EXAMPLE.rst

Example problem (under construction)

Say we have a physical system whose performance, y, depends on three independent variables, x₁, x₂, x₃ that we model as taking values uniformly at random in the interval [-pi, pi]. We model the performance of the system as

y = sin x₁ + a sin ² x₂ + b x₃ ⁴ sin x₁.

This is the Ishigami function.

We would like to know which variable influences the performance of the physical system the most. Why? Perhaps there's a certain range of y that is acceptable, and we would like to set the values of x₁, x₂, x₃ such that y is always in that acceptable range.

One way to understand the relative influence of each input with respect to the output is to perform a sensitivity analysis.

What is sensitivity analysis (SA)?

Saltelli et al. (2004) define sensitivity analysis as "the study of how the uncertainty in the output of a model... can be apportioned to different sources of uncertainty in the model input."

Local (or deterministic) SA is performed around a particular point of interest in the model input space. Global (or probabilistic) SA considers the entire model input space.

We will perform a global variance-based SA, the result of which is a Sobol' index for each of our three inputs.

What is a Sobol' index?

A Sobol' index approximates an exact sensitivity index that results from a global variance-based sensitivity analysis . Sobol' indices are more efficient to compute than exact sensitivity indices. The magnitude of a (normalized) Sobol' index indicates how much of the variance in an output quantity of interest can be attributed to variance in a particular input.

There are different orders of Sobol' index.

• A first-order Sobol' index indicates how much of the variance in the output can be attributed to variance in one

  particular input alone.
• A total-order Sobol' index indicates how much of the output's variance can be attributed to variance in one

  particular input, including all of that input's interactions with other inputs.
• Sobol' indices of orders between 1 and the total number of inputs indicate how much of the output's variance can

  be attributed to variance in one input, including its interactions of the specified order with other inputs.

We will focus on total-order Sobol' indices for this example.

What is an interaction?

An interaction is that part of the response of an output f(x) to the values of x_a, x_b "that cannot be expressed as a superposition of effects separately due to" x_a, x_b (Saltelli et al. 2004).

Why might I care about quantifying a component's importance?

There are four types of reasons we might want to know how influential an input is with respect to an output (Saltelli

et al. 2004):

• factors prioritisation -- by setting which factor (x₁, x :sub:2, or x₃) to a chosen value

  will we reduce the uncertainty in y the most? We might ask this question if we'd like to rank the inputs by their

  importance in terms of influencing the output.
• factors fixing -- which factors are not influential in terms of the output y? We might ask this question if

  we wanted to fix some factors without affecting the variance in y, rather than letting all of the inputs vary over

  their domains.
• variance cutting -- if we wanted to achieve a particular reduction in the variance of y, which minimal subset

  of the inputs would we need to fix to particular values to do so? We might ask such a question if we had a limited

  budget with which to reduce variance in the inputs and wanted to get the most value for our money.
• factors mapping -- which factors are responsible for realisations of y in a particular region of interest? We

  might ask such a question if we were concerned with the safety of the system under investigation, for instance, and

  wanted to understand what input values and combinations of input values lead to safe vs. dangerous performance.

The sample data set addresses a question in the factors prioritisation setting -- by reducing the fragilities of which bridges could we improve the expected road network performance the most?

References

Gitanjali Bhattacharjee and Jack W. Baker. (TBD) Using global variance-based sensitivity analysis to prioritise bridge retrofits in a regional road network subject to seismic hazard. Manuscript under review.

Andrea Saltelli, Stefano Tarantola, Francesca Campolongo, and Marco Ratto. (2004) Sensitivity Analysis in Practice: A Guide to Assessing Scientific Models. John Wiley & Sons, Ltd.

Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, Stefano Tarantola. (2008) Global Sensitivity Analysis: The Primer. John Wiley & Sons, Ltd.