Time series generated from a dynamical source can often be modeled as sample paths of a stochastic differential equation. The time series thus reflects the motion of a particle that flows along the direction provided by a drift/vector field, and is simultaneously scattered by the effect of white noise. The resulting motion can only be described as a random process instead of a solution curve. Due to the non-deterministic nature of this motion, the task of determining the drift from data is quite challenging, since the data does not directly represent the directional information of the flow. This paper describes an interpretation of a drift as a conditional expectation, which makes its estimation feasible via kernel-integral methods. In addition, some techniques are proposed to overcome the challenge of dimensionality if the stochastic differential equations carry some structure enabling sparsity. The technique is shown to be convergent, consistent and permits a wide choice of kernels.

Machine learning models can provide valuable decision support in many real-world applications. However, a model must be interpretable to those using it. This paper explores the use of post-hoc model interpretability methods in combination with an intrinsically interpretable model design to create a model that is interpretable to both a model designer and a model end user. A hierarchical fuzzy rule-based model is trained with a genetic algorithm on a real-world shipbreaking use case and the performance-interpretability trade-off of the model with respect to a random forest model is discussed. Further, an interesting pattern was found using the post-hoc interpretability method SHapley Additive exPlanations (SHAP), with potential implications for the future design of hierarchical fuzzy rule-based models.