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Fundamentals of Model Development

Margaret L. Loper Georgia Tech Research

Institute Atlanta, GA margaret.loper_at_gtri.gatech.e

du

3-Spoke Wheel of Simulation

- Simulation is a tightly coupled and interactive

component process composed of - Model Design
- High level specification often grounded in

predicate logic or mathematical theory - Model Execution
- One model can be executed in many ways
- Execution Analysis
- Perform tests on the data generated from the

model by running specific analysis

Conceptual Models Model Theory Mathematical Models

Model Design

Model Execution

Execution Analysis

Serial Algorithms Parallel Algorithms

I/O Analysis Experimental Design Surface Response

Techniques Visualization of Data Verification

Validation

Course Outline

Part I The Nature of Modeling - Philosophy,

Roles, Uses Part II The Science of Modeling -

Theory, Assumptions, Scope Part III Problem

Formulation - Objectives Part IV Model

Foundations - Components, Variables,

Interactions Part V Model Engineering -

Techniques, Formalisms Part VI Lessons in

Modeling

Part I The Nature of Modeling

What is a Model?

- If I were to ask several people for examples of

models, I would get a variety of responses - Mathematical equations
- Toy trains
- Prototype cars
- Fashion models

- What these very different things have in common

is that they are representations of reality - Equations may represent growth of a population
- Toy train is a representation of a real train
- Prototype is a representation of a future car
- Fashion model is a representation of how clothes

will look when worn

Modeling in its broadest sense is the

cost-effective use of something in place of

something else for some cognitive purpose

Models as Purposeful Representations

- A model represents reality for the given purpose
- The model is an abstraction in the sense that it

cannot represent all aspects of reality - This allows us to deal with the world in a

simplified manner, avoiding the complexity,

danger, and irreversibility of reality

We cannot build a model if we do not know why we

are building it and we cannot criticize or

discuss a model except in terms of its purpose

- Example A street map is a representation of

reality for navigating streets in a particular

city it is useless for driving across country or

even locating traffic jams or construction within

a city other models are needed for those purposes

Models as Problem Solving

- Modeling is one of the most essential activities

of the human mind - Underlies our ability to think and imagine, to

use signs and language, to communicate, to

generalize from experience, to deal with the

unexpected, and to make sense out of the raw

bombardment of our sensations - It allows us to see patterns, to appreciate,

predict, and manipulate processes and things, and

to express meaning and purpose

Learning to model is closely related to the more

general skills of problem solving to solve

problems is to think imaginatively and

purposefully

- In engineering terms, a model is used because it

enables predictions or calculations or in some

other way makes the design process more

convenient - Models are used to assist the designers

thinking, analyze potential designs, realize what

is known or unknown, predict behavior, identify

connections, etc.

Model Representations

- Verbal Models
- Model couched in the language of everyday

discourse - He is tall with red hair and green eyes, his

cheeks are pale and his nose is pimpled. His

left ear is larger than the right and one of his

front teeth is missing.

- Visual Models
- Diagrammatic, a picture is worth a thousand words
- Diagrams, drawings or images
- Can exclude some of the uncertainties in a verbal

description, e.g. is it the persons left ear or

the persons left ear as seen by the observer? - Physical, a reconstruction of the real object at

a smaller scale - Mannequins used in car crash tests
- Airplane model in a wind tunnel

- Formal Models
- Model that uses equations and formulas to

reproduce the physical object - Q mC(t1-t2) a model of heat emitted by a body

of mass m, when cooling from temperature t1 to

temperature t2. C is the heat capacity parameter.

Modeling Approaches

- The issue in modeling is not complexity as much

as it is expressiveness and orientation - Its not enough to learn one method of modeling

any more than there can be one way to model with

clay or paint a picture - With a single modeling method, you can model a

certain way however, that method might block out

an understanding of a subsystem that is better

understood with a different modeling method

- The block model and the equation provide exactly

the same semantics either model can be

translated into the other - However, these are different models because they

reflect different ways of looking at the physical

system - The way you like to view systems will influence

your decision as to which model to choose

The Golden Rule of Modeling

- The Golden Rule of modeling is that no model,

no matter how accurate, has any inherent value of

its own - The value of every model is based entirely upon

the degree to which it solves someones real

world problem - We never build a perfect model a model is built

for a specific purpose and its accuracy is judged

with respect to that purpose

- Project Take a minute and determine how many MM

candies will fit in this jar

- Why do you need to know how many candies fit in

the jar? - To determine if one bag would fill the jar? The

accuracy of the model may be more or less one

bag. - To determine how many bags are needed to fill the

jar? The accuracy of the model may be to within a

couple of handfuls. - To win a contest? The accuracy of the model may

be the exact maximum number of candies. - Purpose is a vital part of any model that you

build

Is a Model Different From Theory?

