Multinomial Distribution in R

This article will guide you through the use of multinomial distribution in R, including its theory, parameters, and practical applications using built-in R functions.

Multinomial Distribution

The multinomial distribution in R describes the probability of obtaining a specific combination of outcomes when conducting multiple trials with more than two possible outcomes. Each trial results in exactly one of several outcomes, and the number of outcomes is fixed.

Let us consider an example where the random variable Y has a multinomial distribution. Then, we can calculate the probability that outcome 1 occurs exactly y₁ times, outcome 2 occurs exactly y₂ times, the outcome 3 occurs exactly y₃ times can be found with the help of the below formula.

Probability = n! * (p1^y₁ * p2^y₂ * … * pk^y_k) /  (y₁! * y₂! … * y_k!)

Here,

• n: It represents the total number of events
• y₁: It signifies that the number of times the outcome 1 will take place
• y₂: It signifies that the number of times the outcome 2 will take place
• y_k: It signifies that the number of times the outcome k will take place
• p₁: It represents the probability of the outcome 1 occurs for a given trial
• p₂: It represents the probability of the outcome 1 occurs for a given trial
• p_k: It represents the probability of the outcome k occurs for a given trial

Multinomial Distribution in R

Multinomial Distribution in R provides built-in functions to work with multinomial distributions via the rmultinom() function. This function can simulate draws from a multinomial distribution. It allows you to generate random samples of a specific size based on the parameters of the distribution.

Simulating a Multinomial Distribution

In Simulating a Multinomial Distribution rmultinom() is used to simulate random samples from a multinomial distribution in R. Its basic syntax is as follows:

rmultinom(n, size, prob)

• n: The number of random samples to generate.
• size: The total number of trials (sum of outcomes).
• prob: A vector of probabilities for each category. The length of the vector represents the number of possible outcomes.

Suppose we are conducting an experiment with three possible outcomes, with probabilities 0.2, 0.3, and 0.5, respectively. We want to simulate the results of 10 trials of this experiment.

# Set parameters
n_trials <- 10
probs <- c(0.2, 0.3, 0.5)

# Simulate one draw from the multinomial distribution
set.seed(123)
result <- rmultinom(n = 1, size = n_trials, prob = probs)

# Display result
result

Output:

     [,1]
[1,]    1
[2,]    5
[3,]    4

In this example, the function returns a vector of three values, representing the number of times each outcome was observed in 10 trials.

• Outcome 1 was observed 3 times.
• Outcome 2 was observed 4 times.
• Outcome 3 was observed 3 times.

Probability Calculations

R does not have a built-in function for calculating the probability of a specific outcome for the multinomial distribution directly (it doesn't have a dmultinom() function). However, we can calculate the probability manually using the formula for the multinomial PMF.

For instance, suppose we want to calculate the probability of observing the counts k1=3, k2=4, and k3=3 for a multinomial distribution with probabilities p1=0.2, p2=0.3, and p3=0.5, and n=10 trials.

# Calculate factorial of a number
factorial_calc <- function(x) {
  if (x == 0) return(1)
  return(prod(1:x))
}

# Set parameters
n_trials <- 10
counts <- c(3, 4, 3)
probs <- c(0.2, 0.3, 0.5)

# Calculate the multinomial probability
multinomial_prob <- (factorial_calc(n_trials) / prod(sapply(counts, factorial_calc))) * 
  prod(probs ^ counts)

# Display the result
multinomial_prob

Output:

[1] 0.03402

The probability of obtaining the specific outcome (3,4,3) for the given multinomial distribution is approximately 0.03402.

Visualizing Multinomial Outcomes

You can visualize the distribution of multinomial outcomes using basic bar plots in R. For example, let's visualize the outcome of a multinomial experiment with 1000 simulations.

# Set parameters
n_simulations <- 1000
n_trials <- 10
probs <- c(0.2, 0.3, 0.5)

# Simulate the multinomial distribution
simulated_data <- rmultinom(n = n_simulations, size = n_trials, prob = probs)

# Sum the outcomes across simulations
outcome_sums <- rowSums(simulated_data)

# Create a barplot of the outcomes
barplot(outcome_sums, names.arg = c("Outcome 1", "Outcome 2", "Outcome 3"),
        main = "Distribution of Multinomial Outcomes",
        ylab = "Frequency of Outcomes", col = "lightblue")

Output:

This plot visualizes the frequencies of each outcome after simulating 1000 multinomial experiments.

Conclusion

The multinomial distribution is a powerful tool for modeling experiments with more than two possible outcomes. In R, you can easily simulate outcomes using the rmultinom() function, calculate probabilities manually, and visualize the results using basic plotting functions. Understanding and using the multinomial distribution can help you model real-world phenomena such as genetic inheritance, text classification, and survey response patterns.

bhuwanesh

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