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A Core Course on Modeling

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A Core Course on Modeling

Week 2- The Art of Omitting

Contents

- The Conceptual Model
- Concepts and Entities
- Properties
- Relations
- Constructing a Conceptual Model
- Formal issues – part of appendix 4
- Quantities
- Units, Scales and Dimensions
- Summary
- References to lecture notes + book
- References to quiz-questions and homework assignments (lecture notes)

A Core Course on Modeling

Week 2- The Art of Omitting

The Conceptual Model

2

A popular myth:

‘Mexicans, when conquered

by Cortez’

horsemen did not know

that horses were animals

… until a soldier fell from his saddle’

The importance of

SEGMENTATION

A Core Course on Modeling

Week 2- The Art of Omitting

The Conceptual Model

3

- Segmentation = separation from the rest
- A natural tendency in humans (even in babies)
- Language: words are instruments for segmentation
- Entity: a segment that can be referred to inter-subjectively

conceived

entities

real

entities

A Core Course on Modeling

Week 2- The Art of Omitting

Concepts and Entities

4

entities

… and from now on, we are not even going to try

concept:

- mentally constructed with a purpose
- is named and (hopefully) explicitly defined
- carries information

‘real’ thing:

- when we talk (think?) about it, it is a concept
- we cannot inter-subjectively know anything from ‘real things’

model

modeled system

A Core Course on Modeling

Week 2- The Art of Omitting

Concepts and Entities

5

concepts

- referred to by words
- typically: substantives, including proper names
(‘wheel’,’friction’,’John’,’P’, …)

A Core Course on Modeling

Week 2- The Art of Omitting

Properties

6

properties

- literally: ‘possesions’
- a property informs about the concept it belongs to
- a property has a name, and a set of values: the type.
- Examples of names are ‘color’, ‘size’, ‘use’;
- Examples of types (in this case) are colors, numbers (perhaps with unit), activities;
- Example of values (in this case) are {red}, {15 … 18} cm, {drinking coffee}.

Dimensioning is an example where properties are used to specify a concept. Every property comes with a name (say, ‘size’), and a type (say, number, or {4.95 … 5.05} cm)

relation

In international sign language, spatial prepositions directly correspond to the spatial relationships or movements of hands

A Core Course on Modeling

Week 2- The Art of Omitting

Relations

7

concept 1

concept 2

- connecting concepts
- sometimes connecting properties
- typically: prepositions, (both spatial and otherwise) or verbs (‘likes’, ‘produces’, …)
- also forms like ‘is-married-to’, ‘reacts-with’, ‘produces’, …

define

formulate

purpose

identify

entities

conceptualize

choose

relations

obtain

values

formalize

relations

formalize

operate

model

obtain

result

execute

present

result

interpret

result

conclude

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

8

The birth of a conceptual model

Previous week’s street lantern problem: ‘how to illuminate a road?’

From definition: what purpose?

remember the modeling process:

very vague: ‘how’ could mean a lot of different things

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

9

The birth of a conceptual model

Previous week’s street lantern problem: ‘how to illuminate a road?’

From definition: what purpose?

seek a sharp formulation, use the list of purposes (see previous week):

verify:

decide:

optimize:

analyse:

control:

(perhaps more …?)

very vague: ‘how’ could mean a lot of different things

could LED lamps do the job?

yes or no adaptive illumination?

what is the best height (or distance or power or …) for lanterns?

how do benefits of adaptive illumination depend on the traffic flow?

for real time managing adaptive switch on/off strategy

define

formulate

purpose

identify

entities

conceptualize

choose

relations

obtain

values

formalize

relations

formalize

operate

model

obtain

result

execute

present

result

interpret

result

conclude

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

10

The birth of a conceptual model

In our conceptual model, all entities will occur as concepts

step 1: which concepts?

remember the modeling process:

problem should also be solved for moonless nights

complicating factor, perhaps first assume no trees

needed for full understanding of the problem

adaptive illumination: traffic dependent involve traffic

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

11

The birth of a conceptual model

step 1: which concepts?

