Chapter 7 MI500: Ordered exploitation results

Here we present the results for best performances found by each selection scheme replicate on the ordered exploitation diagnostic with our base configurations. Best performance found refers to the largest average trait score found in a given population. Note that performance values fall between 0.0 and 100.0. For our base configuration, we execute migrations every 500 generations and there are 4 islands in a ring topology. When migrations occur, we swap two individuals (same position on each island) and guarantee that no solution can return to the same island.

7.1 Analysis dependencies

library(ggplot2)
library(cowplot)
library(dplyr)
library(PupillometryR)

7.2 Truncation selection

Here we analyze how the different population structures affect truncation selection (size 8) on the ordered exploitation diagnostic.

7.2.1 Performance over time

lines = filter(base_over_time, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'TRUNCATION') %>%
  group_by(Structure, Generations) %>%
  dplyr::summarise(
    min = min(pop_fit_max) / DIMENSIONALITY,
    mean = mean(pop_fit_max) / DIMENSIONALITY,
    max = max(pop_fit_max) / DIMENSIONALITY
  )
ggplot(lines, aes(x=Generations, y=mean, group = Structure, fill = Structure, color = Structure, shape = Structure)) +
  geom_ribbon(aes(ymin = min, ymax = max), alpha = 0.1) +
  geom_line(size = 0.5) +
  geom_point(data = filter(lines, Generations %% 2000 == 0), size = 2.5, stroke = 2.0, alpha = 1.0) +
  scale_y_continuous(
    name="Average trait score",
    limits=c(-1, 101),
    breaks=seq(0,100, 20),
    labels=c("0", "20", "40", "60", "80", "100")
  ) +
  scale_x_continuous(
    name="Generations",
    limits=c(0, 50000),
    breaks=c(0, 10000, 20000, 30000, 40000, 50000),
    labels=c("0e+4", "1e+4", "2e+4", "3e+4", "4e+4", "5e+4")

  ) +
  scale_shape_manual(values=SHAPE)+
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  ggtitle("Performance over time") +
  p_theme

7.2.2 Generation satisfactory solution found

First generation a satisfactory solution is found throughout the 50,000 generations.

filter(base_ssf, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'TRUNCATION') %>%
  ggplot(., aes(x = Structure, y = Generations , color = Structure, fill = Structure, shape = Structure)) +
  geom_flat_violin(position = position_nudge(x = .2, y = 0), scale = 'width', alpha = 0.2) +
  geom_point(position = position_jitter(width = .1), size = 1.5, alpha = 1.0) +
  geom_boxplot(color = 'black', width = .2, outlier.shape = NA, alpha = 0.0) +
  scale_y_continuous(
    name="Generation"
  ) +
  scale_x_discrete(
    name="Structure"
  )+
  scale_shape_manual(values=SHAPE)+
  scale_colour_manual(values = cb_palette, ) +
  scale_fill_manual(values = cb_palette) +
  ggtitle('Generation satisfactory solution found')+
  p_theme + coord_flip()

7.2.2.1 Stats

Summary statistics for the first generation a satisfactory solution is found.

ssf = filter(base_ssf, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'TRUNCATION' & Generations < 60000)
ssf %>%
  group_by(Structure) %>%
  dplyr::summarise(
    count = n(),
    na_cnt = sum(is.na(Generations)),
    min = min(Generations, na.rm = TRUE),
    median = median(Generations, na.rm = TRUE),
    mean = mean(Generations, na.rm = TRUE),
    max = max(Generations, na.rm = TRUE),
    IQR = IQR(Generations, na.rm = TRUE)
  )

## # A tibble: 3 x 8
##   Structure count na_cnt   min median   mean   max   IQR
##   <fct>     <int>  <int> <int>  <dbl>  <dbl> <int> <dbl>
## 1 EA          100      0 14684 15546  15554. 16254  492.
## 2 IS          100      0 24669 26780. 26767. 28518 1318.
## 3 NMIS        100      0 26330 27939  27888. 29654  825.

Kruskal–Wallis test provides evidence of difference among selection schemes.

kruskal.test(Generations ~ Structure, data = ssf)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  Generations by Structure
## Kruskal-Wallis chi-squared = 231.88, df = 2, p-value < 2.2e-16

Results for post-hoc Wilcoxon rank-sum test with a Bonferroni correction.

pairwise.wilcox.test(x = ssf$Generations, g = ssf$Structure, p.adjust.method = "bonferroni",
                     paired = FALSE, conf.int = FALSE, alternative = 'g')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  ssf$Generations and ssf$Structure 
## 
##      EA     IS    
## IS   <2e-16 -     
## NMIS <2e-16 <2e-16
## 
## P value adjustment method: bonferroni

7.3 Tournament selection

Here we analyze how the different population structures affect tournament selection (size 8) on the ordered exploitation diagnostic.

