Systems Analysis Model Simul, 2003, Vol. 43, No. 1, pp.

111120
AN INTRODUCTION TO MODELS
GRANINO A. KORN*
Electrical and Computer Engineering Department, University of Arizona
(Received 3 August 2002)
We survey the origins, development, and applications of models and modeling and discuss their dramatic use-
fulness and possible dangers. We go on to present a nontechnical introduction to some basic abstract models
used in mathematics, statistics, and systems engineering. We strongly emphasize the difference between
models and things modeled and touch on the role of modeling in play, art, operations analysis, and
modern physics.
Keywords: Model; Prediction; Control; Decision; Homomorphism; Isomorphism
1. MODELS
In the Beginning
Human survival requires decision-making in a world perceived as an overwhelming
welter of different sense impressions. Of necessity, our decisions can be based only on
selected features of this enormously complicated real environment.
Next, any sort of communication even among animals requires a consensus on
sounds or gestures to be associated with selected features of significant sense inputs
(e.g. food! or enemies in sight!). Experience normally lets us assume that these
sense impressions are more or less similar for different individuals. Thoughts,
sounds, and gestures abstracted from important sense impressions form the first
simple models.
At this primitive level, humans are not conscious of modeling. Nevertheless, impor-
tant decisions flight, say, or attack do begin to be based on models rather than just
on simple reflexes.
Model Construction
Consciously or not, one constructs each model in terms of classes of objects that
abstract selected, currently significant features of the real world. Specifically, each
class of model objects is defined by abstracted relations between objects. Game animals,
*E-mail: GATMKORN@aol.com
ISSN 0232-9298 print: ISSN 1029-4902 online 2003 Taylor & Francis Ltd
DOI: 10.1080/0232929031000116371
for instance, are objects which you (another abstract object) can eat (a relation or
function).
Symbolic Models can be Communicated and Recorded
Millennia of practice refined and improved the early models. We learned to construct
more manageable abstractions for objects and relations from symbols first growls
or grunts, then words and pictures, then higher-order abstractions and mathematics.
Such models can describe essentials of complicated situations so economically that
it becomes much easier to communicate them. We can, moreover, record model
features for later use, or for use elsewhere. Symbolic models are communicable and
portable: you take measurements to buy a new door without taking your house to
the store. Rules and models, in turn, are often refined so as to simplify recording
and communication.
Models can Predict and Control
But there is much more. A model abstracting past experiences may be able to predict
the future, and thus possibly control your future on the basis of past observations.
When you perceive raindrops, your weapons may get soaked; an interest-rate increase
may prevent inflation.
Science
Much later, the scientific method added a giant step. Science validates models by
comparing predicted results with controlled observations (experiments). Scientists
infer simple models that approximate observed data. Then they try to improve
these models by omitting, modifying, or adding model rules. The resulting rebuilt
models very often suggest new experiments, and thus new experience.
Models in Your Life
Symbolic models can be dramatically useful. Social interactions are largely based
on rules for constructing and applying symbolic models (syntax and semantics of
languages). And much of our formal education is, for better or for worse, a descrip-
tion of models. Explanations of physical situations, in effect, simply specify rules
of correspondence that relate features of a model to real-world observations.
Interestingly, repeated exposure to such rules of correspondence causes us to accept
them subconsciously: we then begin to feel that we understand the situation in
terms of the model. This process of exercising rules of correspondence is called learning.
Exploring rules of correspondence can evoke an almost physical sense of satisfaction,
perhaps because it reminds us of past successes. And just as physical exercise becomes
sport, model construction which may or may not relate to real-world situations
can become a pleasurable activity enjoyed for its own sake. Exactly that is the basis
for toys and art (and thus for pure mathematics).
112 G.A. KORN
Models as Useful Property
Throughout history, skill in using models for decisions about agriculture, construction,
health, social conflicts, etc. quickly produced respect, power and wealth. Therefore,
groups of practitioners (e.g. lawyers, priests, physicians, even hairdressers) soon got
together to formalize their models and to defend their models and applications
against all comers. Such professionals often guard their intellectual property by
making their models complex, mysterious, or secret (and sometimes by inflicting
physical harm on competitors). In our own time, professional organizations justify
and defend their monopolies with standards of skill and conduct sufficiently useful
to preempt regulation by society.
Models can be Dangerous
Having come this far, we ought to stop and consider just how we ourselves interact
with different models. Above all, it is absolutely vital to note that every model exists
apart from the real things (source of sense impressions) which it models.
