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Long-lead forecasting for spatio-temporal systems can often entail complexnonlinear dynamics that are difficult to specify it a priori. Currentstatistical methodologies for modeling these processes are often highlyparameterized and thus, challenging to implement from a computationalperspective. One potential parsimonious solution to this problem is a methodfrom the dynamical systems and engineering literature referred to as an echostate network (ESN). ESN models use so-called {\it reservoir computing} toefficiently compute recurrent neural network (RNN) forecasts. Moreover,so-called "deep" models have recently been shown to be successful at predictinghigh-dimensional complex nonlinear processes, particularly those with multiplespatial and temporal scales of variability (such as we often find inspatio-temporal environmental data). Here we introduce a deep ensemble ESN(D-EESN) model. We present two versions of this model for spatio-temporalprocesses that both produce forecasts and associated measures of uncertainty.The first approach utilizes a bootstrap ensemble framework and the second isdeveloped within a hierarchical Bayesian framework (BD-EESN). This more generalhierarchical Bayesian framework naturally accommodates non-Gaussian data typesand multiple levels of uncertainties. The methodology is first applied to adata set simulated from a novel non-Gaussian multiscale Lorenz-96 dynamicalsystem simulation model and then to a long-lead United States (U.S.) soilmoisture forecasting application.

Deep Echo State Networks with Uncertainty Quantification for Spatio-Temporal Forecasting