Pitfalls of Multi Method Modelling: C oncepts and Views
Abstract
In the formal modelling of problems, a variety of techniques and approaches have
been developed. Although at times this diversity has caused misunderstandings
between groups there are now many examples of dialogue between approaches and
fruitful ventures as the result. These experiences are gradually developing the
principles of a multi method form of modelling, which, seeks to analyse, accommodate
and benefit from the features of more than one modelling approach in a single study.
However, different modelling systems have different underlying principles and it is
important to recognise these in order to avoid the unsafe translation of ideas and
prejudices from one system to another. This paper looks at two seductive pitfalls:
Comparing system dynamic models with analytical differential equation models and
treating the concept of ‘randomness’ as if it were the same across modelling systems
The potential for misunderstandings is underlined with a few examples from
established literature.
1. Introduction
Modelling based on sound mathematical principles is well established in what Wigner
(1960) calls its “unreasonable effectiveness”. Modelling when applied appropriately
support the solution of problems and the understanding systems where such qualities
are lacking. Within the sphere of model construction and realisation many approaches
and techniques exist, each suggesting different ways of looking at a given problem.
As such models are now, typically, solved numerically using software they provide a
very direct realisation of the metaphor claiming computers as “bicycles for the mind”;
extending the limits of human reasoning to the questions within our conception, yet
previously beyond our reach.
There are a growing number of examples where different forms of modelling enter a
dialog in search of some fruitful co-operation. Early examples of these conversations
were often sterile or bruising affairs. Recently however, a succession of studies has
explored the relationship between different modelling methods, mapping both their
differences and areas of common interest. While this work has prompted some
interesting discussions, reshaping and transcending methodological boundaries,
modellers would do well to tread carefully. Beyond the variously distinct methods,
literature which tackles abstract modelling in general terms warns that there are
dangers in our assumptions that ideas will translate neatly between those different
worlds. Our basis for discussion will be ‘touchstone texts’, established literature for
the field, rather than the modellers opinions or prejudices.
This paper aims to explore a number of dangers, or pitfalls, which come into play
when comparing and combining modelling approaches. The first pitfall explored is
the differences in models based on their conceptualisation. To illustrate this we
compare a System Dynamics model of the type described by Sterman (2000) and an
Ordinary Differential Equation model of the type described by Boyce & DiPrima
(2005) and present how the different approaches impact what should technically be
equivalent studies. Next we focus on the concept of randomness and how it is
presented in a variety of modelling approaches. Using an inherently stochastic
problem, the tossing of coins, key underlying concepts in each method are exposed.
Additional some examples of incautious comparisons of those falling into the traps,
are examined.
2. Conceptualisation in SD and ODE models
Consider the proposition: “How can System Dynamics (SD) models and Ordinary
Differential Equation (ODE) models be compared?” To many this question is plainly
a tautology. Surely SD models are simply a form of ODE model and those who are
not aware of this are sadly under educated. Indeed, in mean company such a question
may even attract contempt or derision. Reactions also lean the other way too; ODE
and SD models are not made to be compared. One is a simulation with a graphical
form the other is simply an equation to be solved analytically or using a computer
based tool. However on closer examination the issue is not so straight forward.
ODEs as described in texts such as Boyce & DiPrima (2005) or Mesterton-Gibbons
(1995) are a fundamental approach for applied mathematicians they are used to
predict the change in relationships between quantities and are also used across a wide
range of areas including applied sciences. As well as their equation form they may
also be presented in their ‘box diagram’ form.
The system of equations for this Volterra population model above would be
represented in box form as seen in figure 1:
—_| Rit Fy
Figure 1 Box diagram for Volterra population model
Their long heritage means that they are often still solved analytically and in contrast
to SD models guidelines on their formulation are covered only briefly in the literature.
They have a plethora of classifications many related more complex variants. None the
less, Forrester (1961) states that, SD models are essentially systems of ODEs typically
with non linear properties. One might assume therefore that valid ODE models and
valid SD models would be equivalent. Such an assumption would, I suggest, be a
pitfall as it ignores the important role of conceptualisation inherent in the two
different styles.
Consider, for example, the differential equation model for the spread of diabetes in
Morocco of Boutayeb, Twizell et al. (2004) presented as a box diagram in Figure 2.