Unfortunately, such consciousness of abstracting [1] is, even in this twenty-first
century, not really widespread. Instead, important and familiar models tend to acquire
a life of their own. All too often, we are not conscious of abstraction and model construc-
tion and then react to word-models or images exactly as if they were real-world things.
Useful, although really quite abstract, social-interaction models (say true love, sin,
free enterprise) can produce very real and powerful emotional and physical reactions,
very much like a blow to the head does. Think of it: your own very identity as a
person appears as a variety of models, ranging from accounting records, grades, and
self-appraisal to a possible equestrian statue. The Greek philosopher Plato went even
further: he regarded physical experience itself as the mere shadow of a true or ideal
model world.
Even some scientists (and, unfortunately, most schoolteachers) confuse models
with the real world. They may, for instance, state that a theory such as electromagnetic
theory, approximates the true nature of light. But no theory really approximates
underlying physical reality. The theory call it an explanation if you must
merely describes natural phenomena more or less neatly in terms of different models.
Psychologically, a good theoretical description as simple as possible, and with lots
of correspondences between theoretical concepts and real sensations then somehow
creates a feeling of familiarity with the subject. Such understanding becomes espe-
cially compelling when we can describe the world in term of two or more models
(e.g. pictures and equations). The physical world has been explained by the different
model descriptions, but they are something quite different from physical reality.
Truly, the very process of scientific model construction, generalization, and testing
implies that models are not unique and must necessarily differ from the things they repre-
sent. Nevertheless, when a theory say classical dynamics is more or less validated
by careful observations, we often take it very seriously. And then new concepts
such as relativity or quantum mechanics can, and often do, produce a real sensation
of discomfort akin to vertigo. The newer theories were, in turn, very successful
(they resulted in such useful devices as lasers and nuclear weapons), so they are now
being taken very seriously, too. But would they fit real observations taken in a radically
changed environment, say inside of a model-inferred black hole, or during the early
stages of the model-inferred big bang?
LETTERS ON MODELING 113
A dog learns to associate the sound of a bell with food, but probably does little
conscious model construction. We ought to do better; yet the model of a thing may
be so familiar and useful that it is commonly regarded as the thing itself. This is a
dread-awful mistake. Confusing models and things has caused not only literal missile
crashes but also literal insanity [1].
On the other hand, conscious realization of different levels of abstraction is likely to
make model construction more effective (and safer). Depending on your disposition,
distinguishing important models from reality can be a shock: you may feel either
insecure or liberated (the two, in fact, go together). Far beyond this, Zen adepts
train to perceive the world, or parts of it (archery, fencing), directly and immediately,
without any abstraction whatsoever. That is a daunting (and probably impossible) task.
Everyday objectives like public sanitation would, in fact, be hard to achieve without
using models. In any case, the great Lao-tse and the wise Korzybski have said it in
chorus: the model is not the thing.
2. WELL DEFINED GENERAL-PURPOSE MODELS: MATHEMATICS
Some General-purpose Models
Since the dawn of history, abstractions from human experience have suggested special
general-purpose model constructs that seemed to apply to an astonishing variety of
practical situations. Here are a few such general-purpose model concepts; there are
many others.
1. Equivalence Two objects or relations, or actions are equivalent in some context
if they lead to similar decisions in that context. Equivalence is not necessarily the
same as equality. Mathematically, equivalence implies reflexivity (a a), symmetry
(a b implies b a) and transitivity (a b and b c imply a c).
2. Set Theory specifies rules deciding which objects belong to a class of objects.
3. Logic combines relations such as and, or, if. . . then, not. Corresponding formal
mathematical models are Boolean algebras or event algebras whose rules relate
the occurrence of certain events.
4. Countable sets are abstracted from collections of fruit, animals, people, etc.
Counting lets you order 4 shirts for 4 children. Refinements lead to models that
admit not only simple counting but also the practically significant operations of
ordering, addition, subtraction (take away. . .), a null element, simple multiplica-
tion, and simple equations. The resulting very widely applicable model objects are
integers. Counting models can be redefined to count by twos, tens, hundreds,. . .
and lead to the development of the abacus and similar computing machines.
Sophisticated generalizations later produced fractions and then rational, irrational,
and transcendental real numbers, all with vital practical applications.
5. Simple geometry models involve points, distances, directions, and maps. For
example, you can return to a landmark; you can measure a garden plot with a
length of string, and take the string with you to lay out a similar plot elsewhere.
