# 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 lower and upper bounds, as well as
location of peak The location of peak is in percentage of width e.g.

`[1.0, 3.0, 0.5]`

indicates 1.0 to 3.0 with a peak at 2.0A soon-to-be deprecated two-value format assumes the lower bound to be 0 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']
}
```