Concise API Reference#
This page documents the sensitivity analysis methods supported by SALib.
FAST - Fourier Amplitude Sensitivity Test#
- SALib.sample.fast_sampler.sample(problem, N, M=4, seed=None)[source]
Generate model inputs for extended Fourier Amplitude Sensitivity Test.
Returns a NumPy matrix containing the model inputs required by the extended Fourier Amplitude sensitivity test. The resulting matrix contains N * D rows and D columns, where D is the number of parameters.
The samples generated are intended to be used by
SALib.analyze.fast.analyze()
.- Parameters
References
Cukier, R.I., Fortuin, C.M., Shuler, K.E., Petschek, A.G., Schaibly, J.H., 1973. Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. I theory. Journal of Chemical Physics 59, 3873-3878. https://doi.org/10.1063/1.1680571
Saltelli, A., S. Tarantola, and K. P.-S. Chan (1999). A Quantitative Model-Independent Method for Global Sensitivity Analysis of Model Output. Technometrics, 41(1):39-56, doi:10.1080/00401706.1999.10485594.
- SALib.analyze.fast.analyze(problem, Y, M=4, num_resamples=100, conf_level=0.95, print_to_console=False, seed=None)[source]
Perform extended Fourier Amplitude Sensitivity Test on model outputs.
Returns a dictionary with keys ‘S1’ and ‘ST’, where each entry is a list of size D (the number of parameters) containing the indices in the same order as the parameter file.
Notes
- Compatible with:
fast_sampler :
SALib.sample.fast_sampler.sample()
Examples
>>> X = fast_sampler.sample(problem, 1000) >>> Y = Ishigami.evaluate(X) >>> Si = fast.analyze(problem, Y, print_to_console=False)
- Parameters
problem (dict) – The problem definition
Y (numpy.array) – A NumPy array containing the model outputs
M (int) – The interference parameter, i.e., the number of harmonics to sum in the Fourier series decomposition (default 4)
print_to_console (bool) – Print results directly to console (default False)
seed (int) – Seed to generate a random number
References
Cukier, R. I., C. M. Fortuin, K. E. Shuler, A. G. Petschek, and J. H. Schaibly (1973). Study of the sensitivity of coupled reaction systems to uncertainties in rate coefficients. J. Chem. Phys., 59(8):3873-3878 doi:10.1063/1.1680571
Saltelli, A., S. Tarantola, and K. P.-S. Chan (1999). A Quantitative Model-Independent Method for Global Sensitivity Analysis of Model Output. Technometrics, 41(1):39-56, doi:10.1080/00401706.1999.10485594.
Pujol, G. (2006) fast99 - R sensitivity package cran/sensitivity
RBD-FAST - Random Balance Designs Fourier Amplitude Sensitivity Test#
- SALib.sample.latin.sample(problem, N, seed=None)[source]
Generate model inputs using Latin hypercube sampling (LHS).
Returns a NumPy matrix containing the model inputs generated by Latin hypercube sampling. The resulting matrix contains N rows and D columns, where D is the number of parameters.
- Parameters
References
- McKay, M.D., Beckman, R.J., Conover, W.J., 1979.
A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21, 239-245. https://doi.org/10.2307/1268522
- Iman, R.L., Helton, J.C., Campbell, J.E., 1981.
An Approach to Sensitivity Analysis of Computer Models: Part I—Introduction, Input Variable Selection and Preliminary Variable Assessment. Journal of Quality Technology 13, 174-183. https://doi.org/10.1080/00224065.1981.11978748
- SALib.analyze.rbd_fast.analyze(problem, X, Y, M=10, num_resamples=100, conf_level=0.95, print_to_console=False, seed=None)[source]
Performs the Random Balanced Design - Fourier Amplitude Sensitivity Test (RBD-FAST) on model outputs.
Returns a dictionary with keys ‘S1’, where each entry is a list of size D (the number of parameters) containing the indices in the same order as the parameter file.