- A model is related to, but different from, a

theoretical description of the object - The model may be based on theory
- But may include non-ideal behaviors observed in

experiments that are not well explained by theory - Theory may predict certain trends, but empirical

numbers from experiments are included to get the

calculated results to agree with the real results

- The key difference is that a model must behave as

nearly as possible the same way as the real thing - It is not directly important whether the models

behavior is well predicted by theory it is the

results that count - However, a good theoretical basis is good,

because it will likely expand the range of

conditions over which the model will work

Part II The Science of Modeling

Definitions

- System
- A collection of entities that act and interact

together toward the accomplishment of some

logical end - Theory
- A structured body of knowledge about some

phenomenon that allows us to make meaningful

explanatory or predictive statements about it - Some theories can be proven mathematically, while

others require empirical validation through

observation, collection, and analysis of data - Model
- A formalized theory, a stylistic interpretation

of a body of propositions that a theory

represents - Answers a certain class of questions
- Simulation
- The manipulation of a model of a system in such a

way that properties of the system can be studied - The use of a mathematical model to study the

behavior of a system as it operates over time

Modeling Relationships

Theory of Operation

Simulation Language (ACSL, Arena, Simscript)

Formal Modeling

- The logicians definition a possible realization

in which all valid sentences of a theory T are

satisfied is called a model of T - In this context, this suggests that we might

distinguish between - The prototype (physical entity or system being

modeled) - The theory of the model (precise statement of the

assumptions of axioms) - The model itself (the scheme of equations)

- From the logical point of view, the prototype is

a realization in which the valid sentences of the

mathematical model are to some degree satisfied - There may be several sets of assumptions which

lead to the same model, since in general there is

an equivalence class of assumptions which

connects the phenomenon with the mathematical

model - We then use the model to draw conclusions (i.e.

make predictions) this is the deductive process

if the assumptions are true, then the conclusions

must also be true - Discrete math refresher..If P ? Q

A general method in proofs of undecidability A.

Tarski 1953

Assumptions

- Making assumptions is an integral part of

deciding what to take from the real to the model

world - Assumptions one makes depend on both the purpose

of the model and the constraints under which it

is built - Definition of the variables and their

interrelations constitute the assumptions of the

model

- Take a minute and determine how many ping-pong

balls could you fit into this room

- The room is shaped like a box?
- A ping-pong is assumed to be a cube?
- Furniture in the room can be ignored?
- Door spaces and other nooks and crannies can be

ignored?

Model Assumptions

- Just as theories rest on assumptions, so do

models

Assumptions

- As assumptions bound a theory and make it

possible, they structure a model and make it

viable

Model

Assumptions

Assumptions

- The more highly structured a model, the more

numerous its assumptions

Assumptions

Assumptions should be clearly stated, for it is

the assumptions that determine the purpose of a

model, and the credibility that can be ascribed

to its predictions

Scope of a Model

- Scope is determined by the purpose of a model
- The more complex and structured a model becomes,

the less able it is to answer new and unexpected

questions

Assumptions

Low

High

Highly structured High empirical

content Carefully phrased, narrow questions with

more certainty

Simple structure Low empirical content Broader

classes of questions, but with less certainty

The Value of Detail

- A model should only be as detailed as is

necessary to answer the questions at hand - A model designer must consider, from the points

of view of structure and data, the statistical

aspects of the system being modeled - Structural Uncertainties arise from our

imprecise knowledge of how systems function - Inability to separate real and apparent causes
- Confusing correlation and causation
- Data Uncertainties due to difficulties in

parameter estimation - Input data errors (estimation)
- Statistical nature of simulation models

(randomness)

