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A Core Course on Modeling. Week 5 – Roles of Quantities in a Functional Model. ACCEL (continued) a 4 categories model dominance and Pareto optimality strength algorithm Examples. A Core Course on Modeling. Week 5 – Roles of Quantities in a Functional Model. ACCEL: a four-categories model.

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slide1

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

  • ACCEL (continued)
    • a 4 categories model
    • dominance and Pareto optimality
    • strength algorithm
  • Examples
slide2

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

2

  • to-do list keeps track of incomplete expressions
  • to-do list empty: script is compiled
  • script compiles correctly: script starts running
slide3

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

  • quantities are automatically categorized:
    • x=17  constant: cat. III
    • x=slider(3,0,10)  user input: cat I
    • x not in right hand part:  output only: cat. II
    • otherwise:  cat. IV
slide4

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

4

  • Category I:
    • slider (number), checkbox (boolean), button(boolean event), input (arbitrary), cursorX, cursorY, cursorB
    • cannot occur in expressions:

a=slider(10,0,20) *p

    • slider with integer parameters gives integer results
    • slider with 1 float parameter gives float results
slide5

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

5

  • Category I:
    • to use slider for non-numeric input:

r=[ch0, ch1, ch2, …, chn]

myChoice=slider(0,0,n)

p=r[myChoice]

(p can have arbitrary properties)

http://www.gulfdine.com/McDonald\'s_Markiya

slide6

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

6

  • Category II:
    • all cat.-II quantities are given as output
    • dynamic models: p = f( p{1}, q{1} ) :

p is not in cat.-II

    • to enforce a quantity in cat.-II: pp = p
    • visual output with \'plot()\'; this is a function and produces output  cat.-II (usually \'plotOK\')
slide7

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

7

  • Category II:
    • in IO/edit tab: show / hide values: values of all quantities
    • results output: (too …) few decimals
slide8

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

8

  • Category III:
    • cat.-III is automatically detected for numbers or strings
    • Cat-III is detected for expressions with constants only:

X = 3 * sin (7.14 / 5)

    • don\'t use numerical constants in expressions:

x = pricePerUnit * nrUnits

x = 3.546 * nrUnits

x = 2 * PI * r (built-in constants: PI and E)

why not?

slide9

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

9

  • Category IV:
    • Expressionsshouldbesimple as possible:
      • Prefery = x * p, p = z + t over y = x * (z+t)
    • when in doubt: inspect!
      • make temporary cat.-II quantity
      • (even) better trick:

next week

image: http://shyatwow.blogspot.nl/2010/11/bug-day-inspect-bugs.html

slide10

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

10

  • Category IV:
    • efficiency: re-use common sub-expressions
    • consider user defined functions

image: http://mewantplaynow.blogspot.nl/

slide11

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

11

  • Category IV:
    • efficiency: re-use common sub-expressions
    • consider user defined functions

u = a + b*log(c)*sin(d)

v = e + b*log(c)*sin(d)

term = b*log(c)*sin(d)

u=a + term

v=e + term

re-using same value

slide12

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: a four-categories model

12

  • Category IV:
    • efficiency: re-use common sub-expressions
    • consider user defined functions

u = a + b*log(c)*sin(d)

v = e + p*log(q)*sin(r)

term(x,y,z) = x*log(y)*sin(z)

u = a + term(b,c,d)

v = e + term(p,q,r)

re-using same thinking

slide13

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

13

image: http://hellnearyou.blogspot.nl/2010/06/aspria-managers-want-submission-from.html

slide14

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

14

Dominance

  • Ordinal cat.-II quantities:
  • C1dominates C2 C1.qi is better than C2.qi for all qi;
  • ‘better’: ‘<‘ (e.g., waste) or ‘>’ (e.g., profit);
  • more cat.-II quantities: fewer dominated solutions.
slide15

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

15

q2

(e.g., waste)

C3

Dominance

  • Ordinal cat.-II quantities:
  • C1dominates C2 C1.qi is better than C2.qi for all qi;
  • ‘better’: ‘<‘ (e.g., waste) or ‘>’ (e.g., profit);
  • more cat.-II quantities: fewer dominated solutions.

C2

C1 dominates C2

C1

C1 dominates C3

C2,C3: no dominance

q1(e.g., profit)

slide16

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

16

Dominance

  • Only non-dominated solutions are relevant
  • Dominance: prune cat.-I space;
  • More cat.-II quantities: more none-dominated solutions
  •  nr. cat.-II quantities should be small.

image http://ornamentalplant.blogspot.nl/2011/07/trimming-pruning.html

slide17

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

17

Dominance in ACCEL

  • y=paretoMax(expression)  enlist for maximum
  • y=paretoMin(expression)  enlist for minimum
  • To use Pareto algorithm, express all conditions into penalties
  • For inspection of the results: Paretoplot

paretoHor(x)

paretoVer(x)

slide18

A Core Course on Modeling

Week 5 – Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

18

Dominance in ACCEL

myArea=paretoHor(paretoMax(p[myProv].area))

myPop=paretoVer(paretoMin(p[myProv].pop))

p=[Pgr,Pfr,Pdr,Pov,Pgl,Put,Pnh,Pzh,Pzl,Pnb,Pli]

myProv=slider(0,0,11)

myCap=p[myProv].cap

Pfr=[\'cap\':\'leeuwarden\',\'pop\':647239,\'area\':5748.74]

. . .

