Chapter 9 MI500: Multi-path exploration results

Here we present the results for the best performances and activation gene coverage generated by each selection scheme replicate on the multi-path exploration diagnostic. Best performance found refers to the largest average trait score found in a given population. Note that activation gene coverage values are gathered at the population-level. Activation gene coverage refers to the count of unique activation genes in a given population; this gives us a range of integers between 0 and 100.

9.1 Analysis dependencies

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

9.2 Truncation selection

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

9.2.1 Performance

9.2.1.1 Performance over time

lines = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `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"
  ) +
  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

9.2.1.2 Best performance

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

filter(base_best, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TRUNCATION' & 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()

9.2.1.2.1 Stats

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

performance = filter(base_best, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TRUNCATION' & 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   5     58.0  57.0 100.   40.5
## 2 IS          100      0  11     56.0  58.3  99.9  44.5
## 3 NMIS        100      0  22.0   85.9  81.5  99.9  22.4

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 = 57.688, df = 2, p-value = 2.973e-13

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 = 'g')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  performance$VAL and performance$Structure 
## 
##      EA      IS     
## IS   1       -      
## NMIS 4.3e-11 1.3e-10
## 
## P value adjustment method: bonferroni

9.2.1.3 Final performance

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

filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TRUNCATION' & 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()

9.2.1.3.1 Stats

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

performance = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TRUNCATION' & 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   5     58.0  57.0 100.   40.5
## 2 IS          100      0  11     56.0  58.3  99.9  44.5
## 3 NMIS        100      0  22.0   85.9  81.5  99.9  22.4

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 = 57.688, df = 2, p-value = 2.973e-13

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 = 'g')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  performance$pop_fit_max and performance$Structure 
## 
##      EA      IS     
## IS   1       -      
## NMIS 4.3e-11 1.3e-10
## 
## P value adjustment method: bonferroni

9.2.2 Generation satisfactory solution found

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

filter(base_ssf, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TRUNCATION'& 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) +
  scale_shape_manual(values=SHAPE)+
  scale_y_continuous(
    name="Generations"
  ) +
  scale_x_discrete(
    name="Structure"
  ) +
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  p_theme + coord_flip()

9.2.2.1 Stats

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

ssf = filter(base_ssf, Diagnostic == 'MULTIPATH_EXPLORATION' & `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            1      0 15300  15300 15300  15300     0
## 2 IS            2      0 26492  26654 26654  26816   162
## 3 NMIS          5      0 26188  28563 28313. 29384   372

Kruskal–Wallis test provides evidence of no difference among selection schemes.

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

## 
##  Kruskal-Wallis rank sum test
## 
## data:  Generations by Structure
## Kruskal-Wallis chi-squared = 3.3833, df = 2, p-value = 0.1842

9.2.3 Activation gene coverage

Activation gene coverage analysis.

9.2.3.1 Coverage over time

Activation gene coverage over time.

# data for lines and shading on plots
lines = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TRUNCATION') %>%
  group_by(Structure, Generations) %>%
  dplyr::summarise(
    min = min(pop_act_cov),
    mean = mean(pop_act_cov),
    max = max(pop_act_cov)
  )

## `summarise()` has grouped output by 'Structure'. You can override using the
## `.groups` argument.

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 = 1.5, stroke = 2.0, alpha = 1.0) +
  scale_y_continuous(
    name="Coverage"
  ) +
  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('Activation gene coverage over time')+
  p_theme

9.2.3.2 End of 50,000 generations

Activation gene coverage in the population at the end of 50,000 generations.