I(t) vC
———+|
Figure 2 Differential equation model from Boutayeb (2004)
1 is the diabetes incidence rate, D is the population with diabetes, C is the population
with treatable complications. Constant coefficients are used to describe the fractional
rates of exchange between populations; py is the natural mortality rate, A is the rate at
which complications are developed, 5 is the rate at which complications become
severe and untreatable, v is the rate of mortality due to complications, y is that rate at
which people with complications recover through treatment.
ies)
8 Complication
1 Disability
i a a
Diabetes on Sufferers with
Populatsion Sufferers A Complications Complication
Incidence Deaths
Successes
yi — D Natural Deaths C
5) m
Figure 3 Diagram from Boutayeb et al (2004) adapted to system dynamics form
In an experiment undertaken at the 2006 UK System Dynamics Society annual
meeting, participants were asked to look at a System Dynamics model translated
directly from Boutayeb, Twizell et al. (2004) presented in Figure 3. The original
model uses first order differential equations, the same as those underlying an SD
model and therefore the two models can be considered exactly equivalent. The
participants, comments and complaints focussed on the following issuesl:
e Lack of sophistication in segmenting populations.
e Lack of causal relationships accounting for the rates of movement.
e Lack of points of intervention; areas in the model where change could be
affected.
These observations could be explained in terms of the differences such as tractability;
The simplicity of segmentation may be related to tractability issues. Since no
endogenous feedback hypothesis is required the causal structure, that would typically
be included supporting the dynamic hypothesis of a System Dynamics model, is
unnecessary. The missing points of intervention are unnecessary in a differential
equation model as the iterative experimentation cycle, typical in a based method
simulation method is not applied. However perhaps there is something more at the
root of these observations; an implicit assumption about the purpose of modelling.
The model of Boutayeb, Twizell et al. (2004)is adequate for its stated purpose; “to
show that investment in primary health care is a necessary and cost-effective strategy
[in order] to control the incidences of diabetes and its complications”. Its model is
very concise, particularly the equation form and therefore easily communicable to an
audience of mathematicians. The intention of the model is to describe the level of
incidence rather than to investigate the dynamics of the problem or the effects of
policy changes.
The example may demonstrate a relationship between the purpose, what the model is
intended to do, the conceptualisation, which part of the problem are important, and the
process, how the purpose is realised. All three are influenced by the choice of method.
In Boutayeb, Twizell (2004) the segmentation of the Differential Equation model was
addressed in a separate section by redefining the model using partial differential
equations with age as a second independent variable. This approach has no direct
equivalent in System Dynamics and therefore could not be suggested by practitioners
of that method.
On the whole then, although the models appear to be equivalent the guidelines for
using the two different methods, in terms of how you reason about the problem,
collect and manage data differ enough the a good enough model in one world falls
short in the other. The pitfall here is to underestimate the subtle complexity the
established modes of conceptualising the problem have on the finished model. The
next section looks at how five different modelling methods manage to incorporate the
concept of randomness and into their models.
3. Considering the Role of Randomness
A typical division when considering the properties of modelling systems, is made
between the stochastic and the deterministic methods. Conventional wisdom would
suggest that such categories make it easier to choose between methods, or judge
approaches for their compatibility in advance of their use. However the role of
randomness in modelling varies considerably. A closer examination of the issue in
each case reveals some underlying principles different systems of modelling. In this
section we shall consider five different approaches Differential Equation Modelling,
System Dynamics Modelling, Stochastic Process Modelling, Econometric modelling
and Discrete Event Simulation modelling.
Differential Equation (DE) Modelling
A superficial assessment of DE modelling, typically classified as a continuous,
deterministic approach, finds that the approach does not use randomness in models.
This perception may be because classical DE models are often applied to problems
that are intrinsically continuous and deterministic, however differential equations, as a
field of mathematical study, are very diverse. A special class of DE, Stochastic
Differential Equation (SDE) models, include at least one stochastic process term in
their definition. The role of the stochastic term is typically to model error or variation
from the deterministic value of the equation (Gard 1988, @ksendal 2003).
Additionally difference equations, which model similar patterns of behaviour using a
discrete time step, are well established and (Edwards 2001) among others
demonstrates their use although their appearance in modelling texts is less frequently.
The majority of DE models used in teaching, and perhaps in practice, are classical DE
models rather than difference equations or SDEs.
Classical DE modelling is often applied to problems, such as motion, mechanics and
other applications in the physical sciences and was originally, as (Boyce, DiPrima
2005) notes, the main reason for their development. In these applications continuous,
deterministic behaviour is satisfactorily described by continuous deterministic models.
