Author: Mateusz Staniak

# Studying the behaviour of Kendall random walks

#### Mateusz Staniak

We will approximate the distribution of moments when the random walk changes state through simulations.

First, we simulate many paths of Kendall random walk with normal step distribution.

    library(kendallRandomWalks)
    library(dplyr)
    library(ggplot2)
    set.seed(17)
    walks <- simulate_kendall_rw(1000, 1000, rnorm, 0.5, T)
    walks2 <- simulate_kendall_rw(1000, 1000, rcauchy, 0.5, T)

Example trajectory

    plot(walks, max_id = 1)

    plot(walks2, max_id = 1)

Number of unique states

    ggplot(summarise_kendall_rw(walks, n_distinct), aes(x = aggregated), color = "black") +
      theme_bw() +
      geom_density() +
      geom_density(data = summarise_kendall_rw(walks2, n_distinct), color = "blue") +
      ylab("Estimated density") +
      xlab("Number of unique values") +
      scale_color_discrete(guide = "legend")

Jumps

    diffs <- mutate_kendall_rw(walks, function(x) x - lag(x), F)
    plot(diffs, max_id = 1)

    diffs2 <- mutate_kendall_rw(walks2, function(x) x - lag(x), F)
    plot(diffs2, max_id = 1)

Time with no change of state

    diffs3 <- mutate_kendall_rw(diffs, function(x) as.numeric(x != 0), F)
    lengths <- diffs3$simulation %>%
      group_by(sim_id) %>%
      mutate(id = 1:n()) %>%
      filter(sim != 0) %>%
      mutate(previous = ifelse(is.na(lag(id)), 0, lag(id))) %>%
      mutate(length = id - previous)

    diffs4 <- mutate_kendall_rw(diffs2, function(x) as.numeric(x != 0), F)
    lengths2 <- diffs4$simulation %>%
      group_by(sim_id) %>%
      mutate(id = 1:n()) %>%
      filter(sim != 0) %>%
      mutate(previous = ifelse(is.na(lag(id)), 0, lag(id))) %>%
      mutate(length = id - previous)

    ggplot(subset(lengths, sim_id < 5),
           aes(x = length, fill = as.factor(sim_id), group = as.factor(sim_id))) +
      geom_density() +
      theme_bw() +
      ggtitle("Distribution of time with no state-change (by simulation)")

    ggplot(lengths, aes(x = length)) +
      geom_density() +
      theme_bw() +
      xlab("Jump size") +
      ggtitle("Distribution of time with no state-change (aggregated)")

    ggplot(lengths2, aes(x = length)) +
      geom_density() +
      theme_bw() +
      xlab("Jump size") +
      ggtitle("Distribution of time with no state-change (aggregated)")

    ggplot(subset(lengths, sim_id < 10), aes(x = id, y = length, color = as.factor(sim_id))) +
      geom_point() +
      theme_bw() +
      geom_line() +
      guides(color = "none") +
      xlab("Time") +
      ylab("Jump size") +
      ggtitle("Time with no state-change in time")