Handling Assumptions in Automated Modelling

Neil Smith

Abstract

There are many different classes of decision made duringthe development of a model. If models are to be shown tobe appropriate for the specified task, the information and as-sumptions upon which these decisions are based must bemade explicit . This paper describes AIM, an automatedmodeller that makes all its decisions with explicit knowledge .

AIM separates different types of modelling knowl-edge in its knowledge base to prevent the embedding of as-sumptions in the structure of the modelling knowledge.

Italso uses a novel modelling algorithm that does not rcl) oneither external sources of information or the structure of theknowledge base to generate models. We show that AIMgenerates parsimonious models and discuss some of theconstraints on this approach .

IntroductionThe process of modelling is one of creating a suitable ab-straction of the real world. This is done by the modellerregarding the referent system and deciding what simplifi-cations and abstractions to make to produce a model.However, it is all to easy for the modeller to make unwar-ranted assumptions about what abstractions are appropriateduring the production of the model .

If such unwarrantedassumptions are made, the end result is a model that maynot be suitable for the task at hand .

In order to ensure thatsuch inappropriate models are not used, it is important thatall the modelling decisions are explicit during the model-lingprocess . By making all the decisions explicit, the justi-fications for each decision can be given . This allows theuser of the model to have confidence in the final model, asthey are certain that only appropriate (i.e. justified) deci-sions were taken during the development of the model .

Inaddition, by examining the justifications for the modellingdecisions, the domain of validity of the model (Williams &Raiman, 1994) can be estimated, which can allow the reuseofthe model under appropriate circumstances .

In the next section, we examine the different modellingdecisions that need to be made in the development of anymodel and we discuss some of the ways in which unwar-ranted assumptions can creep into models . The main bodyofthe paper consists of a description of AIM, an automatedmodeller that addresses many of these problems . We dis-cuss each of the different types of modelling decision inturn, and show how the architecture and algorithms of AIMensure that all decisions are explicit . In addition, we showthat the models generated are parsimonious . Finally, wediscuss some limitations of AIM's approach .

CSIMG, Cranfield University.Shrivenham . Swindon, SN6 8LA, UK

nei1 .smith (~i', rmcs .cranfield .ac.uk

Types of AssumptionThere are several different types of modelling decision thatneed to be taken when a system is modelled; all of thesemust refer to the intended use of the model, to ensure thatthe generated model is useful . The most basic modellingdecision is the selection of a model representation and rea-soning formalism (Schut & Bredeweg, 1996) which defineshow the referent system is to be regarded by the modeller,e.g . whether a component-centred or a process-centredanalysis is to be undertaken . Once the most appropriateformalism has been selected, two sets of decisions must bemade in parallel : the boundary of the model must be deter-mined, and the referent system must be broken down intoits component parts for modelling. Once the referent sys-tem has been broken down into the appropriate "lumps"(e.g. components or processes), the modeller must decideon the most appropriate ontology to use to represent eachof these lumps (the is the equivalent to selecting assump-tion classes in a model composition system (Nayak & Jo-skowicz, 1996)) . This process provides the information toguide the formation of the model boundary, both in termsof physical extent and the types of phenomena included inthe model. As the model boundary is developed, newlumps are identified that require representation in themodel . Finally, there are a variety of decisions to be maderegarding the various simplifying assumptions that can beapplied to the model (Weld, 1992; Williams & Raiman,1994). Only when all of these decisions have been made isit possible to finally construct the model, which is thenused to address the task in hand .

This analysis shows that there are a great many differenttypes of modelling decision that need to be taken during thedevelopment of a model. Highly abstract decisions aremade at the beginning of the modelling process, with thedecisions becoming more concrete and detailed throughoutthe modelling process. It is necessary that all ofthese deci-sions be made explicitly by the modeller to ensure that themodel's user has confidence in the final model.