Pli=[\'cap\':\'maastricht\',\'pop\':1121483,\'area\':2209.22]

slide19

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: dominance & pareto optimality

19

Dominance in ACCEL

  • Dominated areas: bounded by iso-cat.-II quantitiy lines;
  • Solutions in dominated areas: ignore;
  • Non-dominated solutions: Pareto front.

D

slide20

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

20

image: http://www.usdivetravel.com/T-BolivianAndesExpedition.html

Optimization in practice

  • Find \'best\' concepts in cat.-I space.
  • Mathematical optimization: single-valued functions.
  • The \'mounteneer approach\';
  • Only works for 1 cat.-II quantity.
slide21

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

21

Optimization in practice

  • Eckart Zitzler: Pareto + Evolution.
  • genotype = blueprint of individual (‘cat.-I’);
  • genotype is passed over to offspring;
  • genotype phenotype, determines fitness (‘cat.-II’);
  • variation in genotypes  variation among phenotypes;
  • fitter phenotypes  beter gene-spreading.
slide22

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

22

Optimization in practice

  • Start: population of random individuals (tuples of values for cat.-I quantities);
  • Fitness: fitter when dominated by fewer;
  • Next generation: preserve non-dominated ones;
  • Complete population: mutations and crossing-over;
  • Convergence: Pareto front stabilizes.

image: http://www.freakingnews.com/Mutation-Pictures---2317.asp

slide23

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

23

Optimization in practice: caveats

  • Too large % non-dominated concepts: no progress;
  • Find individuals in narrow niche: problematic;
  • Analytical alternatives may not exist
  • Need guarantee for optimal solution  DON’T use Pareto-Genetic.

image: http://glup.me/epic-fail-pics-serie-196

slide24

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

ACCEL: strength-algorithm

24

Optimization in practice: brute force

  • If anything else fails:
  • local optimization for individual elements of the Pareto-front;
  • Split cat.-I space in sub spaces if model function behaves different in different regimes;
  • Temporarily fix some cat.-IV quantities (pretend that they are in category-III).

http://www.square2marketing.com/Portals/112139/images/the-hulk-od-2003-resized-600.jpg

slide25

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

Examples

25

paretoMax

paretoMax

paretoMin

Optimal province:

spaciousness = area / population

or

area

population

1 cat.-II quantity

2 cat.-II quantities

meaningful quantity, related to purpose

slide26

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

Examples

26

paretoMin

paretoMin

paretoMin

Optimal street lamps:

efficiency = power * penalty

or

power

penalty

not too much light

not too little light

1 cat.-II quantity

2 cat.-II quantities

contrived quantity, not related to purpose

slide27

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

Examples

27

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,3,30)

p=slider(500.1,100,2000)

intPenalty=paretoMin(paretoHor(-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP)))

roadLength=40

roadWidth=15

. . .

problem: too slow to do optimization

slide28

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

Examples

28

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,3,30)

p=slider(500.1,100,2000)

intPenalty=paretoMin(paretoHor(-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP)))

roadLength=40

roadWidth=15

. . .

  • Minimal intensity computed by the model
  • Minimal intensity to see road marks
  • Maximal intensity computed by the model
  • Maximal intensity tnot to be blinded

problem: too slow to do optimization

slide29

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

Examples

29

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,3,30)

p=slider(500.1,100,2000)

intPenalty=paretoMin(paretoHor(-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP)))

roadLength=40

roadWidth=2

. . .

problem: awkward metric in cat.-II space

problem: too slow to do optimization  use symmetry

slide30

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

Examples

30

intPenalty

minInt

maxInt

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,3,30)

p=slider(500.1,100,2000)

intPenalty=paretoMin(paretoHor(log(0.00001-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP))))

roadLength=40

roadWidth=2

. . .

minP

maxP

problem: awkward metric in cat.-II space  scale penalty

problem: border optima ???

slide31

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

Examples

31

Optimal street lamps:

dL=slider(25.5,5,50)

h=slider(5.5,1,30)

p=slider(500.1,50,2000)

intPenalty=paretoMin(paretoHor(log(0.00001-min(minP,minInt)+max(maxP,maxInt)-(maxP-minP))))

roadLength=40

roadWidth=2

. . .

problem: border optima ???  expand cat.-I ranges

slide32

A Core Course on Modeling

Week 5-Roles of Quantities in a Functional Model

Examples

32

Optimal street lamps:

Summary:

  • check if model exploits symmetries
  • check if penalty functions represent intuition
  • check if optima are not on arbitrary borders
  • keep thinking: interpret trends (h 0, l0 … 1D approximation …?)
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