Presentación de soporte de la tesis doctoral

TESIS DOCTORAL

Reliability of Performance Measures in Tree-BasedGenetic ProgrammingMATERIAL DE SOPORTE

David Fernández Barrero

Directores:Dra. María D. R-Moreno

Dr. David Camacho

Departamento de AutomáticaUniversidad de Alcalá

Diciembre 2011

Capítulos 1 y 2Capítulo 3Capítulo 4Capítulo 5Capítulo 6

Summary1 Capítulos 1 y 2

FigurasTablas

2 Capítulo 3FigurasTablasAlgoritmos

3 Capítulo 4FigurasTablas



2 / 68


FigurasTablas

Capítulo 1Figura 1.1: Curvas de Koza

0 10 20 30 40 50

0.0

0.1

0.2

0.3

0.4

0.5

0.6

P(M

,i)Koza’s performance curves

Generation0e

+00

2e+

054e

+05

6e+

058e

+05

1e+

06

I(M

,i,z)

P(M,i)I(M,i,z)

13: 117000

3 / 68


FigurasTablas

Capítulo 2Figura 2.1: Marco de descripción experimental

4 / 68


FigurasTablas

Capítulo 2Figura 2.2: Clasificación Metaheurísticas

5 / 68


FigurasTablas

Capítulo 2Figura 2.3: Funciones de De Jong

x1x2

y

Quartic function with noise

x1x2

y

Rosenbrock

x1x2

y

Sphere

x1x2

y

Step

6 / 68


FigurasTablas

Capítulo 2Figura 2.4: Camino de Santa Fe

7 / 68


FigurasTablas

Capítulo 2Figura 2.5: Clasificación de medidas

8 / 68


FigurasTablas

Capítulo 2Figura 2.6: Eficacia vs. Eficiencia

0 1 2 3 4 5 6

0.1

0.2

0.3

0.4

0.5

Function evaluations

Fun

ctio

n va

lue

Efficiency

Effectivity

9 / 68


FigurasTablas

Tabla 2.1Tabla 2.1: Clasificación de diseños experimentales

Author Publication AdjectivesC.C. McGeoch [?] Dependency study

Robustness studyProving study

A. E. Eiben [?] DesignRepetitiveControl (particular case of repetitive)

J. Derrac [?] Single problem analysisMultiprogram problem analysis

M. Chiarandini [?] UnivariableMultivariable

R. L. Rardin [?] Scientifical/developmentDesign/planning/controlSequential/factorial

P. R. Cohen [?] Exploratory / confirmatoryManipulation / observation

Several [?, ?, ?] Pilot or exploratory

10 / 68


FigurasTablas

Capítulo 2Tabla 2.2: Clasificación de problemas

Author Publication Classes of problemA. E. Eiben [?] Useless

NaturalArtificial

R. L. Rardin [?] Real world datasetsRandom variants of real datasetsPublished librariesRandomly generated instances

T. Bartz-Beielstein [?] Test functionsReal-world problemsRandomly generated test problems

11 / 68


FigurasTablas

Capítulo 2Tabla 2.3: Funciones de De Jong

Function Expression Domain MinimumSphere f1(x) =

Pni=1 x2

i | xi |≤ 5,12 f (0, . . . , 0) = 0Rosenbrock f2(x) =

Pn−1i=1 ((1 − xi )

2 + 100(xi+1 − x2i )2) | xi |≤ 2,048 f (1, . . . , 1) = 0

Step f3(x) = 25 +Pn

i=1bxic | xi |≤ 5,12 f (([−5,12,−5), . . . ,[−512,−5))) = 0

Quartic f4(x) =Pn

i=1(ix4i ) + N(0, 1) | xi |≤ 1,28 f (0, . . . , 0) = 0

Sheckel (2D) f (x1, x2) = 10,02+

P25j=1

1j+

P2i=1(xi−aji )

6| xi |≤ 65,536 f (−32,−32) = 1

Quartic w/n [?] Solution quality

12 / 68


FigurasTablasAlgoritmos

Capítulo 3Figura 3.1: Arquitectura de Searchy

13 / 68



Capítulo 3Figura 3.2: Ejemplo de integración

14 / 68



Capítulo 3Figura 3.2: Ejemplo de despliegue

15 / 68



Capítulo 3Figura 3.3: Ejemplo de codificación

16 / 68



Capítulo 3Figura 3.3: Evolución del valor de fitness y tamaño

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70

Fitn

ess

Generations

EmailPhone

URL 6

8

10

12

14

16

18

20

22

24

0 10 20 30 40 50 60 70

Avg

. chr

omos

ome

leng

th

Generations

URLEmail

Phone

17 / 68



Capítulo 3Figura 3.3: Probabilidad de éxito

0

10

20

30

40

50

60

70

80

90

0 10 20 30 40 50 60 70

Suc

cess

rat

e (%

)

