182 MoNES Berman and Robert L. Schoenfeld 



uncertainty in the values determined for the system parameters. The uncer- 

 tainty in these values arises from the fact that the collected data may not be 

 sufficient to define the system completely and that the collected data have 

 associated fluctuations. 



A method for the quantification of the information in data and the systematic 

 formulation of models consistent with it is presented here. The information 

 content in the data is expressed by a set of invariants, and a concise matrix 

 relation is shown to exist between the invariants of the data and the system 

 parameters. Uncertainties in the data due to incompleteness or fluctuations 

 are mapped into a generalized co-ordinate space which also represents the 

 degrees of freedom of the system parameters and their uncertainty. The 

 uncertainties in the data are expressed in terms of regions in the generalized 

 co-ordinate space in such a way as to suggest a criterion for their quantification 

 with respect to the system. 



II. DATA INVARIANTS AND SYSTEM PARAMETERS 



The response of the system to a tracer injected into any one compartment 

 can be expressed in terms of the amounts of tracer in the various compartments 

 as a function of time. If we define the probability per unit time for a transition 

 from any compartment / to compartment j as A^^, then the kinetics of the 

 tracer in the /th compartment of an n compartmental system can be represented 

 by the following set of differential equations : 



^^ = -K^iit) +lh^qlt) (/ = 1, 2, • • •, n) (1) 



where ^^(0 is the amount of tracer material in the ;th compartment at time t 

 and 



hi ^ i hi (2) 



is the probability per unit time that any molecule in compartment / will leave 

 that compartment. 



The inequality sign expresses the possibility that a molecule may leave the 

 entire system from compartment / as in the case for open systems. 



The solution of the set of differential equations (1) is: 



n 



q,{t) = I A,, e-^' (3) 



i=i 



In a recent paper (2) we have pointed out that data expressed in the form of 

 equation (3) have the following properties: 



(a) There are at most n a^- in the data and these are invariants of the system 

 and independent of the initial conditions or site of measurements. 



(b) The Ay.^ represent n^ independent variables in the data. Specification of 

 the initial conditions reduces the Aj.^ to {n^ — n) independent variables which 

 are a function of the system parameters only. The Aj^^ thus represent {n^ — n) 

 invariants of the system parameters. 



