Integration convergence using Halton sequence: scrambling effect

halton_convergence.py

"""Integration convergence using Halton sequence: scrambling effect.

Compute the convergence rate for integrating functions using Halton low
discrepancy sequence [1]_. We are interested in the effect of scrambling [2]_.

Two sets of functions are considered:

(i) The first set of functions are synthetic examples specifically designed
to verify the correctness of the implementation [4]_.

(ii) The second set is categorized into types A, B and C [3]_. These categories
state how the variables are important with respect to the function output:

- type A, Functions with a low number of important variables,
- type B, Functions with almost equally important variables but with
  low interactions with each other,
- type C, Functions with almost equally important variables and with
  high interactions with each other.

The theoretical integral for these functions in the unit hypercube is 1.

Quality of the integration is computed using the Root Mean Square Error (RMSE).

.. note:: This script relies on Scipy >= 1.7. Pull Request:
          https://github.com/scipy/scipy/pull/10844

References
----------
.. [1] Halton, "On the efficiency of certain quasi-random sequences of
       points in evaluating multi-dimensional integrals", Numerische
       Mathematik, 1960.
.. [2] A. B. Owen. "A randomized Halton algorithm in R",
       arXiv:1706.02808, 2017.
.. [3] Sergei Kucherenko and Daniel Albrecht and Andrea Saltelli. Exploring
   multi-dimensional spaces: a Comparison of Latin Hypercube and Quasi Monte
   Carlo Sampling Techniques. arXiv 1505.02350, 2015.
.. [4] Art B. Owen. On dropping the first Sobol' point. arXiv 2008.08051,
   2020.

---------------------------

MIT License
Copyright (c) 2020 Pamphile Tupui ROY
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"""
import os
from collections import namedtuple

import numpy as np
from scipy.stats import qmc
import matplotlib.pyplot as plt
from matplotlib.container import ErrorbarContainer
from matplotlib.legend_handler import HandlerErrorbar

path = 'halton_convergence/int'
os.makedirs(path, exist_ok=True)
generate = True
n_conv = 50
ns_gen = np.arange(1, 1000, 5)

# Functions definitions
_exp1 = 1 - np.exp(1)


def art_1(sample):
    # dim 5, true value 0
    return np.sum(np.exp(sample) + _exp1, axis=1)


def art_2(sample):
    # dim 5, true value 5/3 + 5*(5 - 1)/4
    return np.sum(sample, axis=1) ** 2


def art_3(sample):
    # dim 3, true value 0
    return np.prod(np.exp(sample) + _exp1, axis=1)


def type_a(sample, dim=30):
    # true value 1
    a = np.arange(1, dim + 1)
    f = 1.
    for i in range(dim):
        f *= (abs(4. * sample[:, i] - 2) + a[i]) / (1. + a[i])
    return f


def type_b(sample, dim=30):
    # true value 1
    f = 1.
    for d in range(1, dim + 1):
        f *= (d - sample[:, d - 1]) / (d - 0.5)
    return f


def type_c(sample, dim=10):
    # true value 1
    f = 2 ** dim * np.prod(sample, axis=1)
    return f


functions = namedtuple('functions', ['name', 'func', 'dim', 'ref'])

benchmark = [
    functions('Art 1', art_1, 5, 0),
    functions('Art 2', art_2, 5, 5 / 3 + 5 * (5 - 1) / 4),
    functions('Art 3', art_3, 3, 0),
    functions('Type A', type_a, 30, 1),
    functions('Type B', type_b, 30, 1),
    functions('Type C', type_c, 10, 1)
]


def conv_method(sampler, func, n_samples, n_conv, ref):
    samples = [sampler(n_samples) for _ in range(n_conv)]
    samples = np.array(samples)

    evals = [np.sum(func(sample)) / n_samples for sample in samples]
    squared_errors = (ref - np.array(evals)) ** 2
    rmse = (np.sum(squared_errors) / n_conv) ** 0.5

    return rmse, 0. # can add another metric


# Analysis
if generate:
    sample_mc_rmse = []
    sample_halton = []
    sample_halton_0 = []

    for ns in ns_gen:
        print(f'-> ns={ns}')

        _sample_mc_rmse = []
        _sample_halton = []
        _sample_halton_0 = []
        for case in benchmark:
            # Monte Carlo
            sampler_mc = lambda x: np.random.random((x, case.dim))
            conv_res = conv_method(sampler_mc, case.func, ns, n_conv, case.ref)
            _sample_mc_rmse.append(conv_res)

            # Halton
            engine = qmc.Halton(d=case.dim, scramble=False)
            conv_res = conv_method(engine.random, case.func, ns, 1, case.ref)
            _sample_halton.append(conv_res)

            # Halton scrambled
            def _sampler_halton_0(ns):
                engine = qmc.Halton(d=case.dim, scramble=True)
                return engine.random(ns)
            conv_res = conv_method(_sampler_halton_0, case.func, ns, n_conv, case.ref)
            _sample_halton_0.append(conv_res)


        sample_mc_rmse.append(_sample_mc_rmse)
        sample_halton.append(_sample_halton)
        sample_halton_0.append(_sample_halton_0)

    np.save(os.path.join(path, 'mc.npy'), sample_mc_rmse)
    np.save(os.path.join(path, 'halton.npy'), sample_halton)
    np.save(os.path.join(path, 'halton_0.npy'), sample_halton_0)
else:
    sample_mc_rmse = np.load(os.path.join(path, 'mc.npy'))
    sample_halton = np.load(os.path.join(path, 'halton.npy'))
    sample_halton_0 = np.load(os.path.join(path, 'halton_0.npy'))

sample_mc_rmse = np.array(sample_mc_rmse)
sample_halton = np.array(sample_halton)
sample_halton_0 = np.array(sample_halton_0)

# Plot
for i, case in enumerate(benchmark):
    func = case.name
    fig, ax = plt.subplots()

    ratio_1 = sample_halton[:, i, 0][0] / ns_gen[0] ** (-1/2)
    ratio_2 = sample_halton_0[:, i, 0][0] / ns_gen[0] ** (-2/2)
    ax.plot(ns_gen, ns_gen ** (-1/2) * ratio_1, ls='-', c='k')
    ax.plot(ns_gen, ns_gen ** (-2/2) * ratio_2, ls='-', c='k')

    ax.plot(ns_gen, sample_halton[:, i, 0],
            ls=':', label="Halton unscrambled")
    ax.plot(ns_gen, sample_halton_0[:, i, 0],
            ls='-', label="Halton scrambled")

    ax.set_xlabel(r'$N_s$')
    ax.set_ylabel(r'$\epsilon$')

    ax.set_xscale('log')
    ax.set_yscale('log')

    ax.set_xticks([1, 5, 10, 50, 100, 500, 1000])
    ax.set_xticklabels([1, 5, 10, 50, 100, 500, 1000])

    fig.legend(labelspacing=0.7, bbox_to_anchor=(0.5, 0.43),
               handler_map={ErrorbarContainer: HandlerErrorbar(xerr_size=0.7)})
    fig.tight_layout()

    #plt.show()
    fig.savefig(os.path.join(path, f'halton_conv_integration_{func}.png'),
                transparent=True, bbox_inches='tight')

For instance, Art2: