Pamphile Roy tupui

## benford.py
"""Benford's law to measure random number generation quality.

.. note:: This script relies on a modified version of scipy. Pull Request:
          https://github.com/scipy/scipy/pull/10844

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

MIT License

Copyright (c) 2020 Pamphile Tupui ROY

## 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]_.

## 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]_.

## perturb_discrepancy.py
from functools import lru_cache
import numpy as np
import copy
import openturns as ot
from statsmodels.tools import sequences

dim = 2
n_samples = 20
doe_init = np.array(ot.SobolSequence(dim).generate(n_samples))
disc_init = sequences.discrepancy(doe_init)

## moment_independent_visually.py
r"""Visual explanation of Moment independent sensitivity analysis.

Moment-based method are based on the whole PDF to mitigate these
issues (Borgonovo2007). Based on the unconditional PDF, a conditional PDF per
parameter is computed. The more the conditional PDF deviates from the
unconditional PDF, the more the parameter has an impact on the quantity of
interest. The same procedure can be done using the Empirical Cumulative
Density Function (ECDF), respectively with the unconditional ECDF.
This visually shows this procedure. Bins of samples (red circles) are used to
compute a conditional PDF of the output. This PDF is compared to the

## sobol_visually.py
r"""Visual explanation of Sobol' indices.

Sobol' indices are metrics to express sensitivity of the output from
perturbations comming from input parameters. First order indices write

.. math:: S_{x_i} = \frac{\mathbb{V}_i(Y)}{\mathbb{V}[Y]} =
          \frac{\mathbb{\mathbb{V}}[\mathbb{E}(Y|x_i)]}{\mathbb{V}[Y]}

The following is using the Ishigami function

## cosi.py
"""Cosine transformation sensitivity indices.

Using a Discrete Cosine Transformation (DCT), it is possible to compute
first order sensitivity indices.

Reference:

Plischke E., How to compute variance-based sensitivity indicators with your
spreadsheet software, Environmental Modelling & Software,
2012. DOI: 10.1016/j.envsoft.2012.03.004

## minimum_spanning_tree.py
"""Minimum Spanning Tree.

MST is used here as a discrepancy criterion.
Comparing two different designs: the higher the mean, the better the design is
in terms of space filling.

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

MIT License

## sobol_saltelli.py
"""Sobol' indices.

Compute Sobol' variance based sensitivity indices.
Use formulations from Saltelli2010.

Reference:

Saltelli et al. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index,
Computer Physics Communications, 2010. DOI: 10.1016/j.cpc.2009.09.018

## pod.py
"""Proper Orthogonal Decomposition.

Demonstrate how to use it.

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

MIT License

Copyright (c) 2018 Pamphile Tupui ROY
	"""Benford's law to measure random number generation quality.

	.. note:: This script relies on a modified version of scipy. Pull Request:
	https://github.com/scipy/scipy/pull/10844

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

	MIT License

	Copyright (c) 2020 Pamphile Tupui ROY

	"""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]_.
	"""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]_.
	from functools import lru_cache
	import numpy as np
	import copy
	import openturns as ot
	from statsmodels.tools import sequences

	dim = 2
	n_samples = 20
	doe_init = np.array(ot.SobolSequence(dim).generate(n_samples))
	disc_init = sequences.discrepancy(doe_init)
	r"""Visual explanation of Moment independent sensitivity analysis.

	Moment-based method are based on the whole PDF to mitigate these
	issues (Borgonovo2007). Based on the unconditional PDF, a conditional PDF per
	parameter is computed. The more the conditional PDF deviates from the
	unconditional PDF, the more the parameter has an impact on the quantity of
	interest. The same procedure can be done using the Empirical Cumulative
	Density Function (ECDF), respectively with the unconditional ECDF.
	This visually shows this procedure. Bins of samples (red circles) are used to
	compute a conditional PDF of the output. This PDF is compared to the
	r"""Visual explanation of Sobol' indices.

	Sobol' indices are metrics to express sensitivity of the output from
	perturbations comming from input parameters. First order indices write

	.. math:: S_{x_i} = \frac{\mathbb{V}_i(Y)}{\mathbb{V}[Y]} =
	\frac{\mathbb{\mathbb{V}}[\mathbb{E}(Y\|x_i)]}{\mathbb{V}[Y]}

	The following is using the Ishigami function
	"""Cosine transformation sensitivity indices.

	Using a Discrete Cosine Transformation (DCT), it is possible to compute
	first order sensitivity indices.

	Reference:

	Plischke E., How to compute variance-based sensitivity indicators with your
	spreadsheet software, Environmental Modelling & Software,
	2012. DOI: 10.1016/j.envsoft.2012.03.004
	"""Minimum Spanning Tree.

	MST is used here as a discrepancy criterion.
	Comparing two different designs: the higher the mean, the better the design is
	in terms of space filling.

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

	MIT License
	"""Sobol' indices.

	Compute Sobol' variance based sensitivity indices.
	Use formulations from Saltelli2010.

	Reference:

	Saltelli et al. Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index,
	Computer Physics Communications, 2010. DOI: 10.1016/j.cpc.2009.09.018
	"""Proper Orthogonal Decomposition.

	Demonstrate how to use it.

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

	MIT License

	Copyright (c) 2018 Pamphile Tupui ROY