How to create a plot of cumulative distribution function in R?

Empirical distribution is a non-parametric method used to estimate the cumulative distribution function (CDF) of a random variable. It is particularly useful when you have data and want to make inferences about the population distribution without making any assumptions about its form. In this article, we will discuss how to create and visualize empirical distributions in R, using a variety of techniques and functions.

Empirical Distribution in R

The empirical distribution is a statistical concept that describes the observed frequencies or proportions of data values within a dataset. Unlike theoretical distributions, which are based on mathematical models and assumptions, the empirical distribution is derived directly from the data itself. It represents the distribution of actual observed values, providing insights into the characteristics, variability, and patterns present in the dataset without assuming any specific mathematical form.

The empirical distribution function F_n(x) is defined as:

F_n(x) = \frac{1}{n} \sum_{i=1}^{n} \mathbf{1}_{x_i \leq x}

Where:

• \mathbf{1}_{x_i \leq x} ​is the indicator function, equaling 1 if \mathbf{}_{x_i \leq x} and 0 otherwise.
• n is the number of observations in the dataset.

Steps for Calculating of Empirical Distribution

Calculating the empirical distribution involves determining the frequencies or proportions of observed data values within a dataset.

• Collect Data: Gather the dataset containing the observations you want to analyze.
• Identify Unique Values: Identify all unique values present in the dataset.
• Count Frequencies: For each unique value, count the number of times it appears in the dataset. This count represents the frequency of that value.

Here’s a step-by-step guide on how to calculate and visualize the empirical distribution:

Step 1: Install and Load Necessary Packages

While base R provides sufficient functions, you might need the ggplot2 package for visualization. If not already installed, you can install it using:

install.packages("ggplot2")
#Load the necessary packages:
library(ggplot2)

Step 2: Generate or Load Data

Generate some sample data or use your dataset. Here’s an example with generated data:

# Generating sample data
set.seed(123)  # For reproducibility
data <- rnorm(100, mean = 50, sd = 10) 

Step 3: Calculate the Empirical Distribution

Use the ecdf function to compute the empirical cumulative distribution function:

# Calculate the empirical cumulative distribution function
ecdf_function <- ecdf(data)

Step 4: Evaluate the ECDF

You can evaluate the ECDF at specific points:

# Evaluate the ECDF at specific points
ecdf_values <- ecdf_function(c(45, 50, 55))
print(ecdf_values)

Output:

[1] 0.25 0.48 0.70

Step 5: Plot the ECDF

Plotting the ECDF in R can be done using both base R and the ggplot2 package. Below, I will show examples of how to generate and plot ECDFs using both methods.

Using base R plotting

Using base R, you can plot the ECDF using the ecdf function and the plot function.

# Plotting the ECDF using base R
plot(ecdf_function, main = "Empirical Cumulative Distribution Function",
     xlab = "Data", ylab = "ECDF", col = "blue", lwd = 2)

Output:

Using ggplot2 for a more refined plot

Using the ggplot2 package provides more flexibility and customization options for plotting.

# Create a data frame for ggplot
ecdf_data <- data.frame(x = sort(data), y = ecdf_function(sort(data)))

# Plotting the ECDF using ggplot2
ggplot(ecdf_data, aes(x = x, y = y)) +
  geom_step(color = "blue") +
  labs(title = "Empirical Cumulative Distribution Function",
       x = "Data", y = "ECDF") +
  theme_minimal()

Output:

Conclusion

In conclusion, the empirical distribution is a representation of the observed frequencies or proportions of data values in a dataset, providing insights into its characteristics without assuming any specific mathematical form. Through techniques like histograms, kernel density estimation, or contour plots, we can visualize and analyze the distribution of data, aiding in tasks such as exploratory data analysis, model assessment, and generating synthetic data. By directly reflecting the observed data, the empirical distribution serves as a valuable tool in understanding and interpreting datasets in statistics and machine learning.

deepkumarpatra

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