As mentioned above, the first step in any UA is the identification and subsequent characterization of the model input/parameter uncertainties (Figure 3). Depending on the uncertainty propagation method used in step 2, this description of input uncertainties can range from only specifying the moments (e.g., mean and variance) of each source of uncertainty to specifying probability density functions (PDFs), or joint PDFs, for
the uncertain model inputs/parameters. It should be noted that the correlation structure between PDFs should also be taken into account, if possible. This is of particular importance for ecological models that are spatially distributed. Accounting for spatial correlations in detail requires use of a more complex multivariate stochastic model, describing the input uncertainty in space, and, sometimes, time. Development of such models requires spatial analysis skills and a knowledge of geostatistics.
The methods for describing input and parameter uncertainties can be very different, depending on whether historical system response data are available for model calibration or not. When no calibration data are available, the description of model input and parameter uncertainties must be based on literature values or experience, or, by fitting probability distributions to case-study-specific historical data on specific uncertain inputs/parameters, if such data are available. If calibration data are available, an ecological model will typically be applied to simulate system response in two cases (or time periods) - the calibration period and the prediction period. When model parameters (and, perhaps, some other model inputs) can be estimated via calibration utilizing historical system response data, it is necessary to condition the description of model input and parameter uncertainties on the calibration data. Otherwise, the resultant sampling of the uncertain inputs for uncertainty propagation could potentially generate model outputs that are completely inconsistent with the historical calibration data. This issue is sometimes overlooked in UA studies. Two common methods that are able to take the calibration data into account when describing model input and parameter uncertainties include the generalized likelihood uncertainty estimation (GLUE) methodology and methods for Bayesian inference.
Formal methods for Bayesian inference (based on Bayes' theorem) include Bayesian Monte Carlo (BMC) and Markov Chain Monte Carlo (MCMC) methods. MCMC methods are typically more efficient than BMC methods. Formal Bayesian approaches, in comparison to GLUE described below, offer a mathematically rigorous approach to quantifying model prediction uncertainty because they utilize a formal statistical likelihood function. However, defining this likelihood function is a nontrivial issue. MCMC samplers utilize optimization concepts to improve sampling efficiency over BMC and are designed to converge to, and then sample from, the joint posterior distribution of parameters, given the observed calibration data. The Metropolis-Hastings algorithm and Gibbs sampler are examples of MCMC algorithms. MCMC results can be used to derive confidence limits for model predictions and other output distribution characteristics.
Although research is ongoing, there are currently some significant limitations to using Bayesian analysis. One issue is that an explicit statistical model ofthe ecological model prediction errors (also referred to as the model inadequacy function) is needed, but this is often difficult to define based on limited available data. In addition, ecological model complexity in terms of the dimensionality ofuncertain parameters/inputs and the computational requirements ofeach ecological model run, combined with the number of samples required to generate an accurate representation of the joint posterior distribution of model inputs/parameters, make the practical application of these techniques extremely challenging.
The GLUE methodology for calibration and UA of environmental and ecological simulation model predictions is conceptually simple and very flexible. GLUE requires that modelers define a 'likelihood' function that monotonically increases as agreement between model predictions and measured calibration data increases. The GLUE likelihood function can be, but is not required to be, a formal statistical likelihood function. In fact, in the vast majority of GLUE applications, the likelihood function is not statistically based and is then best referred to as a pseudo-likelihood function to eliminate confusion with a formal statistical likelihood function. GLUE utilizes Monte Carlo sampling (typically from uncorrelated uniform distributions between prior bounds on the inputs/ parameters considered to be uncertain) to identify acceptable or behavioral input/parameter sets that satisfy a threshold value of the pseudo-likelihood function. These behavioral input/parameter sets are then used to generate a pseudo-likelihood weighted cumulative distribution function of model outputs.
It is important to note that when formal Bayesian methods or GLUE are utilized to describe input/parameter uncertainties, many thousands or more ecological model simulations are required. Ultimately, both ofthese approaches characterize the joint posterior distribution of uncertain inputs with a large sample of uncertain input vectors, rather than an analytical description.
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