Introduction to the time-varying geometric distribution

In tvgeom: The Time-Varying (Right-Truncated) Geometric Distribution

knitr::opts_chunk$set(
  collapse = TRUE,  # collapse all the source and output blocks?
  comment = "#>"  # the prefix to be put before source code output
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The time-varying geometric distribution has parameter $\boldsymbol{\mathbf{\phi}}$ and support from 1 to n + 1, where n =\code{length(p)}. If $x\sim tvgeom(prob)$, $x = n + 1$ when the event does not occur in the first $n$ trials. It has probability mass function

$$ f(x =1 \mid \boldsymbol{\mathbf{\phi}}) = \phi_x $$ $$ f(x = i \mid \boldsymbol{\mathbf{\phi}}, 1<i\leq n ) = \phi_i\prod_{j = 1}^{i-1}(1 - \phi_j) $$ $$ f(x = n + 1 \mid \boldsymbol{\mathbf{\phi}}) = \prod_{j = 1}^{n}(1 - \phi_j) $$

Description of the tvgeom distribution

The time-varying geometric distribution is derived from the geometric distribution, a discrete probability distribution used in econometrics, ecology, etc. Whereas the geometric distribution has a constant probability of success over time and has no upper bound of support, the time-varying geometric distribution has a probability of success that changes over time. Additionally, to accommodate situations in which the event can only occur in $n$ days, after which success can not occur, the time-varying geometric distribution is right-truncated (i.e. it has a maximum possible value determined by the length of $\boldsymbol{\mathbf{\phi}}$ above.

In-depth example

First let's load the packages we need.

library(tvgeom)
library(dplyr)
library(tidyr)
library(purrr)
library(magrittr)
library(ggplot2)
library(ggthemes)
library(gridExtra)

Next, let's define a few functions, some wrappers, and the set of scenarios over which we hope to iterate in order to develop some sort of intuition.

# A logistic curve, which we can use to create a monotonically increasing or
# decreasing probability of success.
logistic <- function(n, x0, L_min, L_max, k, ...) { 
  (L_max - L_min) / (1 + exp(-k * (seq_len(n) - x0))) + L_min
}

# Wrappers.
get_phi <- function(data) {
  data %>% pull(p_success) %>% c(1)
}
draw_from_tvgeom <- function(data, n_samples = 1000) {
  rtvgeom(n_samples, get_phi(data))
}

# The total number of trials.
n_days <- 100

# Create an array of intuition-building scenarios. The time-varying probability 
# of success (based upon which we will draw our samples) will depend entirely on
# the shape-controlling parameters of the curve for each scenario.
scenarios <- crossing(n = n_days, 
                      x0 = 60, 
                      L_min = 0, 
                      L_max = c(.1, .25, .7), 
                      k = c(-.2, 0, .5)
                      ) %>% 
  mutate(scenario = as.character(1:n()))

knitr::kable(scenarios, caption = 'Scenarios...')

Next, let's use some dplyr/purrr magic to develop data we can plot to show the effects of the various scenarios above.

# Calculate the probability of success for each scenario.
d_phi <- scenarios %>% 
  split(.$scenario) %>% 
  map(~ do.call(logistic, .)) %>% 
  bind_cols %>% 
  mutate(day = 1:n()) %>% 
  gather(scenario, p_success, -day) %>% 
  left_join(scenarios)

# On the basis of d_phi, make draws for new y's using rtvgeom.
d_y <- d_phi %>% select(scenario, p_success) %>% split(.$scenario) %>% 
  map(~ draw_from_tvgeom(.)) %>% 
  bind_cols %>% 
  gather(scenario, y) %>% 
  left_join(scenarios)

# Plotting.
plot_param <- function(d_phi, d_y, parameter, subset = NULL) {

  d1 <- d_phi %>% 
    mutate(focal_param = factor(get(parameter))) %>% 
    {`if`(!is.null(subset), filter_(., subset), .)}

  p1 <- ggplot(d1) +
    facet_grid(scenario ~ .) +
    geom_line(aes_string(x = 'day', y = 'p_success', color = 'focal_param'), 
              size = 1.01) +
    theme_hc(base_size = 13) +
    scale_color_hc(name = parameter) +
    labs(x = 'Day', y = expression(phi))

  d2 <- d_y %>% 
    mutate(focal_param = factor(get(parameter))) %>% 
    {`if`(!is.null(subset), filter_(., subset), .)}
  p2 <- ggplot(d2) +
    facet_grid(scenario ~ .) +
    geom_histogram(aes_string(x = 'y', y = '..density..', fill = 'focal_param'), 
                   color = 'black', alpha = .8) +
    theme_hc(base_size = 13) +
    scale_fill_hc(name = parameter) +
    labs(x = 'Day', y = 'Density') 

  grid.arrange(p1, p2, ncol = 2)

}

Plots of the results

plot_param(d_phi, d_y, 'k', 'L_max == min(L_max)')

plot_param(d_phi, d_y, 'L_max', 'k == max(k)')

tvgeom documentation built on Dec. 10, 2019, 5:11 p.m.