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Summary A variety of demographic statistical models exist for studying population dynamics when individuals can be tracked over time. In cases where data are missing due to imperfect detection of individuals, the associated measurement error can be accommodated under certain study designs (e.g. those that involve multiple surveys or replication). However, the interaction of the measurement error and the underlying dynamic process can complicate the implementation of statistical agent‐based models (ABMs) for population demography. In a Bayesian setting, traditional computational algorithms for fitting hierarchical demographic models can be prohibitively cumbersome to construct. Thus, we discuss a variety of approaches for fitting statistical ABMs to data and demonstrate how to use multi‐stage recursive Bayesian computing and statistical emulators to fit models in such a way that alleviates the need to have analytical knowledge of the ABM likelihood. Using two examples, a demographic model for survival and a compartment model for COVID‐19, we illustrate statistical procedures for implementing ABMs. The approaches we describe are intuitive and accessible for practitioners and can be parallelised easily for additional computational efficiency. 

Abstract Long‐lead forecasting for spatio‐temporal systems can entail complex nonlinear dynamics that are difficult to specify a priori. Current statistical methodologies for modeling these processes are often highly parameterized and, thus, challenging to implement from a computational perspective. One potential parsimonious solution to this problem is a method from the dynamical systems and engineering literature referred to as an echo state network (ESN). ESN models usereservoir computingto efficiently compute recurrent neural network forecasts. Moreover, multilevel (deep) hierarchical models have recently been shown to be successful at predicting high‐dimensional complex nonlinear processes, particularly those with multiple spatial and temporal scales of variability (such as those we often find in spatio‐temporal environmental data). Here, we introduce a deep ensemble ESN (D‐EESN) model. Despite the incorporation of a deep structure, the presented model is computationally efficient. We present two versions of this model for spatio‐temporal processes that produce forecasts and associated measures of uncertainty. The first approach utilizes a bootstrap ensemble framework, and the second is developed within a hierarchical Bayesian framework (BD‐EESN). This more general hierarchical Bayesian framework naturally accommodates non‐Gaussian data types and multiple levels of uncertainties. The methodology is first applied to a data set simulated from a novel non‐Gaussian multiscale Lorenz‐96 dynamical system simulation model and, then, to a long‐lead United States (U.S.) soil moisture forecasting application. Across both applications, the proposed methodology improves upon existing methods in terms of both forecast accuracy and quantifying uncertainty. 

Recurrent neural networks (RNNs) are nonlinear dynamical models commonly used in the machine learning and dynamical systems literature to represent complex dynamical or sequential relationships between variables. Recently, as deep learning models have become more common, RNNs have been used to forecast increasingly complicated systems. Dynamical spatio-temporal processes represent a class of complex systems that can potentially benefit from these types of models. Although the RNN literature is expansive and highly developed, uncertainty quantification is often ignored. Even when considered, the uncertainty is generally quantified without the use of a rigorous framework, such as a fully Bayesian setting. Here we attempt to quantify uncertainty in a more formal framework while maintaining the forecast accuracy that makes these models appealing, by presenting a Bayesian RNN model for nonlinear spatio-temporal forecasting. Additionally, we make simple modifications to the basic RNN to help accommodate the unique nature of nonlinear spatio-temporal data. The proposed model is applied to a Lorenz simulation and two real-world nonlinear spatio-temporal forecasting applications.