Using the Sixth Cumulant Correction to Find Valid Power Method Pdfs"
In SimMultiCorrData: Simulation of Correlated Data with Multiple Variable Types

#knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
knitr::opts_chunk$set(fig.width = 6, fig.height = 4.5)

Sometimes a target (continuous) distribution's standardized cumulants yield polynomial transformation constants that generate an invalid power method pdf. This is an important consideration when utilizing a simulated distribution to perform power analysis or hypothesis testing. In the case of Headrick's fifth-order approximation [-@Head2002], @HeadKow proposed that a correction can be used to increase the sixth cumulant's value in order to produce a valid pdf. This is discussed in the help pages for find_constants and the simulation functions (nonnormvar1, rcorrvar, and rcorrvar2). These functions allow the user to input a Six vector of correction values, and then choose the smallest value that generates a valid pdf. When Headrick generated his Table 1 (2002, p.691 - 692) containing the fifth-order polynomial transformation constants for some common symmetrical and asymmetrical theoretical densities, he did not discuss whether the constants generate valid power method pdfs. His 2007 paper with Kowalchuk explained how to verify if a given set of constants generates a valid pdf (for both Fleishman's third-order [-@Fleish] and Headrick's fifth-order method). These checks are implemented in pdf_check, which is called by find_constants.

Example

The following example illustrates the use of the sixth cumulant correction vector when finding the constants for each of Headrick's Table 1 distributions (given in Headrick.dist), and compares distributions generated with the corrections to those generated without the corrections. (Note that the printr @Printr package is invoked to display the tables.)

Step 1) Find the theoretical standardized cumulants for each distribution.

The parameters are stored in H_params. The order of the columns corresponds to the order of the columns in Headrick.dist. The columns of H_params are named as inputs for calc_theory. Please see the appropriate R help page concerning the parameterization of each distribution. We are rounding the calculated standardized cumulants to 10 decimal places to ensure that the symmetric distributions have $\Large \gamma_{1} = 0$ and $\Large \gamma_{3} = 0$.

library("SimMultiCorrData")
library("printr")
H_stcum <- matrix(1, nrow = 4, ncol = ncol(Headrick.dist))
for (i in 1:ncol(H_params)) {
  if (is.na(H_params[2, i])) {
    params <- H_params[1, i]
  } else {
    params <- as.numeric(H_params[, i])
  }
  H_stcum[, i] <- round(calc_theory(Dist = colnames(H_params)[i], 
                                    params = params)[3:6], 10)
}
colnames(H_stcum) <- colnames(Headrick.dist)
rownames(H_stcum) <- c("skew", "skurtosis", "fifth", "sixth")
round(H_stcum[, 1:6], 5)
round(H_stcum[, 7:12], 5)
round(H_stcum[, 13:18], 5)
round(H_stcum[, 19:22], 5)

Note that the standardized cumulants match those found by Headrick, except for the Gamma($\Large \alpha = 10,\ \beta = 10$) distribution. Either there is a mistake in Headrick's table, or he is using a different parameterization.

Step 2) Use the cumulants to find the constants for each distribution.

The sixth cumulant corrections will be chosen based on previous analysis in order to decrease computation time. If the user does not know what a sixth cumulant correction needs to be, a wide range with a small increment in values may be specified, i.e. Six = seq(0.1, 10, 0.1). In situations where the user has a better idea of the necessary correction, a smaller vector should be chosen. In the case of the Triangular distribution, the standardized kurtosis has been changed to the lower kurtosis boundary (found using calc_lower_skurt).

