Integration convergence using Sobol' sequence: removing the first point

sobol_convergence.py

"""Integration convergence using Sobol' sequence: removing the first point.

Compute the convergence rate for integrating functions using Sobol' low
discrepancy sequence [1]_. We are interested in measuring the effect of
removing the first point of the sequence ([0, ...]).

Two sets of functions are considered:

(i) The first set of functions are synthetic examples specifically designed
to verify the correctness of the implementation [3]_.

(ii) The second set is categorized into types A, B and C [2]_. 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] I. M. Sobol. The distribution of points in a cube and the accurate
   evaluation of integrals. Zh. Vychisl. Mat. i Mat. Phys., 7:784-802,
   1967.

.. [2] 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.

.. [3] Art B. Owen. On dropping the first Sobol' point. arXiv 2008.08051,
   2020.

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

MIT License

Copyright (c) 2020 Pamphile Tupui ROY

Permission is hereby granted, free of charge, to any person obtaining a copy
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SOFTWARE.

"""
import os
from collections import namedtuple

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

path = 'sobol_convergence'
os.makedirs(path, exist_ok=True)
generate = True
n_conv = 999
ns_gen = 2 ** np.arange(4, 13)  # 13

# 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 3, 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

    if n_conv > 1:
        ci = np.sqrt(stats.t.interval(0.95, len(squared_errors) - 1,
                                      loc=squared_errors.mean(),
                                      scale=stats.sem(squared_errors)))
        ci = [rmse - ci[0], ci[1] - rmse]
    else:
        ci = [0, 0]

    #c1, c2 = stats.chi2.ppf([0.025, 1 - 0.025], n_conv)
    #ci = [rmse * (1 - np.sqrt(n_conv/c2)), rmse * (np.sqrt(n_conv/c1) - 1)]

    return rmse, ci[0], ci[1]  # 2 * np.std(evals) / np.sqrt(n_conv)


# Analysis
if generate:
    sample_mc_rmse = []
    sample_sobol_0_rmse = []
    sample_sobol_no_0_rmse = []
    sample_sobol_scramble_0_rmse = []
    sample_sobol_scramble_no_0_rmse = []

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

        _sample_mc_rmse = []
        _sample_sobol_0_rmse = []
        _sample_sobol_no_0_rmse = []
        _sample_sobol_scramble_0_rmse = []
        _sample_sobol_scramble_no_0_rmse = []
        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)

            # Sobol' with zero
            engine = qmc.Sobol(d=case.dim, scramble=False)
            conv_res = conv_method(engine.random, case.func, ns, 1, case.ref)
            _sample_sobol_0_rmse.append(conv_res)

            # Sobol' without zero
            def _sampler_sobol_no_0(ns):
                engine = qmc.Sobol(d=case.dim, scramble=False)
                return engine.random(ns + 1)[1:]
            conv_res = conv_method(_sampler_sobol_no_0, case.func, ns - 1, 1, case.ref)
            _sample_sobol_no_0_rmse.append(conv_res)

            # Sobol' scrambled with zero
            def _sampler_sobol_scrambled_0(ns):
                engine = qmc.Sobol(d=case.dim, scramble=True)
                return engine.random(ns)
            conv_res = conv_method(_sampler_sobol_scrambled_0, case.func, ns, n_conv, case.ref)
            _sample_sobol_scramble_0_rmse.append(conv_res)

            # Sobol' scrambled without zero
            def _sampler_sobol_scrambled_no_0(ns):
                engine = qmc.Sobol(d=case.dim, scramble=True)
                return engine.random(ns + 1)[1:]
            conv_res = conv_method(_sampler_sobol_scrambled_no_0, case.func, ns - 1, n_conv, case.ref)
            _sample_sobol_scramble_no_0_rmse.append(conv_res)

        sample_mc_rmse.append(_sample_mc_rmse)
        sample_sobol_0_rmse.append(_sample_sobol_0_rmse)
        sample_sobol_no_0_rmse.append(_sample_sobol_no_0_rmse)
        sample_sobol_scramble_0_rmse.append(_sample_sobol_scramble_0_rmse)
        sample_sobol_scramble_no_0_rmse.append(_sample_sobol_scramble_no_0_rmse)

    np.save(os.path.join(path, 'mc.npy'), sample_mc_rmse)
    np.save(os.path.join(path, 'sobol_0.npy'), sample_sobol_0_rmse)
    np.save(os.path.join(path, 'sobol_no_0.npy'), sample_sobol_no_0_rmse)
    np.save(os.path.join(path, 'sobol_scramble_0.npy'), sample_sobol_scramble_0_rmse)
    np.save(os.path.join(path, 'sobol_scramble_no_0.npy'), sample_sobol_scramble_no_0_rmse)
else:
    sample_mc_rmse = np.load(os.path.join(path, 'mc.npy'))
    sample_sobol_0_rmse = np.load(os.path.join(path, 'sobol_0.npy'))
    sample_sobol_no_0_rmse = np.load(os.path.join(path, 'sobol_no_0.npy'))
    sample_sobol_scramble_0_rmse = np.load(os.path.join(path, 'sobol_scramble_0.npy'))
    sample_sobol_scramble_no_0_rmse = np.load(os.path.join(path, 'sobol_scramble_no_0.npy'))

sample_mc_rmse = np.array(sample_mc_rmse)
sample_sobol_0_rmse = np.array(sample_sobol_0_rmse)
sample_sobol_no_0_rmse = np.array(sample_sobol_no_0_rmse)
sample_sobol_scramble_0_rmse = np.array(sample_sobol_scramble_0_rmse)
sample_sobol_scramble_no_0_rmse = np.array(sample_sobol_scramble_no_0_rmse)