- Any model that is predicted on a certain

structure that has been arrived at through

observation rather than theory runs the risk of

making predictions that must be qualified by

confidence statement - Accepting an uncertain structure is equivalent to

making an assumption, which limits a models

scope and utility

Summary

- Before a model is designed, two important

questions must be asked and answered - What is the purpose of the model?
- What are the requirements for accuracy and

precision? - Answers to these questions determine the

structure of a model, as they demand certain

assumptions to be made, that certain boundaries

be imposed and respected, that certain types of

questions can and cannot be asked, that certain

territories cannot be explored, and that certain

realities cannot be predicted - Without a statement of purpose, there can be no

theory, no model - Since a models value lies in the way in can be

used, detail is necessary only to the extent that

it contributes to the precision of the model

predictions or estimates without limiting the

variety of questions that can be asked

Part III Model Formulation

Building a Model

- Model building involves imagination and skill

giving rules for doing it is like listing rules

for being an artist - At best we can provide a framework around which

to build skills and develop imagination - Formulate the problem
- What do you wish to know? What do you intend to

do with it? - Outline the model
- Things whose effects are neglected or ignored
- Things that affect the model but whose behavior

the model is not designed to study

(exogenous/input/independent variables) - Things the model is designed to study the

behavior of (endogenous/output/dependent

variables) - Specify the interrelations among variables
- Is it useful?
- Can you obtain the needed data and then use it in

the model to make the predictions you want? - Test the model
- Use the model to make predictions that can be

checked against data or common sense.

Problem Formulation

- Every study should begin with a statement of the

problem - Problem formulation is concerned with defining

the questions one is trying to answer with the

simulation study

- Typically sets of questions are defined by policy

makers who intend to use the simulation results

to make decisions four perspectives on the use

of models in decision making Dutton 1987 - Rational
- Models are seen as tools of scientific management

that provide better information for policy making - Partisan
- Models are tools of propaganda and persuasion

rather than information models are used to

legitimate decisions made for other reasons - Technocratic
- Modelers use information technology to baffle, to

impress and to promote their own positions

technocrats try to gain political power by the

authority of their expertise, i.e. models are

just complicated representations of the modelers

personal theory and biases - Consensual
- Models are used as potential tools of interactive

decision-making and negotiation modeling is a

political process but as one that can be useful

in achieving consensus

Model Formulation

- The problems that modelers wish to solve exist in

the real world

- First step is to simplify the real world to

create a model world - The model world leaves out much of the complexity

of the real world

Occams Razor

Interpreting and Testing

- The original question gets translated into a

question involving the model world

Model World

- Next, construct a model of the problem in the

model world

Formulating Model World Problem

Model Results

- The final step is to interpret the answer found

for the model world problem back in the real world

Model

Mathematical Analysis

Occams Razor

- William of Ockham, 14th Century Philosopher and

Franciscan monk (1285-1349) - "Pluralitas non est ponenda sine neccesitate" or

"plurality should not

be posited without necessity" - Other translations
- things should not be multiplied without good

reason - entities must not be multiplied beyond what is

necessary - the simplest theory that fits the facts of a

problem is the one that should be selected - In other words, dont make things harder than

they have to be! - Eliminate all unnecessary information relating to

the problem being analyzed - Exclude details that are irrelevant given the

purpose, or can not be handled given the

constraints - Cut down the world to a manageable size
- Cut too much and the model solutions have nothing

to do with reality - Cut to little and the problem is too difficult to

solve with the available resources

Heuristics

- Asking questions is a crucial part of modeling

and problem solving - The most useful questions are those which are

pragmatic that ask for information in a useful

form or ask what you would do with it if you had

it

- Heuristics tend to be most useful in the early

stages of solving a problem because they

encourage you to think fruitfully about the

problem - Try to feel the problem (Imagine yourself inside

the system and ask what is going on around you) - Anticipate what a solution would look like

(number, graph, table, etc) - Think about how you would solve a simpler version

of the same problem - Rephrase the problem to make sure you understand

it - Draw a simple diagram of what is happening
- Are there any physical laws to consider

(conservation of matter or energy) - Look for ready-made formula for the answer
- Look for simplifying assumptions
- Look for bounds (simple models that would

definitely underestimate or overestimate the

answer)

Is the Model Useful?

