# Exploring Random Walks with TidyDensity in R

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# Introduction

Welcome back, data enthusiasts! Today, we’re diving into the fascinating world of random walks using the TidyDensity R package. If you’re working with time series data, financial modeling, or stochastic processes, understanding random walks is essential. And with TidyDensity, implementing and visualizing these walks has never been easier.

# Random Walks

A random walk is a mathematical object that describes a path consisting of a succession of random steps. It’s a cornerstone concept in fields like physics, economics, and biology. In finance, for example, the random walk hypothesis suggests that stock market prices evolve according to a random walk and thus cannot be predicted.

# TidyDensity and the `tidy_random_walk()`

Function

TidyDensity simplifies the generation and manipulation of random walks with its intuitive `tidy_random_walk()`

function. This function can be used in conjunction with any `tidy_`

distribution function, allowing for flexible and powerful random walk simulations.

# Function Call

tidy_random_walk( .data, .initial_value = 0, .sample = FALSE, .replace = FALSE, .value_type = "cum_prod" )

## Arguments Breakdown

: The dataset from a`.data`

`tidy_`

distribution function. This forms the basis of your random walk.: The starting value of the random walk. The default is 0, but you can set it to any numeric value.`.initial_value`

: A boolean indicating whether to sample the`.sample`

`y`

values from the`tidy_`

distribution. Defaults to`FALSE`

.: If both`.replace`

`.sample`

and`.replace`

are`TRUE`

, sampling is done with replacement. Defaults to`FALSE`

.: Determines how the walk is computed. Options are:`.value_type`

`"cum_prod"`

: Computes the cumulative product of`y`

.`"cum_sum"`

: Computes the cumulative sum of`y`

.

# Practical Examples

Let’s see `tidy_random_walk()`

in action with some practical examples.

## 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

**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.

**Combining Data**: The results are combined into a single data frame, with a new column`type`

to indicate the sampling method used.**Calculating Confidence Intervals**: We calculate the 95% confidence intervals for each step.**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.

# Conclusion

Random walks are a powerful tool for modeling and understanding various phenomena. With TidyDensity and the `tidy_random_walk()`

function, you can easily generate and visualize these processes in R. Whether you’re conducting financial analysis, simulating biological processes, or exploring theoretical concepts, TidyDensity offers a flexible and user-friendly approach.

Stay tuned for more tutorials and deep dives into the capabilities of TidyDensity. Happy coding!

Feel free to try out these examples and explore the versatility of `tidy_random_walk()`

. Share your insights and results with us in the comments below or on social media using #TidyDensity. Until next time, keep experimenting and learning!

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