Study of the Empirical Distribution of Ties in Irrational Numbers

The purpose of this study is describing the distribution of ties in irrational numbers when considering sequences of embedding dimension \(D\) (and \(\tau=1\)). The finding will help building a model for simulating such occurrences and, then, assessing imputation techniques in a Monte Carlo study.

source('Bandt-Pompe.R')
if(!require(ggpubr)){
  install.packages("ggpubr")
  require(ggpubr)
}
if(!require(ggplot2)){
  install.packages("ggplot2")
  require(ggplot2)
}
if(!require(ggthemes)){
  install.packages("ggthemes")
  require(ggthemes)
}

Analysis of the distribution of repeated symbols for sequences with N = 1000

e.data = read.table('../Data/e.txt', stringsAsFactors=FALSE, fileEncoding="latin1")
pi.data = read.table('../Data/pi.txt', stringsAsFactors=FALSE, fileEncoding="latin1")
sqrt2.data = read.table('../Data/sqrt2.txt', stringsAsFactors=FALSE, fileEncoding="latin1")

e.vector = as.numeric(strsplit(as.character(e.data), "")[[1]])[3:100011]
pi.vector = as.numeric(strsplit(as.character(pi.data), "")[[1]])[3:100011]
sqrt2.vector = as.numeric(strsplit(as.character(sqrt2.data), "")[[1]])[3:100030]

n.series = 100
n.elements = 1000
e.series = asplit(array(e.vector, c(n.elements, round(length(e.vector)/n.elements))), MARGIN = 2)
pi.series = asplit(array(pi.vector, c(n.elements, round(length(pi.vector)/n.elements))), MARGIN = 2)
sqrt2.series = asplit(array(sqrt2.vector, c(n.elements, round(length(sqrt2.vector)/n.elements))), MARGIN = 2)

D = 3
Tau = 1
e.percent.D3 = pi.percent.D3 = sqrt2.percent.D3 = rep(0, n.series)

for(i in 1:n.series){
  elements = formationPattern(e.series[[i]], D, Tau, 1)
  e.percent.D3[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(pi.series[[i]], D, Tau, 1)
  pi.percent.D3[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(sqrt2.series[[i]], D, Tau, 1)
  sqrt2.percent.D3[i] = percentual.equalities(elements)
}

data.D3 <- data.frame(
  type = c(rep("e", n.series), rep("pi", n.series), rep("sqrt2", n.series)),
  value = c(e.percent.D3, pi.percent.D3, sqrt2.percent.D3)
)
par(mfrow=c(1,1)) 
data.D3 %>%
    ggplot(aes(x = value, fill = type)) +
    geom_histogram(color="#e9ecef", alpha = 3.6, position = 'identity', bins = n.series/2) +
    scale_fill_manual(values=c("#69b3a2", "#404080", "#d4bf08")) +
    theme_few(base_size = 12, base_family = "serif") +
    labs(fill = "") +
    ggtitle('D = 3') + 
    facet_grid(~type) +  
    theme(plot.title = element_text(hjust=0.5))

D = 4
Tau = 1
e.percent.D4 = pi.percent.D4 = sqrt2.percent.D4 = rep(0, n.series)

for(i in 1:n.series){
  elements = formationPattern(e.series[[i]], D, Tau, 1)
  e.percent.D4[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(pi.series[[i]], D, Tau, 1)
  pi.percent.D4[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(sqrt2.series[[i]], D, Tau, 1)
  sqrt2.percent.D4[i] = percentual.equalities(elements)
}

data.D4 <- data.frame(
  type = c(rep("e", n.series), rep("pi", n.series), rep("sqrt2", n.series)),
  value = c(e.percent.D4, pi.percent.D4, sqrt2.percent.D4)
)
par(mfrow=c(1,1)) 
data.D4 %>%
    ggplot(aes(x = value, fill = type)) +
    geom_histogram(color="#e9ecef", alpha = 3.6, position = 'identity', bins = n.series/2) +
    scale_fill_manual(values=c("#69b3a2", "#404080", "#d4bf08")) +
    theme_few(base_size = 12, base_family = "serif") +
    labs(fill="") +
    ggtitle('D = 4') + 
    facet_grid(~type) +  
    theme(plot.title = element_text(hjust=0.5))