Notes
- Compatible with:
all samplers
Examples
>>> X = latin.sample(problem, 1000) >>> Y = Ishigami.evaluate(X) >>> Si = rbd_fast.analyze(problem, X, Y, print_to_console=False)
- Parameters
problem (dict) – The problem definition
X (numpy.array) – A NumPy array containing the model inputs
Y (numpy.array) – A NumPy array containing the model outputs
M (int) – The interference parameter, i.e., the number of harmonics to sum in the Fourier series decomposition (default 10)
print_to_console (bool) – Print results directly to console (default False)
seed (int) – Seed to generate a random number
References
S. Tarantola, D. Gatelli and T. Mara (2006) Random Balance Designs for the Estimation of First Order Global Sensitivity Indices, Reliability Engineering and System Safety, 91:6, 717-727 https://doi.org/10.1016/j.ress.2005.06.003
- Elmar Plischke (2010)
An effective algorithm for computing global sensitivity indices (EASI), Reliability Engineering & System Safety, 95:4, 354-360. doi:10.1016/j.ress.2009.11.005
- Jean-Yves Tissot, Clémentine Prieur (2012)
Bias correction for the estimation of sensitivity indices based on random balance designs, Reliability Engineering and System Safety, Elsevier, 107, 205-213. doi:10.1016/j.ress.2012.06.010
- Jeanne Goffart, Mickael Rabouille & Nathan Mendes (2015)
Uncertainty and sensitivity analysis applied to hygrothermal simulation of a brick building in a hot and humid climate, Journal of Building Performance Simulation. doi:10.1080/19401493.2015.1112430
Method of Morris#
- SALib.sample.morris.sample(problem: Dict, N: int, num_levels: int = 4, optimal_trajectories: int = None, local_optimization: bool = True, seed: int = None) ndarray [source]
Generate model inputs using the Method of Morris.
Three variants of Morris’ sampling for elementary effects is supported:
Vanilla Morris (see [1]) when
optimal_trajectories
isNone
/False
andlocal_optimization
isFalse
- Optimised trajectories when
optimal_trajectories=True
using Campolongo’s enhancements (see [2]) and optionally Ruano’s enhancement (see [3]) when
local_optimization=True
- Optimised trajectories when
Morris with groups when the problem definition specifies groups of parameters
Results from these model inputs are intended to be used with
SALib.analyze.morris.analyze()
.Notes
Campolongo et al., [2] introduces an optimal trajectories approach which attempts to maximize the parameter space scanned for a given number of trajectories (where optimal_trajectories \(\in {2, ..., N}\)). The approach accomplishes this aim by randomly generating a high number of possible trajectories (500 to 1000 in [2]) and selecting a subset of
r
trajectories which have the highest spread in parameter space. Ther
variable in [2] corresponds to theoptimal_trajectories
parameter here.Calculating all possible combinations of trajectories can be computationally expensive. The number of factors makes little difference, but the ratio between number of optimal trajectories and the sample size results in an exponentially increasing number of scores that must be computed to find the optimal combination of trajectories. We suggest going no higher than 4 levels from a pool of 100 samples with this “brute force” approach.
Ruano et al., [3] proposed an alternative approach with an iterative process that maximizes the distance between subgroups of generated trajectories, from which the final set of trajectories are selected, again maximizing the distance between each. The approach is not guaranteed to produce the most optimal spread of trajectories, but are at least locally maximized and significantly reduce the time taken to select trajectories. With
local_optimization = True
(which is default), it is possible to go higher than the previously suggested 4 levels from a pool of 100 samples.- Parameters
problem (dict) – The problem definition
N (int) – The number of trajectories to generate
num_levels (int, default=4) – The number of grid levels (should be even)
optimal_trajectories (int) – The number of optimal trajectories to sample (between 2 and N)
local_optimization (bool, default=True) – Flag whether to use local optimization according to Ruano et al. (2012) Speeds up the process tremendously for bigger N and num_levels. If set to
False
brute force method is used, unlessgurobipy
is availableseed (int) – Seed to generate a random number
- Returns
sample_morris – Array containing the model inputs required for Method of Morris. The resulting matrix has \((G/D+1)*N/T\) rows and \(D\) columns, where \(D\) is the number of parameters, \(G\) is the number of groups (if no groups are selected, the number of parameters). \(T\) is the number of trajectories \(N\), or optimal_trajectories if selected.
