Accurate estimates of the surface solar radiation (SSR) are a prerequisite for intra-day forecasts of solar resources and photovoltaic power generation. Intra-day SSR forecasts are of interest to power traders and to operators of solar plants and power grids who seek to optimize their revenues and maintain the grid stability by matching power supply and demand. Our study analyzes systematic biases and the uncertainty of SSR estimates derived from Meteosat with the SARAH-2 and HelioMont algorithms at intra-hour and intra-day time scales. The satellite SSR estimates are analyzed based on 136 ground stations across altitudes from 200 m to 3570 m Switzerland in 2018. We find major biases and uncertainties in the instantaneous, hourly and daily-mean SSR. In peak daytime periods, the instantaneous satellite SSR deviates from the ground-measured SSR by a mean absolute deviation (MAD) of 110.4 and 99.6 W/m2 for SARAH-2 and HelioMont, respectively. For the daytime SSR, the instantaneous, hourly and daily-mean MADs amount to 91.7, 81.1, 50.8 and 82.5, 66.7, 42.9 W/m2 for SARAH-2 and HelioMont, respectively. Further, the SARAH-2 instantaneous SSR drastically underestimates the solar resources at altitudes above 1000 m in the winter half year. A possible explanation in line with the seasonality of the bias is that snow cover may be misinterpreted as clouds at higher altitudes.

Mixed-membership (MM) models such as Latent Dirichlet Allocation (LDA) have been applied to microbiome compositional data to identify latent subcommunities of microbial species. However, microbiome compositional data, especially those collected from the gut, typically display substantial cross-sample heterogeneities in the subcommunity composition which current MM methods do not account for. To address this limitation, we incorporate the logistic-tree normal (LTN) model -- using the phylogenetic tree structure -- into the LDA model to form a new MM model. This model allows variation in the composition of each subcommunity around some ``centroid'' composition. Incorporation of auxiliary P\'olya-Gamma variables enables a computationally efficient collapsed blocked Gibbs sampler to carry out Bayesian inference under this model. We compare the new model and LDA and show that in the presence of large cross-sample heterogeneity, under the LDA model the resulting inference can be extremely sensitive to the specification of the total number of subcommunities as it does not account for cross-sample heterogeneity. As such, the popular strategy in other applications of MM models of overspecifying the number of subcommunities -- and hoping that some meaningful subcommunities will emerge among artificial ones -- can lead to highly misleading conclusions in the microbiome context. In contrast, by accounting for such heterogeneity, our MM model restores the robustness of the inference in the specification of the number of subcommunities and again allows meaningful subcommunities to be identified under this strategy.

A decision tree is commonly restricted to use a single hyperplane to split the covariate space at each of its internal nodes. It often requires a large number of nodes to achieve high accuracy, hurting its interpretability. In this paper, we propose convex polytope trees (CPT) to expand the family of decision trees by an interpretable generalization of their decision boundary. The splitting function at each node of CPT is based on the logical disjunction of a community of differently weighted probabilistic linear decision-makers, which also geometrically corresponds to a convex polytope in the covariate space. We use a nonparametric Bayesian prior at each node to infer the community's size, encouraging simpler decision boundaries by shrinking the number of polytope facets. We develop a greedy method to efficiently construct CPT and scalable end-to-end training algorithms for the tree parameters when the tree structure is given. We empirically demonstrate the efficiency of CPT over existing state-of-the-art decision trees in several real-world classification and regression tasks from diverse domains.