Simulate data using Fleishman transformation — eda_sim

Generates random data with the specified skewness and excess kurtosis using the Fleishman transformation method.

Usage

eda_sim_old(
  n = 1,
  skew = 0,
  kurt = 0,
  check = TRUE,
  coefout = FALSE,
  coefin = NULL
)

Arguments

n: An integer specifying the number of random data points to generate.
skew: A numeric value specifying the desired skewness of the simulated data.
kurt: A numeric value specifying the desired excess kurtosis of the simulated data.
check: Boolean determining if the combination of skewness and kurtosis are valid.
coefout: Boolean determining if the Fleishman coefficients should be outputted instead of the simulated values.
coefin: Vector of the four coefficients to be used in Fleishman's equation. This bypasses the need to solve for the parameters.

Value

A numeric vector of simulated data points.

Details

The function uses Fleishman's polynomial transformation of the form:

$$Y = a + bX + cX^2 + dX^3$$

where a, b, c, and d are coefficients determined to approximate the specified skewness and excess kurtosis, and X is a standard normal variable. The coefficients are solved using a numerical optimization approach based on minimizing the residuals of Fleishman's equations. An excess kurtosis is defined as the kurtosis of a Normal distribution (k=3) minus 3.

The function is valid for a skewness range of -3 to 3 and an excess kurtosis greater than -1.13168 + 1.58837 * skew ^ 2. If check = TRUE, the function will warn the user if an invalid combination of skewness and kurtosis are passed to the function. Deviation from the recommended combination will result in a distribution that may not reflect the desired skewness and kurtosis values.

If the proper combination of skewness and kurtosis parameters are passed to the function, the output distribution will have a mean of around 0 and a variance of around 1. But note that a strongly skewed distribution will require a large n to reflect the desired properties due to the disproportionate influence of the tail's extreme values on the various moments of the distribution, particularly higher-order moments like skewness and kurtosis.

Fleishman, A. I. (1978). A method for simulating non-normal distributions. Psychometrika, 43, 521–532.
Wicklin, R. (2013). Simulating Data with SAS (Appendix D: Functions for Simulating Data by Using Fleishman’s Transformation). Cary, NC: SAS Institute Inc. Retrieved from https://tinyurl.com/4tustnph

Examples


# Generate a normal distribution
set.seed(321)
x <- eda_sim_old(1000, skew = 0, kurt = 0)
#> Skew/kurtosis combination is valid.
eda_theo(x) # Check for normality


# Simulate distribution with skewness = 1.15 and kurtosis = 2
# A larger sample size is more likely to reflect the desired parameters
set.seed(653)
x <- eda_sim_old(500000, skew = 1.15, kurt = 2)
#> Skew/kurtosis combination is valid.

# Verify skewness and excess kurtosis of the simulated data
# Mean and variance should be close to 0 and 1 respectively
eda_moments(x)
#>             n          mean           var          skew          kurt 
#>  5.000000e+05 -4.149649e-04  9.995423e-01  1.163226e+00  2.082610e+00 

# Visualize the simulated data
hist(x, breaks = 30, main = "Simulated Data", xlab = "Value")