# Identifying the Influential Inputs for Network Output Variance Using Sparse Polynomial Chaos Expansion

## TL;DR

This paper introduces a sparse polynomial chaos expansion-based model for sensitivity analysis in network-structured input-output relationships, enabling efficient and accurate estimation of input influence on output variance with fewer observations.

## Contribution

It presents a novel method leveraging network structure for Sobol index estimation, reducing sampling requirements compared to traditional Monte Carlo methods.

## Key findings

- Accurately estimates Sobol indices with fewer samples.
- Theoretical convergence of the model is established.
- Empirical results demonstrate superior efficiency in manufacturing processes.

## Abstract

Sensitivity analysis (SA) is an important aspect of process automation. It often aims to identify the process inputs that influence the process output's variance significantly. Existing SA approaches typically consider the input-output relationship as a black-box and conduct extensive random sampling from the actual process or its high-fidelity simulation model to identify the influential inputs. In this paper, an alternate, novel approach is proposed using a sparse polynomial chaos expansion-based model for a class of input-output relationships represented as directed acyclic networks. The model exploits the relationship structure by recursively relating a network node to its direct predecessors to trace the output variance back to the inputs. It, thereby, estimates the Sobol indices, which measure the influence of each input on the output variance, accurately and efficiently. Theoretical analysis establishes the validity of the model as the prediction of the network output converges in probability to the true output under certain regularity conditions. Empirical evaluation on two manufacturing processes shows that the model estimates the Sobol indices accurately with far fewer observations than a state-of-the-art Monte Carlo sampling method.