lantern

road

moon

car

trees

driver

traffic

problem should also be solved for moonless nights

complicating factor, perhaps first assume no trees

needed for full understanding of the problem

adaptive illumination: traffic dependent involve traffic

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

12

The birth of a conceptual model

First inventory of concepts:

- enough concepts to formulate problem
- few as possible: estimate which concepts are crucial, which not
- discover first set of assumptions (such as: assume no tree shadows and no moonlight)

step 1: which concepts?

lantern

road

moon

car

trees

driver

traffic

define

formulate

purpose

identify

entities

conceptualize

choose

relations

obtain

values

formalize

relations

formalize

operate

model

obtain

result

execute

present

result

interpret

result

conclude

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

13

The birth of a conceptual model

step 2: which properties?

remember the modeling process:

- driver
- traffic

- lantern
- road
- car

has-a,

part-of

difference in viewing angle probably small

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

14

The birth of a conceptual model

For every concept:

- what info do we need for that concept?
- distinguish important and less important properties; start with only important ones

step 2: which properties?

- driver
- visual capabilities

- traffic
- density

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

sometimes we realize there must be an additional property to merit the effort of the model

… and therefore there must be an additional concept as well to host this property

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

15

The birth of a conceptual model

step 2: which properties?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

define

formulate

purpose

identify

entities

conceptualize

choose

relations

obtain

values

formalize

relations

formalize

operate

model

obtain

result

execute

present

result

interpret

result

conclude

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

16

The birth of a conceptual model

remember the modeling process:

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

17

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

Some properties represent a free choice: in a design context, these represent decisions

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

18

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

: {100, 2000} W

Some properties represent alternatives for a what-if analysis: ‘can we illuminate the road with lamps of type X?’

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

19

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

: {100, 2000} W

: {14.40} m

Some properties represent invariable constants for the current, unique situation at hand: can be looked up

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

20

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

: {100, 2000} W

: {14.40} m

Some properties are constant, but an additional model may be needed to find their value (perhaps involving an experiment)

: reflectivity

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

21

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

: {100, 2000} W

: {14.40} m

Some properties can be used to test the reliability of the model: you expect the value of car.height to be not very critical

: reflectivity

: {1…3} m

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

22

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

: {100, 2000} W

: {14.40} m

Some properties can be used to test the range of applicability of the model: does our model for an adaptive road illumination system still make sense if cars go very fast?

: reflectivity

: {1…3} m

: {20 …180} km/h

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

23

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

: driverView

Some properties have a value that is a concept on its own right:

- driverView
- minIntensity:…
- maxntensity:…

: {100, 2000} W

: {14.40} m

: reflectivity

: {1…3} m

: {20 …180} km/h

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

24

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

: driverView

: {100, 2000} W

: {30} cars/minute

: {14.40} m

: reflectivity

Some properties require that a collection of data is aggregated

: {1…3} m

: {20 …180} km/h

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

25

The birth of a conceptual model

step 3: which value sets?

- driver
- visual capabilities

- traffic
- density

- authority
- expenses

- lantern
- height
- power

- road
- width
- reflectivity

- car
- height
- speed

:{5.0 … 25.0} m

: driverView

: {100, 2000} W

: {30} cars/minute

: {14.40} m

: reflectivity

: as little as possible

some properties may represent the actual purpose, goal or objective of the model: how cheap can we illuminate this particular road?

: {1…3} m

: {20 …180} km/h

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

26

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 3: which value sets?

define

formulate

purpose

identify

entities

conceptualize

choose

relations

obtain

values

formalize

relations

formalize

operate

model

obtain

result

execute

present

result

interpret

result

conclude

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

27

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

remember the modeling process:

other than has-a, part-of

A Core Course on Modeling

relation involves multiple lanterns (hence n) and only one road (hence 1; may skip the ‘1’, like in a1b2=ab2)

Week 2- The Art of Omitting

Constructing a Conceptual Model

28

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

illuminate(lanternn, road1)

helps to realise that, at any point of the road, the light of multiple lanterns contributes.