- Return type
np.ndarray
References
Morris, M.D., 1991. Factorial Sampling Plans for Preliminary Computational Experiments. Technometrics 33, 161-174. https://doi.org/10.1080/00401706.1991.10484804
Campolongo, F., Cariboni, J., & Saltelli, A. 2007. An effective screening design for sensitivity analysis of large models. Environmental Modelling & Software, 22(10), 1509-1518. https://doi.org/10.1016/j.envsoft.2006.10.004
Ruano, M.V., Ribes, J., Seco, A., Ferrer, J., 2012. An improved sampling strategy based on trajectory design for application of the Morris method to systems with many input factors. Environmental Modelling & Software 37, 103-109. https://doi.org/10.1016/j.envsoft.2012.03.008
- SALib.analyze.morris.analyze(problem: Dict, X: ndarray, Y: ndarray, num_resamples: int = 100, conf_level: float = 0.95, print_to_console: bool = False, num_levels: int = 4, seed=None) Dict [source]
Perform Morris Analysis on model outputs.
Returns a result set with keys
mu
,mu_star
,sigma
, andmu_star_conf
, where each entry corresponds to the parameters defined in the problem spec or parameter file.mu
metric indicates the mean of the distributionmu_star
metric indicates the mean of the distribution of absolutevalues
sigma
is the standard deviation of the distribution
Notes
When applied with groups, the
mu
metric is less reliable as the effect from parameters within a group become averaged out.The
mu_star
metric avoids this issue as it indicates the mean of the absolute values. If the direction of effects is important, Campolongo et al., [2] suggest comparingmu_star
withmu
. Ifmu
is low andmu_star
is high, then the effects are of different signs.sigma
is used as an indicator of interactions between parameters, or groups of parameters.- Compatible with:
morris :
SALib.sample.morris.sample()
Examples
>>> X = morris.sample(problem, 1000, num_levels=4) >>> Y = Ishigami.evaluate(X) >>> Si = morris.analyze(problem, X, Y, conf_level=0.95, >>> print_to_console=True, num_levels=4)
- Parameters
problem (dict) – The problem definition
X (numpy.array) – The NumPy matrix containing the model inputs of dtype=float
Y (numpy.array) – The NumPy array containing the model outputs of dtype=float
num_resamples (int) – The number of resamples used to compute the confidence intervals (default 1000)
conf_level (float) – The confidence interval level (default 0.95)
print_to_console (bool) – Print results directly to console (default False)
num_levels (int) – The number of grid levels, must be identical to the value passed to SALib.sample.morris (default 4)
seed (int) – Seed to generate a random number
- Returns
Si – A dictionary of sensitivity indices containing the following entries.
mu - the mean elementary effect
mu_star - the absolute of the mean elementary effect
sigma - the standard deviation of the elementary effect
mu_star_conf - the bootstrapped confidence interval
names - the names of the parameters
- Return type
References
Morris, M. (1991). Factorial Sampling Plans for Preliminary Computational Experiments. Technometrics, 33(2):161-174, doi:10.1080/00401706.1991.10484804.
Campolongo, F., J. Cariboni, and A. Saltelli (2007). An effective screening design for sensitivity analysis of large models. Environmental Modelling & Software, 22(10):1509-1518, doi:10.1016/j.envsoft.2006.10.004.
Sobol’ Sensitivity Analysis#
- SALib.sample.saltelli.sample(problem: Dict, N: int, calc_second_order: bool = True, skip_values: int = None)[source]
Generates model inputs using Saltelli’s extension of the Sobol’ sequence
The Sobol’ sequence is a popular quasi-random low-discrepancy sequence used to generate uniform samples of parameter space.
Returns a NumPy matrix containing the model inputs using Saltelli’s sampling scheme. Saltelli’s scheme extends the Sobol’ sequence in a way to reduce the error rates in the resulting sensitivity index calculations. If calc_second_order is False, the resulting matrix has
N * (D + 2)
rows, whereD
is the number of parameters. If calc_second_order is True, the resulting matrix hasN * (2D + 2)
rows. These model inputs are intended to be used withSALib.analyze.sobol.analyze()
.Deprecated since version 1.4.6.
Notes
The initial points of the Sobol’ sequence has some repetition (see Table 2 in Campolongo [1]), which can be avoided by setting the skip_values parameter. Skipping values reportedly improves the uniformity of samples. It has been shown that naively skipping values may reduce accuracy, increasing the number of samples needed to achieve convergence (see Owen [2]).
