1. Distributions
Note
A Normal distribution with \(\mu = 0\) and \(\sigma = 1\) is also called a z-Distribution (aka a Standard Normal Distribution)
ggplot(d, aes(x=x, y=y)) +
geom_line()
ggplotdata.frame(x = myvar) |>
ggplot(aes(x=x)) + # No 'y' aesthetic is passed (we let ggplot compute it)
geom_density()
data.frame(x = myvar) |>
ggplot(aes(x=x)) +
geom_histogram(binwidth = 0.35, color="black")
data.frame(x = myvar) |>
ggplot(aes(x=x)) +
geom_histogram(binwidth = 0.1, color="black")
data.frame(x=x, y=y) |>
ggplot(aes(x=x, y=y)) +
geom_line()
x <- seq(-4, 4, length.out = 1000)
y <- dnorm(x, mean = 0, sd = 1)
data.frame(x=x, y=y) |>
ggplot(aes(x=x, y=y)) +
geom_line(color="orange", linewidth=2) +
see::theme_abyss() +
labs(y="Density", x="Values of x")
df_norm <- data.frame()
for(u in c(0, 2.5, 5)) {
for(sd in c(1, 2, 3)) {
dat <- data.frame(x = seq(-10, 10, length.out=200))
dat$p <- dnorm(dat$x, mean = u, sd = sd)
dat$distribution <- "Normal"
dat$parameters <- paste0("mean = ", u, ", SD = ", sd)
df_norm <- rbind(dat, df_norm)
}
}
p <- df_norm |>
ggplot(aes(x=x, y=p, group=parameters)) +
geom_vline(xintercept=0, linetype="dashed") +
geom_line(aes(color=distribution), linewidth=2, color="#2196F3") +
theme_minimal() +
theme(legend.position = "none",
axis.title.y = element_blank(),
plot.title = element_text(size=15, face="bold", hjust=0.5),
plot.tag = element_text(size=20, face="bold", hjust=0.5),
strip.text = element_text(size=15, face="bold"))
p + facet_wrap(~parameters)
rnorm() and dnorm()?
y <- rbinom(10, size=1, prob=0.5)
y [1] 1 1 1 0 1 1 0 0 1 0library(gganimate)
df_binom <- data.frame()
for(p in c(1/3, 1/2, 2/3)) {
dat <- data.frame()
for(i in 1:1000) {
rez <- rbinom(1, size=1, prob=p)
dat2 <- data.frame(Zero = 0, One = 0, i=i)
dat2$Zero[rez==0] <- 1
dat2$One[rez==1] <- 1
dat2$distribution <- "Binomial"
dat2$parameters <- paste0("p = ", insight::format_value(p))
dat <- rbind(dat, dat2)
}
dat$Zeros <- cumsum(dat$Zero)
dat$Ones <- cumsum(dat$One)
df_binom <- rbind(df_binom, dat)
}
df_binom <- pivot_longer(df_binom, c("Zeros", "Ones"), names_to="Outcome", values_to="Count") |>
mutate(Outcome = factor(Outcome, levels=c("Zeros", "Ones")))
anim <- df_binom |>
ggplot(aes(x=Outcome, y=Count, group=parameters)) +
geom_bar(aes(fill=parameters), stat="identity", position="dodge") +
theme_minimal() +
labs(title = 'N events: {frame_time}', tag="Binomial Distribution") +
transition_time(i)
anim_save("img/binomial.gif", animate(anim, height=500, width=1000, fps=10))
# Simulate random walk
n_steps <- 7
decisions <- sample(c(-1, 1), n_steps, replace = TRUE)
decisions[1] -1 1 -1 -1 1 -1 -1x <- c(0, cumsum(decisions)) # Add starting point at 0
x[1] 0 -1 0 -1 -2 -1 -2 -3
x <- random_walk(10)
data <- data.frame(trajectory = x, time = seq(0, length(x)-1))
data trajectory time
1 0 0
2 1 1
3 2 2
4 1 3
5 0 4
6 -1 5
7 -2 6
8 -3 7
9 -4 8
10 -5 9
11 -4 10ggplot(data, aes(x=time, y=trajectory)) +
geom_line() +
coord_flip() # Flip x and y axes
ggplot(data, aes(x=time, y=trajectory, color=iteration)) +
geom_line() +
