SALib.analyze package

Submodules

SALib.analyze.common_args module

SALib.analyze.common_args.create(cli_parser=None)[source]
SALib.analyze.common_args.run_cli(cli_parser, run_analysis, known_args=None)[source]
SALib.analyze.common_args.setup(parser)[source]

SALib.analyze.delta module

SALib.analyze.delta.analyze(problem: Dict, X: numpy.ndarray, Y: numpy.ndarray, num_resamples: int = 100, conf_level: float = 0.95, print_to_console: bool = False, seed: Optional[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

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 10)

  • conf_level (float) – The confidence interval level (default 0.95)

  • print_to_console (bool) – Print results directly to console (default False)

References

1

Borgonovo, E. (2007). “A new uncertainty importance measure.” Reliability Engineering & System Safety, 92(6):771-784, doi:10.1016/j.ress.2006.04.015.

2

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.

Examples

>>> X = latin.sample(problem, 1000)
>>> Y = Ishigami.evaluate(X)
>>> Si = delta.analyze(problem, X, Y, print_to_console=True)
SALib.analyze.delta.bias_reduced_delta(Y, Ygrid, X, m, num_resamples, conf_level)[source]

Plischke et al. 2013 bias reduction technique (eqn 30)

SALib.analyze.delta.calc_delta(Y, Ygrid, X, m)[source]

Plischke et al. (2013) delta index estimator (eqn 26) for d_hat.

SALib.analyze.delta.cli_action(args)[source]
SALib.analyze.delta.cli_parse(parser)[source]
SALib.analyze.delta.sobol_first(Y, X, m)[source]
SALib.analyze.delta.sobol_first_conf(Y, X, m, num_resamples, conf_level)[source]

SALib.analyze.dgsm module

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()

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)

References

1

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.

Examples

>>> X = finite_diff.sample(problem, 1000)
>>> Y = Ishigami.evaluate(X)
>>> Si = dgsm.analyze(problem, Y, print_to_console=False)
SALib.analyze.dgsm.calc_dgsm(base, perturbed, x_delta, bounds, num_resamples, conf_level)[source]

v_i sensitivity measure following Sobol and Kucherenko (2009). For comparison, total order S_tot <= dgsm

SALib.analyze.dgsm.calc_vi_mean(base, perturbed, x_delta)[source]

Calculate v_i mean.

Same as calc_vi_stats but only returns the mean.

SALib.analyze.dgsm.calc_vi_stats(base, perturbed, x_delta)[source]

Calculate v_i mean and std.

v_i sensitivity measure following Sobol and Kucherenko (2009) For comparison, Morris mu* < sqrt(v_i)

Same as calc_vi_mean but returns standard deviation as well.

SALib.analyze.dgsm.cli_action(args)[source]
SALib.analyze.dgsm.cli_parse(parser)[source]

SALib.analyze.fast module

SALib.analyze.fast.analyze(problem, Y, M=4, num_resamples=100, conf_level=0.95, print_to_console=False, seed=None)[source]

Performs the extended Fourier Amplitude Sensitivity Test (eFAST) 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()

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)

References

1

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.

2

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.

3

Pujol, G. (2006) fast99 - R sensitivity package https://github.com/cran/sensitivity/blob/master/R/fast99.R

Examples

>>> X = fast_sampler.sample(problem, 1000)
>>> Y = Ishigami.evaluate(X)
>>> Si = fast.analyze(problem, Y, print_to_console=False)
SALib.analyze.fast.bootstrap(Y, N, M, omega_0, resamples, conf_level)[source]
SALib.analyze.fast.cli_action(args)[source]
SALib.analyze.fast.cli_parse(parser)[source]

Add method specific options to CLI parser.

Parameters

parser (argparse object) –

Returns

Return type

Updated argparse object

SALib.analyze.fast.compute_orders(outputs, N, M, omega)[source]

SALib.analyze.ff module

Created on 30 Jun 2015

@author: will2

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:

ff : SALib.sample.ff.sample()

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

Returns

Si – A dictionary of sensitivity indices, including main effects ME, and interaction effects IE (if second_order is True)

Return type

dict

References

1

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. https://dx.doi.org/10.1002/9780470725184

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)
SALib.analyze.ff.cli_action(args)[source]
SALib.analyze.ff.cli_parse(parser)[source]
SALib.analyze.ff.interactions(problem, Y)[source]

Computes the second order effects

Computes the second order effects (interactions) between all combinations of pairs of input factors

Parameters

  • problem (dict) – The problem definition

  • Y (numpy.array) – The NumPy array containing the model outputs

Returns

  • ie_names (list) – The names of the interaction pairs

  • IE (list) – The sensitivity indices for the pairwise interactions

SALib.analyze.ff.to_df(self)[source]

Conversion method to Pandas DataFrame. To be attached to ResultDict.