These useful models were sometimes appropriated as priestly magic, but they
also served craftsmen and lawgivers for millennia. But consistent mathematical
formulations were approximated only in the 19th century; they are not complete yet.
114 G.A. KORN
More Abstract Models [2]
Actually, numbers and geometry already combine fairly complicated models with
multiple operations derived from many different practical tasks. Modern algebra
abstracts much simpler and more generally applicable models from numbers and
geometry. For example,
1. Groups are simple models each admitting a single class of objects (elements) and a
single operation, which is usually called abstract addition or multiplication.
Examples are groups of translations (with commutative addition), and groups
of rotations (with noncommutative multiplication).
2. Rings, fields, and integral domains have one class of objects and two operations,
usually called addition and multiplication. Integers and real numbers are the most
important examples.
3. Vector spaces, more properly called linear manifolds, admit two types of objects
(vectors and scalars), and two operations called vector addition and multiplication
of vectors by scalars. The most frequently used scalars are real numbers. Finite
translations, which can be added by the familiar parallelogram law, are good
examples of vectors.
All these models involve equality, identity elements (adding 0 or multiplication by 1
leaves the result unchanged) and inverses. These concepts let you solve an equation to
find an unknown object. This is a powerful a innovation, vital for practical applications.
Algebra deals only with finite numbers of operations. More general models,
including real numbers and most of geometry, require us to define limits of infinite
numbers of operations (e.g. sums of infinite series). By analogy with geometry, objects
are then often called points in an abstract space whose topology defines the closeness
of neighboring points. In particular, the definition of a metric space specifies the
distance between two abstract points as a real number.
Many different physical quantities (positions, velocities, momenta, forces, a-c
currents, electric field strengths) can be neatly described as points in metric vector
spaces because they admit definitions of addition, multiplication by real or complex
scalars, a definition of distance, and limits.
More recently, many newly constructed mathematical models such as game theory
have been applied to systems engineering. Because of this need for inventing new
models systems engineers tend to be more conscious of the abstracting process than
other engineers.
General-purpose Models: Representing one Model by Another
Rules of correspondence relate objects and operations of a model to sense impressions.
Different models can describe the same sense impressions, and this leads to rules
of correspondence relating the objects and operations of different models. A homo-
morphism maps the objects and operations of one model on those of another. One
speaks of an isomorphism when the mapping is one-to-one. Psychologically, tracing
such mappings (just like tracing correspondences between model and reality) again
helps us to produce a feeling of familiarity or understanding.
In this way, one model can represent another model. Such correspondences relate,
say, numbers of grain sacks to beads on an abacus. Other representations label
LETTERS ON MODELING 115
model objects like points on a map, or personnel records with sets of numbers. Useful
general-purpose models like integers, pictures, and maps have been known and used
for a long time. But it again required, literally, millennia to evolve the conscious
formulation of more powerful model correspondences or analogies, such as those
used in analytic geometry and real-number theory. In our day, most real-world items
are catalogued in terms of sets of real numbers (vectors, matrices, database entries).
These are based on counts and measurements and are readily manipulated with digital
computers.
Such model interplay and generalization produced theoretical physics (starting
with mechanics) in the 18th and 19th centuries. Better models suggested new experi-
ments in physics. This, in turn, required better numerical analysis (e.g. for finding
planetary orbits) and more general abstract models, starting with calculus and vector
analysis. Interestingly, new mathematical concepts were sometimes invented by
model generalization before physicists applied them. Tensors, for instance, were
invented before they were applied to elasticity and gravitation.
Modeling: Conscious Choice, Construction, Simplification, Generalization
How does one select models? Modeling can involve abstraction from observations, but
many models are derived from earlier models by simplification or generalization.
This involves omission, addition, or modification of definition rules. Higher-order
abstractions use models of models. Models can also be combinations of simpler models.
Many different models can serve any one application. Here are some desirable
features:
1. Simplicity, with consistent (noncontradictory) rules. Convenient representation by
numerical data (sets of measurements, counts).
2. Possibilities for generalizations suggesting new experience.
Similarity to existing models can be useful, but this can be a trap that limits our
imagination.
The consistency of the rules defining a mathematical model (constructive definition)
must ultimately be validated by an existence proof that exhibits an example descriptively
defined in terms of existing models (perhaps suitably related sets of numbers). We note
here that existence proofs for the simplest abstract models are still work-in-progress.