A recommendation adopted here is that both skip_values and N be a power of 2, where N is the desired number of samples (see [2] and discussion in [5] for further context). It is also suggested therein that
skip_values >= N
.The method now defaults to setting skip_values to a power of two that is
>= N
. If skip_values is provided, the method now raises a UserWarning in cases where sample sizes may be sub-optimal according to the recommendation above.- Parameters
problem (dict) – The problem definition
N (int) – The number of samples to generate. Ideally a power of 2 and <= skip_values.
calc_second_order (bool) – Calculate second-order sensitivities (default True)
skip_values (int or None) – Number of points in Sobol’ sequence to skip, ideally a value of base 2 (default: a power of 2 >= N, or 16; whichever is greater)
References
- Campolongo, F., Saltelli, A., Cariboni, J., 2011.
From screening to quantitative sensitivity analysis. A unified approach. Computer Physics Communications 182, 978-988. https://doi.org/10.1016/j.cpc.2010.12.039
- Owen, A. B., 2020.
On dropping the first Sobol’ point. arXiv:2008.08051 [cs, math, stat]. Available at: http://arxiv.org/abs/2008.08051 (Accessed: 20 April 2021).
- Saltelli, A., 2002.
Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications 145, 280-297. https://doi.org/10.1016/S0010-4655(02)00280-1
- Sobol’, I.M., 2001.
Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, The Second IMACS Seminar on Monte Carlo Methods 55, 271-280. https://doi.org/10.1016/S0378-4754(00)00270-6
- Discussion: scipy/scipy#10844
- SALib.sample.sobol.sample(problem: Dict, N: int, *, calc_second_order: bool = True, scramble: bool = True, skip_values: int = 0, seed: Optional[Union[int, Generator]] = None)[source]
Generates model inputs using Saltelli’s extension of the Sobol’ sequence.
The Sobol’ sequence is a popular quasi-random low-discrepancy sequence used to generate uniform samples of parameter space.
Returns a NumPy matrix containing the model inputs using Saltelli’s sampling scheme. Saltelli’s scheme extends the Sobol’ sequence in a way to reduce the error rates in the resulting sensitivity index calculations. If calc_second_order is False, the resulting matrix has
N * (D + 2)
rows, whereD
is the number of parameters. If calc_second_order is True, the resulting matrix hasN * (2D + 2)
rows.These model inputs are intended to be used with
SALib.analyze.sobol.analyze()
.Notes
The initial points of the Sobol’ sequence has some repetition (see Table 2 in Campolongo [1]__), which can be avoided by scrambling the sequence.
Another option, not recommended and available for educational purposes, is to use the skip_values parameter. Skipping values reportedly improves the uniformity of samples. But, it has been shown that naively skipping values may reduce accuracy, increasing the number of samples needed to achieve convergence (see Owen [2]__).
- Parameters
problem (dict,) – The problem definition.
N (int) – The number of samples to generate. Ideally a power of 2 and <= skip_values.
calc_second_order (bool, optional) – Calculate second-order sensitivities. Default is True.
scramble (bool, optional) – If True, use LMS+shift scrambling. Otherwise, no scrambling is done. Default is True.
skip_values (int, optional) – Number of points in Sobol’ sequence to skip, ideally a value of base 2. It’s recommended not to change this value and use scramble instead. scramble and skip_values can be used together. Default is 0.
seed ({None, int, numpy.random.Generator}, optional) – If seed is None the numpy.random.Generator generator is used. If seed is an int, a new
Generator
instance is used, seeded with seed. If seed is already aGenerator
instance then that instance is used. Default is None.
References
Campolongo, F., Saltelli, A., Cariboni, J., 2011. From screening to quantitative sensitivity analysis. A unified approach. Computer Physics Communications 182, 978-988. https://doi.org/10.1016/j.cpc.2010.12.039
Owen, A. B., 2020. On dropping the first Sobol’ point. arXiv:2008.08051 [cs, math, stat]. Available at: http://arxiv.org/abs/2008.08051 (Accessed: 20 April 2021).
Saltelli, A., 2002. Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications 145, 280-297. https://doi.org/10.1016/S0010-4655(02)00280-1
Sobol’, I.M., 2001. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, The Second IMACS Seminar on Monte Carlo Methods 55, 271-280. https://doi.org/10.1016/S0378-4754(00)00270-6
- SALib.analyze.sobol.analyze(problem, Y, calc_second_order=True, num_resamples=100, conf_level=0.95, print_to_console=False, parallel=False, n_processors=None, keep_resamples=False, seed=None)[source]
Perform Sobol Analysis on model outputs.
