Hierarchical Continuous Time Markov Chain Models for Threshold Exceedance

Thresholds have been defined for many water quality indicators (WQIs) which separate the measurement space of the indicator into two states, one of which, the exceedance or violation state, has undesirable consequences. Observations are often made at unevenly spaced intervals, are usually uncoordinated with the timing of state changes, and are usually made asynchronously at multiple locations. These typical observation protocols have hindered estimation of violation - state properties. To address this problem, six hierarchical two - state continuous - time Markov chain (CTMC) models were developed and tested. These allow estimation of duration, frequency, and limiting probabilities from asynchronous, uncoordinated, and unevenly spaced observations. Three of these models were developed for single Markov processes but can be modified to handle multiple processes. Three of the models were developed for multiple processes. Two of the models were homogeneous; the other four were non - homogeneous with sinusoidally varying components. Model parameters were estimated with Bayesian MCMC methods. In each of three experiments, processes were simulated at high - frequency time steps. Asynchronous, infrequent, uncoordinated, and unevenly spaced observations of these processes were then extracted using protocols specified with varying observation Page 6 of 215 period length, quasi - regular observation interval, and violation - state observation probability. Models were estimated from the simulated observations, and compared on nominal parameter value recovery, predictive performance, and frequency and duration distribution error. Effects of process and observation protocol characteristics on recovery, prediction performance and distribution estimation error were measured. In the first experiment, simulated observations of single - chain two - state CTMCs were made and modeled. First, choice of prior distribution model was evaluated. Uniform and Gamma priors were found to be roughly equivalent in terms of performance, and both were found to perform substantially better than a J effrey's prior. Next, recovery, prediction, and distribution estimation error were evaluated. Duration, frequency, and violation - state probability were overestimated. Lower distribution estimation error was associated with longer observation period and more observations. Lower prediction and distribution estimation error was associated with more nonhomogeneous processes. In the second experiment, observation and modeling of multiple correlated WQI processes was simulated by mimicking WQIs with dual correlated two - state continuous time Markov chains. Estimates were made both jointly and individually, using the homogeneous model from the first experiment modified for multiple chains. Duration, frequency, and long - term violation - state probability were overestimated. Joint and individual estimates produced nearly equal results. Positively correlated and relatively low transition - rate processes were more - accurately predicted.

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