However for models where uncertainty, or human agency, is present such as
population growth, epidemics and similar problems, continuous deterministic
descriptions appear inadequate. For example the problem of whether an infectious
illness is caught on exposure appears to be a discrete, probabilistic problem.
The use of classical DE models in such problems rests on what (Dym 1980) call the
“continuum hypothesis’ whereby the model is formulated so that the main subject is
treated as a field or continuum regardless of its observable composition. Using the
example of a traffic flow analysis model (Dym 1980) details some of the requirements
and restrictions the continuum hypothesis places on the formulation of the model. For
example measures of traffic flow, and traffic density are in the model are based on
aggregate values with appropriate scaling to maintain the continuum hypothesis. The
DE model in this case has a necessarily macroscopic outlook. With the continuum
hypotheses established, by virtue of the Strong Law of Large Numbers, a probability
distribution is replaced by a flow equivalent to the mean value of the distribution.
The Coin Tossing Problem
To illustrate the role of randomness in different systems of modelling consider an
inherently stochastic problem, such as tossing coins. Imagine a scenario where coins
arrive at a rate t. The probability of a coin showing heads is p and the probability of
showing tails is g. Those that show heads are placed in basket H and those showing
tails placed in basket T. Models can be made in each system to investigate the
contents of the baskets H and I after a fixed period of time t has elapsed, assuming
that both assuming p and q have the value 0.5.
The DE model for the Coin Tossing problem has the form d in Figure 4:
, Hit)
dH dT :
a ee H(t)= pt T(t)=qt
a} a! (t)=pt T(t)=q
“ TH
Figure 4 Differential equation model of the Coin Tossing Problem
In accordance with the continuum hypothesis, assuming p and q to be 0.5 does not, in
this case, assert that all the coins are fair, rather that on average they are fair.
System Dynamics (SD) Modelling
SD models have a strong relationship with classical DE models and in practice
representing randomness in both approaches is similar. The effect of variations
considered random in the micro scale may he averaged into flows as with DE models.
For example modelling the coin tossing problem using SD constructs (Figure 5)
appears very similar to the DE approach. Both rely on the continuum hypothesis
although in the case of SD implicitly so.
OOH
P
Figure 5 System dynamics model of the coin tossing problem
Randomness is rarely included explicitly in models and the reasons why capture some
important properties of the approach. Because of the way SD models are structured
where, random inputs to the system do occur they are often smoothed out by the effect
of delays and aggregation (Forrester 1961). SD modelling also assumes an
endogenous feedback-based hypothesis about the behaviour of the problem. The
premise is that the behaviour of the problem can be explained by the interaction of
internal variables. Complex system behaviour may be explained through the non-
linear dynamics of a model’s feedback structure whereas such behaviour may be
described as randomness in the absence of any other available explanation. The
endogenous feedback hypothesis favours the inclusion of a closed information
feedback loop in the model, including as many of the causal variables as necessary to
understand the behaviour of the problem. As a result (Sterman 2000) argues that
genuinely random behaviour is uncommon: “Many variables appear to vary
randomly. In most situations, randomness is a measure of our ignorance, not intrinsic
to the system.... When we say there are random Variations in , say, the demand for a
firms product, what we actually mean is that we don’t know the reasons for these
variations ...people tend to call the residual random as if the customers were somehow
rolling dice to decide whether to buy the product”
However, the effect of exogenous randomness is treated differently. After the
feedback properties in the model have been established (Forrester 1961) recommends
the use of randomness or “noise signals” in the system inputs to test the its robustness.
(Sterman 2000) describes how noise can be used to excite the latent dynamics in a
model or unfreeze a system stuck in local optima.
Stochastic Process (SP) Modelling
It may fairly be claimed that randomness is a pre-requisite in order for SP models to
be used effectively. The form and structure of SP models is diverse, discrete and
continuous forms are equally common and SP models can also consider the stochastic
properties of individual occurrences as well as macro level trends. However in all SP
models randomness is the key generating principle by which problems are described,
represented and solved. An integral part of the approach is to characterise and bound
randomness in the behaviour or state of the problem over time. The use of SP
modelling in uncovering deterministic properties of stochastic problems is a common
application. Mesterton-Gibbons (1995) and Bunday (1986) demonstrate the use of
calculus and SP modelling find the underlying stationary distributions of some SP
models.