coord_flip()
color aesthetic needs to be converted to a factor to be interpreted as a discrete variabledata |>
mutate(iteration = as.factor(iteration)) |>
ggplot(aes(x=time, y=trajectory, color=iteration)) +
geom_line() +
coord_flip()
data <- data.frame() # Initialize empty data frame
for(i in 1:20) {
walk_data <- data.frame(
trajectory = random_walk(500),
time = seq(0, 500),
iteration = i
)
data <- rbind(data, walk_data)
}
data |>
mutate(iteration = as.factor(iteration)) |>
ggplot(aes(x=time, y=trajectory, color=iteration)) +
geom_line(linewidth = 0.5) +
labs(y = "Position", x = "Time") +
scale_color_viridis_d(option = "inferno", guide="none") +
see::theme_blackboard() +
coord_flip()
data |>
filter(time == max(time)) |>
ggplot(aes(x=trajectory)) +
geom_histogram()
library(gganimate)
set.seed(1)
data <- data.frame()
all_iter <- data.frame()
distr <- data.frame()
for(i in 1:100) {
walk_data <- data.frame(
trajectory = random_walk(99),
time = seq(0, 99),
group = i,
iteration = NA
)
data <- rbind(data, walk_data)
data$iteration <- i
max_t <- max(walk_data$time)
distr_data <- filter(data, time == max_t)$trajectory |>
bayestestR::estimate_density(precision=256) |>
mutate(
y = datawizard::rescale(y, to=c(max_t, max_t + max_t/3)),
iteration = i)
distr <- rbind(distr, distr_data)
all_iter <- rbind(all_iter, data)
}
df <- all_iter |>
mutate(iteration = as.factor(iteration),
group = as.factor(group))
p <- df |>
# filter(iteration == 100) |>
ggplot() +
geom_line(aes(x=time, y=trajectory, color=group), alpha=0.7, linewidth = 1.3) +
geom_ribbon(data=distr, aes(xmin=max(all_iter$time), xmax=y, y=x), fill="#FF9800") +
geom_point(data=filter(df, time == max(time)), aes(x=time, y=trajectory, color=group), size=5, alpha=0.2) +
labs(y = "Position", x = "Time", title="N random walks: {frame}") +
scale_color_viridis_d(option = "turbo", guide="none") +
see::theme_blackboard() +
theme(plot.title = element_text(hjust = 0.5, size=15, face="bold")) +
coord_flip()
# p
anim <- p + transition_manual(iteration)
gganimate::anim_save("img/random.gif", animate(anim, height=1000, width=1000, fps=7, end_pause=7*4, duration=20))
🤯🤯🤯🤯🤯🤯🤯🤯
A NORMAL DISTRIBUTION
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runif()# Use runif(): random + uniform
data.frame(x = runif(50000, min=-10, max=10)) |>
ggplot(aes(x=x)) +
geom_histogram(bins = 50, color="black") +
coord_cartesian(xlim=c(-15, 15))
shape1 & shape2)1data.frame(x = rbeta(10000, shape1 = 1.5, shape2 = 10)) |>
ggplot(aes(x=x)) +
geom_histogram(bins=50, color="black")
shape1) and \(\beta\) (shape2) parameters control the shape of the distributiondf_beta <- data.frame()
for(shape1 in c(1, 2, 3, 6)) {
for(shape2 in c(1, 2, 3, 6)) {
dat <- data.frame(x = seq(-0.25, 1.25, 0.0001))
dat$p <- dbeta(dat$x, shape1 = shape1, shape2 = shape2)
dat$distribution <- "Beta"
dat$parameters <- paste0("alpha = ", shape1, ", beta = ", shape2)
df_beta <- rbind(dat, df_beta)
}
}
p <- df_beta |>
ggplot(aes(x=x, y=p, group=parameters)) +
geom_vline(xintercept=c(0, 1), linetype="dashed") +
