Invariant Causal Prediction for Nonlinear Models

An important problem in many domains is to predict how a system will respond to interventions. This task is inherently linked to estimating the system's underlying causal structure. To this end, Invariant Causal Prediction (ICP) (Peters et al., 2016) has been proposed which learns a causal model exploiting the invariance of causal relations using data from different environments. When considering linear models, the implementation of ICP is relatively straightforward. However, the nonlinear case is more challenging due to the difficulty of performing nonparametric tests for conditional independence. In this work, we present and evaluate an array of methods for nonlinear and nonparametric versions of ICP for learning the causal parents of given target variables. We find that an approach which first fits a nonlinear model with data pooled over all environments and then tests for differences between the residual distributions across environments is quite robust across a large variety of simulation settings. We call this procedure "invariant residual distribution test". In general, we observe that the performance of all approaches is critically dependent on the true (unknown) causal structure and it becomes challenging to achieve high power if the parental set includes more than two variables. As a real-world example, we consider fertility rate modelling which is central to world population projections. We explore predicting the effect of hypothetical interventions using the accepted models from nonlinear ICP. The results reaffirm the previously observed central causal role of child mortality rates

Ellipsoidal Prediction Regions for Multivariate Uncertainty Characterization

While substantial advances are observed in probabilistic forecasting for power system operation and electricity market applications, most approaches are still developed in a univariate framework. This prevents from informing about the interdependence structure among locations, lead times and variables of interest. Such dependencies are key in a large share of operational problems involving renewable power generation, load and electricity prices for instance. The few methods that account for dependencies translate to sampling scenarios based on given marginals and dependence structures. However, for classes of decision-making problems based on robust, interval chance-constrained optimization, necessary inputs take the form of polyhedra or ellipsoids. Consequently, we propose a systematic framework to readily generate and evaluate ellipsoidal prediction regions, with predefined probability and minimum volume. A skill score is proposed for quantitative assessment of the quality of prediction ellipsoids. A set of experiments is used to illustrate the discrimination ability of the proposed scoring rule for misspecification of ellipsoidal prediction regions. Application results based on three datasets with wind, PV power and electricity prices, allow us to assess the skill of the resulting ellipsoidal prediction regions, in terms of calibration, sharpness and overall skill.Comment: 8 pages, 7 Figures, Submitted to IEEE Transactions on Power System