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

    ratio_1 = sample_sobol_0_rmse[:, i, 0][0] / ns_gen[0] ** (-2/2)
    # ratio_1 = sample_sobol_scramble_rmse[:, i, 0][0] / ns_gen[0] ** (-2 / 2)
    # ratio_2 = sample_sobol_scramble_0_rmse[:, i, 0][0] / (np.log2(ns_gen[0]) * ns_gen[0] ** (-3 / 2))
    ratio_3 = sample_sobol_scramble_0_rmse[:, i, 0][0] / (ns_gen[0] ** (-3 / 2))

    # ax.plot(ns_gen, ns_gen ** (-1 / 2), ls='-', c='k')
    ax.plot(ns_gen, ns_gen ** (-2/2) * ratio_1, ls='-', c='k')
    # ax.plot(ns_gen, np.log2(ns_gen) * ns_gen ** (-3 / 2) * ratio_2, ls='-.')
    ax.plot(ns_gen, ns_gen ** (-3 / 2) * ratio_3, ls='-', c='k')

    #ax.errorbar(ns_gen, sample_mc_rmse[:, i, 0], sample_mc_rmse[:, i, 1],
    #            ls='None', marker='x', label="MC", c='k')
    ax.plot(ns_gen, sample_sobol_no_0_rmse[:, i, 0],
            ls='None', marker='s', label="Sobol' no 0", c='k')
    ax.plot(ns_gen, sample_sobol_0_rmse[:, i, 0],
            ls='None', marker='o', label="Sobol' with 0", c='k')
    ax.errorbar(ns_gen, sample_sobol_scramble_no_0_rmse[:, i, 0],
                yerr=sample_sobol_scramble_no_0_rmse[:, i, 1:3].T.reshape(2, -1),
                ls='None', marker='+', label="Sobol' scrambled no 0", c='k')
    ax.errorbar(ns_gen, sample_sobol_scramble_0_rmse[:, i, 0],
                yerr=sample_sobol_scramble_0_rmse[:, i, 1:3].T.reshape(2, -1),
                ls='None', marker='^', label="Sobol' scrambled with 0", c='k')

    ax.set_xlabel(r'$N_s$')

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

    ax.set_xticks(ns_gen)
    ax.set_xticklabels([fr'$2^{{{ns}}}$' for ns in np.arange(4, 20)])
    ax.set_ylabel(r'$\epsilon$')
    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'sobol_conv_integration_{func}.pdf'),
                transparent=True, bbox_inches='tight')
    

Type A:

Type B:

Type C:

Thank you very much Pamphile for these scripts: they are very helpful!

I tried to use this script, but it produces two errors:

• RuntimeWarning: invalid value encountered in sqrt : ci = np.sqrt(
• ValueError: The number of FixedLocator locations (9), usually from a call to set_ticks, does not match the number of labels (16).

Here are methods to fix them.

Fix the sqrt error

The sqrt error is produced by the lines:

ci = np.sqrt(stats.t.interval(0.95, len(squared_errors) - 1,
    loc=squared_errors.mean(),
    scale=stats.sem(squared_errors)))

This is because the interval sometimes contain negative values, for example for ns=16, I sometimes get (-4.0884010942014476e-05, 0.0018436065100577177). This should not happen in theory, since the sample of squared errors is non-negative.

In order to fix this, I suggest to use the sample quantile. This is potentially less accurate than the original T-based interval, but it has the advantage that it cannot go on the negative side of R.

confidence_level = 0.95
half_length = (1.0 - confidence_level) / 2.0
quantile_025 = np.quantile(squared_errors, half_length)
quantile_975 = np.quantile(squared_errors, 1.0 - half_length)
interval = [quantile_025, quantile_975]
ci = np.sqrt(interval)

Fix the number of labels

The second message is produced by:

ax.set_xticklabels([rf"$2^{{{ns}}}$" for ns in np.arange(4, 20)])

The first problem is that this is not always consistent with the chosen value of ns_gen. Another problem is that, since we display $2^e$, then e must be the base-2 log of the sample size.
One method to fix this is:

ax.set_xticklabels([rf"$2^{{{int(np.log2(ns))}}}$" for ns in ns_gen])

As a side note, I do not fully understand why A. Owen got such a small variance for the scrambled Sobol' sequence. The current script uses R=999 while Owen uses R=10 according to the paper. But when using Scipy's implementation using the same parameters, we get a much larger variability of the RMSE. Also, I do not understand why using the RMSE instead of, for example, the sample mean of the absolute errors.

Edit: Now I understand. The theory states that the variance of a scrambled Sobol' sequence has order $n^{-3} \log(n)^{s - 1}$. This is why the RMSE is used. The sample mean of the absolute errors may have a different order of convergence. Using the variance to estimate the error makes sense, as can be seen in the proof of ¹: it is based on the ANOVA decomposition of the integrand, based on the Haar basis.

Footnotes

1. Owen, A. B. (1997). Scrambled net variance for integrals of smooth functions. The Annals of Statistics, 25(4), 1541-1562. ↩