Generations

EmailPhone

URL

18 / 68



Capítulo 3Tabla 3.1: Parámetros del Algoritmo Genético para la evolución de regex

Parameter Value

Population 50Mutation probability 0.003Crossover probability 1Tournament size 2Elitism 1Initial chromosome length 4 - 40

19 / 68



Capítulo 3Tabla 3.2: Capacidad de extracción

Phone regex URL regex Email regexPh. URL Email F P R F P R F P R

Set 1 99 0 0 1 1 1 - - - - - -Set 2 0 51 0 - - - 0.24 0.14 0.84 - - -Set 3 0 0 862 - - - - - - 0.79 0.51 0.62Set 4 20 77 0 1 1 1 0.27 0.16 1 - - -Set 5 37 686 0 1 1 1 0.20 0.11 0.97 - - -Set 6 24 241 0 1 1 1 0.02 0.01 0.37 - - -Set 7 83 0 88 0.92 1 0.96 - - - 0.92 1 0.96Set 8 0 51 0 - - - 0.63 0.47 0.96 - - -Avg. - - - 0.98 1 0.99 0.27 0.18 0.83 0.85 0.79 0.79

20 / 68



Capítulo 3Tabla 3.3: Ejemplos de expresiones regulares evolucionadas

Evolved regex (Phone) Fitness

\w+ 0\(\d+\) 0.33\(\d+\)\d+ 0.58\(\d+\)\d+-\d+ 1

Evolved regex (URL) Fitness

http://-http://http:// 0/\w+\. 0.55http://\w+\.\w+\ 0.8http://\w+\.\w+\.com 1

Evolved regex (Email) Fitness

\w+\. 0.31\w+\.\w+ 0.49\w+@\w+\.\com 1

21 / 68



Capítulo 3Tabla 3.4: Comparación de Searchy con otras herramientas de integración de información

Platform Agent support Semantic Web Web Services Interdomainsupport

InfoSleuth Yes No No YesSIMS Yes No No YesBuilding Finder No Yes No NoSODIA No Yes Yes YesKnowledge Sifter Yes Yes Yes LimitedSearchy Yes Yes Yes Yes

22 / 68



Capítulo 3Algoritmo 3.1: Algoritmo de selección de alfabeto

1 .- P := Set of positive examples2 .- S := Set of candidate delimiters3 .- D := T := { }4 .-5 .- for each p in P6 .- for each s in S7 .- tokens := split p using s8 .- numberTokens := number of tokens9 .-10.- for each token in tokens11.- occurrence(token) := occurrence(token) + 112.- endfor13.-14.- if (numberTokens > 0) add s to D15.- endfor16.- endfor17.-18.- sort occurrence19.- add n first elements of occurrence to T

23 / 68


FigurasTablas

Capítulo 4Figura 4.1: Histograma de probabilidad de éxito

Number of successes (k)

Den

sity

0.00

0.10

0.20

0.30

0 2 4 6 8 10

n=30Artificial ant

0.00

0.10

0 5 10 15

n=50Artificial ant

0.00

0.05

0.10

5 10 15 20 25

n=100Artificial ant

0.00

0.04

0.08

20 30 40 50

n=250Artificial ant

0.00

0.02

0.04

40 50 60 70 80 90

n=500Artificial ant

0.00

0.02

100 120 140 160

n=1000Artificial ant

0.0

0.2

0.4

0 2 4 6

n=305−Parity

0.0

0.1

0.2

0.3

0 2 4 6 8 10

n=505−Parity

0.00

0.10

0.20

0 5 10

n=1005−Parity

0.00

0.05

0.10

0.15

5 10 15 20 25

n=2505−Parity

0.00

0.04

20 30 40 50

n=5005−Parity

0.00

0.02

0.04

0.06

40 50 60 70 80

n=10005−Parity

0.0

0.2

0.4

0.6

24 26 28 30

n=306−Multiplexer

0.0

0.1

0.2

0.3

42 44 46 48 50

n=506−Multiplexer

0.00

0.10

0.20

90 95 100

n=1006−Multiplexer

0.00

0.10

225 230 235 240 245


0.00

0.04

0.08

465 475 485


0.00

0.04

940 950 960 970 980


0.00

0.10

5 10 15

n=30Regression

0.00

0.05

0.10

0.15

5 10 15 20 25

n=50Regression

0.00

0.04

0.08

15 20 25 30 35 40 45

n=100Regression

0.00

0.02

0.04

50 60 70 80 90 100

n=250Regression

0.00

0.02

120 140 160 180

n=500Regression

0.00

00.

