Multi-Dimensional Walks"
In RandomWalker: Generate Random Walks Compatible with the 'tidyverse'

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.width = 7,
  fig.height = 5
)

library(RandomWalker)
library(dplyr)
library(ggplot2)

RandomWalker supports random walks in 1D, 2D, and 3D space. This guide covers everything you need to know about working with multi-dimensional random walks.

Overview
Generating Multi-Dimensional Walks
Understanding the Data Structure
Visualizing 2D Walks
Visualizing 3D Walks
Distance and Spatial Analysis
Use Cases
Best Practices

Overview

What are Multi-Dimensional Random Walks?

A multi-dimensional random walk extends the concept of a 1D random walk into higher-dimensional spaces:

1D: Movement along a line (left/right, up/down)
2D: Movement on a plane (x and y coordinates)
3D: Movement in space (x, y, and z coordinates)

When to Use Multi-Dimensional Walks

2D Walks: - Particle diffusion in a plane - Animal movement patterns - Robot navigation - 2D cellular automata - Currency exchange rates (two currencies vs reference)

3D Walks: - Particle diffusion in space - Molecular dynamics - Drone or aircraft movement - 3D cellular automata - Financial modeling (multiple asset classes)

Generating Multi-Dimensional Walks

Basic Syntax

All random walk generator functions support the .dimensions parameter:

# 1D walk (default)
walk_1d <- random_normal_walk(.num_walks = 5, .n = 100, .dimensions = 1)

# 2D walk
walk_2d <- random_normal_walk(.num_walks = 5, .n = 100, .dimensions = 2)

# 3D walk
walk_3d <- random_normal_walk(.num_walks = 5, .n = 100, .dimensions = 3)

All Distributions Support Multi-Dimensions

Every continuous distribution generator supports 1D, 2D, and 3D:

# 2D Brownian motion
brownian_motion(.num_walks = 10, .n = 500, .dimensions = 2)

# 3D geometric Brownian motion
geometric_brownian_motion(
  .num_walks = 10,
  .n = 500,
  .initial_value = 100,
  .dimensions = 3
)

# 2D Cauchy walk (heavy tails)
random_cauchy_walk(.num_walks = 10, .n = 200, .dimensions = 2)

# 3D exponential walk
random_exponential_walk(.num_walks = 5, .n = 300, .dimensions = 3)

Understanding the Data Structure

Column Naming Convention

1D Walk Columns:

walk_number, step_number, y, cum_sum_y, cum_prod_y, cum_min_y, cum_max_y, cum_mean_y

2D Walk Columns:

walk_number, step_number, x, y,
cum_sum_x, cum_sum_y,
cum_prod_x, cum_prod_y,
cum_min_x, cum_min_y,
cum_max_x, cum_max_y,
cum_mean_x, cum_mean_y

3D Walk Columns:

walk_number, step_number, x, y, z,
cum_sum_x, cum_sum_y, cum_sum_z,
cum_prod_x, cum_prod_y, cum_prod_z,
(and so on for each dimension)

Inspecting Multi-Dimensional Data

# 2D walk
walk_2d <- random_normal_walk(.num_walks = 3, .n = 100, .dimensions = 2)

# View structure
head(walk_2d, 10)
names(walk_2d)

# 3D walk
walk_3d <- random_normal_walk(.num_walks = 3, .n = 100, .dimensions = 3)

# View structure
head(walk_3d, 10)
names(walk_3d)

Visualizing 2D Walks

Basic 2D Trajectory Plot

# Generate 2D walk
walk_2d <- random_normal_walk(.num_walks = 10, .n = 200, .dimensions = 2)

# Plot trajectories
ggplot(walk_2d, aes(x = cum_sum_x, y = cum_sum_y, color = walk_number)) +
  geom_path(alpha = 0.6, linewidth = 0.5) +
  geom_point(data = walk_2d %>% filter(step_number == 1),
             size = 3, shape = 21, fill = "white") +  # Start points
  geom_point(data = walk_2d %>% group_by(walk_number) %>% slice_max(step_number),
             size = 3, shape = 24, fill = "red") +  # End points
  coord_equal() +
  theme_minimal() +
  labs(
    title = "2D Random Walk Trajectories",
    x = "X Position",
    y = "Y Position",
    color = "Walk"
  )