‘located on’ is another relation between lanterns and road: helps to think about distance between lanterns

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

29

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

illuminate(lanternn, road1)

operatedBy(car, driver)

helps to realize that the location of the viewer (=driver) and the location of the car (=the trigger for the adaptivity) are the same

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

30

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

helps to realize that properties (e.g., density) of ‘traffic’ can be found by aggregating (averaging) properties (e.g., speed) of multiple cars

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

31

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

ridesOn(car, road)

helps to realize that we can make assumptions on the possible location of cars, facilitating the adaptivity and the places where we should assess road visibility

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

32

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

ridesOn(car, road)

sees(driver,lanternn, road)

3-fold relation, tells us what geometric reasoning we need to calculate visibility and/or blinding thresholds

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

33

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

ridesOn(car, road)

sees(driver,lanternn, road)

pays(authority, lanternn)

helps to investigate (a) what else needs to be paid for, and (b) what else might be a concern for authorities

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

34

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

ridesOn(car, road)

sees(driver,lanternn, road)

pays(authority, lanternn)

adjacent(lantern, lantern2)

helps thinking about adaptivity control: lantern communicates to neighbor about presence of car

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

35

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

Given the full list of concepts, it is recommended to check (all?) possible relations to find which might be essential for the model

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

ridesOn(car, road)

sees(driver,lanternn, road)

pays(authority, lanternn)

adjacent(lantern, lantern2)

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

36

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

It usually requires several iterations before the lists of concepts, properties, values and relations are appropriate.

At any time, check against the purpose of the eventual model

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

ridesOn(car, road)

sees(driver,lanternn, road)

pays(authority, lanternn)

adjacent(lantern, lantern2)

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

37

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

Relations in the CM are in natural language, not yet in the form of mathematics, logic or computer ‘language’.

Next two weeks: how to find suitable mathematical expressions (typically: functions) to go from conceptual model to formal model.

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

ridesOn(car, road)

sees(driver,lanternn, road)

pays(authority, lanternn)

adjacent(lantern, lantern2)

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

38

The birth of a conceptual model

lantern

height: {5.0 … 25.0} m

power: {100, 2000}W

road

width: {14.4} m

reflectivity:

car

height: {1…3} m

speed: {20 … 180} km/h

driver

visual capabilities: driverView

traffic

density: {30} cars/minute

authority

expenses: minimal

step 4: which relations?

For all but trivial models, a purely textual list of concepts, properties and values is inapproporiate.

Use complementary (schematic) drawing.

Use ‘standard’ notation, such as Entity Relationships or UML-light.

illuminate(lanternn, road1)

operatedBy(car, driver)

consistsOf(traffic, carn)

ridesOn(car, road)

sees(driver,lanternn, road)

pays(authority, lanternn)

adjacent(lantern, lantern2)

A Core Course on Modeling

Week 2- The Art of Omitting

Constructing a Conceptual Model

39

authority

driver

This CM representation is called: ‘Entity-Relationship Graph’

All occurring entities are concepts

Concepts are boxes

Concept-info in the form of properties

Relations are diamonds

Numbers indicate ‘arity’ of relations

adjacent

1

1

expenses

visual capabilities

1

2

1

pays

lantern

operated by

n

1

height

power

1

n

sees

car

n

speed

height

located on

illuminate

n

1

1

consists of

road

1

1

width

surface reflectance

1

traffic

rides on

density

1

A conceptual model is a collection of concepts (with properties and values) and the relations connecting them, related to the model’s purpose

A Core Course on Modeling

Week 2- The Art of Omitting

Formal Issues

40

concept: a bundle of properties

property: (name, set of values)

name: to distinguish properties

type: set of all possible values for this property

value : to single out a unique concept. A value can be atomic or compound

atomic: cannot be decomposed

compound: a concept, to be composed into further properties

A Core Course on Modeling

Week 2- The Art of Omitting

Formal Issues

41

a type is always a set of values. But if the set contains only one value, and there is no confusion,we may drop the accolades

Notation:

concept: lantern called myLantern

properties: height, power

height

type: {5 … 25}m

power

type: {100,2000}W

abbreviation:

myLantern= [height: 12m, power: 1000W]

allLanterns= [height: {5 … 25}m, power: {100,2000}W]

tallLanterns= [height: {15 … 25}m, power: {2000} W]

oneTallLantern: tallLanterns

myLantern: allLanterns, myLantern:tallLanterns

allLanterns and tallLanterns are sets

one particular lantern: set with only 1 element, abbreviation: no accolades

use set-notation to indicate ranges or other collections of values

oneTallLantern is a subset of the set tallLanterns, hence the ‘:‘ in stead of ‘=‘

A Core Course on Modeling

Week 2- The Art of Omitting

Formal Issues

42

OK. So: conceptual modeling is mainly a large amount of burocracy to make an impressive mess of something completely trivial, right?