### end of run
filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TRUNCATION' & Generations == 50000) %>%
  ggplot(., aes(x = Structure, y = pop_act_cov, color = Structure, fill = Structure, shape = Structure)) +
  geom_flat_violin(position = position_nudge(x = .2, y = 0), scale = 'width', alpha = 0.3) +
  geom_point(position = position_jitter(height = .05, width = .05), size = 1.5, alpha = 0.5) +
  geom_boxplot(color = 'black', width = .2, outlier.shape = NA, alpha = 0.0) +
  scale_shape_manual(values=SHAPE)+
  scale_y_continuous(
    name="Coverage"
  ) +
  scale_x_discrete(
    name="Structure"
  ) +
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  ggtitle('Final activation gene coverage')+
  p_theme + coord_flip()

9.2.3.2.1 Stats

Summary statistics for activation gene coverage.

coverage = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TRUNCATION' & Generations == 50000)
coverage %>%
  group_by(Structure) %>%
  dplyr::summarise(
    count = n(),
    na_cnt = sum(is.na(pop_act_cov)),
    min = min(pop_act_cov, na.rm = TRUE),
    median = median(pop_act_cov, na.rm = TRUE),
    mean = mean(pop_act_cov, na.rm = TRUE),
    max = max(pop_act_cov, na.rm = TRUE),
    IQR = IQR(pop_act_cov, 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     1      2  1.96     3     0
## 2 IS          100      0     1      2  2.01     3     0
## 3 NMIS        100      0     4      6  6.38     8     1

Kruskal–Wallis test provides evidence of difference among activation gene coverage.

kruskal.test(pop_act_cov ~ Structure, data = coverage)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  pop_act_cov by Structure
## Kruskal-Wallis chi-squared = 258.93, df = 2, p-value < 2.2e-16

Results for post-hoc Wilcoxon rank-sum test with a Bonferroni correction on activation gene coverage.

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

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  coverage$pop_act_cov and coverage$Structure 
## 
##      EA     IS    
## IS   0.34   -     
## NMIS <2e-16 <2e-16
## 
## P value adjustment method: bonferroni

9.3 Tournament selection

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

9.3.1 Performance

9.3.1.1 Performance over time

lines = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `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"
  ) +
  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

9.3.1.2 Best performance

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

filter(base_best, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TOURNAMENT' & 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()

9.3.1.2.1 Stats

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

performance = filter(base_best, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TOURNAMENT' & 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   5     60.0  57.5  99.9  45.0
## 2 IS          100      0  12     59.0  57.1  99.9  43.5
## 3 NMIS        100      0  37.0   85.9  81.2  99.8  23.1

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 = 52.543, df = 2, p-value = 3.895e-12

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 = 'g')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  performance$VAL and performance$Structure 
## 
##      EA      IS     
## IS   1       -      
## NMIS 5.9e-09 5.3e-11
## 
## P value adjustment method: bonferroni

9.3.1.3 Final performance

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

filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TOURNAMENT' & 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()

9.3.1.3.1 Stats

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

performance = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TOURNAMENT' & 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   5     60.0  57.5  99.9  45.0
## 2 IS          100      0  12     59.0  57.1  99.9  43.5
## 3 NMIS        100      0  37.0   85.9  81.2  99.8  23.1

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 = 52.543, df = 2, p-value = 3.895e-12

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 = 'g')

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  performance$pop_fit_max and performance$Structure 
## 
##      EA      IS     
## IS   1       -      
## NMIS 5.9e-09 5.3e-11
## 
## P value adjustment method: bonferroni

9.3.2 Generation satisfactory solution found

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

filter(base_ssf, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TOURNAMENT'& 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) +
  scale_shape_manual(values=SHAPE)+
  scale_y_continuous(
    name="Generations"
  ) +
  scale_x_discrete(
    name="Structure"
  ) +
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  p_theme + coord_flip()

9.3.2.1 Stats

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

ssf = filter(base_ssf, Diagnostic == 'MULTIPATH_EXPLORATION' & `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            6      0 25843 26598. 26813  27721  954 
## 2 IS            3      0 33462 34801  34458. 35112  825 
## 3 NMIS          8      0 34401 36612. 36496. 38154  989.

Kruskal–Wallis test provides evidence of no difference among selection schemes.

