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A Core Course on Modeling

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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A Core Course on Modeling

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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 'descartes()'; this is a function and produces output  cat.-II (usually 'plotOK')

  7. 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

  8. 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?

  9. 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

  10. 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/

  11. 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

  12. 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

  13. 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

  14. 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.

  15. 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)

  16. 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

  17. 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)

  18. 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]

  19. 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

  20. 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.

  21. 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.

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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 ???

  31. 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

  32. 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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