Returns

main_effect, inter_effect – A tuple of DataFrames for main effects and interaction effects. The second element (for interactions) will be None if not available.

Return type

tuple

SALib.analyze.hdmr module

SALib.analyze.hdmr.analyze(problem: Dict, X: numpy.ndarray, Y: numpy.ndarray, maxorder: int = 2, maxiter: int = 100, m: int = 2, K: int = 20, R: Optional[int] = None, alpha: float = 0.95, lambdax: float = 0.01, print_to_console: bool = True, seed: Optional[int] = None) → Dict[source]

High-Dimensional Model Representation (HDMR) using B-spline functions.

HDMR is used for variance-based global sensitivity analysis (GSA) with correlated and uncorrelated inputs. This function 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

Returns: - each factor’s first, second, and third order sensitivity coefficient

(separated in total, structural and correlative contributions),

  • an estimate of their 95% confidence intervals (from bootstrap method)

  • the coefficients of the significant B-spline basis functions that govern output,

  • Y (determined by an F-test of the error residuals of the HDMR model (emulator) with/without a given first, second and/or third order B-spline). These coefficients define an emulator that can be used to predict the output, Y, of the original (CPU-intensive) model for any d-vector of model inputs. For uncorrelated model inputs (columns of X are independent), the HDMR sensitivity indices reduce to a single index (= structural contribution), consistent with their values derived from commonly used variance-based GSA methods.

Notes

Compatible with:

all samplers

Contributed by @sahin-abdullah (sahina@uci.edu)

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) – Set a seed value

Returns

Si – Sa: Uncorrelated contribution Sa_conf: Confidence interval of Sa Sb: Correlated contribution Sb_conf: Confidence interval of Sb S: Total contribution of a particular term S_conf: Confidence interval of S ST: Total contribution of a particular dimension/parameter ST_conf: Confidence interval of ST Sa: Uncorrelated contribution select: Number of selection (F-Test) Em: Result set

C1: First order coefficient C2: Second order coefficient C3: Third Order coefficient

Return type

ResultDict,

References

1

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

Examples

>>> X = saltelli.sample(problem, 512)
>>> Y = Ishigami.evaluate(X)
>>> Si = hdmr.analyze(problem, X, Y, **options)
SALib.analyze.hdmr.cli_action(args)[source]
SALib.analyze.hdmr.cli_parse(parser)[source]

SALib.analyze.morris module

SALib.analyze.morris.analyze(problem: Dict, X: numpy.ndarray, Y: numpy.ndarray, num_resamples: int = 100, conf_level: float = 0.95, print_to_console: bool = False, num_levels: int = 4, seed=None) → numpy.ndarray[source]

Perform Morris Analysis on model outputs.

Returns a dictionary with keys ‘mu’, ‘mu_star’, ‘sigma’, and ‘mu_star_conf’, where each entry is a list of parameters containing the indices in the same order as the parameter file.

Notes

Compatible with:

morris : SALib.sample.morris.sample()

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

dict

References

1

Morris, M. (1991). “Factorial Sampling Plans for Preliminary Computational Experiments.” Technometrics, 33(2):161-174, doi:10.1080/00401706.1991.10484804.

2

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.

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)
SALib.analyze.morris.cli_action(args)[source]
SALib.analyze.morris.cli_parse(parser)[source]

SALib.analyze.pawn module

SALib.analyze.pawn.analyze(problem: Dict, X: numpy.ndarray, Y: numpy.ndarray, S: int = 10, print_to_console: bool = False, seed: Optional[int] = None)[source]

Performs PAWN sensitivity analysis.

Calculates the min, mean, median, max, and coefficient of variation (CV).

CV is (standard deviation / mean), and so lower values indicate little change over the slides, and larger values indicate large variations across the slides.

Notes

Compatible with:

all samplers

This implementation ignores all NaNs.

