1 A Core Course on Modeling      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) Week 2- The Art of Omitting

2 A Core Course on Modeling      The Conceptual Model      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 Week 2- The Art of Omitting

3 A Core Course on Modeling      The Conceptual Model      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 Week 2- The Art of Omitting

4 A Core Course on Modeling      Concepts and Entities      entities conceived entities real entities 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’ … and from now on, we are not even going to try modelmodeled system Week 2- The Art of Omitting

5 A Core Course on Modeling concepts referred to by words typically: substantives, including proper names (‘wheel’,’friction’,’John’,’P’, …)      Concepts and Entities      Week 2- The Art of Omitting

6 A Core Course on Modeling 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}.      Properties      Week 2- The Art of Omitting 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)

7 A Core Course on Modeling      Relations      concept 1concept 2 relation connecting concepts sometimes connecting properties typically: prepositions, (both spatial and otherwise) or verbs (‘likes’, ‘produces’, …) also forms like ‘is-married-to’, ‘reacts-with’, ‘produces’, … In international sign language, spatial prepositions directly correspond to the spatial relationships or movements of hands Week 2- The Art of Omitting

8 A Core Course on Modeling      Constructing a Conceptual Model      The birth of a conceptual modelPrevious week’s street lantern problem: ‘how to illuminate a road?’ define conceptualize conclude execute formalize formulate purpose formulate purpose identify entities identify entities choose relations choose relations obtain values obtain values formalize relations formalize relations operate model operate model obtain result obtain result present result present result interpret result interpret result From definition: what purpose? remember the modeling process: very vague: ‘how’ could mean a lot of different things Week 2- The Art of Omitting

9 A Core Course on Modeling The birth of a conceptual modelPrevious 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 …?) 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 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 very vague: ‘how’ could mean a lot of different things      Constructing a Conceptual Model      Week 2- The Art of Omitting

10 A Core Course on Modeling The birth of a conceptual model step 1: which concepts? define conceptualize conclude execute formalize formulate purpose formulate purpose identify entities identify entities choose relations choose relations obtain values obtain values formalize relations formalize relations operate model operate model obtain result obtain result present result present result interpret result interpret result remember the modeling process: In our conceptual model, all entities will occur as concepts      Constructing a Conceptual Model      Week 2- The Art of Omitting

11 A Core Course on Modeling 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 problemadaptive illumination: traffic dependent  involve traffic      Constructing a Conceptual Model      Week 2- The Art of Omitting

12 A Core Course on Modeling 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 problemadaptive illumination: traffic dependent  involve traffic 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 )      Constructing a Conceptual Model      Week 2- The Art of Omitting

13 A Core Course on Modeling The birth of a conceptual model step 2: which properties? lantern road car driver traffic define conceptualize conclude execute formalize formulate purpose formulate purpose identify entities identify entities choose relations choose relations obtain values obtain values formalize relations formalize relations operate model operate model obtain result obtain result present result present result interpret result interpret result remember the modeling process:      Constructing a Conceptual Model      has-a, part-of Week 2- The Art of Omitting

14 A Core Course on Modeling The birth of a conceptual model step 2: which properties? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density For every concept: what info do we need for that concept? distinguish important and less important properties; start with only important ones difference in viewing angle probably small      Constructing a Conceptual Model      Week 2- The Art of Omitting

15 A Core Course on Modeling The birth of a conceptual model step 2: which properties? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses 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      Constructing a Conceptual Model      Week 2- The Art of Omitting

16 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses define conceptualize conclude execute formalize formulate purpose formulate purpose identify entities identify entities choose relations choose relations obtain values obtain values formalize relations formalize relations operate model operate model obtain result obtain result present result present result interpret result interpret result remember the modeling process:      Constructing a Conceptual Model      Week 2- The Art of Omitting

17 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 5.0 … 25.0} m Some properties represent a free choice: in a design context, these represent decisions      Constructing a Conceptual Model      Week 2- The Art of Omitting

18 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 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?’      Constructing a Conceptual Model      Week 2- The Art of Omitting

19 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 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      Constructing a Conceptual Model      Week 2- The Art of Omitting

20 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 5.0 … 25.0} m : {100, 2000} W : reflectivity Some properties are constant, but an additional model may be needed to find their value (perhaps involving an experiment) : {14.40} m      Constructing a Conceptual Model      Week 2- The Art of Omitting

21 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 5.0 … 25.0} m : {100, 2000} W : reflectivity : {1…3} 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 : {14.40} m      Constructing a Conceptual Model      Week 2- The Art of Omitting

22 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 5.0 … 25.0} m : {100, 2000} W : reflectivity : {1…3} m : {20 …180} km/h 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? : {14.40} m      Constructing a Conceptual Model      Week 2- The Art of Omitting

23 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 5.0 … 25.0} m : {100, 2000} W : reflectivity : {1…3} m : driverView : {14.40} m Some properties have a value that is a concept on its own right: driverView ‒ minIntensity:… ‒ maxntensity:… : {20 …180} km/h      Constructing a Conceptual Model      Week 2- The Art of Omitting

