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Sensitivity analysis generally involve an a priori sampling of the input space and a statistical method to analyse the co-variance of the inputs and outputs of the model.

Sensitivity analysis can be done at a global or local level. Global methods provide summary statistics of the effects of inputs variation in the complete input space, whereas local methods focus the effect of inputs variation around a given point of the input space (think of a Jacobian matrix e.g.). The one factor at a time method can be viewed as a local sensitivity method, as only one factor vary, the other remaining fixed at their nominal value.

OpenMOLE implements two classical methods for global sensitivity analysis: Morris and Saltelli.

OpenMOLE Market example

## Contents

## Sensitivity analysis π

Sensitivity analysis correspond to a set of methods capturing how a model reacts to a change in its inputs. The goal of these statistical methods is to measure how variation propagates from the inputs to the outputs. More specifically, sensitivity analysis is defined by (Saltelli et al., 2008) as describing the Β«relative importance of each input in determining [output] variabilityΒ». As a consequence, typical result of such methods is an ordering of its inputs according to their sensitivity.Sensitivity analysis generally involve an a priori sampling of the input space and a statistical method to analyse the co-variance of the inputs and outputs of the model.

Sensitivity analysis can be done at a global or local level. Global methods provide summary statistics of the effects of inputs variation in the complete input space, whereas local methods focus the effect of inputs variation around a given point of the input space (think of a Jacobian matrix e.g.). The one factor at a time method can be viewed as a local sensitivity method, as only one factor vary, the other remaining fixed at their nominal value.

OpenMOLE implements two classical methods for global sensitivity analysis: Morris and Saltelli.

## Morris' method π

### Principle π

Morris method is a statistical method for global sensitivity analysis. This method is of the type "one-factor-at-a-time", and was conceived as a preliminary computational experiment, to grasp the relative influence of each factor. In comparison to LHS screening, it has the advantage to provide information for each factor. The input space is considered as a grid and trajectories are sampled among these points. The method captures output variation when one of the trajectory points is moved to one of its closest neighbors. This variation is called an elementary effect. A certain number of trajectories are generated, in order to observe the consequence of elementary effects anywhere in the input space (trajectories are generated such that given a starting point, any point at fixed distance is equiprobable - note that the method is still subject to the curse of dimensionality for trajectories to fill the input space). Finally, the method summarizes these elementary effect to estimate global sensitivity in the output space.### Results and Interpretation π

Morris' method computes three sensitivity indicators for each model input:- the final sensitivity measure,
**mu**is the average of the elementary effects along the sampled trajectories. It is interpreted as the overall influence of the input_{i}**i**on the model output variability, - a more advanced version of the final sensitivity measure,
**mu**, is computed with the absolute value of the elementary effects, granting robustness against non-monotonic models. It is interpreted the same way as^{*}_{i}**mu**,_{i} -
**sigma**is interpreted as the extent of interaction and non-linear effects of each input. It is based on the variance of previous_{i}**mu**measures._{i}

## Morris' method within OpenMOLE π

### Specific constructor π

The`SensitivityMorris`

constructor is defined in OpenMOLE and can take the following parameters:
`evaluation`

is the task (or a composition of tasks) that uses your inputs, typically your model task,`inputs`

is the list of your model's inputs,`outputs`

is the list of your model's outputs,`repetitions`

is the number of trajectories sampled,`levels`

is the resolution of relative variations (increases in variables will be in +- 1 / levels).

### Hook π

The`hook`

keyword is used to save or display results generated during the execution of a workflow.
The generic way to use it is to write either `hook(workDirectory / "path/of/a/file.csv")`

to save the results in a CSV file, or `hook display`

to display the results in the standard output.
### Use example π

Here is how you can make use of this constructor in OpenMOLE:```
SensitivityMorris(
evaluation = model,
inputs = Seq(
i1 in (0.0, 1.0),
i2 in (0.0, 1.0),
i3 in (0.0, 1.0)),
outputs = Seq(o1, o2),
repetitions = 10,
levels = 10
) hook display
```

### Additional material π

Paper describing method and its evaluation : Campolongo F, Saltelli A, Cariboni, J, 2011, From screening to quantitative sensitivity analysis. A unified approach, Computer Physics Communication. 182 4, pp. 978-988.OpenMOLE Market example

## Saltelli's method π

Saltelli is a statistical method for global sensitivity analysis. It estimates at the first order the relative variance along each input factor.## Saltelli's method within OpenMOLE π

### Specific constructor π

The`SensitivitySaltelli`

constructor is defined in OpenMOLE and can take the following parameters:
`evaluation`

is the task (or a composition of tasks) that uses your inputs, typically your model task,`inputs`

is the list of your model's inputs,`outputs`

is the list of your model's outputs,`samples`

number of samples to draw for the estimation of the relative variances.

### Hook π

The`hook`

keyword is used to save or display results generated during the execution of a workflow.
The generic way to use it is to write either `hook(workDirectory / "path/of/a/file.csv")`

to save the results in a CSV file, or `hook display`

to display the results in the standard output.
### Use example π

Here is how you can make use of this constructor in OpenMOLE:```
SensitivitySaltelli(
evaluation = model,
inputs = Seq(
i1 in (0.0, 1.0),
i2 in (0.0, 1.0),
i3 in (0.0, 1.0)),
outputs = Seq(o1, o2),
samples = 100
) hook display
```