Author: Allison C Fialkowski
Date: 2018-06-28

# Using the Sixth Cumulant Correction to Find Valid Power Method Pdfs

#### Allison C Fialkowski

#### 2018-06-28

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 (2002), Headrick and Kowalchuk (2007) 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 (1978) 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` Xie (2017) 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)

| Gaussian | Logistic | Uniform | Laplace | Triangular | t_7df
--- | --- | --- | --- | --- | --- | ---
skew | 0 | 0.00000 | 0.00000 | 0 | 0.00000 | 0
skurtosis | 0 | 1.20000 | -1.20000 | 3 | -0.60000 | 2
fifth | 0 | 0.00000 | 0.00000 | 0 | 0.00000 | 0
sixth | 0 | 6.85714 | 6.85714 | 30 | 1.71424 | 80

    round(H_stcum[, 7:12], 5)

| t_10df | Chisq_1df | Chisq_2df | Chisq_3df | Chisq_4df | Chisq_8df
--- | --- | --- | --- | --- | --- | ---
skew | 0 | 2.82843 | 2 | 1.63299 | 1.41421 | 1.0
skurtosis | 1 | 12.00000 | 6 | 4.00000 | 3.00000 | 1.5
fifth | 0 | 67.88274 | 24 | 13.06394 | 8.48528 | 3.0
sixth | 10 | 479.99841 | 120 | 53.33333 | 30.00000 | 7.5

    round(H_stcum[, 13:18], 5)

| Chisq_16df | Chisq_32df | Beta_a4b4 | Beta_a4b2 | Beta_a4b1.5 | Beta_a4b1.25
--- | --- | --- | --- | --- | --- | ---
skew | 0.70711 | 0.50000 | 0.00000 | -0.46771 | -0.69388 | -0.84815
skurtosis | 0.75000 | 0.37500 | -0.54545 | -0.37500 | -0.06860 | 0.22105
fifth | 1.06066 | 0.37500 | 0.00000 | 1.40312 | 1.82825 | 1.90702
sixth | 1.87500 | 0.46875 | 1.67832 | -0.42614 | -3.37911 | -5.82705

    round(H_stcum[, 19:22], 5)

| Weibull_a6b10 | Gamma_a10b10 | Rayleigh_a0.5msqrt0.5pi | Pareto_t10a1
--- | --- | --- | --- | ---
skew | -0.37326 | 0.63246 | 0.63111 | 2.81106
skurtosis | 0.03546 | 0.60000 | 0.24509 | 14.82857
fifth | 0.44706 | 0.75895 | -0.31314 | 130.20815
sixth | -1.02207 | 1.20000 | -0.86829 | 1808.89959

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")

    ## Total computation time: 0.019 minutes

    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]

    ## [1] "Uniform"                 "Chisq_1df"
    ## [3] "Weibull_a6b10"           "Rayleigh_a0.5msqrt0.5pi"
    ## [5] "Pareto_t10a1"

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)

| Gaussian | Logistic | Uniform | Laplace | Triangular | t_7df
--- | --- | --- | --- | --- | --- | ---
c0 | 0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000
c1 | 1 | 0.887926 | 1.347438 | 0.782367 | 1.100185 | 0.907394
c2 | 0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000
c3 | 0 | 0.036052 | -0.140177 | 0.067899 | -0.037538 | 0.014980
c4 | 0 | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 0.000000
c5 | 0 | 0.000001 | 0.001808 | 0.000000 | 0.000632 | 0.002780
sixcorr | NA | 1.750000 | 2.000000 | 25.140000 | 0.160000 | NA

    round(H_cons[, 7:12], 6)

| t_10df | Chisq_1df | Chisq_2df | Chisq_3df | Chisq_4df | Chisq_8df
--- | --- | --- | --- | --- | --- | ---
c0 | 0.000000 | -0.397713 | -0.307740 | -0.259037 | -0.227508 | -0.163968
c1 | 0.920482 | 0.621096 | 0.800560 | 0.867102 | 0.900716 | 0.950794
c2 | 0.000000 | 0.416871 | 0.318764 | 0.265362 | 0.231610 | 0.165391
c3 | 0.021453 | 0.068429 | 0.033500 | 0.021276 | 0.015466 | 0.007345
c4 | 0.000000 | -0.006386 | -0.003675 | -0.002108 | -0.001367 | -0.000474
c5 | 0.000830 | 0.000043 | 0.000159 | 0.000092 | 0.000055 | 0.000014
sixcorr | NA | 2.000000 | NA | NA | NA | NA

    round(H_cons[, 13:18], 6)

