Fundamentals of Model Development

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Fundamentals of Model Development
Margaret L. Loper Georgia Tech Research
Institute Atlanta, GA margaret.loper_at_gtri.gatech.e
du
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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
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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
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Part I The Nature of Modeling
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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
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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

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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.

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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.

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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

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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

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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

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Part II The Science of Modeling
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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

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Modeling Relationships
Theory of Operation
Simulation Language (ACSL, Arena, Simscript)
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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
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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?

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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
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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
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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

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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

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Part III Model Formulation
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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.

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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

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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
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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

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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)

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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.

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Part IV Model Foundations
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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
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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

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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
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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

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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

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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

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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

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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
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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

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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

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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

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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
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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

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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

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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

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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

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Part V Modeling Engineering
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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
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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
48
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

49
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

50
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

51
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

52
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

53
Part VI Lessons in Modeling
54
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

55
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!
56
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
57
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

58
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
59
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
60
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

61
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.

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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/