Six <- list(NULL, seq(1.7, 1.8, 0.01), seq(0.5, 2, 0.5), seq(25.1, 25.2, 0.01),
            seq(0.1, 0.3, 0.01), NULL, NULL, seq(0.5, 2, 0.5), 
            NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, 
            seq(0.01, 0.05, 0.01), seq(0.15, 0.2, 0.01), seq(0.5, 2, 0.5), 
            NULL, seq(0.5, 2, 0.5), seq(0.5, 2, 0.5))
H_consol <- list()

start.time <- Sys.time()
for (i in 1:ncol(H_stcum)) {
  skurtsH <- ifelse(colnames(H_stcum)[i] == "Triangular", -0.5856216, 
                    H_stcum[2, i])
  H_consol[[i]] <- find_constants(method = "Polynomial", skews = H_stcum[1, i],
                                  skurts = skurtsH, fifths = H_stcum[3, i],
                                  sixths = H_stcum[4, i], Six = Six[[i]])
}
stop.time <- Sys.time()
Time <- round(difftime(stop.time, start.time, units = "min"), 3)
cat("Total computation time:", Time, "minutes \n")

H_cons <- matrix(1, nrow = 7, ncol = ncol(Headrick.dist))
valid <- numeric(ncol(Headrick.dist))
for (i in 1:ncol(H_stcum)) {
  H_cons[1:6, i] <- H_consol[[i]]$constants
  H_cons[7, i] <- ifelse(is.null(H_consol[[i]]$SixCorr1), NA, 
                         H_consol[[i]]$SixCorr1)
  valid[i] <- H_consol[[i]]$valid
}
colnames(H_cons) <- colnames(Headrick.dist)
rownames(H_cons) <- c("c0", "c1", "c2", "c3", "c4", "c5", "sixcorr")

Step 3) Look at results to see which distributions still have invalid power method pdfs

colnames(H_cons)[valid == FALSE]

Therefore, the Uniform($\Large 0,\ 1$), Chisq($\Large df = 1$), Weibull($\Large \alpha = 6,\ \beta = 10$), Rayleigh($\Large \alpha = 0.5,\ \mu = \sqrt{0.5 * \pi}$), and Pareto($\Large \theta = 10,\ \alpha = 1$) distributions still have invalid power method pdf constants.

Step 4) Look at constants and sixth cumulant corrections

round(H_cons[, 1:6], 6)
round(H_cons[, 7:12], 6)
round(H_cons[, 13:18], 6)
round(H_cons[, 19:22], 6)

The Gaussian, t($\Large df = 10$), Chisq($\Large df = 2, ..., 32$), Beta($\Large \alpha = 4,\ \beta = 4$), Beta($\Large \alpha = 4,\ \beta = 2$), and Gamma($\Large \alpha = 10,\ \beta = 10$) distributions had valid power method pdfs with the original cumulants.
The Logistic($\Large 0, 1$), Triangular($\Large 0, 1$), t($\Large df = 7$), Beta($\Large \alpha = 4,\ \beta = 1.5$), and Beta($\Large \alpha = 4,\ \beta = 1.25$) required relatively small sixth cumulant corrections.
The Laplace($\Large 0, 1$) required the largest correction at 25.14.

Step 5) Simulate distributions

We will choose the Logistic($\Large 0, 1$) and Laplace($\Large 0, 1$) distributions for illustration.

First, simulate without the sixth cumulant correction.

seed <- 1234
Rey <- matrix(c(1, 0.4, 0.4, 1), 2, 2)

# Make sure Rey is within feasible correlation bounds
valid <- valid_corr(k_cont = 2, method = "Polynomial", 
                    means = rep(0, 2), vars = rep(1, 2), 
                    skews = H_stcum[1, c("Logistic", "Laplace")], 
                    skurts = H_stcum[2, c("Logistic", "Laplace")],
                    fifths = H_stcum[3, c("Logistic", "Laplace")], 
                    sixths = H_stcum[4, c("Logistic", "Laplace")], 
                    rho = Rey, seed = seed)

A <- rcorrvar(n = 10000, k_cont = 2, method = "Polynomial", 
              means = rep(0, 2), vars = rep(1, 2), 
              skews = H_stcum[1, c("Logistic", "Laplace")], 
              skurts = H_stcum[2, c("Logistic", "Laplace")], 
              fifths = H_stcum[3, c("Logistic", "Laplace")], 
              sixths = H_stcum[4, c("Logistic", "Laplace")], 
              rho = Rey, seed = seed)