015

260 280 300 320 340

n=1000Regression

24 / 68


FigurasTablas

Capítulo 4Figura 4.2: Diagrama QQ binomialidad

220 240 260 280 300

220

260

300

Santa Fe trail Q−Q Plot

Theoretical Quantiles

Sam

ple

Qua

ntile

s

1880 1900 1920 1940

1880

1910

1940

6−multiplexer Q−Q Plot


Sam

ple

Qua

ntile

s

100 120 140

100

120

140

160

5−parity Q−Q Plot


Sam

ple

Qua

ntile

s

540 580 620 660

540

580

620

660

Regression Q−Q Plot


Sam

ple

Qua

ntile

s

25 / 68


FigurasTablas

Capítulo 4Figura 4.3: Ejemplo 3D de CP y CIW

Pro

babi

lity

(p)

0.0

0.2

0.4

0.6

0.8

1.0

Number of trials (n)

20

40

60

80

100

CP

0.2

0.4

0.6

0.8

Pro

babi

lity

(p)

0.0

0.2

0.4

0.6

0.8

1.0

Number of trials (n)

20

40

60

80

100

CIW

0.2

0.4

0.6

26 / 68


FigurasTablas

Capítulo 4Figura 4.4: CP

27 / 68


FigurasTablas

Capítulo 4Figura 4.5: Comparación de métodos de intervalos binomiales

Wils

on C

P0.

850.

900.

951.

00

n= 20

"Exa

ct"

CP

0.85

0.90

0.95

1.00

Sta

ndar

d C

P0.

850.

900.

951.

00A

gres

ti−C

oull

CP

0.85

0.90

0.95

1.00

0.0 0.2 0.4 0.6 0.8 1.0p

n= 50

0.0 0.2 0.4 0.6 0.8 1.0p

n= 100

0.0 0.2 0.4 0.6 0.8 1.0p

n= 500

0.0 0.2 0.4 0.6 0.8 1.0p

28 / 68


FigurasTablas

Capítulo 4Figura 4.6: CP de intervalos de ejecuciones de GP simulados

0.0 0.2 0.4 0.6 0.8 1.0

0.85

0.95

Wilson

p

CP

0.0 0.2 0.4 0.6 0.8 1.0

0.85

0.95

"Exact"

p

CP

0.0 0.2 0.4 0.6 0.8 1.0

0.85

0.95

Standard

p

CP

0.0 0.2 0.4 0.6 0.8 1.0

0.85

0.95

Agresti−Coull

p

CP

29 / 68


FigurasTablas

Capítulo 4Figuras 4.7 y 4.8: Valores medios de CP y CIW

0 20 40 60 80 100

0.80

0.85

0.90

0.95

1.00

Number of runs (n)

Ave

rage

CP

WilsonExactStandardA−C

0.0 0.2 0.4 0.6 0.8 1.0

0.5

0.6

0.7

0.8

0.9

1.0

Success probability (p)

Ave

rage

CP

0 20 40 60 80 100

0.0

0.2

0.4

0.6

Number of runs (n)

Ave

rage

CIW

WilsonExactStandardA−C

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.1

0.2

0.3

0.4


Ave

rage

CIW

30 / 68


FigurasTablas

Capítulo 4Figuras 4.9 y 4.10: CP de ejecuciones GP

Artificial ant

Number of runs (n)

CP

5 15 27 39 51 63 75 87 99

0.80

0.85

0.90

0.95

1.00

6−multiplexer

Number of runs (n)

CP

5 15 27 39 51 63 75 87 99

0.80

0.85

0.90

0.95

1.00

4−parity

Number of runs (n)

CP

5 15 27 39 51 63 75 87 99

0.80

0.85

0.90

0.95

1.00

Regression

Number of runs (n)

CP

5 15 27 39 51 63 75 87 99

0.80

0.85

0.90

0.95

1.00

20 40 60 80 100

0.80

0.85

0.90

0.95

1.00

Number of runs (n)

CP

p=0.13168

20 40 60 80 100

0.80

0.85

0.90

0.95

1.00

Number of runs (n)

CP

p=0.95629

20 40 60 80 100

0.80

0.85

0.90

0.95

1.00

Number of runs (n)

CP

p=0.061

20 40 60 80 100

0.80

0.85

0.90

0.95

1.00

Number of runs (n)

CP

p=0.29462

Artificial ant

Number of runs (n)

CIW

Wilson

"Exact"

Standard

Agresti

5 15 27 39 51 63 75 87 99

0.05

0.10

0.15

0.20

0.25

0.30

6−multiplexer

Number of runs (n)

CIW

5 15 27 39 51 63 75 87 99

0.05

0.10

0.15

0.20

0.25

0.30

4−parity

Number of runs (n)

CIW

5 15 27 39 51 63 75 87 99

0.05

0.10

0.15

0.20

0.25

0.30

Regression

Number of runs (n)

CIW

5 15 27 39 51 63 75 87 99

0.05

0.10

0.15

0.20

0.25

0.30

20 40 60 80 100

0.05

0.10

0.15

0.20

0.25

0.30

Number of runs (n)