Returns a dictionary with keys ‘S1’, ‘S1_conf’, ‘ST’, and ‘ST_conf’, where each entry is a list of size D (the number of parameters) containing the indices in the same order as the parameter file. If calc_second_order is True, the dictionary also contains keys ‘S2’ and ‘S2_conf’.
Notes
- Compatible with:
saltelli :
SALib.sample.saltelli.sample()
sobol :SALib.sample.sobol.sample()
Examples
>>> X = saltelli.sample(problem, 512) >>> Y = Ishigami.evaluate(X) >>> Si = sobol.analyze(problem, Y, print_to_console=True)
- Parameters
problem (dict) – The problem definition
Y (numpy.array) – A NumPy array containing the model outputs
calc_second_order (bool) – Calculate second-order sensitivities (default True)
num_resamples (int) – The number of resamples (default 100)
conf_level (float) – The confidence interval level (default 0.95)
print_to_console (bool) – Print results directly to console (default False)
parallel (bool) – Perform analysis in parallel if True
n_processors (int) – Number of parallel processes (only used if parallel is True)
keep_resamples (bool) – Whether or not to store intermediate resampling results (default False)
seed (int) – Seed to generate a random number
References
Sobol, I. M. (2001). Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Mathematics and Computers in Simulation, 55(1-3):271-280, doi:10.1016/S0378-4754(00)00270-6.
Saltelli, A. (2002). Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications, 145(2):280-297 doi:10.1016/S0010-4655(02)00280-1.
Saltelli, A., P. Annoni, I. Azzini, F. Campolongo, M. Ratto, and S. Tarantola (2010). Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Computer Physics Communications, 181(2):259-270, doi:10.1016/j.cpc.2009.09.018.
Delta Moment-Independent Measure#
- SALib.analyze.delta.analyze(problem: Dict, X: ndarray, Y: ndarray, num_resamples: int = 100, conf_level: float = 0.95, print_to_console: bool = False, seed: int = None) Dict [source]
Perform Delta Moment-Independent Analysis on model outputs.
Returns a dictionary with keys ‘delta’, ‘delta_conf’, ‘S1’, and ‘S1_conf’, where each entry is a list of size D (the number of parameters) containing the indices in the same order as the parameter file.
Notes
- Compatible with:
all samplers
Examples
>>> X = latin.sample(problem, 1000) >>> Y = Ishigami.evaluate(X) >>> Si = delta.analyze(problem, X, Y, print_to_console=True)
- Parameters
problem (dict) – The problem definition
X (numpy.matrix) – A NumPy matrix containing the model inputs
Y (numpy.array) – A NumPy array containing the model outputs
num_resamples (int) – The number of resamples when computing confidence intervals (default 100)
conf_level (float) – The confidence interval level (default 0.95)
print_to_console (bool) – Print results directly to console (default False)
References
- Borgonovo, E. (2007). “A new uncertainty importance measure.”
Reliability Engineering & System Safety, 92(6):771-784, doi:10.1016/j.ress.2006.04.015.
- Plischke, E., E. Borgonovo, and C. L. Smith (2013). “Global
sensitivity measures from given data.” European Journal of Operational Research, 226(3):536-550, doi:10.1016/j.ejor.2012.11.047.
Derivative-based Global Sensitivity Measure (DGSM)#
- SALib.analyze.dgsm.analyze(problem, X, Y, num_resamples=100, conf_level=0.95, print_to_console=False, seed=None)[source]
Calculates Derivative-based Global Sensitivity Measure on model outputs.
Returns a dictionary with keys ‘vi’, ‘vi_std’, ‘dgsm’, and ‘dgsm_conf’, where each entry is a list of size D (the number of parameters) containing the indices in the same order as the parameter file.
Notes
- Compatible with:
finite_diff :
SALib.sample.finite_diff.sample()
Examples
>>> X = finite_diff.sample(problem, 1000) >>> Y = Ishigami.evaluate(X) >>> Si = dgsm.analyze(problem, Y, print_to_console=False)
- Parameters
problem (dict) – The problem definition
X (numpy.matrix) – The NumPy matrix containing the model inputs
Y (numpy.array) – The NumPy array containing the model outputs
num_resamples (int) – The number of resamples used to compute the confidence intervals (default 1000)
conf_level (float) – The confidence interval level (default 0.95)
print_to_console (bool) – Print results directly to console (default False)
seed (int) – Seed to generate a random number
References
- Sobol, I. M. and S. Kucherenko (2009). “Derivative based global
sensitivity measures and their link with global sensitivity indices.” Mathematics and Computers in Simulation, 79(10):3009-3017, doi:10.1016/j.matcom.2009.01.023.