24 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 5.0 … 25.0} m : {100, 2000} W : reflectivity : {1…3} m : {30} cars/minute : {14.40} m : {20 …180} km/h : driverView Some properties require that a collection of data is aggregated      Constructing a Conceptual Model      Week 2- The Art of Omitting

25 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? lantern −height −power road −width −reflectivity car −height −speed driver −visual capabilities traffic −density authority −expenses :{ 5.0 … 25.0} m : {100, 2000} W : reflectivity : {1…3} m : {30} cars/minute : as little as possible : {14.40} m : {20 …180} km/h : driverView some properties may represent the actual purpose, goal or objective of the model: how cheap can we illuminate this particular road?      Constructing a Conceptual Model      Week 2- The Art of Omitting

26 A Core Course on Modeling The birth of a conceptual model step 3: which value sets? 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      Constructing a Conceptual Model      Week 2- The Art of Omitting

27 A Core Course on Modeling The birth of a conceptual model step 4: which relations? define conceptualize conclude execute formalize formulate purpose formulate purpose identify entities identify entities choose relations choose relations obtain values obtain values formalize relations formalize relations operate model operate model obtain result obtain result present result present result interpret result interpret result remember the modeling process:      Constructing a Conceptual Model      other than has-a, part-of Week 2- The Art of Omitting 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

28 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) 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      Constructing a Conceptual Model      Week 2- The Art of Omitting relation involves multiple lanterns (hence n) and only one road (hence 1; may skip the ‘1’, like in a 1 b 2 =ab 2 ) 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

29 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) 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      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

30 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) helps to realize that properties (e.g., density) of ‘traffic’ can be found by aggregating (averaging) properties (e.g., speed) of multiple cars      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

31 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) 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      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

32 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) ridesOn(car, road) sees(driver,lantern n, road) 3-fold relation, tells us what geometric reasoning we need to calculate visibility and/or blinding thresholds      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

33 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) ridesOn(car, road) sees(driver,lantern n, road) pays(authority, lantern n ) helps to investigate (a) what else needs to be paid for, and (b) what else might be a concern for authorities      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

34 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) ridesOn(car, road) sees(driver,lantern n, road) pays(authority, lantern n ) adjacent(lantern, lantern 2 ) helps thinking about adaptivity control: lantern communicates to neighbor about presence of car      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

35 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) ridesOn(car, road) sees(driver,lantern n, road) pays(authority, lantern n ) adjacent(lantern, lantern 2 ) Given the full list of concepts, it is recommended to check (all?) possible relations to find which might be essential for the model      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

36 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) ridesOn(car, road) sees(driver,lantern n, road) pays(authority, lantern n ) adjacent(lantern, lantern 2 ) 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      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

37 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) ridesOn(car, road) sees(driver,lantern n, road) pays(authority, lantern n ) adjacent(lantern, lantern 2 ) 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.      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

38 A Core Course on Modeling The birth of a conceptual model step 4: which relations? illuminate(lantern n, road 1 ) operatedBy(car, driver) consistsOf(traffic, car n ) ridesOn(car, road) sees(driver,lantern n, road) pays(authority, lantern n ) adjacent(lantern, lantern 2 ) 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.      Constructing a Conceptual Model      Week 2- The Art of Omitting 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

39 A Core Course on Modeling A conceptual model is a collection of concepts (with properties and values) and the relations connecting them, related to the model’s purpose 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 lantern height power road width surface reflectance traffic density car speed height driver visual capabilities rides on consists of operated by sees adjacent n 1 n n 1 1 authority expenses pays n 1 1 illuminatelocated on      Constructing a Conceptual Model      Week 2- The Art of Omitting 1

40 A Core Course on Modeling      Formal Issues      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 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 Week 2- The Art of Omitting

41 A Core Course on Modeling Week 2- The Art of Omitting      Formal Issues      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 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 oneTallLantern is a subset of the set tallLanterns, hence the ‘:‘ in stead of ‘=‘ use set-notation to indicate ranges or other collections of values one particular lantern: set with only 1 element, abbreviation: no accolades 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

42 A Core Course on Modeling 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 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 Week 2- The Art of Omitting      Formal Issues     

43 A Core Course on Modeling Notation for addressing properties: let concept C have property P with value vP Dot notation:C.P = vP Index notation:C[P] = vP Function = vP Subscript notationC P = vP Notation for addressing properties: let concept C have property P with value vP Dot notation:C.P = vP Index notation:C[P] = vP Function = vP Subscript notationC P = vP supported by all computer languages ‘dot’ abbreviation of ‘its’ (=part-of) if type of P is compound: C.P.X. etc. 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: etc. nice for e.g. not supported by computer languages developed from hand writing (few symbols) if P is compound: subsubsubscript etc. not standardized (C P, P C, P C ) Week 2- The Art of Omitting      Formal Issues     

44 A Core Course on Modeling Recapitalizing on notation 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 name : something Week 2- The Art of Omitting      Formal Issues      name = something what value does it have? or: what values do its properties have?

45 A Core Course on Modeling Recapitalizing on notation 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’] 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’] Week 2- The Art of Omitting      Formal Issues      name : something name = something value value set set, no order set with order name of another concept another concept strings that are not concept names come in quotes!

