State-space models for stream networks - Coar, William J., author; Breidt, F. Jay, advisor (Colorado State University. Libraries, 2007)
The natural branching that occurs in a stream network, in which two upstream reaches merge to create a new downstream reach, generates a tree structure. Furthermore, because of the natural flow of water in a stream network, characteristics of a downstream reach may depend on characteristics of upstream reaches. Since the flow of water from reach to reach provides a natural time-like ordering throughout the stream network, we propose a state-space model to describe the spatial dependence in this tree-like structure with ordering based on flow. Developing a state-space formulation permits the use of the well known Kalman recursions. Variations of the Kalman Filter and Smoother are derived for the tree-structured state-space model, which allows recursive estimation of unobserved states and prediction of missing observations on the network, as well as computation of the Gaussian likelihood, even when the data are incomplete. To reduce the computational burden that may be associated with optimization of this exact likelihood, a version of the expectation-maximization (EM) algorithm is presented that uses the Kalman Smoother to fill in missing values in the E-step, and maximizes the Gaussian likelihood for the completed dataset in the M-step. Several forms of dependence for discrete processes on a stream network are considered, such as network analogues of the autoregressive-moving average model and stochastic trend models. Network parallels for first and second differences in time-series are defined, which allow for definition of a spline smoother on a stream network through a special case of a local linear trend model. We have taken the approach of modeling a discrete process, which we see as a building block to more appropriate yet more complicated models. Adaptation of this state-space model and Kalman prediction equations to allow for more complicated forms of spatial and perhaps temporal dependence is a potential area of future research. Other possible directions for future research are non-Gaussian and nonlinear error structures, model selection, and properties of estimators.
Keywords: Kalman filter; stream networks; statistics