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Re: question: fitting a distribution from quantiles
*To*: mathgroup at smc.vnet.net
*Subject*: [mg126471] Re: question: fitting a distribution from quantiles
*From*: Darren Glosemeyer <darreng at wolfram.com>
*Date*: Sat, 12 May 2012 04:57:38 -0400 (EDT)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*References*: <201205110414.AAA23695@smc.vnet.net>
On 5/10/2012 11:14 PM, László Sándor 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
take 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
that 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, 120}}]
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
starting 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
adjust 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
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