Table of Contents | What is a Random Walk? | Simple Example | Real-World Analogies | Types of Random Walks | 1. Simple Random Walk | 2. Random Walk with Drift | 3. Brownian Motion (Wiener Process) | 4. Geometric Brownian Motion | Key Properties | Property 1: Mean Displacement | Property 2: Variance Growth | Property 3: Distance from Origin | Property 4: First Return to Origin | Property 5: Scaling | Mathematical Background | One-Dimensional Random Walk | Brownian Motion | Geometric Brownian Motion | RandomWalker Implementation | How RandomWalker Works | Example: Behind the Scenes | Dimensions | Common Terminology | Terms Used in RandomWalker | Statistical Terms | Probability Distributions | Worked Examples | Example 1: Verify Properties | Example 2: Distribution of Final Position | Example 3: Path Dependency | Next Steps | Further Reading | Academic Resources | Online Resources

Libaray Load | Information | Generate some data | User Item Tibble | K-Means Mapped Tibble | Scree Plot and Data | UMAP List Object | UMAP Plot

Libraries | Data | Data Set | Splits | Initial Recipe | Inspect PCA Output | PCA Transform | Variable Loadings | Variable Variance | PCA Estimates | Jucied and Baked Data | Roatation Data | Variance and Scree Plot | Variable Loading Plots

Overview | General Pattern | Estimation Methods | 1. Maximum Likelihood Estimation (MLE) | 2. Method of Moments Estimation (MME) | 3. Minimum Variance Unbiased Estimation (MVUE) | Basic Usage | Simple Parameter Estimation | Understanding the Output | Visualizing the Fit | Comparing Methods | Example: Normal Distribution | Understanding Differences | Example: Gamma Distribution | Example: Exponential Distribution | Model Selection with AIC | What is AIC? | AIC Functions | Visualization of Fitted Distributions | Basic Visualization | Customizing the Comparison Plot | Multiple Distribution Overlay | Advanced Techniques | 1. Goodness-of-Fit Tests | 2. QQ Plot Validation | 3. Residual Analysis | Statistics Tables | Distribution Statistics | Best Practices | 1. Always Visualize | 2. Check Sample Size | 3. Try Multiple Distributions | 4. Validate Assumptions | 5. Consider Data Characteristics | 6. Document Your Process | Common Issues and Solutions | Issue: Parameters Don't Make Sense | Issue: Large Difference Between MLE and MVUE | Issue: Poor Fit Quality | Available Functions | Continuous Distributions | Discrete Distributions | Next Steps

Data Science and Analytics | Example 1: Exploring Customer Purchase Amounts | Example 2: A/B Test Analysis with Bootstrap | Example 3: Detecting Data Distribution Changes | Statistical Analysis | Example 4: Power Analysis for Sample Size Determination | Example 5: Distribution Goodness-of-Fit Testing | Finance and Risk Management | Example 6: Value at Risk (VaR) Calculation | Example 7: Option Pricing with Monte Carlo Simulation | Quality Control and Manufacturing | Example 8: Process Capability Analysis | Example 9: Control Chart Analysis | Healthcare and Epidemiology | Example 10: Disease Outbreak Modeling | Machine Learning and Simulation | Example 11: Monte Carlo Simulation for Project Planning | Education and Teaching | Example 12: Demonstrating Central Limit Theorem | Tips for Your Own Use Cases | 1. Start with Exploration | 2. Try Multiple Distributions | 3. Validate with Visualization | 4. Use Bootstrap for Uncertainty | Key Takeaways | 1. Distribution Fitting is Essential | 2. Bootstrap Provides Robust Uncertainty | 3. Visualization Validates Analysis | 4. TidyDensity Integrates with Tidyverse | 5. Domain Knowledge Guides Choices

Tidy Data Philosophy | What is Tidy Data? | Why Tidy Data Matters | Benefits of Tidy Format | Probability Distributions | What is a Probability Distribution? | Types of Distributions | Continuous Distributions | Discrete Distributions | Distribution Characteristics | Distribution Functions (d, p, q, r) | 1. Density Function (d) | 2. Probability Function (p) | 3. Quantile Function (q) | 4. Random Generation Function (r) | Visual Comparison | Random Number Generation | Pseudorandom Numbers | Setting Seeds for Reproducibility | Multiple Simulations | Parameter Estimation | What is Parameter Estimation? | Estimation Methods | Maximum Likelihood Estimation (MLE) | Method of Moments Estimation (MME) | Minimum Variance Unbiased Estimation (MVUE) | Model Selection | Statistical Inference | Hypothesis Testing | Confidence Intervals | Power Analysis | Tidyverse Integration | Works with dplyr | Works with ggplot2 | Works with tidyr | Works with purrr | Key Takeaways | 1. Tidy Format Enables Analysis | 2. Four Functions (d, p, q, r) | 3. Multiple Methods Available | 4. Reproducibility Matters | 5. Visualization is Essential