- Establish a clear statement of the deductive

objectives - Objectives indicate the questions to be answered

by simulation and may include metrics to be

collected by the simulation - Throughput, Delay, Efficiency, etc
- Do you want the model to predict the consequences

of various policies or suggest an optimal policy? - Provide the criterion for determining the

deductive viability or tractability of the model

however, it may prove unachievable or different

objectives may emerge as the model develops

- Model Viability
- If the wrong things are ignored, the model is no

good if too much is taken into consideration,

the resulting model will be hopelessly complex

and probably require incredible amounts of data - Identify all significant variables and their

relative importance - Recognize the risk of leaving out significant

factors - Recognize the level of uncertainty associated

with each variable - If a tractable model is obtained, enrich it.

Otherwise, simplify.

Part IV Model Foundations

Definitions Revisited

System Part of some potential reality where we

are concerned with space-time effects and causal

relationships among parts of the system

Model Abstract from reality a description of a

system

Modeling A way of thinking and reasoning about

systems A goal is to come up with a

representation that is easy to use in describing

systems in a consistent manner

The Concepts of Modeling Nance

- A Model is a representation and abstraction of

anything such as a real system, a proposed

system, a futuristic system design, an entity, a

phenomena, or an idea

Model Definition (Formal)

- A unifying formalism that serves to represent a

wide variety of system models comes from

classical systems theory and defines a

deterministic system as ?T, U, Y, Q, ?, ?, ?? - T is the time set. For continuous systems T ?

(real numbers), and for discrete time systems, T

? (integer numbers). - U is the input set containing the possible values

of the input to the system. - Y is the output set.
- Q is the state set.
- ? is the set of admissible (or acceptable) input

functions. This contains a set of input

functions to use during system operation. Often,

due to physical limitations, ? is a subset of all

the possible input functions (T ? U). - ? is the transition function. It is defined as ?

Q x T x T x ? ? Q. If the system is time

invariant, ? Q x ? ? Q - ? is the output function, ? Q ? Y.

System

Model

Output Observable of some part of system

Input Controlling influence on system

System Modeling

- The task of deriving a model of the system may be

divided broadly into two subtasks

- Establish Model Structure
- Components
- The parts from which the model is constructed
- Descriptive Variables
- Describe the conditions of the components at

points in time - Component Interactions
- Rules by which components exert influence on each

other, altering their conditions and so

determining the evolution of the models behavior

over time

- Supply Data
- Provide the values the attributes can have and

defines the relationships involved in the

activities

Model Structure Components

- The parts from which the model is constructed
- States
- State is a collection of variables that describe

the physical system for an interval of time - Discrete State variables assume a discrete set

of values - Continuous State ranges can be represented by

the real numbers - Mixed State both kinds of variables are present
- Changes in the physical system are realized by

updating one or more of the state variables - A state is defined as a simple tuple ?S1,Sn?,

where n is the number of components in the state

vector and Si are the states components

- Events
- An event is an instantaneous occurrence that

changes the state of the system (special kind of

state that has no time duration) - Each event has a time associated with it

indicating when the event occurred an event is a

time tagged state ?t, S1,Sn? - Events are relative to the level of abstraction

for which the system has been defined

Model Structure Components

- Input
- Input is a state that has a controlling influence

on a system which does not contain the input

state - Input is just another kind of state except that

it permits us to place boundaries around what is

considered to be inside and outside a system - An input that is constant (non-time varying) is a

parameter - Output
- An output is a function of the system state and

input

- Time
- Time is denoted by either an integer or a real

number (quantitative) or a nominal variable such

as early, late, or before-lunch (qualitative) - In quantitative systems, time can be represented

as continuous or discrete - Discrete - time changes in increment steps (1

minute, 1 day, 1 year, etc) - State variables change instantaneously at

distinct points in simulation time - Continuous time is specified to flow

continuously through the real numbers - Discrete Class time flows continuously but state

changes occur in discontinuous jumps - Differential Equation Class continuous time and

continuous state, thus the time derivatives are

governed by differential equations

Model Structure Descriptive Variables

- Describe the conditions of the components at

points in time - Quantitative Data values are counts or numerical

measurements - Continuous data
- Measurements can be any value, usually within

some range, e.g. weight - Discrete data
- Measurements are integers, e.g. number of people

in a household

- Qualitative Data values are non-numeric

categories - Also described as discrete since there are a

finite number of categories observations may fall - Color of eyes blue, green, brown etc
- Socio-economic status low, middle or high
- Nominal No natural order between the data, e.g.

