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Angela B. Shiflet and George W. Shiflet:

Introduction to Computational Science

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MODULE 1.2

The Modeling Process

Introduction

The process of making and testing hypotheses about models and then revising designs

or theories has its foundation in the experimental sciences. Similarly, computational

scientists use modeling to analyze complex, real-world problems in order to predict

what might happen with some course of action. For example, Dr. Jerrold Marsden, a

computational physicist at CalTech, models space mission trajectory design (Marsden).

Dr. Julianne Collins, a genetic epidemiologist (statistical genetics) at the Greenwood

Genetics Center, runs genetic analysis programs and analyzes epidemiological studies

using the Statistical Analysis Software (SAS) (Greenwood Genetics Center). Some of

the projects on which she has worked involve analyzing data from a genome scan of

Alzheimer’s disease, performing linkage analyses of X-linked mental retardation fami-

lies, determining the recurrence risk in nonsyndromic mental retardation, analyzing

folic acid levels from a nutritional survey of Honduran women, and researching new

methods to detect genes or risk factors involved in autism. Scientists in areas such as

cognitive psychology and social psychology at the Human-Technology Interaction

Center of The University of Oklahoma perform research on the interaction of people

with modern technologies (Human-Technology Interaction Center). Some of the stud-

ies involve “strategic planning in air trafﬁc control” and “designing interfaces for effec-

tive information retrieval from collections of multimedia.” Buried land mines are a

serious danger in many areas of the world (Weldon et al. 2001). Scientists are using a

combination of mathematics, signal processing, and scientiﬁc visualization to model,

image, and discover land mines. Lourdes Esteva, Cristobal Vargas, and Jorge Velasco-

Hernandez have modeled the oscillating patterns of the disease dengue fever, for which

an estimated 50 to 100 million cases occur globally each year (Esteva and Vargas 1999).

Deﬁnition Modeling is the application of methods to analyze complex,

real-world problems in order to make predictions about what

might happen with various actions.

7 Overview

Model Classiﬁcations

Several classiﬁcation categories for models exist. A system we are modeling ex-

hibits probabilistic or stochastic behavior if an element of chance exists. For ex-

ample, the path of a hurricane is probabilistic. In contrast, a behavior can be deter-

ministic, such as the position of a falling object in a vacuum. Similarly, models can

be deterministic or probabilistic. A probabilistic or stochastic model exhibits ran-

dom effects, while a deterministic model does not. The results of a deterministic

model depend on the initial conditions; and in the case of computer implementation

with particular input, the output is the same for each program execution. As we see

in Module 9.2 on “Simulations” and other modules, we can have a probabilistic

model for a deterministic situation, such as a model that uses random numbers to es-

timate the area under a curve.

Deﬁnitions A system exhibits probabilistic or stochastic behavior if an

element of chance exists. Otherwise, it exhibits deterministic

behavior. A probabilistic or stochastic model exhibits random

effects, while a deterministic model does not.

We can also classify models as static or dynamic. In a static model, we do not

consider time, so that the model is comparable to a snapshot or a map. For example,

a model of the weight of a salamander as being proportional to the cube of its length

has variables for weight and length, but not for time. By contrast, in a dynamic

model, time changes, so that such a model is comparable to an animated cartoon or

a movie. For example, the number of salamanders in an area undergoing develop-

ment changes with time; and, hence, a model of such a population is dynamic. Many

of the models we consider in this text are dynamic and employ a static component as

part of the dynamic model.

Deﬁnitions A static model does not consider time, while a dynamic

model changes with time.

When time changes continuously and smoothly, the model is continuous. If time

changes in incremental steps, the model is discrete. A discrete model is analogous

to a movie. A sequence of frames moves so quickly that the viewer perceives

motion. However, in a live play, the action is continuous. Just as a discrete sequence

of movie frames represents the continuous motion of actors, we often develop dis-

crete computer models of continuous situations (Voinov 2003).

Deﬁnitions

In a continuous model, time changes continuously, while in

a discrete model time changes in incremental steps.

Module 1.2

Steps of the Modeling Pr

ocess

The modeling process is cyclic and closely parallels the scientiﬁc method and the

software life cycle for the development of a major software project. The process is

cyclic because at any step we might return to an earlier stage to make revisions and

continue the process from that point.