Uncertainty, or randomness, in the problem is modelled by random variables in the SP
model. As Edwards (2001) describes, an appropriate pattern or distribution must be
assigned to each variable in the model building process. Once constructed the SP
model may provide incite into three key issues; What are the possible behaviours or
states for the problem? What is the probable behaviour or state for the problen? How
probable is a given state or behaviour?
In discrete models, without the continuum hypothesis restriction, it is relatively easy
to model micro level uncertainty. For example for in the Coin Tossing problem the
behaviour of an individual coin can be described at micro level using a simple
Markov chain model (Figure 6);
Figure 6 Markov chain model for tossing an individual coin
However the Coin Tossing problem as originally presented could be modelled using
the binomial distribution. This is a discrete probability distribution that models events
with two possible outcomes.
H(t) ~B(t,p) T(t) ~B(t.q)
t!
P(H (t) == Fw!
H (1000) =tp=500
p'qh®
Based on the original values of p and q the model expects 500 occurrences of heads
from 1000 instances of tossing the coin. It is possible to test the validity of the model
using statistical methods using samples from the population if necessary. Notably in
this model the assumption is that all the coins are fair. In the case of the Coin Tossing
problem, with certain limits, the continuous normal distribution it is considered an
acceptable approximation to the discrete binomial distribution creating a continuum
hypothesis for the model.
Econometric Modelling
Most principles of economic theory are deterministic, describing a concrete
relationship between variables that behave, ceteris paribus, as described by theory.
Begg, Fischer et al. (2005) provides the example of the Keynesian macro-economic
consumption function relating spending, with household income.
C =A+cY
Where C is total Consumption, A Autonomous consumption, c the marginal
propensity to consume and Y household income.
However econometric modelling and econometric methods, integrating theory with
data and statistical techniques is very diverse with many variations available to the
modeller. All econometric models are based on observed data and the role of
randomness in data analysis is central, if occasionally not fully acknowledged; many
courses in econometric modelling require advanced study of statistical distributions
and probability theory as a prerequisite. Structural models must account for
uncertainty in the relationship between the sample and population data. An analysis of
sampling theory is beyond the scope of this study however Cuthbertson, Taylor et al.
(1992) and Greene (2003) address how properties of sample data affect the estimates
of the B coefficients in regression models.
Structural models also must account for uncertainty, usually called emor or
disturbance, in the accuracy of the observations. Cuthbertson, Taylor et al. (1992)
describes the structure of a Classic Linear Regression Model (CLRM) of the
Keynesian consumption function.
Vi =A + AoX+e
Where the ¢; term represents the quantity of error.
The most common approach in structural models, such as CLRMs, is to consider this
error as stochastic, belonging to a normal distribution with a constant finite variance
and a mean of value of zero. Like the other structural variables error terms are
considered to be independent of each other and non-auto correlated. Greene (2003)
examines the statistical principles that support these assumptions and examines the
issues of auto correlated error in the context of time series.
Cuthbertson, Taylor et al. (1992) describes time series data, used to create
econometric models, as the realisation of a stochastic process. As such, time series
models such as AR, MA and ARMA models, described in Chapter 3, propose a
structure for a stochastic process that produced the time series data. AR models
consider the effect of autoregressive stochastic elements over a specific number of
time periods. MA models consider variation on a mean value plus stochastic elements
over a specific number of time periods. ARMA models include both properties. Terms
in the proposed models still require residual coefficients to be calculated form
observed data.
Although stationary time series models are the most common use of stochastic models
in econometrics texts (Greene 2003) also includes non stationary processes such as
random walks. Few texts advise under what circumstances such models are necessary
however. The use of time series modelling in econometrics is a method for focussing
solely on behaviour rather than cause or correlation they are therefore used analyse
variables with poorly understood or innumerable influences, such as stock prices.
The practical use of randomness in econometric models then is either as a tool to
account for error or as a method to analyse data without proposing a causal structure
for the underlying behaviour. Modelling the Coin Tossing problem using
econometrics requires data observed by tossing a real coin and the proposal of a
model that fits the observed behaviour. (Cuthbertson, Taylor et al. 1992) describes an
equivalent problem where observed data is used to produce the Maximum Likelihood
(LM) of the overall behaviour of the coins.
It is hypothesised that the data belongs to a binomial distribution. The total number of
heads observed is H and the unknown probability of a single observation of heads is
Il. From the definition of the binomial distribution the probability, P, of H
observations in n tosses is given by.
n!
P= _.
H!(n—H)!