7.3.1 Performance over time

lines = filter(base_over_time, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'TOURNAMENT') %>%
  group_by(Structure, Generations) %>%
  dplyr::summarise(
    min = min(pop_fit_max) / DIMENSIONALITY,
    mean = mean(pop_fit_max) / DIMENSIONALITY,
    max = max(pop_fit_max) / DIMENSIONALITY
  )
ggplot(lines, aes(x=Generations, y=mean, group = Structure, fill = Structure, color = Structure, shape = Structure)) +
  geom_ribbon(aes(ymin = min, ymax = max), alpha = 0.1) +
  geom_line(size = 0.5) +
  geom_point(data = filter(lines, Generations %% 2000 == 0), size = 2.5, stroke = 2.0, alpha = 1.0) +
  scale_y_continuous(
    name="Average trait score",
    limits=c(-1, 101),
    breaks=seq(0,100, 20),
    labels=c("0", "20", "40", "60", "80", "100")
  ) +
  scale_x_continuous(
    name="Generations",
    limits=c(0, 50000),
    breaks=c(0, 10000, 20000, 30000, 40000, 50000),
    labels=c("0e+4", "1e+4", "2e+4", "3e+4", "4e+4", "5e+4")

  ) +
  scale_shape_manual(values=SHAPE)+
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  ggtitle("Performance over time") +
  p_theme

7.3.2 Generation satisfactory solution found

First generation a satisfactory solution is found throughout the 50,000 generations.

filter(base_ssf, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'TOURNAMENT') %>%
  ggplot(., aes(x = Structure, y = Generations , color = Structure, fill = Structure, shape = Structure)) +
  geom_flat_violin(position = position_nudge(x = .2, y = 0), scale = 'width', alpha = 0.2) +
  geom_point(position = position_jitter(width = .1), size = 1.5, alpha = 1.0) +
  geom_boxplot(color = 'black', width = .2, outlier.shape = NA, alpha = 0.0) +
  scale_y_continuous(
    name="Generation"
  ) +
  scale_x_discrete(
    name="Structure"
  )+
  scale_shape_manual(values=SHAPE)+
  scale_colour_manual(values = cb_palette, ) +
  scale_fill_manual(values = cb_palette) +
  ggtitle('Generation satisfactory solution found')+
  p_theme + coord_flip()

7.3.2.1 Stats

Summary statistics for the first generation a satisfactory solution is found.

ssf = filter(base_ssf, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'TOURNAMENT' & Generations < 60000)
ssf %>%
  group_by(Structure) %>%
  dplyr::summarise(
    count = n(),
    na_cnt = sum(is.na(Generations)),
    min = min(Generations, na.rm = TRUE),
    median = median(Generations, na.rm = TRUE),
    mean = mean(Generations, na.rm = TRUE),
    max = max(Generations, na.rm = TRUE),
    IQR = IQR(Generations, na.rm = TRUE)
  )

## # A tibble: 3 x 8
##   Structure count na_cnt   min median   mean   max   IQR
##   <fct>     <int>  <int> <int>  <dbl>  <dbl> <int> <dbl>
## 1 EA          100      0 25242 27228. 27172. 28742  921.
## 2 IS          100      0 30589 34356. 34349. 36461 1564.
## 3 NMIS        100      0 33412 35764  35692. 37306 1213

Kruskal–Wallis test provides evidence of difference among selection schemes.

kruskal.test(Generations ~ Structure, data = ssf)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  Generations by Structure
## Kruskal-Wallis chi-squared = 229.49, df = 2, p-value < 2.2e-16

Results for post-hoc Wilcoxon rank-sum test with a Bonferroni correction.

pairwise.wilcox.test(x = ssf$Generations, g = ssf$Structure, p.adjust.method = "bonferroni",
                     paired = FALSE, conf.int = FALSE, alternative = 'g')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  ssf$Generations and ssf$Structure 
## 
##      EA      IS     
## IS   < 2e-16 -      
## NMIS < 2e-16 2.8e-16
## 
## P value adjustment method: bonferroni

7.4 Lexicase selection

Here we analyze how the different population structures affect standard lexicase selection on the ordered exploitation diagnostic.