However, not all automated modelling systems are capa-ble of making all of these decisions. Many existing auto-mated modelling systems, particularly model inductionsystems (e.g . Ironi & Stefanelli, 1994 ; Addanki et al .,1991) have only addressed the problem of selecting thecorrect simplifications to make in the model. Model com-position systems (e.g . Nayak & Joskowicz, 1996) haveaddressed the problem of the selection of the most suitableontologies to use in the model, but at the cost of increasingthe complexity and decreasing the flexibility of their mod-

Smith 123

Finger

124 QR-98

elling knowledge; this has reduced these modeller's abilityto select appropriate simplifications (Smith, 1998).AIM uses a combination of the model composition and

model induction approaches, combining the best aspects ofboth, while eliminating their drawbacks. The expressivepower of model composition techniques is harnessed bothto develop 'he model boundary and to select the ontologiesfor representing "lumps" within this boundary . Model in-duction techniques are used to refine the model within thisboundary : the ability to select the most appropriate combi-nations of simplifying assumptions to make in the modelgives AIM the flexibility to generate parsimonious models.These three facets of modelling are addressed by a combi-nation of novel knowledge architectures and novel algo-rithms using these architectures. AIM does not address theproblem of selecting an appropriate representation andreasoning formalism : AIM is a component-centred model-ling system . Neither does it address the lumping problem,instead relying on the description of the components in thesystem description to determine the different regions withinboth the system and the model .The remainder of this paper describes AIM and how it

addresses some of these problems in modelling. We shallalso discuss the conditions under which AIM is able togenerate models which have behaviours that quantitativelymatch those of the systems being modelled . We shall thengive some results from AIM and draw some conclusions .To show how AIM operates, we shall illustrate the

model generation process by showing how AIM generatesa model of a water-filled syringe (figure 1). The modelwill be used to simulate the response of all variables thatare affected by a force applied by the finger .

Selecting Models of ComponentsThe first problem we shall address concerns how AIM de-cides to represent each component in the model. The dan-ger at this step is that the modeller will be influenced in itschoice of representation by knowledge about how compo-nents are usually used . Such influences represent hiddenmodelling decisions and, if such hidden decisions couldoccur, the user of the model will have a lesser faith in theveracity of the model. In order to prevent such hidden de-cisions, AIM separates entirely knowledge about compo-nents from knowledge about m-components (or models ofcomponents). This separation ensures that the modellingknowledge base cannot contain any implicit links betweenthe components that exist in the referent system and themanner in which these components are modelled .

M-components are similar to model fragments (Falken-hainer & Forbus, 1991) in that they are a partial represen-

Figure 1 : The Syringe

Thumb

tation of a component under a particular set of conditions .A wire could be modelled variously as an electrical-Conductor m-Component, as a mechanical Spring m-component, or as a simple linearmass m-component,depending on the situation in which the wire was found andthe interactions to which it was subjected. Basic m-components always represent the ideal model of a compo-nent ; non-ideal effects are accommodated through the in-clusion of corrections in the model (see below) .

While the total separation between components and m-components prevents implicit decisions, AIM must be ableto select the most appropriate m-component to use to rep-resent a given component in a given situation . This facilityis provided through a function that maps components ontom-components . This function takes account of the type ofcomponent being modelled, the interaction to which it issubjected, as well as the port at which this interaction oc-curs and the component's state (a switch will have differentmodels depending on whether it is open or closed). Thefunction's domain covers all physically possible combina-tions of component type, interaction domain, port, andstate; this ensures that AIM can always determine the mostappropriate m-component to use to represent a componentunder any given circumstances .The behaviour of an m-component is described by a

fragment of a bond graph (Rosenberg & Karnopp, 1983)contained in the m-component. When the model is gener-ated, these bond graph fragments are merged to produce abond graph representation of the model; the bond graph isused to produce the state equations that yield the system'sbehaviour. The various parameters that determine the m-component's behaviour, such as a block's moment of iner-tia, are calculated from physical parameters defined for themodelled component. M-components have other features,and these will be discussed below when appropriate .

Finding the Model BoundaryWith AIM capable of identifying the correct m-componentto use to represent any component in the model, the nextproblem is how to select the components to model. This isthe model boundary identification problem, and AIMsolves it through an incremental expansion of the boundaryfrom the information specified in the task description .The key to the development of the model boundary lies

in knowledge about how effects at one port of a componentaffect the other ports of that component. When an m-component is used in a model to represent a component, aneffect (an interaction in a specified energy domain) at oneport of a component will not necessarily cause similar effects at all other ports of the same component.