Based on experimental observations where 518 occurrences of heads in 1000 trials.
(Cuthbertson, Taylor et al. 1992) suggests that the modeller may experiment with
values of IT as 0.1, 0.2, 0.3 etc in order to find the value that given the highest value of
P based on H=518 n=1000 however demonstrates that the LM, can be found by taking
the first derivative, equating it to zero and simplifying to give.
m40-m#
Discrete Event Simulation (DES) Modelling
For DES models, by contrast, stochastic behaviour is a key generating mechanism.
DES modelling is supported by the assumption, that accurate imitation of the problem
is sufficient to understand its behaviour and randomness, real or metaphorical, has a
direct role in both the intemal description of the model and the overall behaviour of
the problem. Although software is a flexible medium and DES models are able to
include any behaviour that can be encoded, deterministic or stochastic, from the
literature implementing random behaviour is clearly central to DES modelling. The
description and implementation of random behaviour within the DES model is the
subject more than 70 pages in (Banks 2001) and almost 200 pages in(Law, Kelton
1991).
The implementation of randomness in DES models is distinct to that in SP models.
Whereas SP models conceptualise randomness, DES models actually recreate the
random behaviour. DES models reproduce the behaviour rather than just describe it.
The behaviour of the elements within the problem, described by random processes, is
imitated in the model. The generated behaviour is then analysed to understand how
the elements interact to create the overall behaviour of the problem.. In many ways
randomness in DES models could be considered the inversion of the econometric time
series approach; Rather than starting with data, proposing structure and analysing the
relationship in DES stochastic structure is proposed and integrated, data sets are the
then generated and these are analysed, using statistical methods, to understand the
overall behaviour.
A DES model of the coin tossing problem, created using Arena, simulated the tossing
of individual coins, randomly assigning values of heads or tails. Each run of the
model, using a different set of random numbers, produces a different outcome and
over successive runs collecting and statistically summarising the data the original
distribution is confirmed.
Coin Tossing Experiment
Ls
450 4
Occurance of outcomes
Tae je H 518
[ar 482
Figure 7 Discrete event simulation model of the coin tossing problem.
Establishing distributions in the problem and choosing appropriate distributions to
generate behaviour in the DES model is therefore very important. (Law, Kelton 1991)
provided examples of 18 different probability distributions that may be used in
models.
An insufficient knowledge of the workings and literature of other modelling methods
can lead to confusion and increased risk of modelling failure. Unfortunately poorly
formed models produce results just as the well formed ones do, the difference is their
usefulness. Our final excursion looks at the kind of mistakes it is possible to make if
your view of other modelling methods is developed without sufficient reference to
that systems core principles and techniques. They may cause the practitioner to
misfomulate the model misread the results and miss important truths in the
comparison.
4. Conceptual Traps for the Unwary
Modellers of different backgrounds are often prone to misunderstanding other
approaches because of misconceptions of how those methods really work. Most
modellers will have at least a superficial knowledge of other approaches and this of
itself may be the cause of some confusion. (Meadows 1985) suggests that common
misunderstandings, and conflict, in comparing models and modelling approaches
stems from an unconscious tendency to judge the models and methods of one system
according to the principles and assumptions of another. For example, for practitioners
using methods where a significant amount of data is collected prior to modelling,
models formed from conceptualised relationships may seem to lack an evidence base.
For practitioners of methods that emphasise causal mechanisms, models based on
randomness may seem less than convincing.
It is also possible that the full range and ability of an approach is not recognised even
by its practitioners. Structural econometric models, for example, are sometimes
portrayed as proposing linear relationships between the independent variables of the
problem however, as (Greene 2003) makes clear, the underlying theory requires that
only a linear function of the variable is compared, provided that the number of
observations tested is sufficiently large. Other statistical techniques may be used to
relax the assumptions of the simplest forms of linear regression. Such misconceptions
are perhaps understandable in observers from other methods however it s not
uncommon for practitioners to not fully understand the capabilities and assumptions
of their own method. (Meadows 1985) refers to as the “selective blindness” of
working within a particular system.
Flawed Analysis, Misleading Conclusions
In some cases a superficial appreciation of a particular method, coupled with analysis
based on the values of a different one, are the cause of the misleading conclusions.
(Atherton, Borne 1992), a general reference work on modelling and simulation
discuss system dynamics during the entry for ‘Ordinary Differential Equations’.
“Biologists and Sociologists [...] are ofien not well trained in numerical mathematics.
