Distribution#

Here is a brief description of the Distribution type.

In stat_tool, it is a shortname for discrete distributions and is formally represented by the class openalea.stat_tool._stat_tool._DiscreteParametricModel. Usually, it is more convenient for users to define directly usual parametric distributions deriving from Distribution, such as Binomial, Poisson, etc. These distributions can then be combined using Compound, Convolution or Mixtures. The main other feature to be used with distributions is simulation.

Constructor#

Here is how to build distributions. The parameters are specific to each family of distribution. The shift parameter comes first. If a parameter cannot be interpreted as a probability, it comes second and the other parameter, for example a probability parameter, comes third if any.

>>> N = NegativeBinomial(0, 1, 0.1)
>>> B = Binomial(0, 10, 0.5)
>>> P = Poisson(1, 5.2)
>>> U = Uniform(2, 10)
>>> print(U)
UNIFORM   INF_BOUND : 2   SUP_BOUND : 10
mean: 6   median: 6   mode: 6
variance: 6.66667   standard deviation: 2.58199   lower quartile: 4   upper quartile: 8
coefficient of skewness: 0   coefficient of kurtosis: -0.5
mean absolute deviation: 2.22222   coefficient of concentration: 0.246914
information: -2.19722

It is equivalent to use the constructor of Distribution with an additional parameter: the name of the parametric family:

>>> N = Distribution("NEGATIVE_BINOMIAL", 0, 1, 0.1)
>>> B = Distribution("BINOMIAL", 0, 10, 0.5)
>>> P = Distribution("POISSON", 1, 5.2)
>>> U = Distribution("UNIFORM", 2, 10)

In order to display the contents, or to save the data, one uses methods Display() and plot().

Categorical distributions, i.e., multinomials ${\mathcal M}(1; p_1, \ldots, p_k)$ are recovered by the Multinomial class:

>>> M = Multinomial([0.1, 0.6, 0.3])

Note that the constructor has a single parameter, which is the vector of probabilities.

Plotting#

>>> clf()
>>> import openalea.stat_tool.plot
>>> plot.DISABLE_PLOT=True
>>> print(N.display())
NEGATIVE_BINOMIAL   INF_BOUND : 0   PARAMETER : 1   PROBABILITY : 0.1
mean: 9   median: 6   mode: 0
variance: 90   standard deviation: 9.48683   lower quartile: 2   upper quartile: 13
coefficient of skewness: 2.00278   coefficient of kurtosis: 6.01111
mean absolute deviation: 6.90287   coefficient of concentration: 0.523978
information: -3.24418

>>> N.plot()
>>> savefig('user/stat_tool_distribution_plot1.png')

The following figure gathers the probability (mass) function, the cumulative distribution function and the concentration curve.

../_images/stat_tool_distribution_plot1.png

Simulate#

Once you have a Distribution, you can simulate a data set using sample size as an argument:

>>> simulation = N.simulate(400)

This creates an Histogram, formally a openalea.stat_tool._stat_tool._DiscreteDistributionData that is associated with the distribution. Thus, plotting it provides both simulated data and distribution:

>>> clf()
>>> simulation.plot()
>>> savefig('user/nbinom_simul.png')

Estimate#

A Distribution can be estimated from an object of class Vectors or a derived class, such as openalea.stat_tool._stat_tool._DiscreteDistributionData, using openalea.stat_tool.estimate.Estimate(). The parametric family is passed as an argument:

>>> N_est = Estimate(simulation, "NEGATIVE_BINOMIAL")
>>> print(N_est.display())
NEGATIVE_BINOMIAL   INF_BOUND : 0   PARAMETER : 0.953195   PROBABILITY : 0.0943692
mean: 9.1475   median: 6   mode: 0
variance: 96.9331   standard deviation: 9.84546   lower quartile: 2   upper quartile: 13
coefficient of skewness: 2.05103   coefficient of kurtosis: 6.30494
mean absolute deviation: 7.13637   coefficient of concentration: 0.532088

frequency distribution - sample size: 400
mean: 9.1475   median: 6   mode: 0
variance: 96.9331   standard deviation: 9.84546   lower quartile: 2   upper quartile: 13
coefficient of skewness: 1.80722   coefficient of kurtosis: 3.6898
mean absolute deviation: 7.34454   coefficient of concentration: 0.539223

log-likelihood: -1306.02   (normalized: -3.26506)
maximum possible log-likelihood: -1267.79   (information: -3.16947)
deviance: 76.47

chi-square test (36 degrees of freedom)
chi-square value: 49.4593   critical probability: 0.0668912
reference chi-square value: 50.9985   reference critical probability: 0.05