What is Bootstrap? | Concept | How It Works | When to Use Bootstrap | Bootstrap in TidyDensity | Main Function: tidy_bootstrap() | Return Value | Basic Bootstrap Analysis | Simple Bootstrap Example | Visualizing Bootstrap Distribution | Quick Statistics | Bootstrap Statistics | Unnesting Bootstrap Data | Calculating Bootstrap Statistics | Overall Bootstrap Statistics | Confidence Intervals | Bootstrap Percentile Method | Confidence Intervals for Multiple Statistics | Visualizing Confidence Intervals | Bootstrap Augmentation Functions | Augment Density | Augment Probability | Augment Quantile | Advanced Bootstrap Techniques | Bootstrap for Difference of Means | Bootstrap for Correlation | Visualization | Multiple Visualizations | Custom Visualization with CI | Best Practices | 1. Choose Appropriate Number of Simulations | 2. Verify Bootstrap Convergence | 3. Consider Sample Size | 4. Understand Limitations | Troubleshooting | Issue: CI Too Wide | Issue: Long Computation Time

Mixture Models | What are Mixture Models? | Creating Mixture Models | Basic Mixture Creation | Mixture Types | 1. Addition Mixture ("add") | 2. Multiplication Mixture ("multiply") | 3. Subtraction Mixture ("subtract") | 4. Division Mixture ("divide") | Complex Mixture Example | Weighted Mixtures | Different Distribution Types | Empirical Distributions | What are Empirical Distributions? | Multiple Empirical Simulations | Comparing Empirical with Theoretical | Empirical Bootstrap | Multi-Distribution Comparison | Compare Same Distribution with Different Parameters | Compare Different Distributions | Random Walk Generation | Basic Random Walk | Random Walk Visualization | Random Walk Analysis | Distribution Combinations | Combining Multiple Distributions | Multi-Single Distribution Table | Quantile Normalization | What is Quantile Normalization? | Advanced Plotting | Four-Panel Plots | Triangular Distribution Plots | Real-World Examples | Example 1: Modeling Bimodal Data | Example 2: Quality Control | Tips and Tricks | Tip 1: Validate Mixture Models | Tip 2: Debug by Plotting Components Separately | Troubleshooting | Issue: Mixture Doesn't Look Right | Issue: Empirical Distribution Too Noisy | Issue: Multi-Distribution Plots Cluttered

Table of Contents | Summary Statistics | Basic Summary with summarize_walks() | Analyzing Different Values | Understanding Output Columns | Practical Examples | Example 1: Analyzing Stock Price Simulations | Example 2: Comparing Distributions | Cumulative Functions | Available Cumulative Functions | Using Cumulative Functions | Custom Cumulative Functions | Confidence Intervals | Using confidence_interval() | Confidence Intervals for Random Walks | Confidence Intervals for Final Values | Running Quantiles | Using running_quantile() | Visualizing Quantile Evolution | Distance Calculations | Using euclidean_distance() | Distance Statistics | First Passage Time | Subsetting Walks | Using subset_walks() | Finding Specific Walks | Advanced Analysis | Autocorrelation Analysis | Distribution Testing | Variance Ratio Test | Return Analysis (Financial) | Statistical Tests | Comparing Distributions | Testing for Drift | Next Steps

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

Table of Contents | Common Parameters | Discrete Walk | discrete_walk() | Binomial Distribution | random_binomial_walk() | Geometric Distribution | random_geometric_walk() | Hypergeometric Distribution | random_hypergeometric_walk() | Multinomial Distribution | random_multinomial_walk() | Negative Binomial Distribution | random_negbinomial_walk() | Poisson Distribution | random_poisson_walk() | Wilcoxon Tests | random_wilcox_walk() | random_wilcoxon_sr_walk() | Smirnov Distribution | random_smirnov_walk() | Comparison Guide | When to Use Each Distribution | Count Data Selection | Practical Examples | Example 1: Website Traffic | Example 2: Quality Control | Example 3: Customer Service | Best Practices | Choosing Parameters | Validation | Next Steps

General Questions | What is RandomWalker? | Who should use RandomWalker? | Is RandomWalker free to use? | Installation Questions | How do I install RandomWalker? | What R version do I need? | Why am I getting dependency errors? | Usage Questions | How do I generate a simple random walk? | How do I visualize my random walks? | How do I create a custom random walk? | Can I set a seed for reproducibility? | Distribution Questions | Which distribution should I use? | What's the difference between random_normal_walk() and brownian_motion()? | What's the difference between brownian_motion() and geometric_brownian_motion()? | Can I use custom distributions? | Multi-Dimensional Questions | How do I create a 2D random walk? | How do I visualize 2D walks? | What's the difference between x, y in 1D vs 2D walks? | Visualization Questions | How do I make interactive plots? | How do I show only specific panels? | How do I adjust transparency? | How do I export plots? | Can I customize colors? | Statistical Analysis Questions | How do I get summary statistics? | What statistics are included? | How do I subset walks by extremes? | Performance Questions | How many walks can I generate? | My visualization is slow. How do I speed it up? | Can I parallelize generation? | Data Structure Questions | What format does RandomWalker return? | How do I access attributes? | Can I convert to other formats? | Error Messages | "The value to summarize must be provided" | "object 'y' not found" | Integration Questions | Does RandomWalker work with dplyr? | Can I use it in Shiny apps? | Can I use it with ggplot2? | Application Questions | How do I model stock prices? | How do I simulate particle diffusion? | How do I test an algorithm? | Getting Help | Where can I find more examples? | Where do I report bugs? | Where can I ask questions? | How do I cite RandomWalker? | Is there a community? | Contributing | Can I contribute? | How do I suggest a new feature? | Related Questions | What's the difference between RandomWalker and other R packages? | Can RandomWalker handle big data? | Is RandomWalker actively maintained?