Fractional Factorial#
- SALib.sample.ff.sample(problem, seed=None)[source]
Generates model inputs using a fractional factorial sample.
Returns a NumPy matrix containing the model inputs required for a fractional factorial analysis. The resulting matrix has D columns, where D is smallest power of 2 that is greater than the number of parameters. These model inputs are intended to be used with
SALib.analyze.ff.analyze()
.The problem file is padded with a number of dummy variables called
dummy_0
required for this procedure. These dummy variables can be used as a check for errors in the analyze procedure.This algorithm is an implementation of that contained in Saltelli et al [Saltelli et al. 2008]
- Parameters
- Returns
sample
- Return type
numpy.array
References
Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis: The Primer. Wiley, West Sussex, U.K. http://doi.org/10.1002/9780470725184
- SALib.analyze.ff.analyze(problem, X, Y, second_order=False, print_to_console=False, seed=None)[source]
Perform a fractional factorial analysis
Returns a dictionary with keys ‘ME’ (main effect) and ‘IE’ (interaction effect). The techniques bulks out the number of parameters with dummy parameters to the nearest 2**n. Any results involving dummy parameters could indicate a problem with the model runs.
Notes
- Compatible with:
Examples
>>> X = sample(problem) >>> Y = X[:, 0] + (0.1 * X[:, 1]) + ((1.2 * X[:, 2]) * (0.2 + X[:, 0])) >>> analyze(problem, X, Y, second_order=True, print_to_console=True)
- Parameters
problem (dict) – The problem definition
X (numpy.matrix) – The NumPy matrix containing the model inputs
Y (numpy.array) – The NumPy array containing the model outputs
second_order (bool, default=False) – Include interaction effects
print_to_console (bool, default=False) – Print results directly to console
seed (int) – Seed to generate a random number
- Returns
Si – A dictionary of sensitivity indices, including main effects
ME
, and interaction effectsIE
(ifsecond_order
is True)- Return type
References
- Saltelli, A., Ratto, M., Andres, T., Campolongo, F.,
Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S., 2008. Global Sensitivity Analysis: The Primer. Wiley, West Sussex, U.K. http://doi.org/10.1002/9780470725184
PAWN Sensitivity Analysis#
- SALib.analyze.pawn.analyze(problem: Dict, X: ndarray, Y: ndarray, S: int = 10, print_to_console: bool = False, seed: int = None)[source]
Performs PAWN sensitivity analysis.
The PAWN method [1] is a moment-independent approach to Global Sensitivity Analysis (GSA). It is described as producing robust results at relatively low sample sizes (see [2]) for the purpose of factor ranking and screening.
The distribution of model outputs is examined rather than their variation as is typical in other common GSA approaches. The PAWN method further distinguishes itself from other moment-independent approaches by characterizing outputs by their cumulative distribution function (CDF) as opposed to their probability distribution function. As the CDF for a given random variable is typically normally distributed, PAWN can be more appropriately applied when outputs are highly-skewed or multi-modal, for which variance-based methods may produce unreliable results.
PAWN characterizes the relationship between inputs and outputs by quantifying the variation in the output distributions after conditioning an input. A factor is deemed non-influential if distributions coincide at all
S
conditioning intervals. The Kolmogorov-Smirnov statistic is used as a measure of distance between the distributions.This implementation reports the PAWN index at the min, mean, median, and max across the slides/conditioning intervals as well as the coefficient of variation (
CV
). The median value is the typically reported value. As theCV
is (standard deviation / mean), it indicates the level of variability across the slides, with values closer to zero indicating lower variation.Notes
- Compatible with:
all samplers
This implementation ignores all NaNs.
When applied to grouped factors, the analysis is conducted on each factor individually, and the mean of their results are reported.