CIW

p=0.13168

20 40 60 80 100

0.05

0.10

0.15

0.20

0.25

0.30

Number of runs (n)

CIW

p=0.95629

20 40 60 80 100

0.05

0.10

0.15

0.20

0.25

0.30

Number of runs (n)

CIW

p=0.061

20 40 60 80 100

0.05

0.10

0.15

0.20

0.25

0.30

Number of runs (n)

CIW

p=0.29462

31 / 68


FigurasTablas

Capítulo 4Figura 4.11: Tamaño de la muestra para error dado

Anticipated success probability (p0)

Num

ber

of s

ampl

es (

n)

0.01 0.125 0.25 0.375 0.5

020

4060

8010

0

Half interval width (ε)0.01 0.125 0.25 0.375 0.5

110

100

1000

1000

0

StandardAgresti−CoullWilson

32 / 68


FigurasTablas

Capítulo 4Tabla 4.1: Simulación del lanzamiento de una moneda

Experiment Successes pi1 483 0.4832 531 0.5313 594 0.5944 521 0.5215 513 0.513

Total 2642 26425000 = 0,5284

33 / 68


FigurasTablas

Capítulo 4Tabla 4.2: Configuración de los casos de estudio de la estimación estática

Parameter Artificial ant 6-multiplexer 5-parity RegressionPopulation 500 500 4,000 500Generations 50 50 50 50Terminal Set Left, Right, Move, If-

FoodAheadA0, A1, A2, D0, D1, D2,D3, D4, D5

D0, D1, D2, D3, D4 X

Function set Progn2, Progn3,Progn4

And, Or, Not, If And, Or, Nand, Nor Add, Mul, Sub, Div,Sin, Cos, Exp, Log

Success predicate fitness = 0 fitness = 0 fitness = 0 fitness ≤ 0,001Initial depth 5 5 5 5Max. depth 17 17 17 17Selection Tournament (size=7) Tournament (size=7) Tournament (size=7) Tournament (size=7)Crossover 0.9 0.9 0.9 0.9

Reproduction 0.1 0.1 0.1 0.1Elitism size 0 0 0 0Terminals 0.1 0.1 0.1 0.1

Non terminals 0.9 0.9 0.9 0.9Observations Timesteps=600 Even parity No ERC

y = x4 + x3 + x2 + xx ∈ [−1, 1]

34 / 68


FigurasTablas

Capítulo 4Tabla 4.3: Mejor estimación de la probabilidad de éxito

Artificial ant 6-multiplexer 5-parity Regressionpbest 0.13168 0.95629 0.061 0.29462

k 13,168 95,629 305 29,462n 100,000 100,000 5,000 100,000

CI Stdα=0,05 [0.1295842, 0.1337758] [0.9550228, 0.9575572] [0.05436622, 0.06763378] [0.2917945, 0.2974455]CI Stdα=0,01 [0.12892566, 0.1344343] [0.9546247, 0.9579553] [0.05228174, 0.06971826] [0.2909067, 0.2983333]CI ACα=0,05 [0.1295983, 0.1337900] [0.9550051, 0.9575399] [0.05468869, 0.06798535] [0.2918025, 0.2974533]CI ACα=0,01 [0.12894997, 0.1344589] [0.9545939, 0.9579256] [0.05283056, 0.07033299] [0.2909204, 0.2983469]CI Wilα=0,05 [0.1295984, 0.1337899] [0.9550052, 0.9575397] [0.05469723, 0.06797681] [0.2918025, 0.2974533]CI Wilα=0,01 [0.12895008, 0.1344588] [0.9545942, 0.9579253] [0.05284989, 0.07031365] [0.2909204, 0.2983468]

35 / 68


FigurasTablas

Capítulo 4Tabla 4.4: Prueba χ2 de la binomialidad

NSanta Fe 6-Multiplexer 5-Parity Regression

p − val sd diff p − val sd diff p − val sd diff p − val sd diff15 0.2275 0.0032 0.2243 0.2206 0.0789 0.1417 0.2211 0.0042 0.2169 0.2331 0.0152 0.217930 0.2303 0.0243 0.206 0.2242 0.0060 0.2182 0.2279 0.0041 0.2238 0.2425 0.0203 0.222250 0.2374 0.0197 0.2177 0.2293 0.0053 0.224 0.2327 0.0048 0.2279 0.2453 0.0382 0.2071100 0.2355 0.0535 0.182 0.2342 0.0285 0.2057 0.2383 0.0125 0.2258 0.2316 0.0809 0.1507250 0.2397 0.1155 0.1242 0.2420 0.0249 0.2171 0.2300 0.0631 0.1669 0.2132 0.1535 0.0597500 0.1885 0.1479 0.0406 0.2326 0.0756 0.157 0.2348 0.1044 0.1304 0.1303 0.1629 -0.03261000 0.1279 0.1813 -0.0534 0.2109 0.1301 0.0808 0.2041 0.1006 0.1035 0.0407 0.1006 -0.0599

36 / 68


FigurasTablas

Capítulo 5Figura 5.1: Histograma de la generación de éxito

Artificial ant

Generation−to−success

Den

sity

0 10 20 30 40 50

0.00

0.02

0.04

Regression


Den

sity

0 10 20 30 40 50

0.00

0.04

0.08

4−Parity


Den

sity

0 10 20 30 40 50

0.00

0.04

0.08

5−Parity

Generation−to−successD

ensi

ty0 200 400 600 800

0.00

00.

