:10
suggest you spend some time learning what a beta distribution is. Do some plots and manually try to find some parameters that reasonably resemble your data. – george2079 Jan 16 '14 at 20:41
up vote 3 down vote accepted
data = Import["", "TEXT"];
data1 = Transpose[{Rest@Range[0, 1, 1./Length@#], #}] &@ ToExpression@StringSplit[data];
model = PDF[BetaDistribution[a, b], x];
parms = FindFit[data1, model, {a, b}, x]
Show[Graphics[Point@data1, Axes -> True], 
     Plot[model /. parms, {x, 0, 1}, Evaluated -> True, PlotRange -> All],
     AspectRatio -> 1]

Mathematica graphics

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I tried to use this method and got an error about a "gradient" as shown above in the updated question. – Tyler Durden Jan 16 '14 at 17:14
@TylerDurden See Sjoerd's comments under your question – Dr. belisarius Jan 16 '14 at 19:24
I get the same error if I scale the y-values to be less than 1 (see updated screenshot in question) – Tyler Durden Jan 16 '14 at 20:11
@TylerDurden Please post a sample of your data set. Otherwise your problem is impossible to debug – Dr. belisarius Jan 16 '14 at 21:16
How do I post data (it is a .wav file that is about 1 megabyte in size)? – Tyler Durden Jan 16 '14 at 21:37

You mentioned in the comments that you are interested in a general shape of your data. This can be inferred from the nature of your data, and most probably does not follow a beta distribution. You can use a polynomial series of a chosen degree to visualize a more general shape of your data:

data = Import["", "TEXT"];
data1 = Transpose[{Rest@Range[0, 1, 1./Length@#], #}] &@
lp = ListLinePlot[MovingAverage[data1, 50]];

Clear[n, a, b, c, i, x, polynome, polyGauss]
polynome[n_] := 
 NonlinearModelFit[data1, Sum[a[i + 1]*x^i, {i, 0, n}], 
  Flatten[Table[{a[i + 1]}, {i, 0, n}]], x]

Show[{Plot[Evaluate[Table[polynome[n][x], {n, 3, 12, 3}]], {x, 0, 1}, 
   PlotRange -> All, AxesOrigin -> {0, 0}, PlotStyle -> Thick], lp}, ImageSize -> 600]

polynome fit

Or you can use a series of Gaussian bell functions:

polyGauss[n_] := 
 NonlinearModelFit[data1, Sum[a[i]*Exp[-(b[i] x - c[i])^2], {i, n}], 
  Flatten[Table[{{a[i], 1}, {b[i], i/(n + 1)}, c[i]}, {i, n}], 1], x]

Show[{Plot[Evaluate[Table[polyGauss[n][x], {n, 3, 12, 3}]], {x, 0, 1},
    PlotRange -> All, AxesOrigin -> {0, 0}, PlotStyle -> Thick], lp}, ImageSize -> 600]

gaussian fit

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