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 (default 10)

  • print_to_console (bool) – Print results directly to console (default False)

  • seed (int) – Seed value to ensure deterministic results

References

1

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

2

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

3

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

4

Baroni, G., Francke, T., 2020. GSA-cvd Combining variance- and distribution-based global sensitivity analysis https://github.com/baronig/GSA-cvd

Examples

>>> X = latin.sample(problem, 1000)
>>> Y = Ishigami.evaluate(X)
>>> Si = pawn.analyze(problem, X, Y, S=10, print_to_console=False)
SALib.analyze.pawn.cli_action(args)[source]
SALib.analyze.pawn.cli_parse(parser)[source]

SALib.analyze.rbd_fast module

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

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)

References

1

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

2

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

3

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

4

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

Examples

>>> X = latin.sample(problem, 1000)
>>> Y = Ishigami.evaluate(X)
>>> Si = rbd_fast.analyze(problem, X, Y, print_to_console=False)
SALib.analyze.rbd_fast.bootstrap(X_d, Y, M, resamples, conf_level)[source]
SALib.analyze.rbd_fast.cli_action(args)[source]
SALib.analyze.rbd_fast.cli_parse(parser)[source]
SALib.analyze.rbd_fast.compute_first_order(permuted_outputs, M)[source]
SALib.analyze.rbd_fast.permute_outputs(X, Y)[source]

Permute the output according to one of the inputs as in [_2]

References

2

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

SALib.analyze.rbd_fast.unskew_S1(S1, M, N)[source]

Unskew the sensivity indice (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)

SALib.analyze.sobol module

SALib.analyze.sobol.Si_list_to_dict(S_list, D: int, num_resamples: int, keep_resamples: bool, calc_second_order: bool)[source]

Convert the parallel output into the regular dict format for printing/returning

SALib.analyze.sobol.Si_to_pandas_dict(S_dict)[source]

Convert Si information into Pandas DataFrame compatible dict.

Parameters

S_dict (ResultDict) – Sobol sensitivity indices

See also

Si_list_to_dict

Returns

tuple – Total and first order are dicts. Second order sensitivities contain a tuple of parameter name combinations for use as the DataFrame index and second order sensitivities. If no second order indices found, then returns tuple of (None, None)

Return type

of total, first, and second order sensitivities.

Examples

>>> X = saltelli.sample(problem, 512)
>>> Y = Ishigami.evaluate(X)
>>> Si = sobol.analyze(problem, Y, print_to_console=True)
>>> T_Si, first_Si, (idx, second_Si) = sobol.Si_to_pandas_dict(Si, problem)
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()

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)

  • keep_resamples (bool) – Whether or not to store intermediate resampling results (default False)

References

1

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.

2

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.

3

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.

Examples

>>> X = saltelli.sample(problem, 512)
>>> Y = Ishigami.evaluate(X)
>>> Si = sobol.analyze(problem, Y, print_to_console=True)
SALib.analyze.sobol.cli_action(args)[source]
SALib.analyze.sobol.cli_parse(parser)[source]
SALib.analyze.sobol.create_Si_dict(D: int, num_resamples: int, keep_resamples: bool, calc_second_order: bool)[source]

initialize empty dict to store sensitivity indices

SALib.analyze.sobol.create_task_list(D, calc_second_order, n_processors)[source]

Create list with one entry (key, parameter 1, parameter 2) per sobol index (+conf.). This is used to supply parallel tasks to multiprocessing.Pool

SALib.analyze.sobol.first_order(A, AB, B)[source]

First order estimator following Saltelli et al. 2010 CPC, normalized by sample variance

SALib.analyze.sobol.second_order(A, ABj, ABk, BAj, B)[source]

Second order estimator following Saltelli 2002

SALib.analyze.sobol.separate_output_values(Y, D, N, calc_second_order)[source]
SALib.analyze.sobol.sobol_parallel(Z, A, AB, BA, B, r, tasks)[source]
SALib.analyze.sobol.to_df(self)[source]

Conversion method to Pandas DataFrame. To be attached to ResultDict.

Returns

List

Return type

of Pandas DataFrames in order of Total, First, Second

Example

>>> Si = sobol.analyze(problem, Y, print_to_console=True)
>>> total_Si, first_Si, second_Si = Si.to_df()
SALib.analyze.sobol.total_order(A, AB, B)[source]

Total order estimator following Saltelli et al. 2010 CPC, normalized by sample variance

Module contents