002

0.00

40.

006

6−Multiplexer


Den

sity

0 10 20 30 40 50

0.00

0.02

0.04

0.06

0.08

11−Multiplexer


Den

sity

0 200 400 600 800

0.00

00.

001

0.00

20.

003

NormalLognormalWeibullLogistic

37 / 68


FigurasTablas

Capítulo 5Figura 5.2: Diagrama QQ del logaritmo de la generación de éxito vs. lognormal

Normal

log(

gene

ratio

n−to

−su

cces

s)

1

2

3

4

5

6

Artificial ant

−3 −2 −1 0 1 2 3

4−Parity 6−Multiplexer

−3 −2 −1 0 1 2 3

Regression 5−Parity

−3 −2 −1 0 1 2 3

1

2

3

4

5

6

11−Multiplexer

38 / 68


FigurasTablas

Capítulo 5Figura 5.3: Histograma con la fase inicial eliminada


Exp

onen

tial 0

1020

3040

0 10 20 30 40

Artificial ant

010

2030

0 10 20 30

4−Parity

010

2030

40

0 10 20 30 40

6−Multiplexer

010

2030

40

0 10 20 30 40

Regression

020

040

060

080

0

0 200 400 600 800

5−Parity

020

040

060

0

0 200 400 600

11−Multiplexer

39 / 68


FigurasTablas

Capítulo 5Figura 5.4: Diagrama QQ con fase inicial eliminada contra exponencial


Den

sity

0.00

0.01

0.02

0.03

0.04

0.05

0 10 20 30 40 50

Artificial ant

0.00

0.02

0.04

0.06

0.08

0.10

0.12

10 20 30 40 50

4−Parity

0.00

0.02

0.04

0.06

0.08

0 10 20 30 40 50

6−Multiplexer

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0 10 20 30 40 50

Regression

0.00

00.

001

0.00

20.

003

0.00

4

0 200 400 600 800 1000

5−Parity

0.00

00.

001

0.00

20.

003

0 200 400 600 800

11−Multiplexer

40 / 68


FigurasTablas

Capítulo 5Figura 5.5: Tamaño medio del árbol

generation

Ave

rage

Dep

th +

Ave

rage

Nod

es

510

15

0 100 200 300 400

AverageDepth

020

040

060

080

010

00

0 100 200 300 400

AverageNodes

11−Multiplexer4−Parity5−Parity6−MultiplexerArtificial antRegression

41 / 68


FigurasTablas

Capítulo 5Figura 5.6: Histograma de la generación de éxito sin presión selectiva

Artificial ant


Den

sity

0 200 400 600 800 1000

0.00

000.

0010

0.00

200.

0030

0 200 400 600 800 1000

020

040

060

080

010

00


Wei

bull

Regression


Den

sity

0 200 400 600 800 1000

0.00

000.

0010

0.00

200.

0030

NormalLognormalWeibullLogistic

0 200 400 600 800 1000

020

040

060

080

010

00


Wei

bull

42 / 68


FigurasTablas

Capítulo 5Figura 5.7: Modelo de Markov

43 / 68


FigurasTablas

Capítulo 5Figura 5.8: Comparación de P(M, i) y P?(M, i)

0 10 20 30 40 50

0.00

0.10

0.20

Artificial ant

Generation

Suc

cess

pro

babi

lity

StandardLognormal

0 10 20 30 40 50

0.0

0.4

0.8

6−Multiplexer

Generation

Suc

cess

pro

babi

lity

0 10 20 30 40 50

0.00

0.04

0.08

4−Parity

Generation

Suc

cess

pro

babi

lity

0 10 20 30 40 50

0.0

0.1

0.2

0.3

0.4

Regression

Generation

Suc

cess

pro

babi

lity

44 / 68


FigurasTablas

Capítulo 5Figura 5.9: Comparación de P(M, i) y P?(M, i) variando n

Art

ifici

al a

nt0.

00.

20.

4

n= 25

Best est.

Standard

Lognormal

n= 50 n= 100 n= 200

6−M

ultip

lexe

r0.

00.

40.

84−

Par

ity0.

000.

06R

egre

ssio

n0.

00.

20.

