Why Use Models?

In my previous post, I defined a model as one entity that represents another and can predict some properties of the second.  The ability to predict is the most important aspect of a model.  It indicates not only how to use a model, but also to measure how good a model is.

The question of whether to use models is really somewhat silly, because we innately use hundreds of models every day.  There is a model relating to which hand will move and how far, according to neural impulses sent from you brain.  There is a model of what will happen when you turn the tap.  You have a mental model of the roads in your community and how congested they will be as a function of time.  Your world is filled with models. 

So the question in the title is really more about understanding what models are, so one can use them purposefully.  Whether to use a model comes back to the definition: What are you trying to predict?  How will it benefit you to know the answer before you run the experiment? 

There are three main reasons for using a model:

1. To make predictions where direct experimentation is impractical;
2. To generalize the results of experimentation; or
3. To abstract and generalize other models.

These approaches to modelling reflect roughly the disciplines of engineering, science and mathematics, in that order.  An engineer will use a model if it helps him get the job done.  A scientist is in the business of producing models without necessarily concerning himself how it is used.  The mathematician is the same in that respect, but constructs models in the absence of experimental data.

My description of the engineer provides a testable assertion on when one should use a model.  If it is more practical (i.e. cheaper) to make a direct measurement, then developing and using a model would be a waste of resources.  But there are myriad instances where building and using models is cheaper than running the experiment directly.  If someone had thought to put a scale model of the Tacoma Narrows Bridge (rev. 1) in a wind tunnel, that would have saved a pile of money.

Science is interesting, because it produces models without necessarily wanting to use them.  The ability to make predictions is certainly worth money.  By itself, the raw data obtained from experimentation is hopefully worth more than the effort required to obtain it.  Sufficiently dense sampling spanning the region of interest might support the interpolation of any point of interest.  But such a model is verbose and therefore not very efficient.  A terse summarization of experimental data makes it much more useful.  (More on this in "How good is a model?".)

Early mathematicians examined the commonalities between models to produce abstract models unconnected with experimental data.  The discipline evolved to the creation of models for which there may not be any practical application.

The observation that scientists are more-or-less in the business of producing models suggests another important use for models: To convey a description of an entity from one person to another.  Since this involves communication, a secondary criterion for the quality of a model is its conciseness.

We may want to distinguish using a model from merely transferring it: A person receiving a model may want to run the model to make predictions, and that constitutes its actual use.  Merely sending or receiving a model doesn't really amount to using it.

To summarize, the decision of whether to use a model (as opposed to direct measurement) should be driven by economic considerations.  The model is superior if it provides a prediction (i.e. an answer) more cheaply than direct measurement.  In cases where direct measurement is impossible, its cost can be taken as infinite -- in which case using the model is clearly advantageous.

How Do You Use a Model?

A model is one entity which represents another, and enables some predictions about the latter.  Its ability to predict is only realized by exercising it.  A complete model contains both a representation and a process.  The process gives an indication of how the model can be used and also how it can be abused.

My plastic model F-18 is a good dimensional representation of the aircraft.  The knowledge that it is a 1/72nd scale model is an important part of the representation.  But lacking that knowledge, one measurement from the real aircraft would provide the needed calibration.  To make a prediction about the dimensions of a part of the real airplane, I can measure the model with calipers and multiply the result by 72.  The representation part of the model is the glued-together collection of plastic parts.  The procedure for producing a prediction is to measure the part of interest and multiply by the scale factor.

Since the model is dimensional, such geometrical predictions ought to be acceptably accurate.  On the other hand, the model airplane is not intended to be an aerodynamic model.  Launching the plastic model like a paper airplane would lead you to conclude that an F-18 would never get off the ground.  The representation is the same, but the procedure is not right for this particular model.

To obtain good predictions about how an F-18 will fly requires either a better representation or a different procedure.  Perhaps placing the plastic model in a wind tunnel would yield better results.  The faster airflow might more closely simulate the forces the real plane experiences in flight.  It should be clear that models may be easier to abuse than to use correctly.  Understanding how to run a model is at least as important as the model itself.

Using a simple model may be as easy as porting a few measurements from the example to the model, running the model and reading out the result.  The measurements supply calibration or initial conditions that allow the model to make an accurate prediction.  On the other hand, models may be complex enough that they can answer many different questions.  The question you intend to answer must be properly framed for the model's result to be of value.  The model's intended use must be specified before it can be determined whether the chosen model is appropriate or not.