To reflectthis, each m-component has a set of intra-actions, whichspecify which of the m-component's other ports are af-fected by an effect at any one port. Intra-actions do notthemselves indicate any causal direction within the m-component; they describe the possible causal orientationsthe m-component can support. Causal directions within anm-component are only defined when the m-component isplaced in a model and the casual ordering of the whole

v

End 1(fluid)

End 2(fluid)

2a : Intra-actions in a fiuidPipe

2b : Intra-actions in a sample bathtubFigure 2: Intra-actions

model determined . The explicit nature of intra-actions inthe modelling library ensures that the assumptions under-lying the model boundary expansion are explicit in the li-brary.

Intra-actions are normally symmetric, to reflect thesymmetric nature of the relationships between effects in acomponent, i.e . the symmetric relationship between fluidflow rates at either end of a fiuidPipe (figure 2a). How-ever, the relationships between some effects are non-symmetric and the m-component's intra-actions reflect this .For example, the water flow rate from a tap into a bathaffects the flow rate from the plughole, but the converse isnot true (figure 2b). This information is used by AIMwhen it determines the model boundary .The algorithm for the boundary identification process is

shown in figure 3 . It centres on the boundary analysis

function expand boundary (sd, initmodel, baq, cd)sd: system descriptionbag: boundary analysis queuecd : causal direction specified in taskinitmodel: the initial model that is expandedm: the expanded modelbap: boundary analysis portp: portcp, mp : additional portsme : m-component

beginm = initmodelwhile baq $ (Ibap = dequeue (bag)if bap e m

cp = system ports connected to bapenqueue (bag, cp)m=m+cp

endme = correct_m-component (bap, sd)if me e m v -,(3p such that [P E m A

p E intra-actions (mc, bap, -cd)]mp = intra-actions (mc, bap, cd)enqueue (baq, mp)m=m+mc

endendreturn m

end

Figure 3 : Algorithm for finding the model boundary

Input 1(fluid)

W&~PIAOutpuu(fluid)

Input 2(fluid)

~c input 2(fluid)

queue, which stores details of all the ports in the model(with their corresponding effects) that are currently on themodel boundary . However, there can be no ports on a cor-rectly-drawn model boundary as any ports on the boundaryrepresent an under-specified effect : each port forms part ofa junction, and the effects at a junction depend on all theports at that junction . For example, the consideration ofthe current on one lead at an electrical junction requires theconsideration of the electrical properties of all the otherleads at that junction . This means that the boundary analy-sis continues until the boundary analysis queue becomesempty.The boundary identification process starts by identifying

the ports and effects referred to in the task definition andplacing these in the boundary analysis queue. Using thesyringe example (figure 1), the boundary analysis queuewould initially contain the single port/effect combination(finger, end, linearMechanical), representingAIM's interest in the mechanical properties of the finger .

Boundary expansion performed port by port, as shown infigure 3 . When several ports are connected, an effect atone port requires AIM to consider the same effect at all theconnected ports (providing the connection supports theinter-action) . All the connected ports are added to theboundary analysis queue and the connection is added to themodel. For instance, the finger touches the end of theplunger handle, so this latter port will be added to theboundary analysis queue. However, if a port already existsin the model (as part of a connection), the connection mustalready have been processed and AIM does not process itagain.AIM checks whether this port requires further processing

by examining the m-component's intra-actions . If the cur-rent port was placed in the boundary analysis queuethrough the analysis of other ports of this m-component(i .e. if there exists in the model a port of the current m-component whose consideration would have caused thisport to be included), AIM can move on to the next port inthe boundary analysis queue.

If not, details of the in-

component, and the component to which it relates, areadded to the model .The m-component's intra-actions are then used to deter-

mine which of the component's other ports have effectsthat are significant in the model. The causal information inthe task description indicates whether to use the intra-actions entering the port (to find the ports that affect theport in question) or the intra-actions leaving the port (tofind the ports affected by this port). This only becomessignificant where an m-component's intra-actions are not

Smith 125

126 QR-98

Table 1 : M-components of the Syringe

symmetric . All ports mentioned in the relevant intra-actions are added to the boundary analysis queue . In thesyringe, the finger is modelled as an exogenous applicationof force, and so has no intra-actions . The plunger handle,modelled as a rigid bar, has a symmetric intra-action thatrelates mechanical effects at the ends to each other : thefinger pressing at one end of the handle prompts AIM toconsider the effects of the other end of the handle on theplunger .The components found to be inside the model boundary

in the syringe example, together with their m-components,are shown in table 1 . Note that all the components are rep-resented by ideal models, and note that the atmospheresurrounding the syringe is explicitly included in the modelboundary .

