Model Structure Identification From Experimental Data

Abstract

PREFACEIn recent years there has been considerable interest in the development of models for river and lake ecological systems. Much of this interest has been directed toward the development of progressively larger and more complex simulation models. In contrast, relatively little attention has been devoted to the problems of uncertainty and errors in the field data, of inadequate numbers of field data, of uncertainty in the relationships between the important system variables, and of uncertainty in the model parameter estimates. The International Institute for Applied Systems Analysis Resources and Environment Area's Task on Models for Environmental Quality Control and Management addresses problems such as these.The subject of this paper is model calibration. But rather than solving the customary problem of model parameter estimation, given an established structure for the model, the paper attempts to answer the prior question of identifying the dominant relationships between the system inputs, state variables, and output responses. That, then, is the problem that needs to be solved before one can consider how to estimate the model parameter values accurately. And it is a problem because, despite very many laboratory-scale experiments and a number of major field studies, our knowledge of the relationships between the mineral, organic, and microbiological components of an aquatic ecosystem is still quite uncertain.iii SUMMARY This paper is reprinted from the book Theoretical Systems Ecology: Advances and Case Studies, edited by E. Halfon of the Canada Centre for Inland Waters and published by Academic Press, New York. Acknowledging that systems ecology has had a large impact on all aspects of environmental research, the book aims to bridge the gap in communication between theoreticians, modelers, and field ecologists. Three classes of problems are treated in the book. They separate approximately into (a) how the model should be developed and analyzed prior to the collection and use of field data, (b) how the model should be developed or modified when it is evaluated against field data, and (c) the desired properties of the models for control and management purposes. The objective of this paper is to emphasize the fact that the second of these three problem categories is not simply a matter of straightforward model parameter estimation.The basic problem of model calibration -or system identification -is that information about the "externally" observed behavior of a system is required to be translated into information about the model-based description of the system's "internal" mechanisms of behavior. The measured input and output variables represent the system's external description, whereas state variables and parameter values refer to the system's internal description. In other words, during model calibration, and especially in the process of identifying the model's structure, we seek improved understanding of those physical, chemical, and biological phenomena that are thought to govern the system's observed be...

Water quality modeling: A review of the analysis of uncertainty

“…In a qualitative, nonrigorous sense it is these changes of the estimates of unknown but constant parameters that are symptoms of the failure of individual hypotheses [Beck and Young, 1976 ;Beck, 1979a. By analogy with physical engineering structures, there has been a plastic deformation, or collapse, of a structural member (component hypothesis).…”

“…Even for this more narrowly and more easily defined problem for which there are relatively systematic procedures of solution [e.g., Box and Jenkins, 1970;Soderstrom, 1977;Ha/fan et al, 1979;Hipel, 1981;Young et al, 1980], there is not necessarily any truly "objective" indicator of having identified the best model order, nor are these procedures especially effective on the data typical of environmental systems [e.g., Beck, 1979a]. Since model-order estimation is usually associated with a class III model structure, it will not be a primary concern of this review.…”

Uncertainty and arbitrariness in ecosystems modelling: A lake modelling example

“…Since the model structure was fixed a priori in this study we have not addressed the difficult problem of model structure identification (e.g. Beck, 1979Beck, , 1980; the 'calibration' exercise here has essentially been concerned with a 'parameter estimation' problem. Nevertheless, the induced distributions of the parameter values (in the character vector) that result from the analysis can provide not only meaningful insight into the nature of the system's behaviour but also a posteriori information about the adequacy of the model structure.…”

“…For example, in problems with state variables having different physical dimensions some (subjective) form of weighting is required in the formulation of a single-valued loss function. Furthermore, it is not easy to account for uncertainty in the field data, although methods to do this have been attempted (Beck and Young, 1976;Jolankai and Szollosi-Nagy, 1978;Lewis and Nir, 1978;Beck, 1979;Di Toro and van Straten, 1979;Fedra, 1979Fedra, , 1980. Finally, however, it has to be recognized, that the assumption that a single 'best' parameter vector exists is at least questionable, especially if data uncertainty is considered, and in any case experience shows that it is extremely difficult to find such a unique vector if the number of parameters to be estimated is larger than, say, six to ten.…”

Hard or soft environmental systems?

“…related to eachother, and whether information about these relationships can be identified from the in situ field data. In short, the analyst is faced with the problem of model structure identification (Beck, 1979). Put another way, and in a simpler form than the sophisticated example of the study of Saginaw Bay (Bierman et al, 1980): it is a fine idea to estimate the slope and intercept of a straight line drawn through a set of data points (i.e.…”