Not quite. Clean conceptualization matters, because:

- Confusion about naming is a main cause for disaster
- Organization helps against chaos in case of complex models (1000-s of concepts and relations) and serves as a checklist: what relations to incorporate in the model?
- Notation helps making subtle choices explicit and invites to think accurately even prior the formalisation phase
- Computers start playing an essential role even in conceptual modeling, and computers require unambiguous notation (Web 2.0 !)
- When going to the next step, computers come in anyhow: so good naming helps consistency between conceptual model and formal model

A Core Course on Modeling

Week 2- The Art of Omitting

Formal Issues

43

supported by all computer languages

‘dot’ abbreviation of ‘its’ (=part-of)

if type of P is compound: C.P.X. etc.

Notation for addressing properties:

let concept C have property P with value vP

- Dot notation:C.P = vP
- Index notation:C[P] = vP
- Function notation:@(C,P) = vP
- Subscript notationCP = vP

supported by most computer languages

reminiscent of arrays: property name instead of integer index

if type of P is compound: C[P][X] etc.

supported by some computer languages

functions are a natural way to obtain dependent information

if P is compound: @(@C,P),X) etc.

nice for e.g. vectors: @(u+v,x)

not supported by computer languages

developed from hand writing (few symbols)

if P is compound: subsubsubscript etc.

not standardized (CP, PC, PC)

A Core Course on Modeling

Week 2- The Art of Omitting

Formal Issues

44

Recapitalizing on notation

name = something

name : something

what value does it have? or: what values do its properties have?

what type of concept is it?

Notice:

p:{3,4} is the same as p=3 or p=4;

p:{3} is the same as p=3

strings that are not concept names come in quotes!

A Core Course on Modeling

Week 2- The Art of Omitting

Formal Issues

45

Recapitalizing on notation

name = something

name : something

value

radius1 = 14.6 (in some unit; see later)

radius2: real, radius3: {3 … 12}

bandMembers = {‘Paul’,’John’,’George’,’Ringo’}

bandMember : {‘Paul’,’John’,’George’,’Ringo’} or bandMember:bandMembers

monarchs = [‘Willem1’,‘Willem2’,‘Willem3’,‘Emma’,‘Wilhelmina’,‘Juliana’,‘Beatrix’]

Bea=monarchs[6]

poodle : dog

dog : [ sound: bark, skin: fur, food: meat, name:{‘Toby’,’Fifi’}]

myDog=[sound: bark, skin: fur, food: meat, name:’Toby’]

value set

set, no order

set with order

name of another concept

another concept

This picture illustrates the ‘quality vs. quantity’ metaphor. Our term ‘quantity’ is not used as ‘multitude’, however. In Dutch, our term ‘quantity’ translates as ‘grootheid’, not ‘hoeveelheid’.

A Core Course on Modeling

Week 2- The Art of Omitting

Quantities

46

Property: attribute of a concept

Quantity: (name, type, value), disregarding the concept this quantity may be an attribute of. So: quantities can appear to ‘stand alone’

Mathematics is about ‘stand alone’ quantities, where the types are mathematical objects (numbers, functions, vectors, equations …)

Properties always occur in the realm of conceptual modeling

Quantities are useful for re-using mathematical results: area A of a circle with radius r: A=r 2, irrespective what the circle stands for

A Core Course on Modeling

Week 2- The Art of Omitting

Quantities

47

From small to big: quantities, types and various forms of order

Nominal: no ordering

Partial ordering

Total ordering

example: taste, material, car brand, …

example: intervals, comes-before, preference, …

example: Mohs’ scale (hardness), interval scale (oC), ratio scale (K)

A Core Course on Modeling

Week 2- The Art of Omitting

Quantities

48

From small to big: quantities, types and various forms of order

A Core Course on Modeling

Week 2- The Art of Omitting

Units, Scales and Dimensions

49

Properties often relate to measurements units

Measuring starts with counting: ‘how often does a unit element fit in the quantity to be measured?’

