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 door-to-door 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 well-studied 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 TI-83 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 n-1 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. One-way 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.

 

Pollack-Johnson 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. Pollack-Johnson and A. Borchardt, Mathematical Connections, Chapter 1.