Construction of sparse polynomial chaos surrogate models for simulators with mixed continuous and categorical variables

Abstract

Performing uncertainty quantification for engineering systems typically requires a large number of evaluations of the associated simulation model. Due to limited computational resources, however, such an analysis becomes intractable for expensive numerical models. In this respect, surrogate models have received tremendous attention in the last decade, as they allow one to approximate the original model by an easy-to-evaluate proxy built on a limited number of model runs. Among others, polynomial chaos expansions have been successfully developed to emulate simulators by a linear combination of a suitable basis functions. Particularly, large efforts have been made to build PCE for computational models with continuous input variables. However, how to construct a PCE for models involving categorical input variables (factors) remains an important but unresolved question. In this contribution, we encode categorical inputs as quantitative dummy variables. This allows for transforming the problem into the standard regression setup. Therefore, solvers such as ordinary least-squares can be applied with the full basis. It is well-known that PCE suffers from the curse of dimensionality: the number of unknown terms increases dramatically with the input dimension and the maximum polynomial degrees. This problem is even exacerbated when dealing with categorical variables, especially when each qualitative variable has a lot of levels. To overcome this problem, least-angle regressions (LARS) can be used to select only the most important basis functions among a candidate set. However, when applying LARS to regressions involving categorical variables, the results depend on the encoding scheme (i.e., the choice of the dummy variables). To avoid such a dependence, we propose an adaptive algorithm based on group LARS. This method features the sparsity of the resulting PCE at the group level: instead of selecting basis functions one by one, the proposed method selects a group of basis at each iteration. Such a property better reveals the effect of categories. Similar to the algorithm hybrid LAR, we use the leave-one-out cross-validation error for model selections and adaptively choose the best model. The performance of the developed algorithm is compared with LARS on several analytical examples and real case studies.