2D Walk with Step Numbers

# Generate single walk
walk_2d <- random_normal_walk(.num_walks = 1, .n = 100, .dimensions = 2)

# Plot with step labels at intervals
walk_2d_labeled <- walk_2d %>%
  filter(step_number %% 10 == 0 | step_number == 1)

ggplot(walk_2d, aes(x = cum_sum_x, y = cum_sum_y)) +
  geom_path(color = "steelblue", linewidth = 1) +
  geom_point(data = walk_2d_labeled, aes(color = step_number), size = 3) +
  geom_text(data = walk_2d_labeled, aes(label = step_number),
            hjust = -0.3, vjust = -0.3, size = 3) +
  scale_color_viridis_c() +
  coord_equal() +
  theme_minimal() +
  labs(
    title = "2D Random Walk with Step Numbers",
    x = "X Position",
    y = "Y Position",
    color = "Step"
  )

Heat Map of 2D Walk Density

library(ggplot2)

# Generate many walks
walk_2d <- random_normal_walk(.num_walks = 100, .n = 500, .dimensions = 2)

# Create density heat map
ggplot(walk_2d, aes(x = cum_sum_x, y = cum_sum_y)) +
  geom_bin2d(bins = 50) +
  scale_fill_viridis_c(option = "plasma") +
  coord_equal() +
  theme_minimal() +
  labs(
    title = "2D Random Walk Density Heat Map",
    subtitle = "100 walks, 500 steps each",
    x = "X Position",
    y = "Y Position",
    fill = "Count"
  )

Animated 2D Walk

library(ggplot2)
library(gganimate)

# Generate walk
walk_2d <- random_normal_walk(.num_walks = 5, .n = 100, .dimensions = 2)

# Create animation
p <- ggplot(walk_2d, aes(x = cum_sum_x, y = cum_sum_y, color = walk_number)) +
  geom_path(aes(group = walk_number), alpha = 0.5) +
  geom_point(size = 3) +
  coord_equal() +
  theme_minimal() +
  labs(title = "Step: {frame_along}", x = "X", y = "Y") +
  transition_reveal(step_number)

# Render
animate(p, nframes = 100, fps = 10)

Visualizing 3D Walks

3D Scatter Plot

library(plotly)

# Generate 3D walk
walk_3d <- random_normal_walk(.num_walks = 5, .n = 200, .dimensions = 3)

# Create 3D plot
plot_ly(
  data = walk_3d,
  x = ~cum_sum_x,
  y = ~cum_sum_y,
  z = ~cum_sum_z,
  color = ~walk_number,
  type = "scatter3d",
  mode = "lines",
  line = list(width = 2)
) %>%
  layout(
    title = "3D Random Walk Trajectories",
    scene = list(
      xaxis = list(title = "X Position"),
      yaxis = list(title = "Y Position"),
      zaxis = list(title = "Z Position")
    )
  )

3D Interactive with Markers

library(plotly)
library(dplyr)

# Generate 3D walk
walk_3d <- random_normal_walk(.num_walks = 3, .n = 100, .dimensions = 3)

# Mark start and end points
walk_with_markers <- walk_3d %>%
  mutate(
    point_type = case_when(
      step_number == 1 ~ "Start",
      step_number == max(step_number) ~ "End",
      TRUE ~ "Path"
    )
  )

# Create plot
plot_ly(data = walk_with_markers) %>%
  # Add paths
  add_trace(
    data = walk_with_markers %>% filter(point_type == "Path"),
    x = ~cum_sum_x, y = ~cum_sum_y, z = ~cum_sum_z,
    color = ~walk_number,
    type = "scatter3d",
    mode = "lines",
    line = list(width = 2),
    showlegend = FALSE
  ) %>%
  # Add start points
  add_trace(
    data = walk_with_markers %>% filter(point_type == "Start"),
    x = ~cum_sum_x, y = ~cum_sum_y, z = ~cum_sum_z,
    type = "scatter3d",
    mode = "markers",
    marker = list(size = 8, color = "green", symbol = "circle"),
    name = "Start"
  ) %>%
  # Add end points
  add_trace(
    data = walk_with_markers %>% filter(point_type == "End"),
    x = ~cum_sum_x, y = ~cum_sum_y, z = ~cum_sum_z,
    type = "scatter3d",
    mode = "markers",
    marker = list(size = 8, color = "red", symbol = "diamond"),
    name = "End"
  ) %>%
  layout(
    title = "3D Random Walk with Start/End Markers",
    scene = list(
      xaxis = list(title = "X"),
      yaxis = list(title = "Y"),
      zaxis = list(title = "Z")
    )
  )

