| The uses made of aggregation in building simplified dynamic models are surveyed. The motivations for using aggregate, reduced-form models are described. The concept of 'perfect aggregation' is reviewed and extended to cover systems of partial differential equations and (Volterra) integral equations, as well as to aggregations which change the model type. The concept of an 'optimal approximate aggregate model' is critically examined and then expanded upon. Specifically, a variational technique is proposed for constructing optimal aggregation functions. An aggregate model is always problematical, either because it is systematically biased or because information about the original state variables is lost. The nature of such biases is described formally: the equilibria, stability characteristics, sensitivity structure, bifurcation behavior and statistical properties of aggregate models generally do not coincide with those of the underlying models. Furthermore, the errors in aggregate model predictions may grow exponentially with time. However, under certain conditions aggregate models may experience quite limited error growth; these conditions are established formally. It is proved that optimal aggregate models of conservative phenomena are generally not conservative. Several chapter-length applications of the theory are rendered. A global carbon cycle model is simplified by approximate aggregation techniques. Next, for a model for species extinction in ecological systems it is shown that the conditions for extinction propagation in a food web depend on the topology of the web, and that certain regular topologies can be inter-related through aggregation. The next chapter demonstrates how novel finite difference algorithms can be formulated through aggregation of more usual ones. The final application chapter reviews 'lumping' (aggregation) of chemical kinetic relations. A decision-theoretic method for determining the optimal level of model aggregation is described. The cognitive processes active in aggregate model construction are examined. The implications of aggregation theory for the general practice of modeling dynamical systems are drawn. Finally, an appendix treats a semi-dynamical problem of aggregation in applied economics. |