Examples
>>> X = latin.sample(problem, 1000) >>> Y = Ishigami.evaluate(X) >>> Si = pawn.analyze(problem, X, Y, S=10, print_to_console=False)
- Parameters
problem (dict) – The problem definition
X (numpy.array) – A NumPy array containing the model inputs
Y (numpy.array) – A NumPy array containing the model outputs
S (int) – Number of slides; the conditioning intervals (default 10)
print_to_console (bool) – Print results directly to console (default False)
seed (int) – Seed value to ensure deterministic results
References
- Pianosi, F., Wagener, T., 2015.
A simple and efficient method for global sensitivity analysis based on cumulative distribution functions. Environmental Modelling & Software 67, 1-11. https://doi.org/10.1016/j.envsoft.2015.01.004
- Pianosi, F., Wagener, T., 2018.
Distribution-based sensitivity analysis from a generic input-output sample. Environmental Modelling & Software 108, 197-207. https://doi.org/10.1016/j.envsoft.2018.07.019
- Baroni, G., Francke, T., 2020.
An effective strategy for combining variance- and distribution-based global sensitivity analysis. Environmental Modelling & Software, 134, 104851. https://doi.org/10.1016/j.envsoft.2020.104851
- Baroni, G., Francke, T., 2020.
GSA-cvd Combining variance- and distribution-based global sensitivity analysis baronig/GSA-cvd
High-Dimensional Model Representation#
- SALib.analyze.hdmr.analyze(problem: Dict, X: ndarray, Y: ndarray, maxorder: int = 2, maxiter: int = 100, m: int = 2, K: int = 20, R: int = None, alpha: float = 0.95, lambdax: float = 0.01, print_to_console: bool = False, seed: int = None) Dict [source]
Compute global sensitivity indices using the meta-modeling technique known as High-Dimensional Model Representation (HDMR).
HDMR itself is not a sensitivity analysis method but a surrogate modeling approach. It constructs a map of relationship between sets of high dimensional inputs and output system variables [1]. This I/O relation can be constructed using different basis functions (orthonormal polynomials, splines, etc.). The model decomposition can be expressed as
\[\widehat{y} = \sum_{u \subseteq \{1, 2, ..., d \}} f_u\]where \(u\) represents any subset including an empty set.
HDMR becomes extremely useful when the computational cost of obtaining sufficient Monte Carlo samples are prohibitive, as may be the case with Sobol’s method. It uses least-square regression to reduce the required number of samples and thus the number of function (model) evaluations. Another advantage of this method is that it can account for correlation among the model input. Unlike other variance-based methods, the main effects are the combination of structural (uncorrelated) and correlated contributions.
This method uses as input
a N x d matrix of N different d-vectors of model inputs (factors/parameters)
a N x 1 vector of corresponding model outputs
Notes
- Compatible with:
all samplers
Sets an emulate method allowing re-use of the emulator.
Examples
1sp = ProblemSpec({ 2 'names': ['X1', 'X2', 'X3'], 3 'bounds': [[-np.pi, np.pi]] * 3, 4 # 'groups': ['A', 'B', 'A'], 5 'outputs': ['Y'] 6}) 7 8(sp.sample_saltelli(2048) 9 .evaluate(Ishigami.evaluate) 10 .analyze_hdmr() 11) 12 13sp.emulate()
- Parameters
problem (dict) – The problem definition
X (numpy.matrix) – The NumPy matrix containing the model inputs, N rows by d columns
Y (numpy.array) – The NumPy array containing the model outputs for each row of X
maxorder (int (1-3, default: 2)) – Maximum HDMR expansion order
maxiter (int (1-1000, default: 100)) – Max iterations backfitting
m (int (2-10, default: 2)) – Number of B-spline intervals
K (int (1-100, default: 20)) – Number of bootstrap iterations
R (int (100-N/2, default: N/2)) – Number of bootstrap samples. Will be set to length of Y if K is set to 1.