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

## 
##  Kruskal-Wallis rank sum test
## 
## data:  Generations by Structure
## Kruskal-Wallis chi-squared = 12.797, df = 2, p-value = 0.001664

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 exact test 
## 
## data:  ssf$Generations and ssf$Structure 
## 
##      EA    IS   
## IS   0.036 -    
## NMIS 0.001 0.073
## 
## P value adjustment method: bonferroni

9.3.3 Activation gene coverage

Activation gene coverage analysis.

9.3.3.1 Coverage over time

Activation gene coverage over time.

# data for lines and shading on plots
lines = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TOURNAMENT') %>%
  group_by(Structure, Generations) %>%
  dplyr::summarise(
    min = min(pop_act_cov),
    mean = mean(pop_act_cov),
    max = max(pop_act_cov)
  )

## `summarise()` has grouped output by 'Structure'. You can override using the
## `.groups` argument.

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 = 1.5, stroke = 2.0, alpha = 1.0) +
  scale_y_continuous(
    name="Coverage"
  ) +
  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('Activation gene coverage over time')+
  p_theme

9.3.3.2 End of 50,000 generations

Activation gene coverage in the population at the end of 50,000 generations.

### end of run
filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TOURNAMENT' & Generations == 50000) %>%
  ggplot(., aes(x = Structure, y = pop_act_cov, color = Structure, fill = Structure, shape = Structure)) +
  geom_flat_violin(position = position_nudge(x = .2, y = 0), scale = 'width', alpha = 0.3) +
  geom_point(position = position_jitter(height = .05, width = .05), size = 1.5, alpha = 0.5) +
  geom_boxplot(color = 'black', width = .2, outlier.shape = NA, alpha = 0.0) +
  scale_shape_manual(values=SHAPE)+
  scale_y_continuous(
    name="Coverage"
  ) +
  scale_x_discrete(
    name="Structure"
  ) +
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  ggtitle('Final activation gene coverage')+
  p_theme + coord_flip()

9.3.3.2.1 Stats

Summary statistics for activation gene coverage.

coverage = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'TOURNAMENT' & Generations == 50000)
coverage %>%
  group_by(Structure) %>%
  dplyr::summarise(
    count = n(),
    na_cnt = sum(is.na(pop_act_cov)),
    min = min(pop_act_cov, na.rm = TRUE),
    median = median(pop_act_cov, na.rm = TRUE),
    mean = mean(pop_act_cov, na.rm = TRUE),
    max = max(pop_act_cov, na.rm = TRUE),
    IQR = IQR(pop_act_cov, 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     1      2  1.96     3  0   
## 2 IS          100      0     1      2  2.05     3  0   
## 3 NMIS        100      0     3      6  6.22     8  1.25

Kruskal–Wallis test provides evidence of difference among activation gene coverage.

kruskal.test(pop_act_cov ~ Structure, data = coverage)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  pop_act_cov by Structure
## Kruskal-Wallis chi-squared = 264.53, df = 2, p-value < 2.2e-16

Results for post-hoc Wilcoxon rank-sum test with a Bonferroni correction on activation gene coverage.

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

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  coverage$pop_act_cov and coverage$Structure 
## 
##      EA     IS    
## IS   0.019  -     
## NMIS <2e-16 <2e-16
## 
## P value adjustment method: bonferroni

9.4 Lexicase selection

Here we analyze how the different population structures affect standard lexicase selection on the contradictory objectives diagnostic.

9.4.1 Performance

9.4.1.1 Performance over time

lines = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `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"
  ) +
  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

9.4.1.2 Best performance

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

filter(base_best, Diagnostic == 'MULTIPATH_EXPLORATION' & `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()

9.4.1.2.1 Stats

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

performance = filter(base_best, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'LEXICASE' & VAR == 'pop_fit_max')
performance$Structure = factor(performance$Structure, levels=c('EA','NMIS','IS'))
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  83.4   93.2  92.8  98.4  4.80
## 2 NMIS        100      0  66.3   75.9  76.1  86.4  5.66
## 3 IS          100      0  61.0   73.9  74.1  87.4  7.42

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 = 202.16, 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     NMIS  
## NMIS <2e-16 -     
## IS   <2e-16 0.0032
## 
## P value adjustment method: bonferroni

