Stochastic Dynamics Modeling and Reduction for Predicting Phenomena Behavior in a Dynamic, Data-Driven Application System

Abstract

When surveying an environment, it is often beneficial to construct empirical models that realistically define how phenomena might unfold. One reason is that such forecasts can help with the placement of either fixed or mobile sensors in widespread domains. In particular, if investigators can determine locations where crucial measurements can be made to either validate the model or extend its forecasting capabilities, then sensors can be accordingly allocated. It is also possible to use details from such models to limit the overall size of the network. This capability is crucial when the cost of physically deploying sensor nodes is prohibitive. Another reason is that empirical models can help lower the energy consumption of battery-operated sensor networks. If it is known in advance that no informative observations will likely be made in certain regions, then the devices in those regions can be remotely disabled. The selective switching of sensors can improve the lifespan of the network.Due to the utility of empirical, predictive models, a number of modeling approaches have been proposed. One popular approach entails constructing differential-equation-based dynamical systems from available sensor observation ensembles. Such dynamical systems permit the simulation of many types of phenomena. They also routinely allow for principled interpolations and extrapolations of the model forecasts to times and places not examined. Furthermore, statistics from the dynamical systems can offer insight into where the sensors should be positioned to collect meaningful information.A defining trait of these dynamical-systems-based approaches is that they are deterministic: the state of a phenomenon is uniquely determined by the system parameters and by sets of previous states. Since there is no element of randomness to these models, many of the inherent uncertainties associated with sensing and characterizing phenomena are completely ignored. Consequently, it is difficult to determine if the predicted outcome has either a high, moderate, or low chance of occurring in the real world. Phrased another way, investigators have little to no advance knowledge of if they should believe a model's forecasts. Not accounting for uncertainty can additionally yield models that fit the observations well yet provide poor predictions.We develop a novel, empirical modeling approach for constructing stochastic, (non-)linear dynamical systems that describe and predict the evolution of phenomena. Under this model, we assume that the phenomena dynamics is encoded within sensor observations of the environment.We also assume that the sensor observations are modeled by a series of stochastic processes composed of stochastic partial differential equations. These stochastic processes have been designed to capture both continuous and discrete dynamics behaviors. Capturing both types of behaviors helps characterize the type of phenomena that we consider and predict its behaviors.We favor a modeling approach that relies on a series of simple, extensible representations of the dynamics versus a single, complex representation.We assume that the phenomena dynamics can be temporally and spatially segmented into one or more phase space regions. The number of regions is determined automatically by our model in a dynamics-driven manner. That is, phase space regions with simple, slowly-changing dynamics are coarsely partitioned, while regions with complicated, quickly-changing dynamics are finely partitioned.We assign one or more stochastic processes to describe the evolution of the phenomena dynamics for each of these regions. The associated parameters for each of these processes are derived automatically from statistics of the sensor observations. This functionality helps ensure that the model predictions will be accurate. It also relieves the investigator of needing to manually specify model parameter values, which can be a time-consuming an error-prone task.The stochastic processes that we learn typically have several redundant degrees of freedom. Each of these redundancies decreases their evaluation rate and hence how quickly the model can make predictions. To improve the evaluation rate, we propose to uncover reduced-order versions of the stochastic processes. Our simplification strategy is based on non-linearly projecting the stochastic processes onto a domain with fewer degrees of freedom. These projections are derived in a dynamics-sensitive manner. That is, our model ensures that the projections retain as much as possible about the dynamics when removing any unnecessary details.