If a model is an analog of the example, then performing an experiment on the model is analogous in the same way to performing an experiment on the example.  One can consider the framing of a question and the exploration of the model to be an application of the scientific method to the exploration of the model.

The mapping between the example and the model often involves a number of assumptions.  When examining the results of a simulation, and mapping these back onto the example, it is important to bear these assumptions in mind and check their validity.  Assumptions or approximations may reduce the accuracy of the model or its results, so calibration against the example is important.

Therefore, I conclude that the complete description of a model must include the instructions for its use.  I noted that how a model is used can be influenced by the answer one is seeking, so a model may be incomplete until one devises and performs an experiment upon it.

What is a Model?

The term 'model' is used extensively in Computer Science, but is often left undefined by the author.  I intend to write a series of essays discussing the theory of modelling abstractly, so a natural first step is to provide a working definition.

A working definition should comprise a testable description of what constitutes a model.  The description may also suggest how a model may be used.  As we shall see, this description of a working definition also makes it a model.  The definition of a model has become recursive; Computer Scientists should feel right at home.

A dictionary is often a good place to start.  Websters' first two definitions [1] are best suited to my purposes:

1. A miniature representation of a thing, with the several parts in due proportion; sometimes, a facsimile of the same size.
2. Something intended to serve, or that may serve, as a pattern of something to be made; a material representation or embodiment of an ideal; sometimes, a drawing; a plan; as, the clay model of a sculpture; the inventor's model of a machine.

In Wiktionary [2] Definitions 2, 3, and 5 are close to the meaning I ascribe to the word, being rendered respectively as:

•   2. A miniature representation of a physical object;
•   3. A simplified representation used to explain the workings of a real-world system or event; and
•   5. The structure design of a complex system.

These definitions are close to the mark, but they do not come up to my expectations for a working definition, since most are based on example.  The main page for "model" on Wikipedia [3] lists some 46 ramifications, which give domain-specific examples of models.  The entry for "conceptual model" [4] provides a suitably abstract starting-point:

  ... a model is anything used in any way to represent anything else.

That is the first half of my favorite definition for a model, but it lacks an indication of how a model may be used.  The other half is that the model can be used to make a prediction about the entity being represented.  This is essential, because it describes why one would want to have a model in the first place.

With that, I can put forth my definition of a model.  If you trust me, you can stop there.  My intent is to spend the rest of this blog providing examples and defending the accuracy and usefulness of the definition.

A model is one entity which represents another, the first predicting characteristics of the second.

The first thing to point out about this statement is that the two entities are not necessarily distinct.  Often the best way to make a prediction about how something will behave is to just run the experiment.  Development of a model may not so much involve seeking an abstraction among concrete objects as generalizing the characteristics of a well-known exemplar to encompass the larger concrete set.

A second observation is that the statement does not stipulate that the model is simpler than the example.  (For simplicity henceforth, I'll refer to the first entity as "the model" and the second entity as "the example".)  Neither does it require that the example is real or physical or already in existence.  If you imagine a design tool connected to 3-D printer, the model presented by the tool can be far more complex than the part that is produced, and the model certainly exists before the part does.

We also note that the statement leaves aside the notion of quality.  Models can be good or bad; they can be excellent with respect to some characteristics and awful with respect to others.  The quality of a model depends on how well it predicts the characteristics of the example.  The applicability of a model depends on whether the set of characteristics it predicts well matches up with the set of interest.

An amorphous blob of clay is a pretty poor model for a skyscraper.  They both take up space; they might be of similar color; that's about it.  The plastic model I have of an F-18 is an excellent model if my interest is in the relative dimensions of various parts of the airplane.  Aerodynamically, it's a very poor model: It flies like a rock because its scale weight is a dozen times that of the real plane.  It's a poor model structurally as well: all of the interior bracing was omitted in favor of ease-of-manufacture.  A full-scale version could not stand up under its own weight.


This blog presents my working definition of a model.  In the next entry, I'll explore why models are so important.


References:
[1] machaut.uchicago.edu?resourc=Webster's&word=model&use1913=on
[2] en.wiktionary.org/wiki/model
[3] en.wikipedia.org/wiki/Model
[4] en.wikipedia.org/wiki/Conceptual_model