Distribution Fitting
This package provides methods to fit a distribution to a given set of samples. Generally, one may write
d = fit(D, x)This statement fits a distribution of type D to a given dataset x, where x should be an array comprised of all samples. The fit function will choose a reasonable way to fit the distribution, which, in most cases, is maximum likelihood estimation.
|
One can use as the first argument simply the distribution name, like |
```julia
julia> fit(Cauchy{Float32}, collect(-4:4))
Cauchy{Float64}(μ=0.0, σ=2.0)
```
Maximum Likelihood Estimation
The function fit_mle is for maximum likelihood estimation.
Synopsis
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Distributions.fit_mle — Method
fit_mle(D, x)Fit a distribution of type D to a given data set x.
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For univariate distribution, x can be an array of arbitrary size.
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For multivariate distribution, x should be a matrix, where each column is a sample.
#
Distributions.fit_mle — Method
fit_mle(D, x, w)Fit a distribution of type D to a weighted data set x, with weights given by w.
Here, w should be an array with length n, where n is the number of samples contained in x.
Applicable distributions
The fit_mle method has been implemented for the following distributions:
Univariate:
Multivariate:
For most of these distributions, the usage is as described above. For a few special distributions that require additional information for estimation, we have to use a modified interface:
fit_mle(Binomial, n, x) # n is the number of trials in each experiment
fit_mle(Binomial, n, x, w)
fit_mle(Categorical, k, x) # k is the space size (i.e. the number of distinct values)
fit_mle(Categorical, k, x, w)
fit_mle(Categorical, x) # equivalent to fit_mle(Categorical, max(x), x)
fit_mle(Categorical, x, w)Sufficient Statistics
For many distributions, the estimation can be based on (sum of) sufficient statistics computed from a dataset. To simplify implementation, for such distributions, we implement suffstats method instead of fit_mle directly:
ss = suffstats(D, x) # ss captures the sufficient statistics of x
ss = suffstats(D, x, w) # ss captures the sufficient statistics of a weighted dataset
d = fit_mle(D, ss) # maximum likelihood estimation based on sufficient statsWhen fit_mle on D is invoked, a fallback fit_mle method will first call suffstats to compute the sufficient statistics, and then a fit_mle method on sufficient statistics to get the result. For some distributions, this way is not the most efficient, and we specialize the fit_mle method to implement more efficient estimation algorithms.