Advanced Examples

Group sampling (Sobol and Morris methods only)

It is sometimes useful to perform sensitivity analysis on groups of input variables to reduce the number of model runs required, when variables belong to the same component of a model, or there is some reason to believe that they should behave similarly.

Groups can be specified in two ways for the Sobol and Morris methods.

First, as a fourth column in the parameter file:

# name lower_bound upper_bound group_name
P1 0.0 1.0 Group_1
P2 0.0 5.0 Group_2
P3 0.0 5.0 Group_2
...etc.

Or in the problem dictionary:

problem = {
    'groups': ['Group_1', 'Group_2', 'Group_2'],
    'names': ['x1', 'x2', 'x3'],
    'num_vars': 3,
    'bounds': [[-3.14, 3.14], [-3.14, 3.14], [-3.14, 3.14]]
}

groups is a list of strings specifying the group name to which each variable belongs. The rest of the code stays the same:

param_values = saltelli.sample(problem, 1024)
Y = Ishigami.evaluate(param_values)
Si = sobol.analyze(problem, Y, print_to_console=True)

But the output is printed by group:

            ST   ST_conf
Group_1  0.555309  0.084058
Group_2  0.684332  0.057449
            S1   S1_conf
Group_1  0.316696  0.056797
Group_2  0.456497  0.079049
                        S2   S2_conf
(Group_1, Group_2)  0.238909  0.127195

The output can then be converted to a Pandas DataFrame for further analysis.

total_Si, first_Si, second_Si = Si.to_df()

Generating alternate distributions

In Essential basic functionality, we generate a uniform sample of parameter space.

from SALib.sample import saltelli
from SALib.analyze import sobol
from SALib.test_functions import Ishigami
import numpy as np

problem = {
    'num_vars': 3,
    'names': ['x1', 'x2', 'x3'],
    'bounds': [[-3.14159265359, 3.14159265359],
               [-3.14159265359, 3.14159265359],
               [-3.14159265359, 3.14159265359]]
}

param_values = saltelli.sample(problem, 1024)

SALib is also capable of generating alternate sampling distributions by specifying a dists entry in the problem specification.

As implied in the basic example, a uniform distribution is the default.

When an entry for dists is not ‘unif’, the bounds entry does not indicate parameter bounds but sample-specific metadata.

bounds definitions for available distributions:

  • unif: uniform distribution

    e.g. [-np.pi, np.pi] defines the lower and upper bounds

  • triang: triangular with width (scale) and location of peak.

    Location of peak is in percentage of width. Lower bound assumed to be zero.

    e.g. [3, 0.5] assumes 0 to 3, with a peak at 1.5

  • norm: normal distribution with mean and standard deviation

  • lognorm: lognormal with ln-space mean and standard deviation

An example specification is shown below:

problem = {
    'names': ['x1', 'x2', 'x3'],
    'num_vars': 3,
    'bounds': [[-np.pi, np.pi], [1.0, 0.2], [3, 0.5]],
    'groups': ['G1', 'G2', 'G1'],
    'dists': ['unif', 'lognorm', 'triang']
}