Real world systems evolve in continuous-time according to their underlying causal relationships, yet their dynamics are often unknown. Existing approaches to learning such dynamics typically either discretize time ---leading to poor performance on irregularly sampled data--- or ignore the underlying causality. We propose CADYT, a novel method for causal discovery on dynamical systems addressing both these challenges. In contrast to state-of-the-art causal discovery methods that model the problem using discrete-time Dynamic Bayesian networks, our formulation is grounded in Difference-based causal models, which allow milder assumptions for modeling the continuous nature of the system. CADYT leverages exact Gaussian Process inference for modeling the continuous-time dynamics which is more aligned with the underlying dynamical process. We propose a practical instantiation that identifies the causal structure via a greedy search guided by the Algorithmic Markov Condition and Minimum Description Length principle. Our experiments show that CADYT outperforms state-of-the-art methods on both regularly and irregularly-sampled data, discovering causal networks closer to the true underlying dynamics.

Active learning of physical systems must commonly respect practical safety constraints, which restricts the exploration of the design space. Gaussian Processes (GPs) and their calibrated uncertainty estimations are widely used for this purpose. In many technical applications the design space is explored via continuous trajectories, along which the safety needs to be assessed. This is particularly challenging for strict safety requirements in GP methods, as it employs computationally expensive Monte Carlo sampling of high quantiles. We address these challenges by providing provable safety bounds based on the adaptively sampled median of the supremum of the posterior GP. Our method significantly reduces the number of samples required for estimating high safety probabilities, resulting in faster evaluation without sacrificing accuracy and exploration speed. The effectiveness of our safe active learning approach is demonstrated through extensive simulations and validated using a real-world engine example.

Learning precise surrogate models of complex computer simulations and physical machines often require long-lasting or expensive experiments. Furthermore, the modeled physical dependencies exhibit nonlinear and nonstationary behavior. Machine learning methods that are used to produce the surrogate model should therefore address these problems by providing a scheme to keep the number of queries small, e.g. by using active learning and be able to capture the nonlinear and nonstationary properties of the system. One way of modeling the nonstationarity is to induce input-partitioning, a principle that has proven to be advantageous in active learning for Gaussian processes. However, these methods either assume a known partitioning, need to introduce complex sampling schemes or rely on very simple geometries. In this work, we present a simple, yet powerful kernel family that incorporates a partitioning that: i) is learnable via gradient-based methods, ii) uses a geometry that is more flexible than previous ones, while still being applicable in the low data regime. Thus, it provides a good prior for active learning procedures. We empirically demonstrate excellent performance on various active learning tasks.

Learning the kernel parameters for Gaussian processes is often the computational bottleneck in applications such as online learning, Bayesian optimization, or active learning. Amortizing parameter inference over different datasets is a promising approach to dramatically speed up training time. However, existing methods restrict the amortized inference procedure to a fixed kernel structure. The amortization network must be redesigned manually and trained again in case a different kernel is employed, which leads to a large overhead in design time and training time. We propose amortizing kernel parameter inference over a complete kernel-structure-family rather than a fixed kernel structure. We do that via defining an amortization network over pairs of datasets and kernel structures. This enables fast kernel inference for each element in the kernel family without retraining the amortization network. As a by-product, our amortization network is able to do fast ensembling over kernel structures. In our experiments, we show drastically reduced inference time combined with competitive test performance for a large set of kernels and datasets.

Multi-output regression problems are commonly encountered in science and engineering. In particular, multi-output Gaussian processes have been emerged as a promising tool for modeling these complex systems since they can exploit the inherent correlations and provide reliable uncertainty estimates. In many applications, however, acquiring the data is expensive and safety concerns might arise (e.g. robotics, engineering). We propose a safe active learning approach for multi-output Gaussian process regression. This approach queries the most informative data or output taking the relatedness between the regressors and safety constraints into account. We prove the effectiveness of our approach by providing theoretical analysis and by demonstrating empirical results on simulated datasets and on a real-world engineering dataset. On all datasets, our approach shows improved convergence compared to its competitors.

Despite recent advances in automated machine learning, model selection is still a complex and computationally intensive process. For Gaussian processes (GPs), selecting the kernel is a crucial task, often done manually by the expert. Additionally, evaluating the model selection criteria for Gaussian processes typically scales cubically in the sample size, rendering kernel search particularly computationally expensive. We propose a novel, efficient search method through a general, structured kernel space. Previous methods solved this task via Bayesian optimization and relied on measuring the distance between GP's directly in function space to construct a kernel-kernel. We present an alternative approach by defining a kernel-kernel over the symbolic representation of the statistical hypothesis that is associated with a kernel. We empirically show that this leads to a computationally more efficient way of searching through a discrete kernel space.

In many real-world planning applications, e.g. dynamic design of experiments, autonomous driving and robot manipulation, it is necessary to evaluate candidate movement paths with respect to a safety cost function. Here, the continuous candidate paths need to be discretized first and, subsequently, evaluated onthe discretization points. The resulting quality of planned paths, thus, highly depends on the definition of the safety cost functions, and the resolution of the discretization. In this paper, we propose an approach for evaluating continuous candidate paths by employing an adaptive discretization scheme, with a probabilistic cost function learned from observations. The obtained path is then guaranteed to be epsilon-safe, i.e. the remaining risk of still finding an unsafe point on the trajectory is smaller than epsilon. The proposed approach is investigated theoretically, as well as empirically validated on several robotic path planning scenarios.