Toy Models
Toys are models mainly designed for fun. But toys provide fascinating insights into
the modeling process. Dolls, playhouses, and model airplanes clearly abstract signi-
ficant features of their real-life counterparts. Many different levels of abstraction
are used. Toy cars come in many different sizes, can be stationary, have wheels, add
steering, add model engines, lights; the choice of such features depends on what the
model will be used for (locomotion, table-top play, learning mechanics, etc.).
Surely, the most interesting toys are construction kits, starting with sets of blocks,
plain or in colors. Here are your model objects defined by rules about their relations
and functions. Preschoolers have much more time than adults do, and their block-
modeling activities are not so different from simple mathematics. More advanced
construction kits have interlocking blocks and then fasteners, wheels, axles, and electric
motors, all defining different modeling rules. They permit representation of
116 G.A. KORN
real-world devices on many different levels of abstraction. But even the simplest
construction kits also permit wholly nonobjective constructions like impossible
towers, dynamic sculptures, models created simply for fun a good definition of art.
3. STATISTICAL MODELS PREDICT THE UNPREDICTABLE
The well-established fact that many relatively simple models predict real-world events
(dropped china plates, planetary orbits) so remarkably well is, really, astonishing
(one person expressing such astonishment was Albert Einstein). One might say that
models have evolved to fit physical phenomena; but still, models are relatively simple
and physical phenomena are not. There are, however, also many situations where
currently available simple models do not work. We often badly need to make decisions
and cannot ordinary models simply will not do. This is, regrettably, true when
1. Measurements, and thus model predictions, are corrupted by noise. The noise may
be natural, or it can be deliberately introduced, as in gambling.
2. Currently existing models are too complicated (too many people to predict what
a population will do; too many molecules to predict what a volume of gas will
do; too many possible physical events to predict local weather or an individuals
death).
3. The rules for ordinary models at least those we have been able to construct seem
to contradict experience (atomic physics).
In such situations, statistical models can often extend our decision-making ability into
totally unforeseen realms.
Statistics
Statistics are simply numerical functions of repeated measurements (measurements
include counts). A sample is a set of n sample measurements x[1], x[2], . . . , x[n] of
the same quantity x; each sample value x[i] can also be a multidimensional set like
(x[i], y[i ], . . .). The most important statistics are sample averages (sample mean values)
hxi x1 x2 xn=n
and statistical relative frequencies (see below). All statistics can be derived from either
one of these. In particular, the sample variance
s
2
hx hxi
2
i hx
2
i hxi
2
measures the mean squared deviation of a sample of measurements from their
sample mean and therefore the spread or unpredictability of the sample. Note that
all statistics are themselves measurements, so that one can define samples of statistics
and then new statistics computed from such samples.
The epoch-making importance of statistics is based on the discovery of a remarkable
law of nature. The Empirical Law of Large Numbers states that, for most practical
measurements, variances of sample statistics decrease when the sample size n increases.
That means that suitably chosen statistics can be predictable when simple measurements
are not. Happily, statistics such as averages are often useful in their own right. They
LETTERS ON MODELING 117
determine, in particular, the actual long-term profits of entrepreneurs such as profes-
sional gamblers and insurance companies; they were the earliest direct beneficiaries
of statistics. More generally, physical measurements are usually averaged to produce
more reliable results. And modern physics does not even pretend to predict anything
except averages.
At this point, we must emphasize strongly that the Empirical Law of Large Numbers
is a physical law based on real-world observations. It is NOT derived from or by
mathematics! There exists, indeed, a Mathematical Law of Large Numbers derived
from the rules of probability theory, an abstract construct designed to model statistics
(see below). This mathematical law neatly validates the probability model but the two
laws (which are very often confused) are NOT the same.
Probability Theory Models Statistics
With increasing sample sizes, statistics like averages, mean squares, etc. seem to
approximate properties of a population (of measurements) from which the samples
were drawn. For large sample sizes, sample averages were therefore inferred to
approximate population averages; and the statistical relative frequency (the
number of times an event occurs in n trials, divided by n) hinted at a population statistic
called a probability. The early probability theory based on idealized large-sample
statistics yielded useful results. But the required limiting processes make this model,
at best, barely consistent. Modern probability theory, introduced by Kolmogorov in
the 1920s, is not defined in terms of large-sample statistics.
Probabilities are, instead, defined constructively by a set of abstract rules relating
the probabilities of combined events. The defining properties of probabilities assigned
to events A, B, . . . , are chosen so that they exactly model corresponding properties of
statistical relative frequencies. For example, if A, B, . . . are events symbolizing outcomes
of independent experiments, and AB symbolizes their joint outcome, then
ProbfA B g ProbfAg ProbfBg
Simple counting shows that statistical relative frequencies have exactly the same
property.