Look at the maximum correlation error:

cat(paste("The maximum correlation error is ", round(A$maxerr, 5), ".", 
          sep = ""))

Look at the interquartile-range of correlation errors:

Acorr_error = round(A$correlations - Rey, 6)
cat(paste("The IQR of correlation errors is [", 
          round(quantile(as.numeric(Acorr_error), 0.25), 5), ", ",
          round(quantile(as.numeric(Acorr_error), 0.75), 5), "].", sep = ""))

Second, simulate with the sixth cumulant correction.

Six <- list(H_cons["sixcorr", "Logistic"], H_cons["sixcorr", "Laplace"])

# Make sure Rey is within feasible correlation bounds
valid2 <- valid_corr(k_cont = 2, method = "Polynomial", 
                     means = rep(0, 2), vars = rep(1, 2), 
                     skews = H_stcum[1, c("Logistic", "Laplace")], 
                     skurts = H_stcum[2, c("Logistic", "Laplace")],
                     fifths = H_stcum[3, c("Logistic", "Laplace")], 
                     sixths = H_stcum[4, c("Logistic", "Laplace")], 
                     Six = Six, rho = Rey, seed = seed)

B <- rcorrvar(n = 10000, k_cont = 2, method = "Polynomial", 
              means = rep(0, 2), vars = rep(1, 2), 
              skews = H_stcum[1, c("Logistic", "Laplace")], 
              skurts = H_stcum[2, c("Logistic", "Laplace")], 
              fifths = H_stcum[3, c("Logistic", "Laplace")], 
              sixths = H_stcum[4, c("Logistic", "Laplace")], Six = Six, 
              rho = Rey, seed = seed)

Look at the maximum correlation error:

cat(paste("The maximum correlation error is ", round(B$maxerr, 5), ".", 
          sep = ""))

Look at the interquartile-range of correlation errors:

Bcorr_error = round(B$correlations - Rey, 6)
cat(paste("The IQR of correlation errors is [", 
          round(quantile(as.numeric(Bcorr_error), 0.25), 5), ", ",
          round(quantile(as.numeric(Bcorr_error), 0.75), 5), "].", sep = ""))

In both cases, the correlation errors are small, indicating that the error loop does not need to be used.

Now compare the results numerically.

Target distributions:

as.matrix(round(A$summary_targetcont, 5), nrow = 2, ncol = 7, byrow = TRUE)

Without the sixth cumulant correction:

as.matrix(round(A$summary_continuous[, c("Distribution", "mean", "sd", "skew", 
                               "skurtosis", "fifth", "sixth")], 5), nrow = 2, 
          ncol = 7, byrow = TRUE)
A$valid.pdf

With the correction:

as.matrix(round(B$summary_continuous[, c("Distribution", "mean", "sd", "skew", 
                               "skurtosis", "fifth", "sixth")], 5), nrow = 2,
          ncol = 7, byrow = TRUE)
B$valid.pdf

The distributions simulated with the sixth cumulant corrections are closer to the target distributions.

Compare the results graphically.

Logistic Distribution:

plot_sim_pdf_theory(sim_y = A$continuous_variables[, 1], 
                    title = "Logistic Pdf", Dist = "Logistic", 
                    params = H_params$Logistic)
plot_sim_pdf_theory(sim_y = B$continuous_variables[, 1], 
                    title = "Corrected Logistic Pdf", Dist = "Logistic", 
                    params = H_params$Logistic)

Laplace Distribution:

plot_sim_pdf_theory(sim_y = A$continuous_variables[, 2], 
                    title = "Laplace Pdf", Dist = "Laplace", 
                    params = H_params$Laplace)
plot_sim_pdf_theory(sim_y = B$continuous_variables[, 2], 
                    title = "Corrected Laplace Pdf", Dist = "Laplace", 
                    params = H_params$Laplace)