The steps of the modeling process are as follows:

Analyze the problem

We must ﬁrst study the situation sufﬁciently to identify the problem pre-

cisely and understand its fundamental questions clearly. At this stage, we de-

termine the problem’s objective and decide on the problem’s classiﬁcation,

such as deterministic or stochastic. Only with a clear, precise problem identi-

ﬁcation can we translate the problem into mathematical symbols and develop

and solve the model.

Formulate a model

In this stage, we design the model, forming an abstraction of the system we

are modeling. Some of the tasks of this step are as follows:

Gather data

We collect relevant data to gain information about the system’s behavior.

Make simplifying assumptions and document them

In formulating a model, we should attempt to be as simple as reason-

ably possible. Thus, frequently we decide to simplify some of the fac-

tors and to ignore other factors that do not seem as important. Most

problems are entirely too complex to consider every detail, and doing

so would only make the model impossible to solve or to run in a rea-

sonable amount of time on a computer. Moreover, factors often exist

that do not appreciably affect outcomes. Besides simplifying factors,

we may decide to return to Step 1 to restrict further the problem under

investigation.

Determine variables and units

We must determine and name the variables. An independent variable

is the variable on which others depend. In many applications, time is an

independent variable. The model will try to explain the dependent

variables. For example, in simulating the trajectory of a ball, time is an

independent variable; and the height and the horizontal distance from

the initial position are dependent variables whose values depend on the

time. To simplify the model, we may decide to neglect some variables

(such as air resistance), treat certain variables as constants, or aggre-

gate several variables into one. While deciding on the variables, we

must also establish their units, such as days as the unit for time.

Establish relationships among variables and submodels

If possible, we should draw a diagram of the model, breaking it into

submodels and indicating relationships among variables. To simplify

the model, we may assume that some of the relationships are simpler

than they really are. For example, we might assume that two variables

are related in a linear manner instead of in a more complex way.

9 Overview

Determine equations and functions

While establishing relationships between variables, we determine

equations and functions for these variables. For example, we might de-

cide that two variables are proportional to each other, or we might es-

tablish that a known scientiﬁc formula or equation applies to the

model. Many computational science models involve differential equa-

tions, or equations involving a derivative, which we introduce in Mod-

ule 2.3 on “Rate of Change.”

Solve the model

This stage implements the model. It is important not to jump to this step be-

fore thoroughly understanding the problem and designing the model. Other-

wise, we might waste much time, which can be most frustrating. Some of the

techniques and tools that the solution might employ are algebra, calculus,

graphs, computer programs, and computer packages. Our solution might

produce an exact answer or might simulate the situation. If the model is too

complex to solve, we must return to Step 2 to make additional simplifying

assumptions or to Step 1 to reformulate the problem.

Verify and interpret the model’s solution

Once we have a solution, we should carefully examine the results to make

sure that they make sense (veriﬁcation) and that the solution solves the origi-

nal problem (validation) and is usable. The process of veriﬁcation deter-

mines if the solution works correctly, while the process of validation estab-

lishes if the system satisﬁes the problem’s requirements. Thus, veriﬁcation

concerns “solving the problem right,” and validation concerns “solving the

right problem.” Testing the solution to see if predictions agree with real data

is important for veriﬁcation. We must be careful to apply our model only in

the appropriate ranges for the independent data. For example, our model

might be accurate for time periods of a few days but grossly inaccurate when

applied to time periods of several years. We should analyze the model’s solu-

tion to determine its implications. If the model solution shows weaknesses,

we should return to Step 1 or 2 to determine if it is feasible to reﬁne the

model. If so, we cycle back through the process. Hence, the cyclic modeling

process is a trade-off between simpliﬁcation and reﬁnement. For reﬁne-

ment, we may need to extend the scope of the problem in Step 1. In Step 2,

while reﬁning, we often need to reconsider our simplifying assumptions, in-

clude more variables, assume more complex relationships among the vari-

ables and submodels, and use more sophisticated techniques.