For such individuals Forrester developed his method of rates and levels”.
This is apparently a misunderstanding; arguably rates and levels have a more
significant role in system dynamics modelling than simply to make up for a lack of
mathematical training on the part of the modeller. Many relationships have non-linear
properties which require considerable skill to conceptualise and solve mathematically.
Rates and levels provide a form of conceptualisation that allows the modeller to focus
on the qualities of the problem rather than the mathematical detail of the model. They
also enable actors in the problem, who may not be trained in mathematics, to
contribute to the construction and verification of the model.
On reproducing the method from (Forrester 1961) for calculating level equations
(Atherton, Bome 1992) states “This is obviously nothing but a reformulation of
Eulers’ integration. However, persons with weak mathematical background seem to
be more at ease with the terms rate and level than with the term differential equation”.
The failure of (Atherton, Borne 1992) to recognise any benefit of the approach, aside
of an avoidance of complicated mathematics, is apparent. The use of Euler’s
integration, as the simplest and most well known numerical approximation, appears to
play a role in this reasoning. Students of differential equations are well aware of its
weaknesses and the relative strengths of more accurate alternatives such as Runge-
Kutta (h5). (Boyce, DiPrima 2005) devote a whole chapter to comparing Euler and
other numerical solutions to differential equations. Although it is not certain (A therton,
Bore 1992) may be reflecting on the choice of an elementary numerical
approximation as evidence of mathematical naivety. However, the position of
(Atherton, Bome 1992) overlooks that many system dynamics modellers are aware of
other approximations and indeed the short comings of the Euler method. They offer
the following reasoning for its use “Jn models of social and human systems the errors
in initial conditions, parameters and especially model specification are large and the
data [the model may be compared with] are often corrupted by significant
measurement error. In such cases Euler’s errors are inconsequential” (Sterman
2000). The Euler method is not integral to the system dynamics approach and most
modem SD software offers a choice of numerical algorithms including both Euler and
Runge-Kutta (h5). In the choice between methods it is left to the modeller to prioritise
model execution time or accuracy of calculation.
On the limitations of the Euler method (Sterman 2000) states “Euler integration is
simple and adequate for many applications. However there are some systems and
some model purposes, particularly in engineering and physics where Euler is not
appropriate”. The comments of (Atherton, Borne 1992) may be based on the
assumption that physical problems are the primary application area for both systems
of modelling however this is not the case.
Conflicting Assumptions
In discussing the problem of using of expected or mean values in a model rather than
considering the stochastic behaviour (Bartholomew 1973) states “Such calculations
tell us, in an average sense what would happen if the model were allowed to operate
and in this sense may be said to simulate the process...extensive use has been made of
[such simulation techniques] by Forrester and his colleagues at the Massachusetts
Institute of Technology”. The author appears to overlook two critical assumptions of
in system dynamics modelling; Firstly, the Strong Law of Large numbers is deemed
to apply and on that premise a distribution may be substituted by its mean. Secondly,
in system dynamics modelling the basis of the dynamic behaviour is the complex
causal relationships rather than stochastics. It is assumed, implicitly, that the
phenomena of interest are behaviours produced by the interaction of causal trends and
therefore simulation “in an average sense” is an appropriate way to examine them.
The methods of (Bartholomew 1973) by contrast implicitly assume that phenomena of
interest are produced by the interaction of stochastic behaviours in the problem.
The cases cited above demonstrate that, whatever the facts unguarded comments may
live on to haunt their owners, if nothing else live on. The essence of the multi method
approach is that understanding concepts from other methods improves one’s own
practice. Rash comments are as much a pitfall as misunderstanding the problem in this
case.
5. Conclusions
The opening section discussed the promise of increased fruitful dialogue between
methods developing to establish a diverse skill set for modellers based on multiple
perspectives The key role of conceptualisation was identified as a discriminating
factor in transferring knowledge, skills and good practice. Expanding the range of
techniques we saw how random features of a problem were incorporated into different
models, suggesting different uses according to the properties of the problem. Finally
we looked at how even the most experience and senior practitioners can be caught out
if now fully aware of the quality of the comparisons they are making. Although multi
method approaches are still gaining credibility there are already some guidelines we
can identify to avoid the most obvious mistakes.
Future work in this area would include the exploration other areas such as how
different systems, use data and the role of iteration in the development of models.
Techniques currently gaining popularity such as agent based models would also be
interesting for comparison.
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