4

0 20 40Generation

0 20 40Generation

0 20 40Generation

0 20 40Generation

45 / 68


FigurasTablas

Capítulo 5Tabla 5.1: Configuración de los problemas bajo estudio

Parameter Artificial ant 6/11-multiplexer 4/5-parity RegressionPopulation 500 500 4,000 500Generations 50 50 800 50Terminal Set Left, Right, Move, If-

FoodAheadA0, A1, A2, D0, D1,D2, D3, D4, D5

D0, D1, D2, D3, D4 X

Function set Progn2, Progn3,Progn4

And, Or, Not, If And, Or, Nand, Nor Add, Mul, Sub, Div,Sin, Cos, Exp, Log

Success predicate fitness = 0 fitness = 0 fitness = 0 fitness ≤ 0,001Initial depth 5 5 5 5Max. depth 17 17 17 17Selection Tour. (size=7) Tour. (size=7) Tour. (size=7) Tour. (size=7)Crossover 0.9 0.9 0.9 0.9

Reproduction 0.1 0.1 0.1 0.1Elitism size 0 0 0 0Terminals 0.1 0.1 0.1 0.1

Non terminals 0.9 0.9 0.9 0.9Observations Timesteps=600 Even parity No ERC

Santa Fe trail y = x4 + x3 + x2 + xx ∈ [−1, 1]

46 / 68


FigurasTablas

Capítulo 5Tabla 5.2: Probabilidad de éxito

Artificial ant 6-Multiplexer 11-Multiplexer 4-Parity 5-Parity Regression

n 100,000 100,000 1,000 400 5,000 100,000k 13,168 95,629 333 299 305 29,462p 0.132 0.956 0.333 0.747 0.061 0.295Lp 0.1296 0.9550 0.3045 0.7027 0.0547 0.2918Up 0.1338 0.9575 0.3628 0.7876 0.0680 0.2975

47 / 68


FigurasTablas

Capítulo 5Tabla 5.3: Estimaciones de los parámetros de la lognormal y exponencial

µ σ λ t0

Artificial ant 2.73 0.595 0.085 9

Regression 2.289 0.44 0.150 5

4-Parity 3.03 0.284 0.146 17

5-Parity 5.004 0.8 0.005 7

6-Multiplexer 2.464 0.425 0.149 7

11-Multiplexer 5.367 0.797 0.004 60

48 / 68


FigurasTablas

Capítulo 6Figura 6.1: Variación de I (M, i , z)

0 10 20 30 40 50

0e+

004e

+05

8e+

05

Sante Fe Trail

Generation

I(M

,i,z)

0 10 20 30 40 50

0e+

004e

+05

8e+

05

6−Multiplexer

Generation

I(M

,i,z)

0 10 20 30 40 50

0e+

004e

+07

8e+

07

5−Parity

Generation

I(M

,i,z)

0 10 20 30 40 50

0e+

004e

+05

8e+

05

Regression

Generation

I(M

,i,z)

49 / 68


FigurasTablas

Capítulo 6Figura 6.2: Histograma del esfuerzo computacional

Computational effort

Freq

uenc

y0

500

1000

1500

2000

0e+00 1e+06 2e+06 3e+06 4e+06

Artificial antn=50

050

010

00

0 10000 20000 30000 40000

6−Multiplexern=50

020

040

060

080

0

1e+07 2e+07 3e+07 4e+07

5−Parityn=50

050

010

0015

0020

00

1e+05 3e+05 5e+05

Regressionn=50

050

010

00

200000 600000 1000000

Artificial antn=200

050

010

0015

00

0 10000 20000 30000

6−Multiplexern=200

050

010

0015

0020

00

2e+07 4e+07 6e+07 8e+07

5−Parityn=200

020

040

060

080

010

00

100000 150000 200000

Regressionn=200

020

040

060

080

010

00

400000 600000 800000 1000000

Artificial antn=500

020

040

060

080

010

00

20000 22000 24000 26000 28000

6−Multiplexern=500

020

040

060

080

010

00

1.0e+07 2.0e+07 3.0e+07

5−Parityn=500

020

040

060

080

010

00

80000 100000 120000 140000 160000

Regressionn=500

50 / 68


FigurasTablas

Capítulo 6Figura 6.3: Error máximo de redondeo como función de ε y comparación de R y Rc

0.0 0.2 0.4 0.6 0.8 1.0

050

100

150

Maximum relative ceiling error

Success probability

Rel

ativ

e ce

iling

err

or (

%)

ε=1−z=0.05ε=1−z=0.01ε=1−z=0.001

0.0 0.2 0.4 0.6 0.8 1.00

510

1520

2530

Comparison of R with and without ceiling

Success probability

R

R (with ceiling)Rc (without ceiling)