Selecting Simplifying AssumptionsOnce the model boundary has been determined and thecorrect representation for all the components within thatboundary has been found, attention can be focussed onwhich simplifying assumptions to include in the model .Basic m-components represent the ideal model of a com-ponent, i.e. a model made under all possible simplifyingassumptions . However, not all of these assumptions arevalid in all modelling circumstances : the viscosity of fluidin a pipe will be negligible if the fluid flows slowly and themodel is only used to represent a small time scale . In othersituations, the viscosity might be significant, and the basicfluidPipe m-component must be augmented to includethe effects of fluid viscosity .

This is achieved in AIMthrough the mechanism of corrections, which can be addedto m-components to reflect the retraction of these simpli-fying assumptions . All corrections in AIM are examples offitting approximations (Weld, 1992) .

Corrections have a very similar structure to m-components . Corrections contain a bond graph fragmentthat describes how the correction affects the behaviour ofthe m-component to which they attach . The parameters inthis bond graph fragment are defined by the physical pa-rameters of the component to which it relates . Each cor-rection has at least one port, the attachment port, that con-trols how the correction is added to the m-component(though note that corrections attach to an m-component'sbody, rather than one of its ports) .

Some corrections (e.g.

heating in an electrical resistor) introduce additional portsand intra-actions to the m-component to which they attach .The inclusion of such corrections can expand the modelboundary, but a detailed examination of such corrections isoutside the scope ofthis paper .

As each correction represents the retraction of a simpli-fying assumption, a correction can only be applied once toa particular instantiated m-component . However, the samecorrection can be applied to an arbitrary number of differ-ent instantiations of the same m-component, and can evenbe applied to different types of m-component . The appli-cation of corrections to one m-component is independent ofthe application ofthat correction to any other m-componentin the model . The correction candidates of a model are allthose corrections that can validly be added to the model,i. e . are not already present in the model .

This architecture for the modelling knowledge used inAIM allows for the combination of the benefits of themodel composition and model induction approaches . Thecomponent-centred approach to modelling, combined withthe m-components' intra-actions, provides the sophistica-tion required to accurately determine the model boundaryin response to a specified task definition . The separation ofthe corrections from the m-components allows AIM thefreedom to select only the simplifying assumptions that areappropriate in this model .

The initial model, found during the identification of themodel boundary, is likely not to be sufficient to address thetask specified . The model is made adequate by the addi-tion of corrections . Corrections can be added for two rea-sons : to ensure coherence in the model, or to reduce thebehavioural difference between the model and the referentsystem .A model is said to be incoherent if the it does not yield a

complete set of state equations ; this normally reflects aphysically impossible situation in the model . The initialmodel of the syringe, shown in table 1, would predict aninfinite velocity for the ram, due to the lack of friction orother similar phenomena in the model . Such errors in themodel are removed by the addition of corrections . Thefunction coherent, described by Smith (1998), identifiesthe corrections that could potentially eliminate the errorand selects the one that has the greatest effect on themodel's behaviour . In the syringe example, this correctionis the friction between the plunger and the cylinder .The main reason for including corrections in a model is

to reduce the behavioural difference between the modeland the referent system . We take the view that models arealways intended, at some level, to explain the behaviour ofthe referent system . This allows us to define a model asadequate for a task if the behaviour of the model is notsignificantly different from the behaviour of interest of thereferent system .

If we assume that AIM's libraries contain all possiblecorrections (thus ignoring any closed world assumption),the behaviour ofa model can be brought as close as desiredto that of the referent system by the inclusion ofcorrectionsin the model .

