Code in Chapter 04: Logit

In discrtr: A Companion Package for the Book "Discrete Choice Analysis with 'R'"

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"}

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)) %>% # The function dgumbel() is the EV Type I distribution mutate(f = dgumbel(x, # Location parameter loc = mu, # Scale parameter scale = sigma))

Plot

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"}

Define parameters for the distribution

Location

mu <- 0

Scale

sigma <- 1

Define an upper limit for calculating the probability;

This equivalent to V_i - V_j.

Negative values represent V_i < V_j, and positive values are V_j > V_i;

when V_j = V_k, then X = 0:

x <- -2

Create data frames for plotting

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))

Plot distribution function and the area under the curve

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 

discrtr documentation built on March 7, 2023, 5:36 p.m.