P(M,i)=0.95

R=1

51 / 68


FigurasTablas

Capítulo 6Figura 6.4: Error de redondeo máximo

−1.0

−0.5

0.0

0.5

1.0

0.0

0.2

0.4

0.6

0.8

1.0

−1000000

−500000

0

500000

εest

P

εestI

52 / 68


FigurasTablas

Capítulo 6Figura 6.5: Error de estimacion relativo

−0.6 −0.4 −0.2 0.0 0.2 0.4 0.6

−60

0−

500

−40

0−

300

−20

0−

100

010

0

Estimation error and I(M,i,z)

Estimation error (εest)

Rel

ativ

e es

timat

ion

erro

r (ε

est

I)(%

)

P = 0.1P = 0.25P = 0.5P = 0.75P = 0.9

53 / 68


FigurasTablas

Capítulo 6Figura 6.6: Máximo error de estimación de P(M, io)

Maximum estimation error (εestα )

Success probability (P)

Num

ber

of r

uns

(n)

0.0 0.2 0.4 0.6 0.8 1.0

2040

6080

100

120

54 / 68


FigurasTablas

Capítulo 6Figura 6.7: Máximo error relativo de I (M, i , z) con CIs

Maximum I(M,i,z) estimation error, max(εestI (%))

Success probability (P)

Num

ber

of r

uns

(n)

0.2 0.4 0.6 0.8

2040

6080

100

120

140

55 / 68


FigurasTablas

Capítulo 6Figura 6.8: Representación de E?

1

2

3

4

0.2

0.4

0.6

0.81.0

1e+05

2e+05

3e+05

4e+05

5e+05

µσ

E

M=1000

1

2

3

4

0.2

0.4

0.6

0.81.0

1e+05

2e+05

3e+05

4e+05

5e+05

µσ

E

M=500

56 / 68


FigurasTablas

Capítulo 6Figura 6.9: Error de medida de E

50 100 200 300 400 500

020

0040

0060

0080

00

6−multiplexer

Number of runs

Err

or o

f com

puta

tiona

l effo

rt

StandardLognormal

Mean EMean E*

100 200 300 400 500

−1.

0−

0.5

0.0

0.5

Artificial ant

Number of runs

(E−

E*)

/Ebe

st

100 200 300 400 500

−0.

2−

0.1

0.0

0.1

0.2

6−Multiplexer

Number of runs

(E−

E*)

/Ebe

st

100 200 300 400 500

−0.

6−

0.4

−0.

20.

00.

2

5−Parity

Number of runs

(E−

E*)

/Ebe

st

100 200 300 400 500

−0.

4−

0.2

0.0

0.2

Regression

Number of runs

(E−

E*)

/Ebe

st

57 / 68


FigurasTablas

Capítulo 6Figura 6.10: Caracterización del error de E

1

23

4

0.2

0.4

0.6

0.81.0

0.1

0.2

0.3

0.4

0.5

µσ

ε

30

1

23

4

0.2

0.4

0.6

0.81.0

0.1

0.2

0.3

0.4

0.5

µσ

ε

50

1

23

4

0.2

0.4

0.6

0.81.0

0.1

0.2

0.3

0.4

0.5

µσ

ε

100

1

23

4

0.2

0.4

0.6

0.81.0

0.1

0.2

0.3

0.4

0.5

µσ

ε

500

58 / 68


FigurasTablas

Capítulo 6Figura 6.11: Error de redondeo experimental

0 10 20 30 40 50

5e+

057e

+05

9e+

05

Artificial ant

Generations

Num

ber

of in

divi

dual

sWith ceiling

Without ceiling

0 10 20 30 40 50

2e+

046e

+04

1e+

05

6−Multiplexer

Generations

Num

ber

of in

divi

dual

s

0 10 20 30 40 50

0e+

004e

+07

8e+

07

5−Parity

Generations

Num

ber

of in

divi

dual

s

0 10 20 30 40 50

1e+

053e

+05

5e+

05

Regression

Generations

Num

ber

of in

divi

dual

s

59 / 68


FigurasTablas

Capítulo 6Figura 6.12: Comparación error de reondeo teórico y experimental

0 10 20 30 40 50

050

0010

000

2000

0

Artificial ant

Generations

Abs

olut

e ce

iling

err

or(ε

ceil

I)

Maximum error

Measured error

0 10 20 30 40 50

050

0010

000

2000

0

6−Multiplexer

Generations

Abs

olut

e ce

iling

err

or(ε

ceil

I)

0 10 20 30 40 50

050

000

1000

0015

0000

2000

00

5−Parity

Generations

Abs

olut

e ce

iling

err

or(ε

ceil

I)

0 10 20 30 40 50

050

0010

000

2000

0

Regression

Generations

Abs

olut

e ce

iling

err

or(ε

ceil

I)

0 10 20 30 40 50

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Artificial ant

Generations

Rel

ativ

e ce

iling

err

or(ε

ceil

I)(%

) Maximum error

Measured error

0 10 20 30 40 50

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

5−Parity

Generations

Rel

ativ

e ce

iling

err

or(ε

ceil

I)(%

)