9.4.1.3 Final performance

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

filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `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()

9.4.1.3.1 Stats

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

performance = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'LEXICASE' & Generations == 50000)
performance$Structure = factor(performance$Structure, levels=c('EA','NMIS','IS'))
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  74.4   91.3  90.6  97.2  6.69
## 2 NMIS        100      0  64.4   73.9  73.8  83.8  5.84
## 3 IS          100      0  57.7   69.5  70.6  87.4  8.30

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 = 198.85, 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      NMIS   
## NMIS < 2e-16 -      
## IS   < 2e-16 1.6e-05
## 
## P value adjustment method: bonferroni

9.4.2 Activation gene coverage

Activation gene coverage analysis.

9.4.2.1 Coverage over time

Activation gene coverage over time.

# data for lines and shading on plots
lines = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'LEXICASE') %>%
  group_by(Structure, Generations) %>%
  dplyr::summarise(
    min = min(pop_act_cov),
    mean = mean(pop_act_cov),
    max = max(pop_act_cov)
  )

## `summarise()` has grouped output by 'Structure'. You can override using the
## `.groups` argument.

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 = 1.5, stroke = 2.0, alpha = 1.0) +
  scale_y_continuous(
    name="Coverage"
  ) +
  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('Activation gene coverage over time')+
  p_theme

9.4.2.2 End of 50,000 generations

Activation gene coverage in the population at the end of 50,000 generations.

### end of run
filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'LEXICASE' & Generations == 50000) %>%
  ggplot(., aes(x = Structure, y = pop_act_cov, color = Structure, fill = Structure, shape = Structure)) +
  geom_flat_violin(position = position_nudge(x = .2, y = 0), scale = 'width', alpha = 0.3) +
  geom_point(position = position_jitter(height = .05, width = .05), size = 1.5, alpha = 0.5) +
  geom_boxplot(color = 'black', width = .2, outlier.shape = NA, alpha = 0.0) +
  scale_shape_manual(values=SHAPE)+
  scale_y_continuous(
    name="Coverage"
  ) +
  scale_x_discrete(
    name="Structure"
  ) +
  scale_colour_manual(values = cb_palette) +
  scale_fill_manual(values = cb_palette) +
  ggtitle('Final activation gene coverage')+
  p_theme + coord_flip()

9.4.2.2.1 Stats

Summary statistics for activation gene coverage.

coverage = filter(base_over_time, Diagnostic == 'MULTIPATH_EXPLORATION' & `Selection\nScheme` == 'LEXICASE' & Generations == 50000)
coverage$Structure = factor(coverage$Structure, levels=c('EA','NMIS','IS'))
coverage %>%
  group_by(Structure) %>%
  dplyr::summarise(
    count = n(),
    na_cnt = sum(is.na(pop_act_cov)),
    min = min(pop_act_cov, na.rm = TRUE),
    median = median(pop_act_cov, na.rm = TRUE),
    mean = mean(pop_act_cov, na.rm = TRUE),
    max = max(pop_act_cov, na.rm = TRUE),
    IQR = IQR(pop_act_cov, 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    24     31  31.2    41     5
## 2 NMIS        100      0    24     30  30.3    44     4
## 3 IS          100      0    12     17  17.3    26     3

Kruskal–Wallis test provides evidence of difference among activation gene coverage.

kruskal.test(pop_act_cov ~ Structure, data = coverage)

## 
##  Kruskal-Wallis rank sum test
## 
## data:  pop_act_cov by Structure
## Kruskal-Wallis chi-squared = 201.31, df = 2, p-value < 2.2e-16

Results for post-hoc Wilcoxon rank-sum test with a Bonferroni correction on activation gene coverage.

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

## 
##  Pairwise comparisons using Wilcoxon rank sum test with continuity correction 
## 
## data:  coverage$pop_act_cov and coverage$Structure 
## 
##      EA     NMIS  
## NMIS 0.077  -     
## IS   <2e-16 <2e-16
## 
## P value adjustment method: bonferroni