3D Projections

library(ggplot2)
library(patchwork)

# Generate 3D walk
walk_3d <- random_normal_walk(.num_walks = 5, .n = 200, .dimensions = 3)

# X-Y projection
p_xy <- ggplot(walk_3d, aes(x = cum_sum_x, y = cum_sum_y, color = walk_number)) +
  geom_path(alpha = 0.6) +
  coord_equal() +
  theme_minimal() +
  labs(title = "X-Y Projection")

# X-Z projection
p_xz <- ggplot(walk_3d, aes(x = cum_sum_x, y = cum_sum_z, color = walk_number)) +
  geom_path(alpha = 0.6) +
  coord_equal() +
  theme_minimal() +
  labs(title = "X-Z Projection")

# Y-Z projection
p_yz <- ggplot(walk_3d, aes(x = cum_sum_y, y = cum_sum_z, color = walk_number)) +
  geom_path(alpha = 0.6) +
  coord_equal() +
  theme_minimal() +
  labs(title = "Y-Z Projection")

# Combine
p_xy / (p_xz | p_yz) +
  plot_layout(guides = "collect") +
  plot_annotation(title = "3D Random Walk - All Projections")

Distance and Spatial Analysis

Euclidean Distance from Origin

library(dplyr)

# 2D walk
walk_2d <- random_normal_walk(.num_walks = 10, .n = 500, .dimensions = 2)

# Calculate distance
walk_with_distance <- walk_2d %>%
  euclidean_distance(.x = x, .y = y)

# Visualize distance over time
library(ggplot2)

ggplot(walk_with_distance, aes(x = step_number, y = distance, color = walk_number)) +
  geom_line(alpha = 0.7) +
  # Add theoretical expectation
  geom_line(aes(y = sqrt(2 * step_number)), color = "black", linetype = "dashed", linewidth = 1) +
  theme_minimal() +
  labs(
    title = "Distance from Origin in 2D Random Walk",
    subtitle = "Dashed line shows theoretical E[distance] = sqrt(2n)",
    x = "Step",
    y = "Euclidean Distance"
  )

3D Distance Analysis

# 3D walk
walk_3d <- random_normal_walk(.num_walks = 100, .n = 1000, .dimensions = 3)

# Calculate distance
walk_with_distance <- walk_3d %>%
  euclidean_distance(.x = x, .y = z)

# Analyze distance distribution at specific steps
library(dplyr)

distance_at_steps <- walk_with_distance %>%
  filter(step_number %in% c(100, 250, 500, 1000)) %>%
  group_by(step_number) %>%
  reframe(
    mean_dist = mean(distance),
    sd_dist = sd(distance),
    theoretical_mean = sqrt(3 * step_number)
  )

print(distance_at_steps)

Radial Distribution Function

library(dplyr)
library(ggplot2)

# Generate many 2D walks
walk_2d <- random_normal_walk(.num_walks = 500, .n = 200, .dimensions = 2)

# Get final positions
final_positions <- walk_2d %>%
  group_by(walk_number) %>%
  slice_max(step_number) %>%
  euclidean_distance(.x = x, .y = y)

# Plot radial distribution
ggplot(final_positions, aes(x = distance)) +
  geom_histogram(aes(y = after_stat(density)), bins = 50, fill = "steelblue", alpha = 0.7) +
  geom_density(color = "darkblue", linewidth = 1) +
  theme_minimal() +
  labs(
    title = "Radial Distribution of Final Positions (2D)",
    subtitle = "500 walks, 200 steps each",
    x = "Distance from Origin",
    y = "Density"
  )

Convex Hull (2D)

library(dplyr)
library(ggplot2)

# Generate 2D walks
walk_2d <- random_normal_walk(.num_walks = 20, .n = 200, .dimensions = 2)

# Get final positions
final_positions <- walk_2d %>%
  group_by(walk_number) %>%
  slice_max(step_number)

# Calculate convex hull
hull <- chull(final_positions$cum_sum_x, final_positions$cum_sum_y)
hull_points <- final_positions[c(hull, hull[1]), ]  # Close the polygon