eye color - Ordinal An ordering exists among data values,

e.g. exam results pass or fail

- Random Variables Data values defined according

to a probability distribution - Deterministic No random variables appear
- Probabilistic or Stochastic Contains at least

one random variable - Probability distributions
- Uniform, Normal, Poisson, etc

Qualitative Simulation

- There are many situations in which it is not

possible to quantify the attributes in a way that

has any meaning or validity or even if it can be

quantified, it is not appropriate for the

particular study - Higher pay must increase incentives, and so

increase productivity and profits - Mixes actual quantities (e.g., pay) with

quantities such as incentives, which cannot

actually be quantified, but for which at least

more or less can be imagined - Qualitative simulation was developed to describe

complex physical phenomena in the absence of good

quantitative information - Modeling a robot that makes a cup of coffee what

variable to use to describe the cup? - A distinguishing characteristic is coverage
- Qualitative simulations simulate all possible

threads or environments - When it determines the next possible state it can

determine that there are several next possible

states due to the imprecise nature of the data - Executes each of these possible next states
- Resulting envisionments include all possible

event sequences

Qualitative Physics Past, Present, and

Future, Exploring Artificial Intelligence, K.D.

Forbus, 1988

Model Structure Component Interactions

- Rules by which components exert influence on each

other, altering their conditions and so

determining the evolution of the models behavior

over time - Rules of Interaction
- An interaction is an explicit action taken by an

object that may have some effect or impact on

another object - Rules specifying object interaction determine the

manner in which the state variables change over

time - Time Invariant rules of interaction are stated

entirely in terms of the values that the

descriptive variables can assume - Time Varying time is an argument of the rules

of interaction and may thus appear to be

different at different times - Interactions do not persist in time, however they

can affect the state of a persistent object

- History
- Whether responses are influenced by past history
- Models that make use of such remembered

information are called adaptive in recognition of

their ability to learn from previous experience

Model Structure Component Interactions

- Environment
- The environment in which the object exists and

operates has impact upon the outcomes of every

operation - Autonomous
- The system is cut off from all influences of its

environment (closed) - Data collected under specific conditions
- Integrated into the objects and interactions

Simulated environment conditions under which

data collected - Non-Autonomous
- The system is influenced by the environment via

input variables (open) - Need to collect and manage large volume of data

terrain surface, natural and cultural features,

atmosphere, sea surface, sub-surface, and ocean

floor

Model Data

- The next step to building a model of a system is

to gather data associated with that system - The real or virtual system that we are interested

in modeling is viewed as a source of observable

data - Systems differ with regard to how much data is

available to populate the system database - Data-rich, data is abundant from prior

experiments or obtained from measurements - Data-poor, meager amounts of historical data or

low-quality data - In some cases it is impossible to acquire better

data (e.g., combat) in others it is too expensive

(topography and vegetation of a forest) - Types of data
- Numeric Form
- Physical Sensors (thermocouple, stress gauge)
- Human Sensors (eye sight, tactile feedback)
- Nominal Data (interviewing methods, knowledge

acquisition techniques developed to obtain

qualitative knowledge) - Symbolic Form
- Extends standard data to the case of variables

not restricted to be numerical - Includes data having internal variations such as

value distributions background knowledge can be

added as input such as ontology, taxonomies,

rules, metadata - Language

Levels of Description

- To build a model there are many decisions that

must be made, either explicitly or implicitly

each is a continuum rather than a discrete choice

These are meta-modeling questions there is no

rigorous ways to make these choices, but there

are rigorous ways to use them once the decision

has been made

Classifications of Models

- Abstract or Physical Models
- Abstract Model is one in which symbols and logic

constitute the model. The symbolism used can be

a language or mathematical notation. - A verbal or written description in English is an

abstract model - A mathematical model is described in the

language of mathematical symbols - Physical Model takes the form of a physical

replica, often on a reduced scale, of the system

it represents. A physical model looks like the

object it represents and is also called an Iconic

Model. - A model of an airplane (scaled down), a model of

the atom (scaled up), a map, a model car

- Static or Dynamic Models
- Static Model describes relationships that do not

change with respect to time - An architectural model of a house which helps us

visualize floor plans and space relationships, A

business profit-and-loss spreadsheet, Modeling of

fractal structures, An equation relating the

lengths and weights on each side of a playground

seesaw - Dynamic Model describes time-varying

relationships - A wind tunnel which shows the aerodynamic

characteristics of proposed aircraft designs,

Prediction of the trajectory of a space craft,

Prediction of the effect of a tax cut on an

economy, Equations of motion of the planets

around the sun constitute a dynamic mathematical

model of the solar system

Classifications of Models

- Analytical or Numerical Mathematical Models
- Analytical Model solved by using the deductive

