[SciPy-User] How to fit parameters of beta distribution?

[josef.pktd at gmail.com]

On Fri, Jun 24, 2011 at 8:37 AM, John Reid <j.reid at mail.cryst.bbk.ac.uk> wrote:
> Thanks for the information. Just out of interest, this is what I get on
> scipy 0.7 (no warnings)
>
> In [1]: import scipy.stats
>
> In [2]: scipy.stats.beta.fit([.5])
> Out[2]:
> array([  1.87795851e+00,   1.81444871e-01,   2.39026963e-04,
>          4.99760973e-01])
>
> In [3]: scipy.__version__
> Out[3]: '0.7.0'
>
> Also I have (following your advice):
>
> In [7]: scipy.stats.beta.fit([.5], floc=0., fscale=1.)
> Out[7]:
> array([  1.87795851e+00,   1.81444871e-01,   2.39026963e-04,
>          4.99760973e-01])
>
> which just seems wrong, surely the loc and scale in the output should be
> what I specified in the arguments? In any case from your example, it
> seems like it is fixed in 0.9

floc an fscale where added in scipy 0.9, extra keywords on 0.7 were just ignored

>
> I'm assuming fit() does a ML estimate of the parameters which I think is
> fine to do for a beta distribution and one data point.

You need at least as many observations as parameters, and without
enough observations the estimate will be very noisy. With fewer
observations than parameters, you cannot identify the parameters.

Josef

>
> Thanks,
> John.
>
>
> On 24/06/11 12:20, Christoph Deil wrote:
>>
>> On Jun 24, 2011, at 11:26 AM, John Reid wrote:
>>
>>> Hi,
>>>
>>> I can see a instancemethod scipy.stats.beta.fit. I can't work out from
>>> the docs how to use it. From trial&  error I got the following:
>>>
>>> In [12]: scipy.stats.beta.fit([.5])
>>> Out[12]:
>>> array([  1.87795851e+00,   1.81444871e-01,   2.39026963e-04,
>>>           4.99760973e-01])
>>>
>>> What are the 4 values output by the method?
>>>
>>> Thanks,
>>> John.
>>
>> Hi John,
>>
>> the short answer is (a, b, loc, scale), but you probably want to fix loc=0 and scale=1 to get meaningful a, b estimates.
>>
>> It takes some time to learn how scipy.stats.rv_continuous works, but this is a good starting point:
>> http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html#distributions
>>
>> There you'll see that every rv_continuous distribution (e.g. norm, chi2, beta) has two parameters loc and scale,
>> which shift and stretch the distribution like this:
>> (x - loc) / scale
>>
>> E.g. from the docstring of scipy.stats.norm, you can see that norm uses these two parameters and has no extra "shape parameters":
>> Normal distribution
>>      The location (loc) keyword specifies the mean.
>>      The scale (scale) keyword specifies the standard deviation.
>>      normal.pdf(x) = exp(-x**2/2)/sqrt(2*pi)
>>
>> You can draw a random data sample and fit it like this:
>> data = scipy.stats.norm.rvs(loc=10, scale=2, size=100)
>> scipy.stats.norm.fit(data) # returns loc, scale
>> # (9.9734277669649689, 2.2125503785545551)
>>
>> The beta distribution you are interested in has two shape parameters a and b, plus in addition the loc and scale parameters every rv_continuous has:
>> Beta distribution
>>      beta.pdf(x, a, b) = gamma(a+b)/(gamma(a)*gamma(b)) * x**(a-1) * (1-x)**(b-1)
>>      for 0<  x<  1, a, b>  0.
>>
>> In your case you probably want to fix loc=0 and scale=1 and only fit the a and b parameter, which you can do like this:
>> data = scipy.stats.beta.rvs(2, 5, size=100) # a = 2, b = 5 (can't use keyword arguments)
>> scipy.stats.beta.fit(data, floc=0, fscale=1) # returns a, b, loc, scale
>> # (2.6928363303187393, 5.9855671734557454, 0, 1)
>>
>> I find that the splitting of parameters into "location and scale" and "shape" makes rv_continuous usage complicated:
>> - it is uncommon that the beta or chi2 or many other distributions have a loc and scale parameter
>> - the auto-generated docstrings are confusing at first
>> But if you look at the implementation it does avoid some repetitive code for the developers.
>>
>> Btw., I don't know how you can fit multiple parameters to only one measurement [.5] in your example.
>> You must have executed some code before that line, otherwise you'll get a bunch of RuntimeWarnings and a different return value from the one you give (I use on scipy 0.9)
>> In [1]: import scipy.stats
>> In [2]: scipy.stats.beta.fit([.5])
>> Out[2]: (1.0, 1.0, 0.5, 0.0)
>>
>> Christoph
>
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