However, many of these corrections will

Component Modelled as

Finger Source of force

Thumb Not modelled

Ram Rigid, massless bar

Plunger Watertight, rigid, frictionless plunger

Cylinder Rigid container of inviscid fluid

Nozzle Rigid container of inviscid fluid

Atmosphere Source of (zero) pressure

function find-parsimonious model (sd, voi, cd, e)sd: system descriptionvoi : variables of interestcd: causal direction specified in taske : significance threshold valuem, m', m n ,, X , : modelsR(m) : correction candidates ofmr: correction

beginm = coherent (expand boundary (sd, 0, voi, cd))repeat

R(m) = all valid correction candidates for m

endendif Jm~ 4k k

% Jma, is significantm = mnc,v,

Figure 4 : AIM's Algorithm

J,,ax = 0for each r E R(m)

m' = coherent (expand boundary (sd, m + r,intra-actions(r, attach-port, cd), cd))

J= 4(m, m')

% difference in behaviourifJ>J�,u,,

Jmax =Jin,,, = m,

enduntil J�,,, < ;~ v R(m) _if R(m) = { }

return nil

%cannot guarantee an adequate modelelse

return mend

end

have little or no effect on the behaviour of the model .Normally, only a few of the possible corrections will havesignificant effects on the model's behaviour ; the objectiveof modelling is to identify which corrections fall into thiscategory . A model that contains all the significant correc-tions will be adequate for the specified task . An adequatemodel that contains no other corrections is parsimonious .

The problem is how to assess the effect of each correc-tion candidate on the behavioural difference between themodel and the referent system. Model induction systems(e.g. Addanki et al., 1991) do this by generating the be-haviour of the model and comparing it to a trace of thereferent system's behaviour, provided as part of the prob-lem specification . This requires that the referent system'sbehaviour be known .

Alternatively, AIM could produce a "most complex"model to produce a behaviour trace against which othermodels are compared . Unfortunately, this model would beformed under an arbitrary closed world assumption . Addi-tionally, a model that contains all the corrections availableunder this closed world assumption might not be coherent .To make it coherent, AIM would have to arbitrarily elimi-

nate corrections to ensure the model's coherence . Finally,increasing the knowledge in the modeller would increasethe complexity of the initial model . If the additional com-plexity is not needed in the final model, including it onlyserves to increase the cost of developing the parsimoniousmodel .

Instead of the above approaches, AIM takes the novelstep ofusing the behaviour of an existing model as the baseagainst which refinements are compared . The algorithm isshown in figure 4 . When a model is generated, its behav-iour is found . When each correction candidate is assessed,the model is revised to include this correction and the be-haviours of the base model and the revised model are com-pared . This revision and comparison is repeated for all themodel's correction candidates . If no correction causes asignificant change in the model's behaviour, the model isdeemed adequate and modelling stops .

If the largestchange in behaviour is significant, then the correction thatcaused that change is included in the model . This revisedmodel becomes the current model and the process repeats .This approach relies on the effects of corrections on themodel's behaviour being nearly independent, and on theinsignificant correction assumption (see below) .AIM's search strategy through the space of possible

models is similar to a steepest ascent hill climbing search .If the effects of corrections on the model's behaviour weretruly independent, the order of inclusion of the significantcorrections would be irrelevant . However, while correc-tions' effects are nearly independent, they are not trulyindependent . This has the result that if AIM is given achoice between two correction candidates, both of whichhave a significant effect on the model's behaviour but withone having a larger effect than the other, AIM should firstinclude the correction candidate with the largest effect andreassess the model .

Whichever correction candidate is chosen, there is achance that the remaining correction candidate will nothave a significant effect on the modified model . If the cor-rections' effects are nearly independent, a correction can-didate with a large effect will continue to have a large, sig-nificant effect, ensuring that this correction is included in alater refinement of the model . However, the small effect ofthe other correction candidate might become insignificantin a later model, meaning that this correction might not beincluded ; in this case, this correction is not necessary toform an adequate model, and should not be included in thefinal, parsimonious model . This consideration leads toAIM's steepest ascent hill climbing search strategy . Byincluding early the corrections with the largest effects onbehaviour, AIM defers decisions on correction candidateswith smaller effects until their impact becomes clearer .In AIM's current implementation, behaviours are generatedby simple numerical methods and compared using an ex-tension of the integral-absolute error performance index(Palm, 1983) . However, any method of generating andcomparing behaviours will suffice for AIM's algorithm towork so long as behavioural differences can be found, or-dered, and tested for significance .

