knitr::opts_chunk$set(echo = TRUE)
print("Hello, Prof. Train")
library(dplyr) # A Grammar of Data Manipulation library(evd) # Functions for Extreme Value Distributions library(ggplot2) # Create Elegant Data Visualisations Using the Grammar of Graphics
```rExtreme Value Type I distribution"}
mu <- 0
sigma <- 1
df <- data.frame(x =seq(from = -5,
to = 5,
by = 0.01)) %>%
# The function dgumbel()
is the EV Type I distribution
mutate(f = dgumbel(x,
# Location parameter
loc = mu,
# Scale parameter
scale = sigma))
ggplot(data = df, aes(x = x, y = f)) + geom_area(fill = "orange", alpha = 0.5) + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + ylab("f(x)")
```rComparison of the logistic (blue) and normal (grey) distributions"} # Define parameters for the distribution # Location mu <- 0 # Scale sigma <- 1 # Create a data frame for plotting df <- data.frame(x =seq(from = -5, to = 5, by = 0.01)) %>% # Add columns with the values of the # `dlogis()` and `dnorm()` distributions mutate(logistic = dlogis(x, # Location parameter location = mu, # Scale parameter scale = sigma), normal = dnorm(x, # The location parameter of the normal # distribution is the mean mean = mu, # The scale parameter of the normal # distribution is the standard deviation sd = sigma)) # Plot ggplot() + # Add geometric object of type area to plot the # logistic distribution geom_area(data = df, aes(x = x, y = logistic), # The fill color of the logistic distribution fill = "blue", alpha = 0.5) + # Add geometric object of type area to plot the # normal distribution geom_area(data = df, aes(x = x, y = normal), # The fill color of the normal distribution fill = "black", alpha = 0.5) + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + ylab("f(x)") # Label the y axis
```rLogit probability"}
mu <- 0
sigma <- 1
x <- -2
df <- data.frame(x =seq(from = -6 + mu, to = 6 + mu, by = 0.01)) %>% mutate(y = dlogis(x, location = mu, scale = sigma)) df_p <- data.frame(x =seq(from = -6, to = x, by = 0.01)) %>% mutate(y = dlogis(x, location = mu, scale = sigma))
ggplot(data = df, aes(x, y)) + geom_area(fill = "orange", alpha = 0.5) + geom_area(data = df_p, fill = "orange", alpha = 1) + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + xlab(expression(paste(V[i] - V[j] - mu))) + ylab("f(x)")
```rLinear cumulative distribution function"} # Define parameters for the distribution # Location mu <- 0 # Scale sigma <- 1 # Create a data frame for plotting df <- data.frame(x =seq(from = -5 + mu, to = 5 + mu, by = 0.01)) %>% mutate(f = plogis(x, location = mu, scale = 1)) # Plot the cumulative distribution function logit_plot <- ggplot(data = df, aes(x = x, y = f)) + geom_line(color = "orange") + ylim(c(0, 1)) + geom_hline(yintercept = 0) + geom_vline(xintercept = 0) logit_plot + xlab(expression(paste(V[i] - V[j] - mu))) + ylab(expression(paste(P[i])))
```rImplication of the sigmoid shape"} logit_plot + xlab(expression(paste(V[transit] - V[car] - mu))) + ylab(expression(paste(P[transit]))) + annotate("segment", x = -3.75, xend = -2.5, y = 0.024, yend = 0.024, colour = "blue", linetype = "solid") + annotate("segment", x = -2.5, xend = -2.5, y = 0.024, yend = 0.075, colour = "blue", linetype = "solid") + annotate("segment", x = 0, xend = 1.25, y = 0.5, yend = 0.5, colour = "red", linetype = "dashed") + annotate("segment", x = 1.25, xend = 1.25, y = 0.5, yend = 0.77, colour = "red", linetype = "dashed")
```r V_j <- -4 V_k <- 8 theta <- 0.8 theta * V_j - theta * V_k
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