Antithetic draws

Calculation of a simple integral using Monte-Carlo integration. It illustrates how to use antithetic draws.

author:

Michel Bierlaire, EPFL

date:

Thu Apr 13 20:48:02 2023

import numpy as np
import pandas as pd
import biogeme.database as db
import biogeme.biogeme as bio
from biogeme import draws
from biogeme.expressions import exp, bioDraws, MonteCarlo

We create a fake database with one entry, as it is required to store the draws.

df = pd.DataFrame()
df['FakeColumn'] = [1.0]
database = db.Database('fake_database', df)

def halton13_anti(sample_size: int, number_of_draws: int) -> np.array:
    """The user can define new draws. For example, antithetic Halton
    draws with base 13, skipping the first 10 draws.

    :param sample_size: number of observations for which draws must be
                       generated.
    :param number_of_draws: number of draws to generate.

    """

    # We first generate half of the number of requested draws.
    the_draws = draws.getHaltonDraws(
        sample_size, int(number_of_draws / 2.0), base=13, skip=10
    )
    # We complement them with their antithetic version.
    return np.concatenate((the_draws, 1 - the_draws), axis=1)

mydraws = {
    'HALTON13_ANTI': (
        halton13_anti,
        'Antithetic Halton draws, base 13, skipping 10',
    )
}
database.setRandomNumberGenerators(mydraws)

Integrate with antithetic pseudo-random number generator.

integrand = exp(bioDraws('U', 'UNIFORM_ANTI'))
simulatedI = MonteCarlo(integrand)

Integrate with antithetic Halton draws, base 13.

integrand_halton13 = exp(bioDraws('U_halton13', 'HALTON13_ANTI'))
simulatedI_halton13 = MonteCarlo(integrand_halton13)

Integrate with antithetic MLHS.

integrand_mlhs = exp(bioDraws('U_mlhs', 'UNIFORM_MLHS_ANTI'))
simulatedI_mlhs = MonteCarlo(integrand_mlhs)

True value

trueI = exp(1.0) - 1.0

Number of draws.

R = 20000

error = simulatedI - trueI

error_halton13 = simulatedI_halton13 - trueI

error_mlhs = simulatedI_mlhs - trueI

simulate = {
    'Analytical Integral': trueI,
    'Simulated Integral': simulatedI,
    'Error             ': error,
    'Simulated Integral (Halton13)': simulatedI_halton13,
    'Error (Halton13)             ': error_halton13,
    'Simulated Integral (MLHS)': simulatedI_mlhs,
    'Error (MLHS)             ': error_mlhs,
}

biosim = bio.BIOGEME(database, simulate)
biosim.modelName = 'b03antithetic'
results = biosim.simulate(theBetaValues={})
results

	Analytical Integral	Simulated Integral	Error	Simulated Integral (Halton13)	Error (Halton13)	Simulated Integral (MLHS)	Error (MLHS)
0	1.718282	1.718272	-0.00001	1.718282	-4.449617e-09	1.718282	-1.322897e-11

Reorganize the results.

print(f"Analytical integral: {results.iloc[0]['Analytical Integral']:.6f}")
print(
    f"         \t{'Uniform (Anti)':>15}\t{'Halton13 (Anti)':>15}\t{'MLHS (Anti)':>15}"
)
print(
    f"Simulated\t{results.iloc[0]['Simulated Integral']:>15.6g}\t"
    f"{results.iloc[0]['Simulated Integral (Halton13)']:>15.6g}\t"
    f"{results.iloc[0]['Simulated Integral (MLHS)']:>15.6g}"
)
print(
    f"Error\t\t{results.iloc[0]['Error             ']:>15.6g}\t"
    f"{results.iloc[0]['Error (Halton13)             ']:>15.6g}\t"
    f"{results.iloc[0]['Error (MLHS)             ']:>15.6g}"
)

Analytical integral: 1.718282
                 Uniform (Anti) Halton13 (Anti)     MLHS (Anti)
Simulated               1.71827         1.71828         1.71828
Error              -9.65275e-06    -4.44962e-09     -1.3229e-11