7.4.1 Performance over time

lines = filter(base_over_time, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'LEXICASE') %>%
  group_by(Structure, Generations) %>%
  dplyr::summarise(
    min = min(pop_fit_max) / DIMENSIONALITY,
    mean = mean(pop_fit_max) / DIMENSIONALITY,
    max = max(pop_fit_max) / DIMENSIONALITY
  )
ggplot(lines, aes(x=Generations, y=mean, group = Structure, fill = Structure, color = Structure, shape = Structure)) +
  geom_ribbon(aes(ymin = min, ymax = max), alpha = 0.1) +
  geom_line(size = 0.5) +
  geom_point(data = filter(lines, Generations %% 2000 == 0), size = 2.5, stroke = 2.0, alpha = 1.0) +
  scale_y_continuous(
    name="Average trait score",
    limits=c(-1, 101),
    breaks=seq(0,100, 20),
    labels=c("0", "20", "40", "60", "80", "100")
  ) +
  scale_x_continuous(
    name="Generations",
    limits=c(0, 50000),
    breaks=c(0, 10000, 20000, 30000, 40000, 50000),
    labels=c("0e+4", "1e+4", "2e+4", "3e+4", "4e+4", "5e+4")

  ) +
  scale_shape_manual(values=SHAPE)+
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  ggtitle("Performance over time") +
  p_theme

7.4.2 Best performance

First generation a satisfactory solution is found throughout the 50,000 generations.

filter(base_best, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'LEXICASE' & VAR == 'pop_fit_max') %>%
  ggplot(., aes(x = Structure, y = VAL / DIMENSIONALITY, color = Structure, fill = Structure, shape = Structure)) +
  geom_flat_violin(position = position_nudge(x = .2, y = 0), scale = 'width', alpha = 0.2) +
  geom_point(position = position_jitter(width = .1), size = 1.5, alpha = 1.0) +
  geom_boxplot(color = 'black', width = .2, outlier.shape = NA, alpha = 0.0) +
  scale_y_continuous(
    name="Average trait score"
  ) +
  scale_x_discrete(
    name="Structure"
  )+
  scale_shape_manual(values=SHAPE)+
  scale_colour_manual(values = cb_palette, ) +
  scale_fill_manual(values = cb_palette) +
  ggtitle('Best performance')+
  p_theme + coord_flip()

7.4.2.1 Stats

Summary statistics for the first generation a satisfactory solution is found.

performance = filter(base_best, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'LEXICASE' & VAR == 'pop_fit_max')
performance %>%
  group_by(Structure) %>%
  dplyr::summarise(
    count = n(),
    na_cnt = sum(is.na(VAL)),
    min = min(VAL, na.rm = TRUE) / DIMENSIONALITY,
    median = median(VAL, na.rm = TRUE) / DIMENSIONALITY,
    mean = mean(VAL, na.rm = TRUE) / DIMENSIONALITY,
    max = max(VAL, na.rm = TRUE) / DIMENSIONALITY,
    IQR = IQR(VAL, na.rm = TRUE) / DIMENSIONALITY
  )

## # A tibble: 3 x 8
##   Structure count na_cnt   min median  mean   max    IQR
##   <fct>     <int>  <int> <dbl>  <dbl> <dbl> <dbl>  <dbl>
## 1 EA          100      0  99.7   99.8  99.8  99.8 0.0291
## 2 IS          100      0  99.5   99.6  99.6  99.7 0.0465
## 3 NMIS        100      0  99.5   99.6  99.6  99.7 0.0535

Kruskal–Wallis test provides evidence of difference among selection schemes.

kruskal.test(VAL ~ Structure, data = performance)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  VAL by Structure
## Kruskal-Wallis chi-squared = 235.04, df = 2, p-value < 2.2e-16

Results for post-hoc Wilcoxon rank-sum test with a Bonferroni correction.

pairwise.wilcox.test(x = performance$VAL, g = performance$Structure, p.adjust.method = "bonferroni",
                     paired = FALSE, conf.int = FALSE, alternative = 'l')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  performance$VAL and performance$Structure 
## 
##      EA     IS    
## IS   <2e-16 -     
## NMIS <2e-16 <2e-16
## 
## P value adjustment method: bonferroni

7.4.3 Final performance

First generation a satisfactory solution is found throughout the 50,000 generations.

filter(base_over_time, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'LEXICASE' & Generations == 50000) %>%
  ggplot(., aes(x = Structure, y = pop_fit_max / DIMENSIONALITY, color = Structure, fill = Structure, shape = Structure)) +
  geom_flat_violin(position = position_nudge(x = .2, y = 0), scale = 'width', alpha = 0.2) +
  geom_point(position = position_jitter(width = .1), size = 1.5, alpha = 1.0) +
  geom_boxplot(color = 'black', width = .2, outlier.shape = NA, alpha = 0.0) +
  scale_y_continuous(
    name="Average trait score"
  ) +
  scale_x_discrete(
    name="Structure"
  )+
  scale_shape_manual(values=SHAPE)+
  scale_colour_manual(values = cb_palette, ) +
  scale_fill_manual(values = cb_palette) +
  ggtitle('Final performance')+
  p_theme + coord_flip()

7.4.3.1 Stats

Summary statistics for the first generation a satisfactory solution is found.