Normality test for N = 1000 with Shapiro-wilk

cat(' (D = 3 & N = 1000) -> p-values: ', 
    shapiro.test(data.D3[1:100,]$value)$p.value, " ", 
    shapiro.test(data.D3[101:200,]$value)$p.value, " ", 
    shapiro.test(data.D3[201:300,]$value)$p.value, '\n',
    '(D = 4 & N = 1000) -> p-values: ', 
    shapiro.test(data.D4[1:100,]$value)$p.value, " ", 
    shapiro.test(data.D4[101:200,]$value)$p.value, " ", 
    shapiro.test(data.D4[201:300,]$value)$p.value)

 (D = 3 & N = 1000) -> p-values:  0.02877825   0.7478345   0.07641253 
 (D = 4 & N = 1000) -> p-values:  0.0455737   0.2775077   0.7623933

Analysis of the distribution of repeated symbols for sequences with N = 10000

n.series = 10
n.elements = 10000
e.series = asplit(array(e.vector, c(n.elements, round(length(e.vector)/n.elements))), MARGIN = 2)
pi.series = asplit(array(pi.vector, c(n.elements, round(length(pi.vector)/n.elements))), MARGIN = 2)
sqrt2.series = asplit(array(sqrt2.vector, c(n.elements, round(length(sqrt2.vector)/n.elements))), MARGIN = 2)

D = 3
Tau = 1
e.percent.D3 = pi.percent.D3 = sqrt2.percent.D3 = rep(0, n.series)

for(i in 1:n.series){
  elements = formationPattern(e.series[[i]], D, Tau, 1)
  e.percent.D3[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(pi.series[[i]], D, Tau, 1)
  pi.percent.D3[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(sqrt2.series[[i]], D, Tau, 1)
  sqrt2.percent.D3[i] = percentual.equalities(elements)
}

data.D3 <- data.frame(
  type = c(rep("e", n.series), rep("pi", n.series), rep("sqrt2", n.series)),
  value = c(e.percent.D3, pi.percent.D3, sqrt2.percent.D3)
)
data.D3 %>%
    ggplot(aes(x = value, fill = type)) +
    geom_histogram(color="#e9ecef", alpha = 3.6, position = 'identity', bins = n.series) +
    scale_fill_manual(values=c("#69b3a2", "#404080", "#d4bf08")) +
    theme_few(base_size = 12, base_family = "serif") +
    labs(fill = "") +
    ggtitle('D = 3') + 
    facet_grid(~type) +  
    theme(plot.title = element_text(hjust=0.5))

D = 4
Tau = 1
e.percent.D4 = pi.percent.D4 = sqrt2.percent.D4 = rep(0, n.series)

for(i in 1:n.series){
  elements = formationPattern(e.series[[i]], D, Tau, 1)
  e.percent.D4[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(pi.series[[i]], D, Tau, 1)
  pi.percent.D4[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(sqrt2.series[[i]], D, Tau, 1)
  sqrt2.percent.D4[i] = percentual.equalities(elements)
}

data.D4 <- data.frame(
  type = c(rep("e", n.series), rep("pi", n.series), rep("sqrt2", n.series)),
  value = c(e.percent.D4, pi.percent.D4, sqrt2.percent.D4)
)
data.D4 %>%
    ggplot(aes(x = value, fill = type)) +
    geom_histogram(color="#e9ecef", alpha = 3.6, position = 'identity', bins = n.series) +
    scale_fill_manual(values=c("#69b3a2", "#404080", "#d4bf08")) +
    theme_few(base_size = 12, base_family = "serif") +
    labs(fill="") +
    ggtitle('D = 4') + 
    facet_grid(~type) +  
    theme(plot.title = element_text(hjust=0.5))

