Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships

One of the main goals of current systems neuroscience is to understand how neuronal populations integrate sensory information to inform behavior. However, estimating stimulus or behavioral information that is encoded in high-dimensional neuronal populations is challenging. We propose a method based on parametric copulas which allows modeling joint distributions of neuronal and behavioral variables characterized by different statistics and timescales. To account for temporal or spatial changes in dependencies between variables, we model varying copula parameters by means of Gaussian Processes (GP). We validate the resulting Copula-GP framework on synthetic data and on neuronal and behavioral recordings obtained in awake mice. We show that the use of a parametric description of the high-dimensional dependence structure in our method provides better accuracy in mutual information estimation in higher dimensions compared to other non-parametric methods. Moreover, by quantifying the redundancy between neuronal and behavioral variables, our model exposed the location of the reward zone in an unsupervised manner (i.e., without using any explicit cues about the task structure). These results demonstrate that the Copula-GP framework is particularly useful for the analysis of complex multidimensional relationships between neuronal, sensory and behavioral variables

Probabilistic forecasts of the distribution grid state using data-driven forecasts and probabilistic power flow

The uncertainty associated with renewable energies creates challenges in the operation of distribution grids. One way for Distribution System Operators to deal with this is the computation of probabilistic forecasts of the full state of the grid. Recently, probabilistic forecasts have seen increased interest for quantifying the uncertainty of renewable generation and load. However, individual probabilistic forecasts of the state defining variables do not allow the prediction of the probability of joint events, for instance, the probability of two line flows exceeding their limits simultaneously. To overcome the issue of estimating the probability of joint events, we present an approach that combines data-driven probabilistic forecasts (obtained more specifically with quantile regressions) and probabilistic power flow. Moreover, we test the presented method using data from a real-world distribution grid that is part of the Energy Lab 2.0 of the Karlsruhe Institute of Technology and we implement it within a state-of-the-art computational framework

Appropriate, accessible and appealing probabilistic graphical models

Appropriate - Many multivariate probabilistic models either use independent distributions or dependent Gaussian distributions. Yet, many real-world datasets contain count-valued or non-negative skewed data, e.g. bag-of-words text data and biological sequencing data. Thus, we develop novel probabilistic graphical models for use on count-valued and non-negative data including Poisson graphical models and multinomial graphical models. We develop one generalization that allows for triple-wise or k-wise graphical models going beyond the normal pairwise formulation. Furthermore, we also explore Gaussian-copula graphical models and derive closed-form solutions for the conditional distributions and marginal distributions (both before and after conditioning). Finally, we derive mixture and admixture, or topic model, generalizations of these graphical models to introduce more power and interpretability. Accessible - Previous multivariate models, especially related to text data, often have complex dependencies without a closed form and require complex inference algorithms that have limited theoretical justification. For example, hierarchical Bayesian models often require marginalizing over many latent variables. We show that our novel graphical models (even the k-wise interaction models) have simple and intuitive estimation procedures based on node-wise regressions that likely have similar theoretical guarantees as previous work in graphical models. For the copula-based graphical models, we show that simple approximations could still provide useful models; these copula models also come with closed-form conditional and marginal distributions, which make them amenable to exploratory inspection and manipulation. The parameters of these models are easy to interpret and thus may be accessible to a wide audience. Appealing - High-level visualization and interpretation of graphical models with even 100 variables has often been difficult even for a graphical model expert---despite visualization being one of the original motivators for graphical models. This difficulty is likely due to the lack of collaboration between graphical model experts and visualization experts. To begin bridging this gap, we develop a novel "what if?" interaction that manipulates and leverages the probabilistic power of graphical models. Our approach defines: the probabilistic mechanism via conditional probability; the query language to map text input to a conditional probability query; and the formal underlying probabilistic model. We then propose to visualize these query-specific probabilistic graphical models by combining the intuitiveness of force-directed layouts with the beauty and readability of word clouds, which pack many words into valuable screen space while ensuring words do not overlap via pixel-level collision detection. Although both the force-directed layout and the pixel-level packing problems are challenging in their own right, we approximate both simultaneously via adaptive simulated annealing starting from careful initialization. For visualizing mixture distributions, we also design a meaningful mapping from the properties of the mixture distribution to a color in the perceptually uniform CIELUV color space. Finally, we demonstrate our approach via illustrative visualizations of several real-world datasets.Computer Science

Data-driven scenario generation for two-stage stochastic programming

Optimisation under uncertainty has always been a focal point within the Process Systems Engineering (PSE) research agenda. In particular, the efficient manipulation of large amount of data for the uncertain parameters constitutes a crucial condition for effectively tackling stochastic programming problems. In this context, this work proposes a new data-driven Mixed-Integer Linear Programming (MILP) model for the Distribution & Moment Matching Problem (DMP). For cases with multiple uncertain parameters a copula-based simulation of initial scenarios is employed as preliminary step. Moreover, the integration of clustering methods and DMP in the proposed model is shown to enhance computational performance. Finally, we compare the proposed approach with state-of-the-art scenario generation methodologies. Through a number of case studies we highlight the benefits regarding the quality of the generated scenario trees by evaluating the corresponding obtained stochastic solutions

Bayesian Structural Learning with Parametric Marginals for Count Data: An Application to Microbiota Systems

High dimensional and heterogeneous count data are collected in various applied fields. In this paper, we look closely at high-resolution sequencing data on the microbiome, which have enabled researchers to study the genomes of entire microbial communities. Revealing the underlying interactions between these communities is of vital importance to learn how microbes influence human health. To perform structural learning from multivariate count data such as these, we develop a novel Gaussian copula graphical model with two key elements. Firstly, we employ parametric regression to characterize the marginal distributions. This step is crucial for accommodating the impact of external covariates. Neglecting this adjustment could potentially introduce distortions in the inference of the underlying network of dependences. Secondly, we advance a Bayesian structure learning framework, based on a computationally efficient search algorithm that is suited to high dimensionality. The approach returns simultaneous inference of the marginal effects and of the dependence structure, including graph uncertainty estimates. A simulation study and a real data analysis of microbiome data highlight the applicability of the proposed approach at inferring networks from multivariate count data in general, and its relevance to microbiome analyses in particular. The proposed method is implemented in the R package BDgraph