| State-Space Models for Stream Networks
| William J. Coar
Ph.D. Candidate, Colorado State University, Department of Statistics
Monday, March 26, 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. This state-space model includes a
state vector that evolves from reach to reach as a function of upstream
reaches, and a measurement vector that depends on the state and allows for
general spatial-temporal dependence among measurements on a reach.
Current methods of estimation and prediction on a stream network are based on
Universal Kriging, where the covariance function is defined in terms of
distance between measurement locations. However, because of the branching
structure, the class of valid covariance functions becomes more restrictive
than the general class available for spatially correlated data.
Application of a state-space model over other tree structures has been
studied, but in a very different context. Areas such as multiscale resolution
and Gaussian directed trees are similar topologically, but model assumptions
for these networks are not always applicable to stream networks.
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
The methods developed here are applied to data available from Maryland's
Department of Environmental Protection. A Moving Average is fit to a measure
of instream cover in fish habitat data in a study that determines that
autocorrelation can be removed by using appropriate spatial covariates. A
smoothing spline is obtained to describe water chemistry data on this same
network. Maximum Likelihood estimators are found for all unknown parameters.
Lastly, an example of a more general nonstationary model with parameters that
depend on a surrogate for flow is presented. Simulation results for the exact
likelihood, an EM algorithm, and a simplified EM algorithm are obtained.
Maximum likelihood estimates and Monte Carlo standard errors for this two
parameter estimation problem are presented.
The proposed models describe a discrete process, and can be used as a building
block for continuous processes on a network. 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
non-linear error structures, computational complexities, model selection, and
properties of estimators.