📖 What is RandomWalker? | 🚀 Quick Navigation | Getting Started | Function Guides | Advanced Topics | Reference | Contributing | 💡 Key Features | 🎲 27+ Distribution Types | 📐 Multi-Dimensional Support | 📊 Rich Visualizations | 📈 Statistical Analysis | 🔧 Tidyverse Compatible | 📦 Package Information | 🔗 External Links | 📚 Learning Path | 🎯 Common Use Cases | 🤝 Getting Help | 🌟 Citation | Example: Quick Start

Overview | The rw30() Function | Basic Usage | What rw30() Does | Output Structure | Understanding the Output | Walk Structure | Random Walk Behavior | Attributes | Common Usage Patterns | Pattern 1: Quick Visualization | Pattern 2: Statistical Analysis | Pattern 3: Finding Extremes | Pattern 4: Filtering and Subsetting | Pattern 5: Teaching Demonstrations | Pattern 6: Comparing to Theory | When to Use rw30() | ✅ Use rw30() When: | ❌ Don't Use rw30() When: | Limitations | Fixed Parameters | Only Normal Distribution | Only 1D | Alternatives to rw30() | For Custom Parameters | For Different Distributions | For Multi-Dimensional | Complete Examples | Example 1: Teaching Random Walk Properties | Example 2: First Passage Time | Example 3: Maximum Excursion | Next Steps

Table of Contents | Common Parameters | Normal Distribution | random_normal_walk() | random_normal_drift_walk() | Brownian Motion | brownian_motion() | geometric_brownian_motion() | Beta Distribution | random_beta_walk() | Cauchy Distribution | random_cauchy_walk() | Chi-Squared Distribution | random_chisquared_walk() | Exponential Distribution | random_exponential_walk() | F Distribution | random_f_walk() | Gamma Distribution | random_gamma_walk() | Log-Normal Distribution | random_lognormal_walk() | Logistic Distribution | random_logistic_walk() | Student's t Distribution | random_t_walk() | Uniform Distribution | random_uniform_walk() | Weibull Distribution | random_weibull_walk() | Comparison Guide | When to Use Each Distribution | Tail Behavior | Symmetry | Next Steps

API Reference | Quick Navigation | Automatic Random Walks | rw30() | Continuous Distribution Generators | random_normal_walk() | random_normal_drift_walk() | brownian_motion() | geometric_brownian_motion() | random_beta_walk() | random_cauchy_walk() | random_chisquared_walk() | random_exponential_walk() | random_f_walk() | random_gamma_walk() | random_lognormal_walk() | random_logistic_walk() | random_t_walk() | random_uniform_walk() | random_weibull_walk() | Discrete Distribution Generators | discrete_walk() | random_binomial_walk() | random_geometric_walk() | random_hypergeometric_walk() | random_multinomial_walk() | random_negbinomial_walk() | random_poisson_walk() | random_wilcox_walk() | random_wilcoxon_sr_walk() | random_smirnov_walk() | Custom Walks | custom_walk() | random_displacement_walk() | Visualization Functions | visualize_walks() | Statistical Functions | summarize_walks() | subset_walks() | Vector Functions | confidence_interval() | running_quantile() | euclidean_distance() | Cumulative Functions | Utility Functions | rand_walk_helper() | convert_snake_to_title_case() | get_attributes() | Data Structure | Return Format | Attributes | Package Information | Function Index | By Category | See Also

Random Walks | Installation | Example Usage | Attributes | Visualizing Random Walks | Future Direction | References

Introduction | Thanks | Installation | Examples

Introduction | The Function | Funcation and Parameters | Example | Data | Plot Data | Run Function | Output | Output Data | data | error_data | input_vector | maximum_harmonic_tbl | differenced_value_tbl | dff_tbl | ts_obj | Output Plots | harmonic_plt | diff_plot | max_har_plot | harmonic_plotly | max_har_plotly | Output Parameters | parameters | Output Model | m

Data | Use the function | Function Output | Auto-ML Object | The Best Model | Scree Plot

Example

Libaray Load | Generate Sample Data | Plot the Time Series