Generating Correlated Random Numbers in R from Scratch

Need random data with specific correlation patterns for your simulations? This post shows you how to generate correlated random numbers in R using a simple matrix approach – perfect for testing algorithms or creating realistic synthetic datasets.

The Cholesky Method in Four Steps

# 1. Define your target correlation matrix
cor_mat <- matrix(c(1, 0.3, 
                   0.3, 1), nrow = 2, byrow = TRUE)

# 2. Apply Cholesky decomposition
chol_mat <- chol(cor_mat)

# 3. Generate uncorrelated random numbers
old_random <- matrix(rnorm(2000), ncol = 2)

# 4. Transform to create correlation
new_random <- old_random %*% chol_mat

# Verify the correlation
cor(new_random)

          [,1]      [,2]
[1,] 1.0000000 0.3533963
[2,] 0.3533963 1.0000000

That’s it! The new_random matrix now contains values with approximately your target correlation structure. This technique uses Cholesky decomposition to create a transformation matrix that induces the desired correlation when applied to uncorrelated data.

Watch Out for These Pitfalls

1. Start with Truly Random Data

Your input data must be uncorrelated for this method to work correctly:

# What happens with already correlated input?
simulate_correlation <- function(input_correlation, target = 0.3) {
  results <- replicate(1000, {
    # Create input with specified correlation
    x <- rnorm(1000)
    y <- input_correlation * x + rnorm(1000, sd = sqrt(1 - input_correlation^2))
    
    # Apply our method
    old_random <- cbind(x, y)
    chol_mat <- chol(matrix(c(1, target, target, 1), ncol = 2))
    new_random <- old_random %*% chol_mat
    
    # Return resulting correlation
    cor(new_random)[1,2]
  })
  return(results)
}

# Compare results with different input correlations
par(mfrow = c(1, 2))
hist(simulate_correlation(0.8), main = "Starting with Correlated Data",
     xlim = c(0, 1), col = "salmon")
hist(simulate_correlation(0.001), main = "Starting with Random Data",
     xlim = c(0, 1), col = "lightblue")

When your input data already has correlation patterns, the Cholesky method can’t properly override them to create your target correlation.

2. Use the Same Distribution for All Variables

# Different distributions cause problems
set.seed(123)
x1 <- rchisq(1000, df = 3)  # Chi-squared (skewed)
y1 <- rnorm(1000)           # Normal (symmetric)
old_mixed <- cbind(x1, y1)

# Same distribution works better
x2 <- rchisq(1000, df = 3)
y2 <- rchisq(1000, df = 3)
old_same <- cbind(x2, y2)

# Apply the same transformation to both
chol_mat <- chol(matrix(c(1, 0.7, 0.7, 1), ncol = 2))
new_mixed <- old_mixed %*% chol_mat
new_same <- old_same %*% chol_mat

# Compare results
cat("Target correlation: 0.7\n")

Target correlation: 0.7

cat("Mixed distributions result:", round(cor(new_mixed)[1,2], 3), "\n")

Mixed distributions result: 0.915 

cat("Same distribution result:", round(cor(new_same)[1,2], 3))

Same distribution result: 0.699

Mixing different distributions (like normal and chi-squared) can lead to unexpected correlation patterns after transformation.

3. Distribution Properties Can Change

# Original positive-only distribution
x <- rchisq(1000, df = 3)  # Always positive
y <- rchisq(1000, df = 3)  # Always positive
old_random <- cbind(x, y)

# Apply negative correlation
chol_mat <- chol(matrix(c(1, -0.7, -0.7, 1), ncol = 2))
new_random <- old_random %*% chol_mat

# Check what happened
cat("Original data range:", round(range(old_random), 2), "\n")

Original data range: 0.02 19.93 

cat("Transformed data range:", round(range(new_random), 2), "\n")

Transformed data range: -12.81 19.93 

cat("Negative values in result:", sum(new_random < 0), "out of", length(new_random))

Negative values in result: 488 out of 2000

The Cholesky transformation can fundamentally change your data’s properties - like introducing negative values into a previously positive-only distribution.

The Easy Way: Using mvtnorm

For most real applications, the mvtnorm package offers a simpler solution:

# Load the package
library(mvtnorm)

# Define means and covariance matrix
means <- c(10, 20)  # Mean for each variable
sigma <- matrix(c(4, 2,   # Covariance matrix
                  2, 3), ncol = 2)

# See the implied correlation
cov2cor(sigma)

          [,1]      [,2]
[1,] 1.0000000 0.5773503
[2,] 0.5773503 1.0000000

# Generate correlated normal data in one step
x <- rmvnorm(n = 1000, mean = means, sigma = sigma)

# Verify the result
round(cor(x), 3)

      [,1]  [,2]
[1,] 1.000 0.613
[2,] 0.613 1.000

When to Use Each Method

Use the Cholesky method when: - You need to understand the mathematical principles - You’re working with non-normal distributions - You need to create custom correlation structures

Use mvtnorm when: - You need multivariate normal data quickly - You want precise control over means and variances - You’re working with many variables