The development and application of environmental models to help understand natural systems, and support decision making, is commonplace. A difficulty encountered in the development of such models is determining which physical and chemical processes to simulate, and on what temporal and spatial scale(s). Modern computing capabilities enable the incorporation of more processes, at increasingly refined scales, than at any time previously. However, the simulation of a large number of fine scale processes has undesirable consequences: first, the execution time of many environmental models has not declined despite advances in processor speed and solution techniques; and second, such complex models incorporate a large number of parameters, for which values must be assigned. Compounding these problems is the recognition that since the inverse problem in groundwater modeling is non-unique the calibration of a single parameter set does not assure the reliability of model predictions. Practicing modelers are, then, faced with complex models that incorporate a large number of parameters whose values are uncertain, and that make predictions that are prone to an unspecified amount of error. In recognition of this, there has been considerable research into methods for evaluating the potential for error in model predictions arising from errors in the values assigned to model parameters. Unfortunately, some common methods employed in the estimation of model parameters, and the evaluation of the potential error associated with model parameters and predictions, suffer from limitations in their application that stem from an emphasis on obtaining an over-determined, parsimonious, inverse problem. That is, common methods of model analysis exhibit artifacts from the propagation of subjective a-priori parameter parsimony throughout the calibration and predictive error analyses. This thesis describes theoretical and practical developments that enable the estimation of a large number of parameters, and the evaluation of the potential for error in predictions made by highly parameterized models. Since the focus of this research is on the use of models in support of decision making, the new methods are demonstrated by application to synthetic applications, where the performance of the method can be evaluated under controlled conditions; and to real-world applications, where the performance of the method can be evaluated in terms of trade-offs in computational effort versus calibration results and the ability to rigorously yet expediently investigate predictive error. The applications suggest that the new techniques are applicable to a range of environmental modeling disciplines. Mathematical innovations described in this thesis focus on combining complementary regularized inversion (calibration) techniques with novel methods for analyzing model predictive error. Several of the innovations are founded on explicit recognition of the existence of the calibration solution and null spaces – that is, that with the available observations there are some (combinations of) parameters that can be estimated; and there are some (combinations of) parameters that cannot. The existence of a non-trivial calibration null space is at the heart of the non-uniqueness problem in model calibration: this research expands upon this concept by recognizing that there are combinations of parameters that lie within the calibration null space yet possess non-trivial projections onto the predictive solution space, and these combinations of parameters are at the heart of predictive error analysis. The most significant contribution of this research is the attempt to develop a framework for model analysis that promotes computational efficiency in both the calibration and the subsequent analysis of the potential for error in model predictions. Fundamental to this framework is the use of a large number of parameters, the use of Tikhonov regularization, and the use of subspace techniques. Use of a large number of parameters enables parameter detail to be represented in the model at a scale approaching true variability; the use of Tikhonov constraints enables the modeler to incorporate preferred conditions on parameter values and/or their variation throughout the calibration and the predictive analysis; and, the use of subspace techniques enables model calibration and predictive analysis to be undertaken expediently, even when undertaken using a large number of parameters. This research focuses on the inability of the calibration process to accurately identify parameter values: it is assumed that the models in question accurately represent the relevant processes at the relevant scales so that parameter and predictive error depend only on parameter detail not represented in the model and/or accurately inferred through the calibration process. Contributions to parameter and predictive error arising from incorrect model identification are outside the scope of this research.

The research in this thesis has been undertaken to develop statistical tools for monitoring adverse events in hospitals that adjust for varying patient risk. The studies involved a detailed literature review of risk adjustment scores for patient mortality following cardiac surgery, comparison of institutional performance, the performance of risk adjusted CUSUM schemes for varying risk profiles of the populations being monitored, the effects of uncertainty in the estimates of expected probabilities of mortality on performance of risk adjusted CUSUM schemes, and the instability of the estimated average run lengths of risk adjusted CUSUM schemes found using the Markov chain approach. The literature review of cardiac surgical risk found that the number of risk factors in a risk model and its discriminating ability were independent, the risk factors could be classified into their "dimensions of risk", and a risk score could not be generalized to populations remote from its developmental database if accurate predictions of patients' probabilities of mortality were required. The conclusions were that an institution could use an "off the shelf" risk score, provided it was recalibrated, or it could construct a customized risk score with risk factors that provide at least one measure for each dimension of risk. The use of report cards to publish adverse outcomes as a tool for quality improvement has been criticized in the medical literature. An analysis of the report cards for cardiac surgery in New York State showed that the institutions' outcome rates appeared overdispersed compared to the model used to construct confidence intervals, and the uncertainty associated with the estimation of institutions' out come rates could be mitigated with trend analysis. A second analysis of the mortality of patients admitted to coronary care units demonstrated the use of notched box plots, fixed and random effect models, and risk adjusted CUSUM schemes as tools to identify outlying hospitals. An important finding from the literature review was that the primary reason for publication of outcomes is to ensure that health care institutions are accountable for the services they provide. A detailed review of the risk adjusted CUSUM scheme was undertaken and the use of average run lengths (ARLs) to assess the scheme, as the risk profile of the population being monitored changes, was justified. The ARLs for in-control and out-of-control processes were found to increase markedly as the average outcome rate of the patient population decreased towards zero. A modification of the risk adjusted CUSUM scheme, where the step size for in-control to out-of-control outcome probabilities were constrained to no less than 0.05, was proposed. The ARLs of this "minimum effect" CUSUM scheme were found to be stable. The previous assessment of the risk adjusted CUSUM scheme assumed that the predicted probability of a patient's mortality is known. A study of its performance, where the estimates of the expected probability of patient mortality were uncertain, showed that uncertainty at the patient level did not affect the performance of the CUSUM schemes, provided that the risk score was well calibrated. Uncertainty in the calibration of the risk model appeared to cause considerable variation in the ARL performance measures. The ARLs of the risk adjusted CUSUM schemes were approximated using simulation because the approximation method using the Markov chain property of CUSUMs, as proposed by Steiner et al. (2000), gave unstable results. The cause of the instability was the method of computing the Markov chain transition probabilities, where probability is concentrated at the midpoint of its Markov state. If probability was assumed to be uniformly distributed over each Markov state, the ARLs were stabilized, provided that the scores for the patients' risk of adverse outcomes were discrete and finite.