D = 5
Tau = 1
e.percent.D5 = pi.percent.D5 = sqrt2.percent.D5 = rep(0, n.series)

for(i in 1:n.series){
  elements = formationPattern(e.series[[i]], D, Tau, 1)
  e.percent.D5[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(pi.series[[i]], D, Tau, 1)
  pi.percent.D5[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(sqrt2.series[[i]], D, Tau, 1)
  sqrt2.percent.D5[i] = percentual.equalities(elements)
}

data.D5 <- data.frame(
  type = c(rep("e", n.series), rep("pi", n.series), rep("sqrt2", n.series)),
  value = c(e.percent.D5, pi.percent.D5, sqrt2.percent.D5)
)
data.D5 %>%
    ggplot(aes(x = value, fill = type)) +
    geom_histogram(color="#e9ecef", alpha = 3.6, position = 'identity', bins = n.series) +
    scale_fill_manual(values=c("#69b3a2", "#404080", "#d4bf08")) +
    theme_few(base_size = 12, base_family = "serif") +
    labs(fill="") +
    ggtitle('D = 5') + 
    facet_grid(~type) +  
    theme(plot.title = element_text(hjust=0.5))

D = 6
Tau = 1
e.percent.D6 = pi.percent.D6 = sqrt2.percent.D6 = rep(0, n.series)

for(i in 1:n.series){
  elements = formationPattern(e.series[[i]], D, Tau, 1)
  e.percent.D6[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(pi.series[[i]], D, Tau, 1)
  pi.percent.D6[i] = percentual.equalities(elements)
}
for(i in 1:n.series){
  elements = formationPattern(sqrt2.series[[i]], D, Tau, 1)
  sqrt2.percent.D6[i] = percentual.equalities(elements)
}

data.D6 <- data.frame(
  type = c(rep("e", n.series), rep("pi", n.series), rep("sqrt2", n.series)),
  value = c(e.percent.D6, pi.percent.D6, sqrt2.percent.D6)
)
data.D6 %>%
    ggplot(aes(x = value, fill = type)) +
    geom_histogram(color="#e9ecef", alpha = 3.6, position = 'identity', bins = n.series) +
    scale_fill_manual(values=c("#69b3a2", "#404080", "#d4bf08")) +
    theme_few(base_size = 12, base_family = "serif") +
    labs(fill="") +
    ggtitle('D = 6') + 
    facet_grid(~type) +  
    theme(plot.title = element_text(hjust=0.5))

Normality test for N = 10000 with Shapiro-wilk

cat(' (D = 3 & N = 10000) -> p-values: ', 
    shapiro.test(data.D3[1:10,]$value)$p.value, " ", 
    shapiro.test(data.D3[11:20,]$value)$p.value, " ", 
    shapiro.test(data.D3[21:30,]$value)$p.value, '\n',
    '(D = 4 & N = 10000) -> p-values: ', 
    shapiro.test(data.D4[1:10,]$value)$p.value, " ", 
    shapiro.test(data.D4[11:20,]$value)$p.value, " ", 
    shapiro.test(data.D4[21:30,]$value)$p.value, '\n',
    '(D = 5 & N = 10000) -> p-values: ', 
    shapiro.test(data.D5[1:10,]$value)$p.value, " ", 
    shapiro.test(data.D5[11:20,]$value)$p.value, " ", 
    shapiro.test(data.D5[21:30,]$value)$p.value, '\n',
    '(D = 6 & N = 10000) -> p-values: ', 
    shapiro.test(data.D6[1:10,]$value)$p.value, " ", 
    shapiro.test(data.D6[11:20,]$value)$p.value, " ", 
    shapiro.test(data.D6[21:30,]$value)$p.value)

 (D = 3 & N = 10000) -> p-values:  0.2182204   0.6359071   0.5956063 
 (D = 4 & N = 10000) -> p-values:  0.7717133   0.0474167   0.2127638 
 (D = 5 & N = 10000) -> p-values:  0.7599577   0.02953326   0.05893071 
 (D = 6 & N = 10000) -> p-values:  0.3857358   0.1359803   0.2876493