Report on the model

Reporting on a model is important for its utility. Perhaps the scientiﬁc report

will be written for colleagues at a laboratory or will be presented at a scien-

tiﬁc conference. A report contains the following components, which parallel

the steps of the modeling process:

Analysis of the problem

Usually, assuming that the audience is intelligent but not aware of the

situation, we need to describe the circumstances in which the problem

arises. Then, we must clearly explain the problem and the objectives of

the study.

Module 1.2

Model design

The amount of detail with which we explain the model depends on the

situation. In a comprehensive technical report, we can incorporate

much more detail than in a conference talk. For example, in the former

case, we often include the source code for our programs. In either

case, we should state the simplifying assumptions and the rationale for

employing them. Usually, we will present some of the data in tables or

graphs. Such ﬁgures should contain titles, sources, and labels for

columns and axes. Clearly labeled diagrams of the relationships

among variables and submodels are usually very helpful in under-

standing the model.

Model solution

In this section, we describe the techniques for solving the problem and

the solution. We should give as much detail as necessary for the audi-

ence to understand the material without becoming mired in technical

minutia. For a written report, appendices may contain more detail, such

as source code of programs and additional information about the solu-

tions of equations.

Results and conclusions

Our report should include results, interpretations, implications, recom-

mendations, and conclusions of the model’s solution. We may also in-

clude suggestions for future work.

Maintain the model

As the model’s solution is used, it may be necessary or desirable to

make corrections, improvements, or enhancements. In this case, the modeler

again cycles through the modeling process to develop a revised solution.

Deﬁnitions The process of veriﬁcation determines if the solution works

correctly, while the process of validation establishes if the sys-

tem satisﬁes the problem’s requirements.

Although we described the modeling process as a sequence or series of steps, we

may be developing two or more steps simultaneously. For example, it is advisable to

be compiling the report from the beginning. Otherwise, we can forget to mention

signiﬁcant points, such as reasons for making certain simplifying assumptions or for

needing particular reﬁnements. Moreover, within modeling teams, individuals or

groups frequently work on different submodels simultaneously. Having completed a

submodule, a team member might be verifying the submodule while others are still

working on solving theirs.

The modeling process is a creative, scientiﬁc endeavor. As such, a problem we

are modeling usually does not have one correct answer. The problems are complex,

and many models provide good, although different, solutions. Thus, modeling is a

challenging, open-ended, and exciting venture.

Overview 11

Exer

cises

1. Compare and contrast the modeling process with the scientiﬁc method:

Make observations; formulate a hypothesis; develop a testing method for the

hypothesis; collect data for the test; using the data, test the hypothesis; ac-

cept or reject the hypothesis.

2. Compare and contrast the modeling process with the software life cycle:

Analysis, design, implementation, testing, documentation, maintenance.

References

Esteva, Lourdes, and Cristobal Vargas. 1999. “A Model for Dengue Disease with

Variable Human Population,” J. Math. Biol., 38: 220–240. http://147.46.94.112/

e_journals/pdf_full/journal_j/j28_380302.pdf

Giordano, Frank R., Maurice D. Weir, and William P. Fox. 2003. A First Course in

Mathematical Modeling. 3rd ed. Paciﬁc Grove, Calif.: Brooks/Cole-Thompson

Learning.

Greenwood Genetics Center. “Welcome to the GGC.” http://www.ggc.org/

Human-Technology Interaction Center. “HTIC.” The University of Oklahoma.

http://www.ou.edu/HTIC/

Marsden, Jerrold. “Control and Dynamical Systems.” California Institute of Tech-

nology. http://www.cds.caltech.edu/~marsden/

Voinov, Alexey Arkady. “Model Classiﬁcations.” Course notes for CS 295 Computer

Simulation and Modeling, University of Vermont, Gund Institute for Ecological

Economics, 2003. http://www.uvm.edu/giee/AV/CS/class2.1.html (accessed

March 20, 2005).

Weldon, T. P., Y. A. Gryazin, and M. V. Kibanov. 2001. “Novel Inverse Methods in

Land Mine Imaging.” IEEE International Conference on Acoustics Speech and

Signal Processing 2001 (ICASSP 2001). Salt Lake City, UT, May 7–11, 2001.

http://wws2.uncc.edu/tpw/papers/icassp01.pdf