Smith 127

Rather than compare the behaviours of all variables inthe model, AIM selects a subset of variables to track, de-pending on the structure of the model and the causal direc-tion in the task definition (if the model is to explain howthe specified variables are affected, only these variables aretracked; ifthe effect of the specified variables on the rest ofthe model is to be found, all state variables and outputs aretracked) . If there are n tracked variables in the base model,of which v,(t) is the ith tracked variable in the base modeland v,'(t) is the corresponding variable in the modifiedmodel, the difference in behaviour over the period [0, r] isgiven by:

12 8 QR-98

T

f I v;'(t) - VP) I dt0

T

f I v,(t) I dtn

where the performance index is normalised to give a meas-ure of the relative difference in behaviour.To show how AIM includes these corrections in the

model, consider the simplest coherent model of the syringe(the model shown in table l, with the addition of theplunger friction correction to ensure model coherence) . Ifthe cylinder is made of steel, the remaining correction withthe largest effect is the inertia of the water in the nozzle :including this correction gives a performance index of0.0 1218 . Given a significance threshold value of 0 .1, thisis an insignificant effect . Therefore, the simplest model,excluding this correction, is deemed adequate . However, ifthe cylinder is made from soft rubber, the deformation ofthe cylinder is has the largest effect of all the correctionswith a significant performance index of 0 .2186. Followingthe algorithm in figure 4, this means that this correction isadded to the model and refinement must continue .

The Insignificant Correction Assumption

AIM uses the insignificant correction assumption to deter-mine when modelling is to stop. Modelling stops when thecurrent model, m, is adequate . m is adequate iff the per-formance index between m and the referent system 5 isinsignificant, i.e. 9(m, 5) « 1 (.x « 1 f-> x < s, where E: isthe significance threshold value) .

Turning attention to the mythical most complex modelm,, (for which 9(m.� , 5) = 0), there are some corrections rin m�, for which 4(m, - r, m.r) << 1 . Such corrections aretermed insignificant, as they have no significant effect onthe behaviour of the most complex model . I is the set ofall such insignificant corrections :

I = {r E R : 9(m �,- r, m,,) << 1 }

where R is the set of all corrections .If we assume that effects of all corrections on the be-

haviour of the model are independent, it is true that :

dates m:

9(m, - I, m~) << I

Vr E I, 9(m, m + r) « 1

9(m,, - I, m,) « 1 /~ .

which means that m is adequate when :

Results

r T

which means that m, - I is an adequate model .The insignificant correction assumption states that no

insignificant correction has a significant effect on the be-haviour of any model :

Therefore, given R(m) is the set of valid correction candi-for

Vr E R(m), 9(m, m + r) < 1 --> R(m) c IHmDM~-I--> 9(m, ma,) < IE-> 9(m, 5)< I

This allows AIM to detect an adequate model by examin-ing the effects of the remaining corrections on the behav-iour of that model. If no correction candidate has a signifi-cant effect on the behaviourofthe model, all the correctioncandidates must be insignificant ; therefore, the model isadequate .

However, the assumption that all corrections have totallyindependent effects on the model's behaviour is not whollytrue . In linear systems, corrections will generally haveindependent effects, but in order to allow for synergisticeffects between corrections, the independence relation isweakened to :

Vr E R(m), 9(m, m + r) « ~.

The value to use for k seems to be problem dependent, butan investigation of the combined effects of corrections(Smith, 1998) suggests that using k = % is reasonable .

AIM's libraries contain approximately 40 components, 40m-components, and 20 corrections . AIM has been used togenerate models for several systems, including a cascadedtank system, a heat exchanger, and a simple tachometer,each containing about a dozen components . Each systemwas modelled under a variety of different conditions ; theresults are described by Smith (1998) . However, only thesyringe system (figure 3) is discussed here .

In order to accurately describe the effect of the forceapplied by the finger, the basic model of the syringe (table1) is augmented with various corrections ; these are shownin table 2 . However, if the physical parameters in the sys-tem description are altered, different corrections will beappropriate in different circumstances . Table 2 shows howthe necessary corrections change depending on whether thefinger exerts a constant or rapidly varying force, andwhether the syringe cylinder is made from steel or soft rub-

Table 2 : Corrections added to the SyringeCorrection added to ensure model coherence .