alpha (float (0.5-1)) – Confidence interval F-test
lambdax (float (0-10, default: 0.01)) – Regularization term
print_to_console (bool) – Print results directly to console (default: False)
seed (bool) – Seed to generate a random number
- Returns
Si – Sa : Uncorrelated contribution of a term
Sa_conf : Confidence interval of Sa
Sb : Correlated contribution of a term
Sb_conf : Confidence interval of Sb
- STotal contribution of a particular term
Sum of Sa and Sb, representing first/second/third order sensitivity indices
S_conf : Confidence interval of S
ST : Total contribution of a particular dimension/parameter
ST_conf : Confidence interval of ST
select : Number of selection (F-Test)
- EmEmulator result set
C1: First order coefficient C2: Second order coefficient C3: Third Order coefficient
- Return type
References
Rabitz, H. and Aliş, Ö.F., “General foundations of high dimensional model representations”, Journal of Mathematical Chemistry 25, 197-233 (1999) https://doi.org/10.1023/A:1019188517934
Genyuan Li, H. Rabitz, P.E. Yelvington, O.O. Oluwole, F. Bacon, C.E. Kolb, and J. Schoendorf, “Global Sensitivity Analysis for Systems with Independent and/or Correlated Inputs”, Journal of Physical Chemistry A, Vol. 114 (19), pp. 6022 - 6032, 2010, https://doi.org/10.1021/jp9096919
Regional Sensitivity Analysis#
- SALib.analyze.rsa.analyze(problem: Dict, X: ndarray, Y: ndarray, bins: int = 20, target: str = 'Y', print_to_console: bool = False, seed: int = None)[source]
Perform Regional Sensitivity Analysis (RSA), also known as Monte Carlo Filtering.
In a usual RSA, a desirable region of output space is defined. Outputs which fall within this region is categorized as being “behavioral” (\(B\)), and those outside are described as being “non-behavioral” (\(\bar{B}\)). The input factors are also partitioned into behavioral and non-behavioral subsets, such that \(f(X_{i}|B) \rightarrow (Y|B)\) and \(f(X_{i}|\bar{B}) \rightarrow (Y|\bar{B})\). The distribution between the two sub-samples are compared for each factor. The greater the difference between the two distributions, the more important the given factor is in driving model outputs.
The approach implemented in SALib partitions factor or output space into \(b\) bins (default: 20) according to their percentile values. Output space is targeted for analysis by default (
target="Y"
), such that \((Y|b_{i})\) is mapped back to \((X_{i}|b_{i})\). In other words, we treat outputs falling within a given bin (\(b_{i}\)) corresponding to their inputs as behavioral, and those outside the bin as non-behavioral. This aids in answering the question “Which \(X_{i}\) contributes most toward a given range of outputs?”. Factor space can also be assessed (target="X"
), such that \(f(X_{i}|b_{i}) \rightarrow (Y|b_{i})\) and \(f(X_{i}|b_{\sim i}) \rightarrow (Y|b_{\sim i})\). This aids in answering the question “where in factor space are outputs most sensitive to?”The $k$-sample Anderson-Darling test is used to compare distributions. Results of the analysis are normalized so that values will be \(\in [0, 1]\), and indicate relative sensitivity across factor/output space. Larger values indicate greater dissimilarity (thus, sensitivity).
Notes
- Compatible with:
all samplers
When applied to grouped factors, the analysis is conducted on each factor individually, and the mean of the results for a group are reported.
Increasing the value of \(S\) increases the granularity of the analysis (across factor space), but necessitates larger sample sizes.
This analysis will produce NaNs, indicating areas of factor space that did not have any samples, or for which the outputs were constant.
- Parameters
problem (dict) – The problem definition
X (numpy.array) – A NumPy array containing the model inputs
Y (numpy.array) – A NumPy array containing the model outputs
bins (int) – The number of bins to use (default: 20)
target (str) – Assess factor space (“X”) or output space (“Y”) (default: “Y”)
print_to_console (bool) – Print results directly to console (default False)
seed (int) – Seed value to ensure deterministic results Unused, but defined to maintain compatibility.
References
1. Pianosi, F., K. Beven, J. Freer, J. W. Hall, J. Rougier, D. B. Stephenson, and T. Wagener. 2016. Sensitivity analysis of environmental models: A systematic review with practical workflow. Environmental Modelling & Software 79:214-232. https://dx.doi.org/10.1016/j.envsoft.2016.02.008
2. Saltelli, A., M. Ratto, T. Andres, F. Campolongo, J. Cariboni, D. Gatelli, M. Saisana, and S. Tarantola. 2008. Global Sensitivity Analysis: The Primer. Wiley, West Sussex, U.K. https://dx.doi.org/10.1002/9780470725184 Accessible at: http://www.andreasaltelli.eu/file/repository/Primer_Corrected_2022.pdf