The theory can equally well be based on properties of idealized averages called
expected values of measurements. But most textbooks define expected values in terms
of probabilities. For an experiment with possible x-measurement values X[1],
X[2], . . . , the expected value of x is
Efxg X1 ProbfX1g X2 ProbfX2g
This definition is readily generalized for experiments with a continuous range of
possible measurement outcomes.
We can now investigate averages and expected values of descriptive-statistics
sample averages (such sample averages are also physical measurements, derived
from a sample of samples). Assuming that samples x[1], x[2], . . . drawn from
the same population have equal expected values, we derive the expected value of the
sample average
118 G.A. KORN
Efhxig Efx1g Efx2g =n Efxg
and the expected value of the sample variance
Efs
2
g Efhx hxi
2
ig s
2
=n
Abstract probability theory thus shows that the expected sample variance decreases
with the sample size n. This theoretical result (Mathematical Law of Large Numbers)
mirrors the corresponding Empirical Law of Large Numbers, which says that measured
sample variances usually decrease with n. This is a truly remarkable validation of the
abstract probability model.
Probability Without Statistics
The earliest practical applications of probability (gambling, insurance) predict large-
sample statistics. But probability models are also applied in situations that do not
admit any possibility of statistical validation. Military and business operations analysts,
in particular, routinely select tactics or strategy on the basis of probability models
that cannot possibly be tested with a large sample of actual military or marketing
campaigns. In effect, the analyst imagines a fictitious population (ensemble) of possible
campaigns and then accepts results based on ensemble probabilities even though these
probabilities predict ensemble statistics, which can never be measured! The results
are often plausible and even compelling; but the decision is not based on real statistics.
The probability model is simply an algorithm used because there is currently no
other way to arrive at a decision. One accepts such models when more or less similar
predictions have, in some sense, turned out to be profitable in the past.
Ensemble-probability models, though, have been dramatically successful in physics,
because ensemble-probability data like particle wave functions often bear on accurately
measurable population statistics like light spectra.
Statistical Models in Physics
The earliest statistical theories (e.g. kinetic gas theory, MaxwellBoltzmann statistics)
extended thermodynamics by representing macroscopic physical quantities like
pressure and temperature as sample averages over a large sample of microscopic
molecules or atoms. Probability simplifies the manipulation of such models, but they
are still basically descriptive-statistics models.
Quantum mechanics is fundamentally different. The properties of physical quantities
(positions, momenta, energy) are now modeled mathematically as expected values
(ensemble averages) over an abstract hypothetical population of possible outcomes of
an experiment. The required probability distributions can be selected to fit measured
results of atomic-particle experiments as well as classical experiments with large
bodies. Concurrently derived ensemble variances over physical quantities indicate
that certain of them (e.g. position and momentum) cannot be accurately measured
at the same time (uncertainty principles). But other model-derived quantities, such
as spectral frequencies, can be predicted with near-incredible accuracy.
LETTERS ON MODELING 119
Limits of Confidence
Statistical decision theory computes, in effect, the probability of making either a correct
yes/no decision, or the probability that the estimate of a quantity is within specified
limits, on the basis of a sample of (hopefully independent) experiments. Decision
theory is the cutting edge of statistical modeling, and it is interesting to study the
curiously tenuous connection between model and reality. Necessarily, we are once
again basing a real-world decision on a probability, say the false-alarm probability
associated with the measured value of a radar signal. This time, though, we can
repeat the experiment and measure the statistical relative frequency of false alarms.
If that is somewhere near our computed probability we are reasonably happy.
But should we, perhaps, repeat our measurement of relative frequency n times and
compute the probability that this validating measurement is between acceptable
limits? And then again measure a corresponding relative frequency, and then. . .? It is
remarkable how well our decision-making works! We conclude this essay by saying
that our acceptance of any model is a subjective decision, hopefully based on profitable
experience with the possible consequences. Inscriptions carved in stone on the
architrave of every science building ought to say, in appropriately large letters: IT
AINT NECESSARILY SO.
References
[1] A. Korzybski (1933). Science and Sanity, 5th ed., Institute of General Semantics, Brooklyn, NY, 1933.
[2] G.A. Korn and T.M. Korn (2000). Mathematical Handbook for Scientists and Engineers, 2nd Edn.
(revised). Dover, New York.
120 G.A. KORN