Re: question: fitting a distribution from quantiles

*To*: mathgroup at smc.vnet.net*Subject*: [mg126453] Re: question: fitting a distribution from quantiles*From*: LÃszlÃ SÃndor <sandorl at gmail.com>*Date*: Sat, 12 May 2012 04:51:24 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201205110414.AAA23695@smc.vnet.net> <4FAD3253.3030007@wolfram.com>

Thank you, Darren! I realized soon (much before the delay cause my the moderation of the list) that I could fit a CDF. This even works with a ridiculously aggregated data, e.g. only two (inverse) quantiles for a Pareto distribution. However, FindFit did not work with Mathematica's representation of the CDF (conditions?), only a hard-coded one. But before I paste my output (with a lengthy error message) below, let me ask another question: What exactly are the benefits of keeping a distribution object in the background? Am I just as well off with a (smoothed) CDF and plugging it or transformations and integrals everywhere? Basically, I want to use an empirical distribution in three ways: -- "keep it as it is" (though it must be smoothed / approximated) as I do need PDFs even though, as any real data, it comes discrete -- fit a parametric distribution and use that everywhere where I would have used the empirical -- fit a mixture of parametric distributions (actually, it might be a special mixture: I might concatenate two different (truncated) CDF for different parts -- real incomes have a Pareto right tail but an obviously non-Pareto bottom. Is this good idea to try to keep these as distributions, or as most of my calculation will need to numeric anyway, I can give up early and use the CDFs? Thanks! Now the output for yesterday: originalecdf = {{500000,0.0182},{1000000,0.1003},{1500000,0.2487},{2000000,0.3871},{400000= 0,0.6802}} ecdf = {{2000000,0.3871},{4000000,0.6802}} FindFit[ecdf,CDF[ParetoDistribution[k,a]],{k,a},x] FindFit::nrlnum: The function value {-0.3871+Function[\[FormalX],\[Piecewise] 1. +Times[<<2>>] \[FormalX]>=k 0. True ,Listable],-0.6802+Function[\[FormalX],\[Piecewise] 1. +Times[<<2>>] \[FormalX]>=k 0. True ,Listable]} is not a list of real numbers with dimensions {2} at {k,a} = {1.,1.}. >> FindFit[ecdf,1-(x / k)^(-a),{k,a},x] {k->1.18709*10^6,a->0.938482} On Fri, May 11, 2012 at 11:37 AM, Darren Glosemeyer <darreng at wolfram.com>wr= ote: > On 5/10/2012 11:14 PM, L=C3=A1szl=C3=B3 S=C3=A1ndor wrote: > >> Hi all, >> >> I have a project (with Mathematica 8) where the first step would be to >> get the distribution describing my "data" which actually only have >> quantiles (or worse: frequencies for arbitrary bins). >> EstimatedDistribution[] looks promising, but I don't know how to feed in >> this kind of data. Please let me know if you know a fast way. >> >> Thank! >> >> >> > There isn't enough information in your data for the types of estimation > done by EstimatedDistribution. > > The type of information you have in your data would lend itself well to a > least squares fit to the cdf of the distribution. As an example, let's ta= ke > this data: > > > In[1]:= data = BlockRandom[SeedRandom[1234]; > RandomVariate[**GammaDistribution[5, 8], 100]]; > > We can use Min and Max to see the range of values and then bin within tha= t > range to construct cutoff and frequency data. > > In[2]:= {Min[data], Max[data]} > > Out[2]= {13.7834, 112.429} > > > Here, xvals are the cutoffs and counts are the bin frequencies. > > In[3]:= {xvals, counts} = HistogramList[data, {{0, 15, 20, 50, 100, 1= 20}}] > > Out[3]= {{0, 15, 20, 50, 100, 120}, {1, 6, 55, 37, 1}} > > > We can get the accumulated probabilities as follows. > > In[4]:= probs = Accumulate[counts]/Length[**data] > > 1 7 31 99 > Out[4]= {---, ---, --, ---, 1} > 100 100 50 100 > > > The analogue of your quantile values would be the right endpoints, > Rest[xvals]. > > In[5]:= quantiles = Rest[xvals] > > Out[5]= {15, 20, 50, 100, 120} > > > Now we can use the quantiles as the x values and the cdf values as the y > values for a least squares fitting to the CDF (parameters may need starti= ng > values in general, but defaults worked fine in this case): > > In[6]:= FindFit[Transpose[{quantiles, probs}], CDF[GammaDistribution[a, > b], x], {a, b}, x] > > Out[6]= {a -> 5.24009, b -> 8.88512} > > > Given that we know that the data don't extend to the right limit of a > gamma's support (gammas can be any positive values), we may want to adjus= t > the cdf values a bit. The following will shift all the cdf values by > 1/(2*numberOfDataPoints) in this particular case: > > In[7]:= FindFit[Transpose[{quantiles, probs - 1/(2 Length[data])}], > CDF[GammaDistribution[a, b], x], {a, b}, x] > > Out[7]= {a -> 5.3696, b -> 8.73319} > > > Darren Glosemeyer > Wolfram Research >

**References**:**question: fitting a distribution from quantiles***From:*László Sándor <sandorl@gmail.com>

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**Re: question: fitting a distribution from quantiles**

**Re: question: fitting a distribution from quantiles**