Fit a distribution by maximum likelihoodΒΆ

In this example we are going to estimate the parameters of a parametric by generic numerical optimization of the likelihood.

The likelihood of a sample X according to a parametric density function p_{\underline{\theta}} is:

likelihood(\underline{x}_1, \dots, \underline{x}_n,\underline{\theta}) = \prod_{i=1}^n p_{\underline{\theta}}(\underline{x}_i)

[1]:

from __future__ import print_function
import openturns as ot
import math as m

[2]:

# Create data from a gaussian pdf with mu=4, sigma=1.5
sample = ot.Normal(4.0, 1.5).getSample(200)

[3]:

# Create the search interval of (mu, sigma): the constraint is sigma>0
lowerBound = [-1.0, 1.0e-4]
upperBound = [-1.0, -1.0]
finiteLowerBound = [False, True]
finiteUpperBound = [False, False]
bounds = ot.Interval(lowerBound, upperBound, finiteLowerBound, finiteUpperBound)

[4]:

# Create the starting point of the research
# For mu : the first point
# For sigma : a value evaluated from the two first points
mu0 = sample[0][0]
sigma0 = m.sqrt((sample[1][0] - sample[0][0]) * (sample[1][0] - sample[0][0]))
startingPoint = [mu0, sigma0]
ot.Point(startingPoint)

[4]:

[4.9123,2.81156]

[5]:

# Create the estimator from a parametric pdf
pdf = ot.Normal()
factory = ot.MaximumLikelihoodFactory(pdf)
factory.setOptimizationBounds(bounds)

[6]:

# Set the starting point via the solver
solver = factory.getOptimizationAlgorithm()
solver.setStartingPoint(startingPoint)
factory.setOptimizationAlgorithm(solver)

[7]:

# Estimate the parametric model
distribution = factory.build(sample)
str(distribution)

[7]:

'Normal(mu = 3.94055, sigma = 1.48893)'

[8]:

# Retrieve the estimated parameters
distribution.getParameter()

[8]:

[3.94055,1.48893]