Different Models for Different Purposes
Introduction
There are many ways to
make models of the real world, mathematics being only one of them. A road map of a city is a model. It can show many features of a city. It can show the number of parks and libraries. It can show locations of schools. It is useful for planning walking or driving
routes through the city. It would be
almost useless to an individual doing doortodoor sales. A different kind of model, with demographic
information about social and economic characteristics of neighborhoods, would
be needed.
In what sense is a
model airplane a model of an airplane?
What real world phenomenon is modeled by a zoo? A photograph?
Models are constructed
to emphasize the visibility of the characteristics that are to be studied. Some factors will be omitted while others are
stressed. The kind of information one
wishes to obtain must be the primary considerations in selecting the type of
model to use.
How can mathematical models be used?
Science begins with
some observations about the real world.
It then wishes to draw some conclusions about the situation
observed. This can be accomplished by
conducting experiments and recording observations by measurements.
But if reliable
patterns appear in the measurements, it may happen that these patterns can be
abstracted to a mathematical model. Logical arguments can be made about the
statements in the mathematical model; the conclusions about the mathematical
statements can be translated to statements about what might be observed in the
real world. The conclusions about the
model can then be used to forecast and predict and explain what has happened
and might happen in the real world.
Schematic diagram of the modeling process[1]
To be useful, the
mathematical system should predict conclusions about the world that are
actually observed when appropriate experiments are performed. Frequently, some predictions of the model are
supported by experimental observations while others are not; this feedback is
then used to modify the model to improve its accuracy. Sometimes, incorrect predictions suggest ways
of rethinking the assumptions of the model.
Modeling can therefore be a process that produces increasingly accurate
tools for description and prediction.
Different types of
mathematical models[2]
Data driven,
difference and differential models are deterministic
models. The basic assumption behind
a deterministic model is that the entire future behavior of a system is exactly
and explicitly determined by the present status of the system and the forces
acting on it. If the states
of each of the component variables and the relationships between them are
known, then the behavior of the system at any instant in the future can be
predicted. This assumption is not well
supported in detailed study of social systems, from many viewpoints. But the certainty assumption is even a
drawback in the study of physical processes, due to the Heisenberg uncertainty
principle. Heisenberg showed that it is
impossible, even in theory, to know the exact state of a physical system; the
act of observation itself changes the system.
We can conclude from deterministic models only
the average behavior of the components of a system.
This led to the
introduction of probabilistic models. These are also predictive models. The basic assumption of such models is that
the system under investigation can occupy one of several different possible
states at each moment, with different probabilities (or a continuum of states
with a known probability density function).
If the probability distributions governing the system at the present
moment and the forces acting on the system are known, then the probability
distribution for the system at subsequent times can be predicted.
Since deterministic
models often provide good approximate predictions, and since they employ the
wellstudied tools of calculus and differential equations, they still are
widely used in the physical as well as life and social sciences. However, calculus was largely developed to
help solve physical problems; there is no intrinsic reason why it should be the
appropriate tool for the investigation of biological systems, even if a deterministic
approach is assumed. Other mathematical
tools, such as linear programming and graph theory, have been used in recent
decades to analyze such systems.
The deterministic and
probabilistic approaches are both predictive in nature; they both aim to say
something about the state of a system at a time when it is not being
measured. These types of models are
contrasted with primarily descriptive or axiomatic
models. Examples of such models are
common in geometry and decision theory (models for voting that satisfy certain
fairness restrictions).
Some guidelines for
developing and refining simple deterministic models
A mathematical model
can be thought of as consisting of three parts:
·
Definitions  verbal descriptions of the
variables;
·
Functions  mathematical definition of
the relationship(s) between the variables;
·
Assumptions  the usual assumptions of
certainty (the model gives the exact relationship between the variables) and
continuity (the variables assume a continuum of values).
When fitting a model,
always consider first the sense of the data and the purpose of the model. The reasoning one applies to expectations of
the relationships between the data give a best first approximation for a model.
Technology has
afforded wonderful tools for finding a best fit curve of a given type. Hand held calculators such as the TI83
provide best fit analyses for linear, quadratic, cubic, quartic, exponential,
logarithmic, power, logistic, and trigonometric functions. Simple graphics programs for the PC provide
these as well. Any spreadsheet will do a
best fit regression analysis to show a trendline of various types as well.
In making decisions
based solely on the data, choose the simplest model that seems appropriate to
the situation. Any set of n data point coordinate pairs will
determine a unique polynomial function of degree n1 exactly. (The exact fit
of the polynomial does not confer optimality of it as a model!)
In selecting a model
based on a set of data point coordinate pairs[3],
·
begin with an autoscaled scatter
plot. The plot will often reveal general
characteristics that point the way toward an appropriate model.
·
If the scatter plot does not appear to be
linear, consider the suggested concavity.
Oneway concavity (up or down) often indicates a quadratic; concave up
often suggests an exponential model. If
input values are evenly spaced, then second differences that are nearly
constant indicate a quadratic model, while percentage changes that are nearly
constant suggest an exponential model.
·
When a single change in concavity seems
apparent, think in terms of cubic or logistic models. Remember that logistic models tend to become
flat on each end, whereas cubics do not.
Also, logistic models are suggested in cases where the data is
cumulative over time,
with bounds on the extent of the accumulation.
Never consider a cubic or logistic model if you cannot identify an
inflection point.
PollackJohnson and
Borchardt[4]
provide the following table for selecting a model based on scatter plot.
Seven
Basic Types of Models
Model
Type 
General
Shape 
Concavity 
Inflection
Point 
constant 
General
Form 
Linear 

None 
0 
1st
Diff. 
_{} 
Quadratic 

Up
or down 
0 
2nd
Diff. 
_{} 
Exponential 

Up 
0 
%
chg. 
_{} 
Power 

Up
or Down 
0 
%
chg y / % chg x 
_{} 
Cubic 

Up
& Down 
1 
3rd
Diff. 
_{} 
Logistic 

Up
& Down 
1 
NA 
_{} 
Quartic 

Three

£2 
(4th
Diff.) 
_{} 
[1] adapted with admiration from M. Olinick, An Introduction to Mathematical Models in the Social and Life Sciences, Chapter 1.
[2]adapted with admiration from M. Olinick, An Introduction to Mathematical Models in the Social and Life Sciences, Chapter 1.
[3] from LaTorre et al, Calculus Concepts,Chapter 2.
[4] B. PollackJohnson and A. Borchardt, Mathematical Connections, Chapter 1.