Arguments Breakdown

• .data: The dataset from a tidy_ distribution function. This forms the basis of your random walk.
• .initial_value: The starting value of the random walk. The default is 0, but you can set it to any numeric value.
• .sample: A boolean indicating whether to sample the y values from the tidy_ distribution. Defaults to FALSE.
• .replace: If both .sample and .replace are TRUE, sampling is done with replacement. Defaults to FALSE.
• .value_type: Determines how the walk is computed. Options are:
• "cum_prod": Computes the cumulative product of y.
• "cum_sum": Computes the cumulative sum of y.

Example 1: Simple Random Walk with Cumulative Sum

First, let’s create a simple random walk using a normal distribution and compute the cumulative sum.

library(TidyDensity)

set.seed(123)
tidy_normal(.num_sims = 25, .n = 100) |>
  tidy_random_walk(.value_type = "cum_sum") |>
  tidy_random_walk_autoplot()

In this example, we generate 25 simulations of 100 points each from a normal distribution. The tidy_random_walk() function then computes the cumulative sum of these points, simulating a simple random walk. The tidy_random_walk_autoplot() function is used to visualize the random walk.

Example 2: Random Walk with Sampling

Next, we’ll explore a random walk where values are sampled.

set.seed(123)
tidy_normal(.num_sims = 25, .n = 100) |>
  tidy_random_walk(.value_type = "cum_sum", .sample = TRUE) |>
  tidy_random_walk_autoplot()

Here, setting .sample to TRUE ensures that each step in the random walk is taken by randomly sampling from the original dataset. This can introduce additional variability and randomness to the walk.

Example 3: Random Walk with Sampling and Replacement

Finally, let’s create a random walk with sampling and replacement.

set.seed(123)
tidy_normal(.num_sims = 25, .n = 100) |>
  tidy_random_walk(
    .value_type = "cum_sum", 
    .sample = TRUE, 
    .replace = TRUE
    ) |>
  tidy_random_walk_autoplot()

In this example, setting both .sample and .replace to TRUE ensures that values are sampled with replacement. This can be useful in bootstrapping scenarios or when simulating more complex stochastic processes.

Bonus Section: Comparing Different Random Walk Sampling Methods

To wrap up, let’s combine multiple random walks and visualize them using ggplot2. This bonus section will show you how different sampling methods impact the random walks.

library(ggplot2)
library(dplyr)

set.seed(123)
df <- rbind(
  tidy_normal(.num_sims = 25, .n = 100) |>
    tidy_random_walk(.value_type = "cum_sum") |>
    mutate(type = "No_Sample"),
  tidy_normal(.num_sims = 25, .n = 100) |>
    tidy_random_walk(.value_type = "cum_sum", .sample = TRUE) |>
    mutate(type = "Sample_No_Replace"),
  tidy_normal(.num_sims = 25, .n = 100) |>
    tidy_random_walk(.value_type = "cum_sum", .sample = TRUE, .replace = TRUE) |>
    mutate(type = "Sample_Replace")
) |>
  select(sim_number, x, random_walk_value, type) |>
  mutate(
    low_ci = -1.96 * sqrt(x),
    hi_ci = 1.96 * sqrt(x)
  )

atb <- attributes(df)

df |>
  ggplot(aes(
    x = x, 
    y = random_walk_value, 
    group = sim_number, 
    color = factor(type))
  ) +
  geom_line(aes(alpha = 0.382)) +
  geom_line(aes(y = low_ci, group = sim_number), 
            linetype = "dashed", size = 0.6, color = "black") +
  geom_line(aes(y = hi_ci, group = sim_number), 
            linetype = "dashed", size = 0.6, color = "black") +
  theme_minimal() +
  theme(legend.position="none") +
  facet_wrap(~type) +
  labs(
    x = "Time",
    y = "Random Walk Value",
    title = "Random Walk with Different Sampling Methods",
    subtitle = paste0("Simulations: ", atb$all$.num_sims, 
                      " | Steps: ", atb$all$.n,
                      " | Distribution: ", atb$all$dist_with_params
                      )
  )

Code Explanation

1. Generating Data: We generate three sets of random walks using different sampling methods:

• No sampling.
• Sampling without replacement.
• Sampling with replacement.

Each set consists of 25 simulations of 100 steps.

2. Combining Data: The results are combined into a single data frame, with a new column type to indicate the sampling method used.

3. Calculating Confidence Intervals: We calculate the 95% confidence intervals for each step.

4. Plotting: Using ggplot2, we plot the random walks, coloring by sampling method and adding dashed lines to indicate the confidence intervals. We also facet the plot by type to separate the different sampling methods visually.