# Plot
ggplot(walk_2d, aes(x = cum_sum_x, y = cum_sum_y, color = walk_number)) +
  geom_path(alpha = 0.3) +
  geom_point(data = final_positions, size = 3) +
  geom_polygon(data = hull_points, aes(x = cum_sum_x, y = cum_sum_y),
               fill = NA, color = "black", linewidth = 1) +
  coord_equal() +
  theme_minimal() +
  labs(
    title = "2D Random Walks with Convex Hull",
    subtitle = "Black polygon shows convex hull of final positions",
    x = "X Position",
    y = "Y Position"
  )

Use Cases

Case 1: Particle Diffusion (2D)

# Simulate particle diffusion in a petri dish
particles <- brownian_motion(
  .num_walks = 50,
  .n = 1000,
  .delta_time = 0.1,
  .dimensions = 2
)

# Visualize
particles %>%
  euclidean_distance(.x = x, .y = y) %>%
  ggplot(aes(x = step_number, y = distance, color = walk_number)) +
  geom_line(alpha = 0.3) +
  stat_summary(aes(group = 1), fun = mean, geom = "line",
               color = "red", linewidth = 1.5) +
  theme_minimal() +
  labs(
    title = "Particle Diffusion in 2D",
    subtitle = "Red line shows mean distance",
    x = "Time Step",
    y = "Distance from Origin"
  )

Case 2: Drone Flight Path (3D)

# Simulate drone wandering in 3D space
drone_path <- brownian_motion(
  .num_walks = 1,
  .n = 500,
  .delta_time = 0.5,
  .initial_value = 100,  # Start at 100m altitude
  .dimensions = 3
)

# 3D visualization
library(plotly)

plot_ly(
  data = drone_path,
  x = ~cum_sum_x,
  y = ~cum_sum_y,
  z = ~cum_sum_z,
  type = "scatter3d",
  mode = "lines+markers",
  marker = list(
    size = 2,
    color = ~step_number,
    colorscale = "Viridis",
    showscale = TRUE
  ),
  line = list(width = 2, color = "steelblue")
) %>%
  layout(
    title = "Drone Flight Path (3D Random Walk)",
    scene = list(
      xaxis = list(title = "X (meters)"),
      yaxis = list(title = "Y (meters)"),
      zaxis = list(title = "Altitude (meters)")
    )
  )

Case 3: Animal Movement (2D)

# Simulate animal foraging behavior
# Using Cauchy walk for heavy tails (occasional long jumps)
animal_movement <- random_cauchy_walk(
  .num_walks = 1,
  .n = 200,
  .scale = 1,
  .dimensions = 2
)

# Add "home" location
animal_movement <- animal_movement %>%
  mutate(
    distance_from_home = sqrt(cum_sum_x^2 + cum_sum_y^2)
  )

# Plot
ggplot(animal_movement, aes(x = cum_sum_x, y = cum_sum_y)) +
  geom_path(color = "darkgreen", linewidth = 0.8, alpha = 0.6) +
  geom_point(aes(color = distance_from_home), size = 2) +
  geom_point(x = 0, y = 0, size = 5, color = "red", shape = 17) +  # Home
  scale_color_viridis_c(option = "magma") +
  coord_equal() +
  theme_minimal() +
  labs(
    title = "Animal Foraging Path (2D Cauchy Walk)",
    subtitle = "Red triangle marks home location",
    x = "X Position",
    y = "Y Position",
    color = "Distance\nfrom Home"
  )

Best Practices

Performance Considerations

For many walks or long walks in 3D:

# Reduce number of dimensions if not needed
walk_2d <- random_normal_walk(.num_walks = 100, .n = 1000, .dimensions = 2)

# Instead of
# walk_3d <- random_normal_walk(.num_walks = 100, .n = 1000, .dimensions = 3)

For visualization:

# Sample walks or steps for large datasets
walk_large <- random_normal_walk(.num_walks = 1000, .n = 1000, .dimensions = 2)

# Sample walks
walk_sample <- walk_large %>%
  filter(walk_number %in% sample(levels(walk_number), 50))

# Or downsample steps
walk_downsample <- walk_large %>%
  filter(step_number %% 10 == 0)

Coordinate Systems

Ensure equal aspect ratios for spatial plots:

# Good - maintains true spatial relationships
ggplot(walk_2d, aes(x = cum_sum_x, y = cum_sum_y)) +
  geom_path() +
  coord_equal()  # Important!

