如何先生成不相关变量，再将其转化为指定相关性的变量？

Got it, let's walk through how to take your uncorrelated normal variables and transform them to have the exact correlations you want. Here's a step-by-step breakdown that builds on your existing code:

Step 1: Start with your uncorrelated variables #

First, let's confirm your initial code generates uncorrelated standard normals:

# Generate 1000 uncorrelated standard normal observations for each variable
a <- rnorm(1000)
b <- rnorm(1000)
c <- rnorm(1000)
x <- matrix(c(a, b, c), ncol = 3)

# Check initial correlations (should be close to 0)
cor(x)

Step 2: Define your target correlation matrix #

Next, create a matrix representing the correlations you want between each pair of variables:

# Desired correlation matrix (rows/columns correspond to a, b, c)
target_cor <- matrix(
  c(1, 0.4, 0.3,  # Correlations for a with a, b, c
    0.4, 1, 0.5,  # Correlations for b with a, b, c
    0.3, 0.5, 1), # Correlations for c with a, b, c
  nrow = 3, ncol = 3
)

Step 3: Use Cholesky decomposition to transform the variables #

The key here is using the Cholesky decomposition of your target correlation matrix. This breaks down the correlation matrix into a lower triangular matrix L such that L %*% t(L) = target_cor. Multiplying your uncorrelated variables by L will give you variables with the desired correlation structure:

# Compute Cholesky decomposition of the target correlation matrix
L <- chol(target_cor)

# Transform the uncorrelated variables
# We transpose twice to ensure matrix multiplication works correctly (rows = observations, columns = variables)
transformed_x <- t(L %*% t(x))

# Rename columns for clarity
colnames(transformed_x) <- c("a_transformed", "b_transformed", "c_transformed")

Step 4: Verify the result #

Finally, check that the transformed variables have the correlations you wanted:

# Check correlations of the transformed variables
cor(transformed_x)

You should see values very close to your target (0.4, 0.3, 0.5) — minor differences are due to random sampling.

Note on non-normal variables #

If you were starting with uniform variables instead of normals, you’d first need to convert them to standard normals using the inverse normal CDF:

# Example for uniform variables
a_unif <- runif(1000)
a_norm <- qnorm(a_unif) # Convert uniform to standard normal

Then follow the same steps above, since this transformation method relies on the variables being multivariate normal.

内容的提问来源于stack exchange，提问作者rook1996