Measure, say, some volume of dough. With unit u1find: x1 times; with u2 find: x2 times.

What is the realvolume ????

A Core Course on Modeling

Complication: x was ‘a number of times’, thus: an integer (counting!). Integers, however, are not closed under division.

Way out: have a series of units (m,dm,cm,mm,…) and assume that xQ. (physicists even assume xR …)

Week 2- The Art of Omitting

Units, Scales and Dimensions

50

What is the real volume ????

Suppose that u2 fits p12 times in u1.

If both measure ‘the same thing’, x1u1=x2u2, or

x1/x2=u2/u1=p21

So we never know the ‘real’ value; we only know x/p with unknown p, plus the ratios p21 for various pairs of units u1, u2.

Scaling means: move from one to another unit (with known ratio)

A Core Course on Modeling

Restricts adding or subtracting quantities.

To get 1/p ‘out of brackets’, they have to be equal in expressions such as (x1 p13+x2 p23)=?

If p23=qp13 this is fine:

(x1 p13+x2 qp13 )=(x1+x2 q) p13

Otherwise, additions (and therefore also sine, cosine, exponent etc – being power series - ) don’t work! Therefore:

ONLY DO MATH WITH UNIT-LESS (=dimensionless) QUANTITIES.

Week 2- The Art of Omitting

Units, Scales and Dimensions

51

Measuring is multiplying:

The expression ‘12.5 cm’ is not merely a notational convention

It really is an algebraic multiplication of two numbers (’12.5’ and ‘1/pcm’) where pcm is an unknown constant

Indeed: 12.5 cm = 125 mm because pcm=0.1 pmm

So 1 m=100 cm but also m=100 cm

Same as 1 x a = a

Consequence: mathematical operations (multiply, divide … ) on quantities SAME mathematical operations on units

‘Constant’ means: not depending on the measurement

A Core Course on Modeling

Week 2- The Art of Omitting

Units, Scales and Dimensions

52

For units u1, u2 with constant ratio , the x1 and x2 can be converted

A set of convertible units defines a dimension (equivalence class)

{m, inch, light year, yard, m, …} defines ‘length’ [L];

similar {sec, year, day, …} defines ‘time’ [T].

There are units that canot be converted (e.g., ‘nr. sheep’). They give rise to dimensions on their own [SHEEP]

A Core Course on Modeling

Week 2- The Art of Omitting

Units, Scales and Dimensions

53

Since units ‘go with the math’, so do dimensions.

Different dimensions cannot be added or subtracted, and therefore also not equated (a=b a-b =0).

Every dimension should match left and right from ‘=‘

Dimensional analysis: check a formula for this rule

Dimensional synthesis: construct a formula using this rule

A Core Course on Modeling

Week 2- The Art of Omitting

Units, Scales and Dimensions

54

Example dimensional synthesis:

How does oscillation time T [T] of pendulum depend on g [LT-2], m[M], l[L]?

T=mlg

[T]=[M][L][LT-2]

=[M][L]+[T]-2

So: M: =0; L: +=0; T:1= -2

or T2/(l/g) is a dimensionless unit: T2 is proportional to l/g.

A Core Course on Modeling

Week 2- The Art of Omitting

Summary

55

- Conceptual model consists of concepts, some represent entities;
- Concept: bundle of properties, each consisting of a nameand a set of values (type):
- Concepts + relations = conceptual model (entity-relationship graph)
- establish concepts;
- establish properties;
- establish types of properties;
- establish relations.

- Quantitiesare properties, disregarding the concept they are a property of;
- Mathematical operations on quantities: ordering
- Nominal(no order), partial orderingor total ordering, interval scale, ratio scale;
- Measuring=counting the number of units in the measured item.
- Sets of units with fixed ratio: dimensionis an equivalence class on units;
- Using dimensions, the form of a mathematical relationships can often be derived.