reasoning of mathematical theory - An M/M/1 queueing model, a Linear Programming

model, a Mixed Integer Linear Programming model,

a nonlinear optimization model - Numerical Model solved by applying computational

procedures - Finding the roots of a nonlinear algebraic

equation, f(x) 0, using the method of Interval

Halving or Simple Iteration - System Simulation is considered to be a numerical

computational technique

- Stochastic or Deterministic Models
- Stochastic Model picks up the response from a

set of possible responses according to a fixed

probability distribution - Behavior of systems under random conditions,

Gamblers Ruin - Deterministic Model generates the response to a

given input by one fixed law - Particle interactions of classical physics,

Positions of buildings and local land features,

Virtual circuits

Classifications of Models

- Individual or Aggregate Models
- Individual Models detailed representations of

individual entities or objects and their behavior - Urban combat, Ecological systems, Social and

Economical processes - Aggregate Models collective representation of

entities or objects in the system - Pedestrian travel patterns, Climate change,

Economic growth, Military wargames

- Continuous or Discrete Models
- Continuous Model changes occur continuously
- Painting, smooth lines and color changes,

Differential equations in partial derivatives,

Storm water management, Variational optic flow - Discrete Model changes occur at fixed time

intervals - Mosaic, discrete uniform segments that change

their shape and color in a stepwise manner,

Finite elements or difference schemes,

Manufacturing

Classifications of Models

- Qualitative or Quantitative Models
- Qualitative Models non-numeric models
- Natural language and cognition, Biological

networks, Fuzzy systems, Molecular function - Quantitative Models numeric models
- Supply chain management, Organic chemistry,

Communication networks

- Causal or Correlational Models
- Causal Models reflect the cause-effect

relationships between entities in the system - Reasoning, Physical and Chemical processes, Time
- Correlational Models observed phenomena are not

the cause of others - Weather forecasting, Fractal Gaussian noise,

International politics

- Linear or Nonlinear Mathematical Models
- Linear Model is one which describes

relationships in linear form - The equation y 3x 4z 1 is a linear model
- Nonlinear Model is one which describes

relationships in nonlinear form - The equation F (2x 4z 2) / (3y x) is a

nonlinear model

Part V Modeling Engineering

Concepts of Modeling Osman Balci

On the one hand, a model should not contain

unnecessary details and become needlessly complex

and difficult to analyze

Modeling is an artful balancing of opposites

On the other hand, it should not exclude the

essential details of what it represents

Mathematical Modeling

- What is a mathematical model?
- The formulation in mathematical terms of the

assumptions and their consequences believed to

underlie a real world problem - An abstract, simplified, mathematical construct

related to a part of reality and created for a

particular purpose

- Fundamental steps in developing a mathematical

model

Mathematics in Nature, John Adams 2003

Mathematical Methods

- Three kinds of mathematical methods are discussed

in the literature - Numerical
- Numerical methods involve the actual data that

appear in tables - Working directly with the numbers and trying to

understand relationships - Uses direct operation on numbers as its main tool
- Graphical
- Graphical methods use pictorial or spatial

representations to communicate relationships - Most useful in conveying qualitative

relationships or approximate data which involve

only a few variables - A graphical approach to a problem is most likely

to be useful when not much information is

available or when it is given in imprecise form - Theoretical
- Theoretical methods are also called analytic or

symbolic methods - Theoretical methods employ mathematical tools for

expressing and manipulating relationships or

patterns these tools typically involve variables

and algebra as well as an understanding of how

different objects in the mathematical universe

behave

More on Mathematical Methods

- Observations
- The three methods complement each other and

contribute to understanding and insight - Each method helps us think about the problem in a

special way - In all methods, assumptions are not exactly

correct, however we simplify the problem by

ignoring the differences between the real data

and the equation and this allows us to proceed

and make predictions - Numerical
- More correct result, but obtained via calculator

trial-and-error (tedious) - Graphical
- Get a rough idea of the situation limited

accuracy - The results may only be partial correct, but

provides an overall picture of the original data - Theoretical
- Most correct results quickly applied and yields

