# Theoretical and Applied Data Science (TADS) Lunch-n-Learn

**Presenter**: Dr. Max Morris

**Title**: Decomposing Functional Model Inputs for Variance-Based Sensitivity Analysis

**Abstract**: Variance-based sensitivity analysis, developed by Sobol' (1993) and further elaborated by others including Saltelli, Chan and Scott (2000) is a popular technique for assessing the importance of model inputs when there are natural or meaningful probability distributions associated with each input. This approach can be used when some of the model inputs are functions rather than scalar-valued (e.g. Iooss and Ribatet, 2009, and Jacques et al., 2006), but may be somewhat less useful in this case because it does not addresss the nature of the relationships between functional inputs and model outputs. We consider the option of separating a random function-valued input, represented by a vector of relatively high dimension, into one or a few scalar-valued summaries that are suggested by the context of the modeling exercise, and an independent, high dimensional ``residual''. The first case we discuss is for inputs that are realizations of Gaussian processes, where the summary statistics are linear functionals of the input, and the residual can always be statistically independent of these. The second case is for input functions that might be described as ``pulses'' occurring in simulated time as a Poission process, where the summary statistic is the number of such pulses and all other details form the residual. The third case involves periodic input functions for which the largest Fourier coefficient is taken as the scalar-valued summary. We conclude by describing a graphical technique that may help to identify useful low-dimensional function summaries. When the model output is more sensitive to the low dimensional summaries than to the residuals, this is useful information concerning the nature of model sensitivity, and also provides a route to constructing model surrogates with scalar-valued indices that accurately represent most of the variation in the output.