| Chisq_16df | Chisq_32df | Beta_a4b4 | Beta_a4b2 | Beta_a4b1.5 | Beta_a4b1.25
--- | --- | --- | --- | --- | --- | ---
c0 | -0.116936 | -0.083017 | 0.000000 | 0.108304 | 0.162966 | 0.200396
c1 | 0.975534 | 0.987806 | 1.093437 | 1.104253 | 1.089984 | 1.075051
c2 | 0.117433 | 0.083192 | 0.000000 | -0.123347 | -0.187329 | -0.231504
c3 | 0.003573 | 0.001762 | -0.035711 | -0.045284 | -0.044950 | -0.044208
c4 | -0.000166 | -0.000058 | 0.000000 | 0.005014 | 0.008121 | 0.010369
c5 | 0.000004 | 0.000001 | 0.000752 | 0.001285 | 0.001445 | 0.001640
sixcorr | NA | NA | NA | NA | 0.030000 | 0.180000

    round(H_cons[, 19:22], 6)

| Weibull_a6b10 | Gamma_a10b10 | Rayleigh_a0.5msqrt0.5pi | Pareto_t10a1
--- | --- | --- | --- | ---
c0 | 0.065524 | -0.104760 | -0.107593 | -0.345859
c1 | 0.969217 | 0.980451 | 1.022480 | 0.712411
c2 | -0.065172 | 0.105115 | 0.091123 | 0.346926
c3 | 0.027783 | 0.002843 | -0.003307 | 0.028078
c4 | -0.000117 | -0.000118 | 0.005490 | -0.000356
c5 | -0.003879 | 0.000002 | -0.002025 | 0.003962
sixcorr | 2.000000 | NA | 2.000000 | 2.000000

1. 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.

2. 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.

3. 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)

    ##
    ##  Constants: Distribution  1
    ##
    ##  Constants: Distribution  2
    ## All correlations are in feasible range!

    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)

    ##
    ##  Constants: Distribution  1
    ##
    ##  Constants: Distribution  2
    ##
    ## Constants calculation time: 0 minutes
    ## Intercorrelation calculation time: 0 minutes
    ## Error loop calculation time: 0 minutes
    ## Total Simulation time: 0 minutes

Look at the maximum correlation error:

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

    ## The maximum correlation error is 0.00149.

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 = ""))

    ## The IQR of correlation errors is [-0.00149, 0].

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)

    ##
    ##  Constants: Distribution  1
    ##
    ##  Constants: Distribution  2
    ## All correlations are in feasible range!

    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)

    ##
    ##  Constants: Distribution  1
    ##
    ##  Constants: Distribution  2
    ##
    ## Constants calculation time: 0 minutes
    ## Intercorrelation calculation time: 0 minutes
    ## Error loop calculation time: 0 minutes
    ## Total Simulation time: 0.001 minutes

Look at the maximum correlation error:

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

    ## The maximum correlation error is 0.00101.

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 = ""))

    ## The IQR of correlation errors is [-0.00101, 0].

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)

| Distribution | mean | sd | skew | skurtosis | fifth | sixth
--- | --- | --- | --- | --- | --- | --- | ---
Logistic | 1 | 0 | 1 | 0 | 1.2 | 0 | 6.85714
Laplace | 2 | 0 | 1 | 0 | 3.0 | 0 | 30.00000

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)

| Distribution | mean | sd | skew | skurtosis | fifth | sixth
--- | --- | --- | --- | --- | --- | --- | ---
X1 | 1 | -0.00043 | 0.99694 | -0.01936 | 1.03693 | 0.13868 | 5.22781
X2 | 2 | 0.00141 | 0.99907 | 0.06174 | 2.86625 | 1.36530 | 22.26331

    A$valid.pdf

    ## [1] "FALSE" "FALSE"

With the correction:

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

| Distribution | mean | sd | skew | skurtosis | fifth | sixth
--- | --- | --- | --- | --- | --- | --- | ---
X1 | 1 | -0.00042 | 0.99694 | -0.01820 | 1.02581 | 0.14315 | 6.01048
X2 | 2 | 0.00133 | 0.99847 | 0.06998 | 2.68915 | 1.91011 | 27.97101

    B$valid.pdf

    ## [1] "TRUE" "TRUE"

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)

## References

Fleishman, A I. 1978. “A Method for Simulating Non-Normal Distributions.” Psychometrika 43: 521–32. doi:10.1007/BF02293811.

Headrick, T C. 2002. “Fast Fifth-Order Polynomial Transforms for Generating Univariate and Multivariate Non-Normal Distributions.” Computational Statistics and Data Analysis 40 (4): 685–711. doi:10.1016/S0167-9473(02)00072-5.

Headrick, T C, and R K Kowalchuk. 2007. “The Power Method Transformation: Its Probability Density Function, Distribution Function, and Its Further Use for Fitting Data.” Journal of Statistical Computation and Simulation 77: 229–49. doi:10.1080/10629360600605065.

Xie, Yihui. 2017. Printr: Automatically Print R Objects to Appropriate Formats According to the ’Knitr’ Output Format. https://CRAN.R-project.org/package=printr.