answer with perfect accuracy - The theoretical method has greater power and

generality - Easily extended to related problems by analyzing

the steps used to solve a problem, it is possible

to formulate procedures or rules that can be

applied to a large number of closely related

problems

Types of Mathematical Models

- Analytical
- Analytical models are mathematical models that

have a closed form solution, i.e. the solution to

the equations used to describe changes in a

system can be expressed as a mathematical

analytic function - Can deduce everything there is to know about a

system the cost is limited applicability much

of the world is too complicated to be described

this way - Ordinary differential and difference equations,

partial differential equations, variational

methods and stochastic processes - Numerical
- Numerical models are mathematical models that use

some sort of numerical time-stepping procedure to

obtain the models behavior over time - Numerical models use approximations to solve

differential equations - Require the model domain and time be discretized
- Model domain is represented by a network of grid

cells or elements, and the time of the simulation

is represented by time steps - Finite differences for ODES and PDES, finite

elements, cellular automata - Observational
- Model inference based on observations and

measured data - Used to characterize and classify data,

generalize from measurements in order to make

predictions, or learn about the rules underlying

the observed behavior - Function fitting, data transformations, search

techniques, density estimation, filtering and

state estimation, linear and nonlinear time series

The Sound of a Violin

- There are many levels of description choosing

among them depends on your goal and on available

tools - Analytical You could try to use an analytical

(pencil-and-paper) solution to the governing

equations in return for some large

approximations you may be able to find a useful

explicit solution, but it might not sound very

good - Numerical You could use a numerical model based

on first-principles description you match your

model parameters to measurements on a real

instrument, and change parameters between a

Stradivarius and Guarneri however running it in

real time would require a super computer and the

effort to find good parameters for the model is

almost as much work as building the real violin - Observational You could forget about the

underlying governing equations entirely and

experimentally try to find an effective

description of how the players actions are

related to the sound made by the instrument

Modeling Formalisms

- Perhaps the hardest general problem is

determining the exact method that one should use

to create a model - There are many modeling formalisms, but which

technique is best under what conditions?

- Differential Equations
- Difference Equations
- Finite Elements
- Linear Programming
- Nonlinear Programming
- Algebraic Equations
- Predicate Calculus
- Statistical Models
- Geometry
- System Dynamics
- Fuzzy Logic

- Graph Theory
- Monte Carlo Methods
- Case-Based Reasoning
- Neural Networks
- Cellular Automata
- Petri Nets
- Finite State Automata
- Queueing Models
- Markov Chains
- Genetic and Evolutionary Algorithms
- Qualitative Modeling

Part VI Lessons in Modeling

Approximations

- It should be remembered at all times that models

and simulations are all approximations of reality - They may use simplifying assumptions
- Unknown effects can not be included
- Equations may be solved by numerical methods,

which do not yield exact results - Often, models are only valid over a specific

range of conditions, especially if they are

semi-empirical (use measured data)

- The engineer must understand
- The theory, models, and techniques on which the

solution is based - Nature of the approximations used in the model
- The situations for which the technique is valid

- There is no substitute to experience with a

particular modeling tool - Often engineers know when a particular tool gives

good or bad results

Problems with Models

- There seems to be three broad sets of problems
- Modeling can be done more or less badly, so much

so that it can become dangerous - The modeling of situations involving human choice

raises all sorts of difficulties that do not

arise in modeling the behavior of inanimate

objects - Models once created can be used in dubious ways

- Problems arise from a confusion between types of

models and the grounds we have for believing in

them - Some models can be checked for consistency

(Newtonian mechanics) while others are based on

purely ad hoc assumptions, the only check being

against empirical observations (economic models

or risk analysis)

Models based on good theory can compensate for

lack of data, and models based on broad evidence

can compensate for lack of theory, but models

alone can hardly compensate for the lack of both!

Examples from History

Bad designs result from errors of judgment, which

is not reducible to science or mathematics

- Tacoma Narrows Bridge collapse, November 7, 1940
- The designer omitted the vertical stiffening

trusses recommended for the deck of his

suspension-bridge. - the unwisdom of allowing a particular

profession to become to inward looking and so

screened from relevant knowledge growing up in

other fields around it.

