ParamBoost: Gradient Boosted Piecewise Cubic Polynomials

Generalized Additive Models (GAMs) can be used to create non-linear glass-box (i.e. explicitly interpretable) models, where the predictive function is fully observable over the complete input space. However, glass-box interpretability itself does not allow for the incorporation of expert knowledge from the modeller. In this paper, we present ParamBoost, a novel GAM whose shape functions (i.e. mappings from individual input features to the output) are learnt using a Gradient Boosting algorithm that fits cubic polynomial functions at leaf nodes. ParamBoost incorporates several constraints commonly used in parametric analysis to ensure well-refined shape functions. These constraints include: (i) continuity of the shape functions and their derivatives (up to C2); (ii) monotonicity; (iii) convexity; (iv) feature interaction constraints; and (v) model specification constraints. Empirical results show that the unconstrained ParamBoost model consistently outperforms state-of-the-art GAMs across several real-world datasets. We further demonstrate that modellers can selectively impose required constraints at a modest trade-off in predictive performance, allowing the model to be fully tailored to application-specific interpretability and parametric-analysis requirements.

RUMBoost: Gradient Boosted Random Utility Models

This paper introduces the RUMBoost model, a novel discrete choice modelling approach that combines the interpretability and behavioural robustness of Random Utility Models (RUMs) with the generalisation and predictive ability of deep learning methods. We obtain the full functional form of non-linear utility specifications by replacing each linear parameter in the utility functions of a RUM with an ensemble of gradient boosted regression trees. This enables piece-wise constant utility values to be imputed for all alternatives directly from the data for any possible combination of input variables. We introduce additional constraints on the ensembles to ensure three crucial features of the utility specifications: (i) dependency of the utilities of each alternative on only the attributes of that alternative, (ii) monotonicity of marginal utilities, and (iii) an intrinsically interpretable functional form, where the exact response of the model is known throughout the entire input space. Furthermore, we introduce an optimisation-based smoothing technique that replaces the piece-wise constant utility values of alternative attributes with monotonic piece-wise cubic splines to identify non-linear parameters with defined gradient. We demonstrate the potential of the RUMBoost model compared to various ML and Random Utility benchmark models for revealed preference mode choice data from London. The results highlight the great predictive performance and the direct interpretability of our proposed approach. Furthermore, the smoothed attribute utility functions allow for the calculation of various behavioural indicators and marginal utilities. Finally, we demonstrate the flexibility of our methodology by showing how the RUMBoost model can be extended to complex model specifications, including attribute interactions, correlation within alternative error terms and heterogeneity within the population.