# Bad - distorts spatial relationships
ggplot(walk_2d, aes(x = cum_sum_x, y = cum_sum_y)) +
  geom_path()  # No coord_equal()

Boundary Conditions

Implement reflecting or absorbing boundaries:

# Reflecting boundary (bounce back)
walk_2d <- random_normal_walk(.num_walks = 10, .n = 500, .dimensions = 2)

walk_bounded <- walk_2d %>%
  mutate(
    cum_sum_x = pmin(pmax(cum_sum_x, -50), 50),  # Bound between -50 and 50
    cum_sum_y = pmin(pmax(cum_sum_y, -50), 50)
  )

# Visualize
ggplot(walk_bounded, aes(x = cum_sum_x, y = cum_sum_y, color = walk_number)) +
  geom_path(alpha = 0.6) +
  geom_rect(xmin = -50, xmax = 50, ymin = -50, ymax = 50,
            fill = NA, color = "black", linewidth = 1) +
  coord_equal() +
  theme_minimal() +
  labs(title = "2D Walk with Reflecting Boundaries")

Next Steps

For more information on RandomWalker, explore these resources:

Getting Started - Introduction to random walks
Basic Concepts - Core ideas and terminology
Continuous Distribution Generators - Walks based on continuous distributions
Discrete Distribution Generators - Walks based on discrete distributions
FAQ - Frequently asked questions
API Reference - Complete function documentation Need more examples? Check out the package documentation and vignettes for additional use cases and applications!

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RandomWalker documentation built on Sept. 8, 2026, 1:06 a.m.

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Note that we can't provide technical support on individual packages. You should contact the package authors for that.

RandomWalker
Generate Random Walks Compatible with the 'tidyverse'

Multi-Dimensional Walks"
In RandomWalker: Generate Random Walks Compatible with the 'tidyverse'

Table of Contents

Overview

What are Multi-Dimensional Random Walks?

When to Use Multi-Dimensional Walks

Generating Multi-Dimensional Walks

Basic Syntax

All Distributions Support Multi-Dimensions

Understanding the Data Structure

Column Naming Convention

Inspecting Multi-Dimensional Data

Visualizing 2D Walks

Basic 2D Trajectory Plot

2D Walk with Step Numbers

Heat Map of 2D Walk Density

Animated 2D Walk

Visualizing 3D Walks

3D Scatter Plot

3D Interactive with Markers

3D Projections

Distance and Spatial Analysis

Euclidean Distance from Origin

3D Distance Analysis

Radial Distribution Function

Convex Hull (2D)

Use Cases

Case 1: Particle Diffusion (2D)

Case 2: Drone Flight Path (3D)

Case 3: Animal Movement (2D)

Best Practices

Performance Considerations

Coordinate Systems

Boundary Conditions

Next Steps

Try the RandomWalker package in your browser

R Package Documentation

Browse R Packages

We want your feedback!

RandomWalker Generate Random Walks Compatible with the 'tidyverse'

Multi-Dimensional Walks" In RandomWalker: Generate Random Walks Compatible with the 'tidyverse'

Table of Contents

Overview

What are Multi-Dimensional Random Walks?

When to Use Multi-Dimensional Walks

Generating Multi-Dimensional Walks

Basic Syntax

All Distributions Support Multi-Dimensions

Understanding the Data Structure

Column Naming Convention

Inspecting Multi-Dimensional Data

Visualizing 2D Walks

Basic 2D Trajectory Plot

2D Walk with Step Numbers

Heat Map of 2D Walk Density

Animated 2D Walk

Visualizing 3D Walks

3D Scatter Plot

3D Interactive with Markers

3D Projections

Distance and Spatial Analysis

Euclidean Distance from Origin

3D Distance Analysis

Radial Distribution Function

Convex Hull (2D)

Use Cases

Case 1: Particle Diffusion (2D)

Case 2: Drone Flight Path (3D)

Case 3: Animal Movement (2D)

Best Practices

Performance Considerations

Coordinate Systems

Boundary Conditions

Next Steps

Try the RandomWalker package in your browser

R Package Documentation

Browse R Packages

We want your feedback!

RandomWalker
Generate Random Walks Compatible with the 'tidyverse'

Multi-Dimensional Walks"
In RandomWalker: Generate Random Walks Compatible with the 'tidyverse'