0 10 20 30 40 50

010

2030

4050

6070

6−Multiplexer

Generations

Rel

ativ

e ce

iling

err

or(ε

ceil

I)(%

)

0 10 20 30 40 50

02

46

Regression

Generations

Rel

ativ

e ce

iling

err

or(ε

ceil

I)(%

)

60 / 68


FigurasTablas

Capítulo 6Figura 6.13: Error experimental absoluto de I (Mi , z)

50 200 400

5.0e

+06

1.5e

+07

2.5e

+07

Artificial ant

Number of runs (n)

Abs

olut

e es

timat

ion

erro

r

50 200 4003950

0000

4100

0000

4250

0000 6−Multiplexer

Number of runs (n)

Abs

olut

e es

timat

ion

erro

r

50 200 400

6e+

078e

+07

1e+

08

Regression

Number of runs (n)

Abs

olut

e es

timat

ion

erro

r

61 / 68


FigurasTablas

Capítulo 6Figura 6.13: Error experimenta relativo de I (M, i , z)

0.00 0.05 0.10 0.15 0.20

−20

00

100

Artificial Ant


Rel

ativ

e es

timat

ion

erro

r (%

)

0.0 0.2 0.4 0.6 0.8 1.0

−20

0−

100

010

0

6−Multiplexer


Rel

ativ

e es

timat

ion

erro

r (%

)

0.00 0.04 0.08 0.12

−20

00

100

5−Parity


Rel

ativ

e es

timat

ion

erro

r (%

)

0.0 0.1 0.2 0.3 0.4

−15

0−

500

50

Regression


Rel

ativ

e es

timat

ion

erro

r (%

)

62 / 68


FigurasTablas

Capítulo 6Figura 6.14: Error experimental absoluto de redondeo

50 200 400

050

0015

000

Artificial ant

Number of runs

E−

Ec

50 200 400

020

0060

00

6−Multiplexer

Number of runs

E−

Ec

50 200 400

050

000

1500

00

5−Parity

Number of runs

E−

Ec

50 200 400

020

0060

00

Regression

Number of runs

E−

Ec

63 / 68


FigurasTablas

Capítulo 6Figura 6.15: Error experimental relativo de E

50 200 400−10

00−

600

−20

0

Artificial ant

Number of runs

Est

imat

ion

erro

r of

E (

%)

With ceilingWithout ceiling

50 200 400

−80

−40

020

6−Multiplexer

Number of runs

Est

imat

ion

erro

r of

E (

%)

50 200 400

−50

0−

300

−10

0

5−Parity

Number of runs

Est

imat

ion

erro

r of

E (

%)

50 200 400

−10

00

50

Regression

Number of runs

Est

imat

ion

erro

r of

E (

%)

64 / 68


FigurasTablas

Capítulo 6Tabla 6.1: Mejores estimaciones

Artificial ant 6-Multiplexer 5-Parity Regression

n 100,000 100,000 5,000 100,000

k 13,168 95,629 305 29,462

Pbest(M, G) 0.13168 0.95629 0.061 0.29462

E best 490,000 24,000 14,800,000 117,000

E bestc 487,276 22,805 14,633,571 116,468

Difference 2,724 (0.5 %) 1,195 (4.98 %) 166,429 (1.13 %) 536 (0.49 %)

65 / 68


FigurasTablas

Capítulo 6Tabla 6.2: Diferencia de E y E?

|Ebest − E | sd |Ebest − E | |Ebest − E ?| sd |Ebest − E ?|50 3627.9770 2562.9722 3482.4215 2910.1751100 2585.3189 1961.6505 2416.7762 1900.6281200 1634.6052 1264.5999 1572.8789 1204.3831300 1368.5800 1013.5731 1300.1529 991.7907400 1158.0780 888.1902 1118.1963 860.9419500 1073.2470 790.5500 1012.5622 766.0491

66 / 68


FigurasTablas

Capítulo 6Tabla 6.3: ANOVA de la diferencia Ec − E

Problem df Sum. sq. Mean sq. F-value p-valueArtificial ant 5 2445 489.03 0.9689 0.4376-Multiplexer 5 5145 1029 5.1462 0.0001529*

5-Parity 5 236380 47276 4.9595 0.0002247*Regression 5 4349 869.79 2.9602 0.01266

67 / 68


FigurasTablas

Capítulo 6Tabla 6.3: ANOVA del error de estimación

Problem df Sum sq Mean sq F value p-valueArtificial Ant 5 126.44 25.2874 12.579 1.035e-106-Multiplexer 5 95.09 19.0180 14.654 3.272e-12

5-Parity 5 98.72 19.7441 8.4623 4.308e-07*Regression 5 106.12 21.2248 12.264 3.414e-10

68 / 68

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