performance = filter(base_over_time, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'LEXICASE' & Generations == 50000)
performance %>%
  group_by(Structure) %>%
  dplyr::summarise(
    count = n(),
    na_cnt = sum(is.na(pop_fit_max)),
    min = min(pop_fit_max / DIMENSIONALITY, na.rm = TRUE),
    median = median(pop_fit_max / DIMENSIONALITY, na.rm = TRUE),
    mean = mean(pop_fit_max / DIMENSIONALITY, na.rm = TRUE),
    max = max(pop_fit_max / DIMENSIONALITY, na.rm = TRUE),
    IQR = IQR(pop_fit_max / DIMENSIONALITY, na.rm = TRUE)
  )

## # A tibble: 3 x 8
##   Structure count na_cnt   min median  mean   max    IQR
##   <fct>     <int>  <int> <dbl>  <dbl> <dbl> <dbl>  <dbl>
## 1 EA          100      0  99.7   99.8  99.8  99.8 0.0291
## 2 IS          100      0  99.5   99.6  99.6  99.7 0.0465
## 3 NMIS        100      0  99.5   99.6  99.6  99.7 0.0535

Kruskal–Wallis test provides evidence of difference among selection schemes.

kruskal.test(pop_fit_max ~ Structure, data = performance)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  pop_fit_max by Structure
## Kruskal-Wallis chi-squared = 235.02, df = 2, p-value < 2.2e-16

Results for post-hoc Wilcoxon rank-sum test with a Bonferroni correction.

pairwise.wilcox.test(x = performance$pop_fit_max, g = performance$Structure, p.adjust.method = "bonferroni",
                     paired = FALSE, conf.int = FALSE, alternative = 'l')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  performance$pop_fit_max and performance$Structure 
## 
##      EA     IS    
## IS   <2e-16 -     
## NMIS <2e-16 <2e-16
## 
## P value adjustment method: bonferroni

7.4.4 Generation satisfactory solution found

First generation a satisfactory solution is found throughout the 50,000 generations.

lex_fail = filter(base_ssf, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'LEXICASE' & GENERATIONS < Generations)
lex_fail$Generations = 55000
lex_fail$Structure <- factor(lex_fail$Structure, levels = MODEL)

filter(base_ssf, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'LEXICASE'& Generations <= GENERATIONS) %>%
  ggplot(., aes(x = Structure, y = Generations, color = Structure, fill = Structure, shape = Structure)) +
    geom_flat_violin(position = position_nudge(x = .2, y = 0), scale = 'width', alpha = 0.2) +
  geom_point(position = position_jitter(width = .1), size = 1.5, alpha = 1.0) +
  geom_boxplot(color = 'black', width = .2, outlier.shape = NA, alpha = 0.0) +
  geom_point(data = lex_fail, aes(x = Structure, y = Generations, color = Structure, fill = Structure, shape = Structure),position = position_jitter(width = .05), size = 2.5) +
  scale_shape_manual(values=SHAPE)+
  scale_y_continuous(
    name="Generations",
    limits=c(30000, 55000),
    breaks=c(30000, 40000, 50000, 55000),
    labels=c("3e+4", "4e+4", "5e+4", "Fail")
  ) +
  scale_x_discrete(
    name="Structure"
  ) +
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  p_theme + coord_flip()

7.4.4.1 Stats

Summary statistics for the first generation a satisfactory solution is found.

ssf = filter(base_ssf, Diagnostic == 'ORDERED_EXPLOITATION' & `Selection\nScheme` == 'LEXICASE' & Generations < 60000)
ssf %>%
  group_by(Structure) %>%
  dplyr::summarise(
    count = n(),
    na_cnt = sum(is.na(Generations)),
    min = min(Generations, na.rm = TRUE),
    median = median(Generations, na.rm = TRUE),
    mean = mean(Generations, na.rm = TRUE),
    max = max(Generations, na.rm = TRUE),
    IQR = IQR(Generations, na.rm = TRUE)
  )

## # A tibble: 2 x 8
##   Structure count na_cnt   min median   mean   max   IQR
##   <fct>     <int>  <int> <int>  <dbl>  <dbl> <int> <dbl>
## 1 EA          100      0 34272 38848  38795. 42983  2814
## 2 IS           18      0 46454 48378. 48402. 49847  1929

Kruskal–Wallis test provides evidence of difference among selection schemes.

kruskal.test(Generations ~ Structure, data = ssf)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  Generations by Structure
## Kruskal-Wallis chi-squared = 45.378, df = 1, p-value = 1.624e-11

Results for post-hoc Wilcoxon rank-sum test with a Bonferroni correction.

pairwise.wilcox.test(x = ssf$Generations, g = ssf$Structure, p.adjust.method = "bonferroni",
                     paired = FALSE, conf.int = FALSE, alternative = 'g')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  ssf$Generations and ssf$Structure 
## 
##    EA     
## IS 8.3e-12
## 
## P value adjustment method: bonferroni