How Engineers Lose Touch, Eugene Ferguson,

Invention Technology, Vol. 8 No. 3, 1993

Examples from History contd

- Hartford Civic Center roof collapse, January 18,

1978 - The modern space-frame roof collapsed under a

heavy snow load - To design a space frame with a slide rule or

mechanical calculator was a laborious process

with too many uncertainties for nearly any

engineer, so they were seldom built before

computer programs were available - The computers apparent precision (to six or

seven significant figures) can give engineers an

unwarranted confidence in the validity of the

resulting numbers.

- The roof design was extremely susceptible to

buckling which was a mode of failure not

considered in the computer analysis and,

therefore, left undiscovered

Examples from History contd

- NASA Hubble space telescope, April 24, 1990
- Controlling computer program had been based on an

outdated star chart introducing a pointing error

that prevented it from being pointed accurately

at stars and planets - An unanticipated cycle of expansion and

contraction of the solar panel supports caused

the panels to sway. This confused the the

program which stabilized the spacecraft and

caused it to take corrective measures that

exacerbated the vibration - These blunders resulted not from mistaken

calculations but from the inability to visualize

realistic conditions. Although a great deal of

hard thinking may have been done, the ability to

imagine the mundane things that can go wrong

remained deficient

- Aegis Air Defense System, USS Vincennes, July 3,

1988 - Shot down an Iranian civilian airliner
- Designers underestimated the demands that their

designs would place on operators, who often lack

the knowledge of the idiosyncrasies and

limitations built into the system - Hubris and an absence of common sense in the

design process set the conditions that produce

the confusingly overcomplicated tasks that the

equipment demands of operators

Students have been taught to rely far too

completely on computer models, and their lack of

old-fashioned, direct hands-on experience can be

disastrous

Causes of Simulation Failure

- The 10 most frequent causes of simulation

analysis failure - Failure to define an achievable goal
- Incomplete mix of essential skills
- Inadequate level of user participation
- Inappropriate level of detail
- Poor communication
- Wrong computer language
- Obsolete or nonexistent documentation
- Using an unverified or invalid model
- Failure to use modern tools and techniques to

manage the development of a large complex

computer program - Poor presentation of results

Anini and Russell, INTERFACES 11(3)59-63, 1981

Morals About Modeling

- In the very nature of a model, it is a restricted

and simplified representation of reality - This makes the problem of what a model does in

circumstances for which it was not designed or

which were not foreseen much more problematic - Therefore it is necessarily open to surprise a

surprise can kill

- Computational models can help us resolve some

complex questions, but by no means all - Where data and relationships are clear cut, but

are too extensive and complex for our minds to

manage, they are at their best - Where complexity arises from an uncertainty as to

how to analyze the problem at all, models may at

best be suggestive

- Computational models have taught us that

complexity itself is more complicated than we

thought - Very simple models can generate very complex and

unpredictable behavior simplicity is not the

same as determinism

References

- Simulation Model Design and Execution, Paul

Fishwick, Prentice Hall, 1995. - Theory of Modeling and Simulation, Bernard

Zeigler, 1976. - Learning with Artificial Worlds Computer Based

Modelling in the Curriculum, Mellar (editor), The

Farmer Press, 1994. - A Course in Mathematical Modeling, Douglas Mooney

and Randall Swift, The Mathematical Association

of America, 1999. - Elementary Mathematical Modeling, Dan Kalman, The

Mathematical Association of America, 1997. - Mathematical Modelling Techniques, Rutherford

Aris, Dover Publications, 1994. - How to Model It Problem Solving for the Computer

Age, Anthony Starfield, Karl Smith, and Andrew

Bleloch, Interaction Book Company, 1994. - Qualitative Simulation and Analysis, Paul

Fishwick and Paul Luke (editors),

Springer-Verlag, 1991. - The Nature of Mathematical Modeling, Neil

Gershenfeld, Cambridge University Press, 1999. - Simulation Modeling Analysis, Law and Kelton,

McGraw Hill, 1991.

References

- Digital Computer Simulation Modeling Concepts,

P.J. Kiviat, RAND Report, 1967. - On the Art of Modeling, William Morris,

Management Science, Vol. 13, No. 12, August 1967. - The Nature of Modeling, Jeff Rothenberg, RAND

Report, November 1989. - CS 4214 Course Notes, Richard Nance and Osman

Balci, VA Tech, http//courses.cs.vt.edu/cs4214/

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