geom_line(aes(color=distribution), linewidth=2, color="#4CAF50") +
theme_minimal() +
theme(legend.position = "none",
axis.title.y = element_blank(),
plot.title = element_text(size=15, face="bold", hjust=0.5),
plot.tag = element_text(size=20, face="bold", hjust=0.5),
strip.text = element_text(size=15, face="bold")) +
facet_wrap(~parameters, scales="free_y")
p
data.frame(x = rgamma(100000, shape = 2, scale = 10)) |>
ggplot(aes(x=x)) +
geom_histogram(bins=100, color="black")
shape) and \(\theta\) (scale) parameters control the shape of the distributiondf_gamma <- data.frame()
for(shape in c(0.5, 1, 2, 4)) {
for(scale in c(0.1, 0.5, 1, 2)) {
dat <- data.frame(x = seq(0.001, 25, length.out=1000))
dat$p <- dgamma(dat$x, shape = shape, scale = scale)
dat$distribution <- "Beta"
dat$parameters <- paste0("k = ", shape, ", theta = ", scale)
df_gamma <- rbind(dat, df_gamma)
}
}
p <- df_gamma |>
ggplot(aes(x=x, y=p, group=parameters)) +
# geom_vline(xintercept=c(0, 1), linetype="dashed") +
geom_line(aes(color=distribution), linewidth=2, color="#9C27B0") +
theme_minimal() +
theme(legend.position = "none",
axis.title.y = element_blank(),
plot.title = element_text(size=15, face="bold", hjust=0.5),
plot.tag = element_text(size=20, face="bold", hjust=0.5),
strip.text = element_text(size=15, face="bold")) +
facet_wrap(~parameters, scales="free_y")
p
means <- c()
for(i in 1:10000) {
x <- rbeta(100, shape1 = 10, shape2 = 1.5)
means <- c(means, mean(x))
}
data.frame(x = means) |>
ggplot(aes(x=x)) +
geom_histogram(bins=40, color="black")
🤯🤯🤯🤯🤯🤯🤯🤯
A NORMAL DISTRIBUTION
🤯🤯🤯🤯🤯🤯🤯🤯
df <- data.frame(x = seq(-5, 5, length.out=1000))
df$Normal <- dnorm(df$x, mean = 0, sd = 1)
df$Cauchy <- dcauchy(df$x, location = 0, scale = 1)
df$Student_1.5 <- dt(df$x, df = 1.5)
df$Student_2 <- dt(df$x, df = 2)
df$Student_3 <- dt(df$x, df = 3)
df$Student_5 <- dt(df$x, df = 5)
df$Student_10 <- dt(df$x, df = 10)
df$Student_20 <- dt(df$x, df = 20)
df |>
pivot_longer(starts_with("Student"), names_to="Distribution", values_to="PDF") |>
separate(Distribution, into=c("Distribution", "df"), sep="_") |>
mutate(label = paste0("Student (df=", df, ")")) |>
ggplot(aes(x=x)) +
geom_line(data=df, aes(y=Normal, color="Normal (0, 1)"), linewidth=2) +
geom_line(data=df, aes(y=Cauchy, color="Cauchy (0, 1)"), linewidth=2) +
geom_line(aes(y=PDF, color=label)) +
scale_color_manual(values=c(`Cauchy (0, 1)`="#F44336",
`Student (df=1.5)`="#FF5722",
`Student (df=2)`="#FF9800",
`Student (df=3)`="#FFC107",
`Student (df=5)`="#FFEB3B",
`Student (df=10)`="#CDDC39",
`Student (df=20)`="#8BC34A",
`Normal (0, 1)`="#2196F3"),
breaks=c("Normal (0, 1)",
"Student (df=20)",
"Student (df=10)",
"Student (df=5)",
"Student (df=3)",
"Student (df=2)",
"Student (df=1.5)",
"Cauchy (0, 1)"
)) +
see::theme_modern() +
theme(axis.title.x = element_blank(),
axis.title.y = element_blank(),
axis.ticks.y = element_blank(),
plot.title = element_text(size=15, face="bold", hjust=0.5),
strip.text = element_text(size=15, face="bold")) +
labs(title="Normal, Cauchy, and Student's t-Distribution", color=NULL)Thank you!