46 A Core Course on Modeling      Quantities      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 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 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’. Week 2- The Art of Omitting

47 A Core Course on Modeling      Quantities      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 ( o C), ratio scale (K) Week 2- The Art of Omitting

48 A Core Course on Modeling      Quantities      From small to big: quantities, types and various forms of order OK to compute nominalordinalintervalratio frequency distribution yes median noyes add, subtract no yes multiply, divide no yes Week 2- The Art of Omitting

49 A Core Course on Modeling      Units, Scales and Dimensions      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 u 1 find: x 1 times; with u 2 find: x 2 times. What is the real volume ???? 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 u 1 find: x 1 times; with u 2 find: x 2 times. What is the real volume ???? Week 2- The Art of Omitting

50 A Core Course on Modeling      Units, Scales and Dimensions      What is the real volume ???? Suppose that u 2 fits p 12 times in u 1. If both measure ‘the same thing’, x 1 u 1 =x 2 u 2, or x 1 /x 2 =u 2 /u 1 =p 21 So we never know the ‘real’ value; we only know x/p with unknown p, plus the ratios p 21 for various pairs of units u 1, u 2. Scaling means: move from one to another unit (with known ratio) What is the real volume ???? Suppose that u 2 fits p 12 times in u 1. If both measure ‘the same thing’, x 1 u 1 =x 2 u 2, or x 1 /x 2 =u 2 /u 1 =p 21 So we never know the ‘real’ value; we only know x/p with unknown p, plus the ratios p 21 for various pairs of units u 1, u 2. Scaling means: move from one to another unit (with known ratio) Week 2- The Art of Omitting 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 x  Q. (physicists even assume x  R …)

51 A Core Course on Modeling      Units, Scales and Dimensions      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/p cm ’) where p cm is an unknown constant Indeed: 12.5 cm = 125 mm because p cm =0.1 p mm 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 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/p cm ’) where p cm is an unknown constant Indeed: 12.5 cm = 125 mm because p cm =0.1 p mm 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 Week 2- The Art of Omitting Restricts adding or subtracting quantities. To get 1/p ‘out of brackets’, they have to be equal in expressions such as (x 1 p 13 +x 2 p 23 )=? If p 23 =qp 13 this is fine: (x 1 p 13 +x 2 qp 13 )=(x 1 +x 2 q) p 13 Otherwise, additions (and therefore also sine, cosine, exponent etc – being power series - ) don’t work! Therefore: ONLY DO MATH WITH UNIT-LESS (=dimensionless) QUANTITIES.

52 A Core Course on Modeling      Units, Scales and Dimensions      For units u 1, u 2 with constant ratio, the x 1 and x 2 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 ] For units u 1, u 2 with constant ratio, the x 1 and x 2 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 ] ‘Constant’ means: not depending on the measurement Week 2- The Art of Omitting

53 A Core Course on Modeling      Units, Scales and Dimensions      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 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 Week 2- The Art of Omitting

54 A Core Course on Modeling      Units, Scales and Dimensions      Example dimensional synthesis: How does oscillation time T [ T ] of pendulum depend on g [ LT -2 ], m[ M ], l[ L ]? T=m  l  g  [ T ]=[ M ]  [ L ]  [ LT -2 ]  =[ M ]  [ L ]  +  [ T ] -2  So: M :  =0; L :  +  =0; T :1= -2  or T 2 /(l/g) is a dimensionless unit: T 2 is proportional to l/g. Example dimensional synthesis: How does oscillation time T [ T ] of pendulum depend on g [ LT -2 ], m[ M ], l[ L ]? T=m  l  g  [ T ]=[ M ]  [ L ]  [ LT -2 ]  =[ M ]  [ L ]  +  [ T ] -2  So: M :  =0; L :  +  =0; T :1= -2  or T 2 /(l/g) is a dimensionless unit: T 2 is proportional to l/g. Week 2- The Art of Omitting

55 A Core Course on Modeling      Summary      Conceptual model consists of concepts, some represent entities; Concept: bundle of properties, each consisting of a name and a set of values (type): Concepts + relations = conceptual model (entity-relationship graph) establish concepts; establish properties; establish types of properties; establish relations. Quantities are properties, disregarding the concept they are a property of; Mathematical operations on quantities: ordering Nominal (no order), partial ordering or total ordering, interval scale, ratio scale; Measuring =counting the number of units in the measured item. Sets of units with fixed ratio: dimension is an equivalence class on units; Using dimensions, the form of a mathematical relationships can often be derived. Conceptual model consists of concepts, some represent entities; Concept: bundle of properties, each consisting of a name and a set of values (type): Concepts + relations = conceptual model (entity-relationship graph) establish concepts; establish properties; establish types of properties; establish relations. Quantities are properties, disregarding the concept they are a property of; Mathematical operations on quantities: ordering Nominal (no order), partial ordering or total ordering, interval scale, ratio scale; Measuring =counting the number of units in the measured item. Sets of units with fixed ratio: dimension is an equivalence class on units; Using dimensions, the form of a mathematical relationships can often be derived. Week 2- The Art of Omitting