The seasonal epidemic of influenza costs thousands of lives each year in the US. While influenza epidemics occur every year, timing and size of the epidemic vary strongly from season to season. This complicates the public health efforts to adequately respond to such epidemics. Forecasting techniques to predict the development of seasonal epidemics such as influenza, are of great help to public health decision making. Therefore, the US Center for Disease Control and Prevention (CDC) has initiated a yearly challenge to forecast influenza-like illness. Here, we propose a new framework based on Gaussian process (GP) for seasonal epidemics forecasting and demonstrate its capability on the CDC reference data on influenza like illness: our framework leads to accurate forecasts with small but reliable uncertainty estimation. We compare our framework to several state of the art benchmarks and show competitive performance. We, therefore, believe that our GP based framework for seasonal epidemics forecasting will play a key role for future influenza forecasting and, lead to further research in the area.

Each year in the United States, influenza causes illness in 9.2 to 35.6 million individuals and is responsible for 12,000 to 56,000 deaths. The U.S. Centers for Disease Control and Prevention (CDC) tracks influenza activity through a national surveillance network. These data are only available after a delay of 1 to 2 weeks, and thus influenza epidemiologists and transmission modelers have explored the use of other data sources to produce more timely estimates and predictions of influenza activity. We evaluated whether data collected from a national commercial network of influenza diagnostic machines could produce valid estimates of the current burden and help to predict influenza trends in the United States. Quidel Corporation provided us with de-identified influenza test results transmitted in real-time from a national network of influenza test machines called the Influenza Test System (ITS). We used this ITS dataset to estimate and predict influenza-like illness (ILI) activity in the United States over the 2015-2016 and 2016-2017 influenza seasons. First, we developed linear logistic models on national and regional geographic scales that accurately estimated two CDC influenza metrics: the proportion of influenza test results that are positive and the proportion of physician visits that are ILI-related. We then used our estimated ILI-related proportion of physician visits in transmission models to produce improved predictions of influenza trends in the United States at both the regional and national scale. These findings suggest that ITS can be leveraged to improve "nowcasts" and short-term forecasts of U.S. influenza activity.

Learning time-series models is useful for many applications, such as simulation and forecasting. In this study, we consider the problem of actively learning time-series models while taking given safety constraints into account. For time-series modeling we employ a Gaussian process with a nonlinear exogenous input structure. The proposed approach generates data appropriate for time series model learning, i.e. input and output trajectories, by dynamically exploring the input space. The approach parametrizes the input trajectory as consecutive trajectory sections, which are determined stepwise given safety requirements and past observations. We analyze the proposed algorithm and evaluate it empirically on a technical application. The results show the effectiveness of our approach in a realistic technical use case.

Stochastic transmission dynamic models are especially useful for studying the early emergence of novel pathogens given the importance of chance events when the number of infectious individuals is small. However, methods for parameter estimation and prediction for these types of stochastic models remain limited. In this manuscript, we describe a calibration and prediction framework for stochastic compartmental transmission models of epidemics. The proposed method, Multiple Shooting for Stochastic systems (MSS), applies a linear noise approximation to describe the size of the fluctuations, and uses each new surveillance observation to update the belief about the true epidemic state. Using simulated outbreaks of a novel viral pathogen, we evaluate the accuracy of MSS for real-time parameter estimation and prediction during epidemics. We assume that weekly counts for the number of new diagnosed cases are available and serve as an imperfect proxy of incidence. We show that MSS produces accurate estimates of key epidemic parameters (i.e. mean duration of infectiousness, R0, and Reff) and can provide an accurate estimate of the unobserved number of infectious individuals during the course of an epidemic. MSS also allows for accurate prediction of the number and timing of future hospitalizations and the overall attack rate. We compare the performance of MSS to three state-of-the-art benchmark methods: 1) a likelihood approximation with an assumption of independent Poisson observations; 2) a particle filtering method; and 3) an ensemble Kalman filter method. We find that MSS significantly outperforms each of these three benchmark methods in the majority of epidemic scenarios tested. In summary, MSS is a promising method that may improve on current approaches for calibration and prediction using stochastic models of epidemics.

MOTIVATION: Computational modeling is widely used for deepening the understanding of biological processes. Parameterizing models to experimental data needs computationally efficient techniques for parameter estimation. Challenges for parameter estimation include in general the high dimensionality of the parameter space with local minima and in specific for stochastic modeling the intrinsic stochasticity. RESULTS: We implemented the recently suggested multiple shooting for stochastic systems (MSS) objective function for parameter estimation in stochastic models into COPASI. This MSS objective function can be used for parameter estimation in stochastic models but also shows beneficial properties when used for ordinary differential equation models. The method can be applied with all of COPASI's optimization algorithms, and can be used for SBML models as well. AVAILABILITY AND IMPLEMENTATION: The methodology is available in COPASI as of version 4.15.95 and can be downloaded from http://www.copasi.org CONTACT: [email protected] or [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.