Related Work

ber. The corrections are shown in the order in which theywere added to the model (i.e. the most significant correc-tion first) . All models were built under the same signifi-cance threshold value (E) of 0.1 .To determine whether these models are adequate, a very

complex model was built manually and the behaviours ofthe generated models were compared to this complexmodel . The results are shown in the third column of table3 . In addition, the precursors of these models were alsocompared to the complex model and the results of thesecomparisons can be found in the fourth column of table 3 .As can be seen from these results, the models produced

by AIM are adequate (S(m, m,) < 1, i. e. 4(m, md,) < E)while the precursor models are not (a(m_, m,) ~'- 0 .1) . Thismeans that AIM produced the simplest adequate, i .e . par-simonious, models .

Nayak & Joskowicz (1996) describe a typical model com-position system, whose modelling knowledge base sharesmany features with AIM . Model fragments are organisedinto hierarchies to allow the reuse of modelling knowledge,and articulation rules (similar to AIM's intra-actions) areused to infer what effects are induced in a component inresponse to a given effect . However, the modelling algo-rithm is entirely reliant on the structure of the modellinglibrary to guide the modelling process . This is achieved atthe cost of embedding modelling assumptions throughoutthe library . This is most obvious in the arbitrary nature ofthe preconditions and coherence constraints, relating toboth the behavioural and structural context ofa component,that control which model fragments can used in a model .As with all model composition systems, the limited numberof model fragments within an assumption class limits the

Table 3 : Performance Indices for Syringe Models

modeller's flexibility in selecting the most appropriate setof simplifying assumptions . Smith (1998a) shows howAIM manages to generate parsimonious models whileavoiding most of these embedded assumptions .DME (Iwasaki & Levy, 1994) uses relevance reasoning

to guide modelling . Each model fragment contains a set ofmodelling assumptions under which it is valid . As themodel boundary expands, these assumptions are assertedand retracted, which can prompt the modeller to reviseearlier decisions . DME's performance depends criticallyon the correctness of these assumptions . Iwasaki & Levydo not guarantee this is the case, and instead offer the li-brary coherence assumption . AIM's simpler knowledgebase structure obviates the need to make any such coher-ence assumptions when developing the libraries, and AIMensures the coherence of models as they are being built.

In contrast, MM (Amsterdam, 1992) is a typical modelinduction system that depends wholly on behavioural con-siderations . However, MM compares qualitative behav-iours, which greatly reduces MM's ability to resolvequalitatively similar but quantitatively different behaviours .AIM's quantitative method of behavioural comparison al-lows it to produce models more appropriate to the specifiedtask. In addition, while MM is able to correctly identifythe different behaviourally significant regions within thesystem (the lumping problem), little attention is paid to thedetermination of the model boundary.

Williams & Raiman (1994) have produced Charicatures,a radically different modelling system based on the simpli-fication of the equations that represent the most complexmodel . Their major contribution is their exploration of theconcept of a model's domain of validity, which describesthe situations in which the model can be used with confi-dence . This is used to indicate when model transitions arerequired . However, focusing solely on the equations is avery shallow approach as it ignores the physics that under-lies the algebraic formulation . This restricts the applicationof Charicatures to situations that have already been mod-elled as algebraic systems .

ConclusionsWe have described AIM, an automated modelling systemthat uses a novel architecture to provide both power andflexibility during the modelling process. AIM implementsan algorithm that does not rely on external sources of in-formation or fine structure in the modelling knowledge toguide and halt modelling . Instead, AIM compares the be-haviours of successive models, including corrections thatcause significant changes in the model's behaviour . Wehave shown that this approach to modelling usually gener-ates parsimonious models .

However, this approach relies on the insignificant cor-rection assumption and assumptions about the independ-ence of corrections . Tests on other systems have shownthat, occasionally, these assumptions are not sophisticatedenough to always ensure parsimonious models . In addi-tion, AIM's reliance on fitting approximations restricts thetypes of corrections that can be made .

Smith 129

Cylinder Force CorrectionsSteel Constant Plunger fricti on'

Plunger friction' Nozzle fluid inertiaSteel Varying

Plunger leakage

Rubber Constant Plunger friction' Cylinder deformationNozzle fluid dragNozzle fluid inertiaPlunger friction * Cylinder deformation

FRubber I VaryingNozzle fluid dragNozzle flu i d inertia

Cylinder Finger Force a(m, ms ) 9(m_, mz )Steel Constant 0.089516 -Steel Varying 0 .023991 0.451486Rubber Constant 0.035248 0 .728074Rubber Varying 0.031973 1 .01271

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