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Overall Workflow for Data Simulation

Allison C Fialkowski

2018-06-28

A step-by-step guideline for data simulation is as follows:

1. Obtain the distribution parameters for the desired variables.

   1. Continuous variables: these are mean, variance, skewness and standardized kurtosis (kurtosis - 3) for Fleishman (1978)’s third-order method, plus standardized fifth and sixth cumulants for Headrick (2002)’s fifth-order method. If the goal is to simulate a theoretical distribution (i.e. Gamma, Beta, Logistic, etc.), these values can be obtained using calc_theory. If the goal is to mimic an empirical data set, these values can be found using calc_moments (using the method of moments) or calc_fisherk (using Fisher’s k-statistics). If the standardized cumulants are obtained from calc_theory, the user may need to use rounded values as inputs (i.e. skews = round(skews, 8)). Due to the nature of the integration involved in calc_theory, the results are approximations. Greater accuracy can be achieved by increasing the number of subdivisions (sub) used in the integration process. For example, in order to ensure that skew is exactly 0 for symmetric distributions. For some sets of cumulants, it is either not possible to find power method constants or the calculated constants do not generate valid power method pdfs. In these situations, adding a value to the sixth cumulant may provide solutions (see find_constants). If simulation results indicate that a continuous variable does not generate a valid pdf, the user can try find_constants with various sixth cumulant correction vectors to determine if a valid pdf can be found.
   2. Ordinal variables (\(\Large r \ge 2\) categories): these are the cumulative marginal probabilities and support values (if desired). The probabilities should be combined into a list of length equal to the number of ordinal variables. The i-th element is a vector of the cumulative probabilities defining the marginal distribution of the i-th variable. If the variable can take r values, the vector will contain r - 1 probabilities (the r-th is assumed to be 1). For binary variables, the probability should be the probability of achieving the \(\Large 1^{st}\) support value. The support values should be combined into a separate list. The i-th element is a vector containing the r ordered support values. If not provided, the default is for the i-th element to be the vector 1, …, r.
   3. Poisson variables: the lambda (mean) values should be given as a vector (see stats::dpois).
   4. Negative Binomial variables: the sizes (target number of successes) and either the success probabilities or the means should be given as vectors (see see stats::dnbinom). The variable represents the number of failures which occur in a sequence of Bernoulli trials before the target number of successes is achieved.

2. If continuous variables are desired, verify that the standardized kurtoses are greater than the lower kurtosis bounds. These bounds can be calculated using calc_lower_skurt, given the skewness (for method = “Fleishman”) and standardized fifth and sixth cumulants (for method = “Polynomial”, referring to Headrick’s method) for each variable. Different seeds should be examined to see if a lower boundary can be found. If a lower bound produces power method constants that yield an invalid pdf, the user may wish to provide a Skurt vector of kurtosis corrections. In this case, calc_lower_skurt will attempt to find the smallest value that produces a kurtosis which yields a valid power method pdf. In addition, if method = “Polynomial”, a sixth cumulant correction vector (Six) may be used to facilitate convergence of the root-solving algorithm. Since this step can take considerable computation time, the user may instead wish to perform this check after simulation if any of the variables have invalid power method pdfs.

3. Check if the target correlation matrix falls within the feasible correlation bounds, given the parameters for the desired distributions. The ordering of the variables in the correlation matrix MUST be 1st ordinal, 2nd continuous, 3rd Poisson, and 4th Negative Binomial. These bounds can be calculated using either valid_corr (correlation method 1) or valid_corr2 (correlation method 2). Note that falling within these bounds does not guarantee that the target correlation can be achieved. However, the check can alert the user to pairwise correlations that obviously fall outside the bounds.

4. Generate the variables using either correlation method 1 and rcorrvar or correlation method 2 and rcorrvar2. The user may want to try both to see which gives a better approximation to the variables and correlation matrix. The accuracy and simulation time will vary by situation. Again, the ordering of the variables in the correlation matrix MUST be 1st ordinal, 2nd continuous, 3rd Poisson, and 4th Negative Binomial. In addition, the error loop can minimize the correlation errors in most situations.

5. Summarize the results numerically. The functions rcorrvar and rcorrvar2 provide data.frames containing summaries by variable type and the maximum error between the final and target correlation matrices. Additional summary functions include: sim_cdf_prob (to calculate a cumulative probability up to a given continuous y value), power_norm_corr (to calculate the correlation between a continuous variable and the generating standard normal variable), stats_pdf (to calculate the 100 * alpha percent symmetric trimmed mean, median, mode, and maximum height of a valid power method pdf).

6. Summarize the results graphically. Comparing the simulated data to the target distribution demonstrates simulation accuracy. The graphing functions provided in this package can be used to display simulated data values, pdfs, or cdfs. The target distributions (either by theoretical distribution name or given an empirical data set) can be added to the data value or pdf plots. Cumulative probabilities can be added to the cdf plots (for continuous variables).

library("SimMultiCorrData")
library("printr")

# Turn off scientific notation
options(scipen = 999)

# Set seed and sample size
seed <- 11
n <- 10000

# Continuous Distributions
Dist <- c("Gaussian", "Chisq", "Beta")

# Calculate standardized cumulants
# Those for the normal distribution are rounded to ensure the correct values 
# are obtained.
M1 <- round(calc_theory(Dist = "Gaussian", params = c(0, 1)), 8)
M2 <- calc_theory(Dist = "Chisq", params = 4)
M3 <- calc_theory(Dist = "Beta", params = c(4, 2))
M <- cbind(M1, M2, M3)

# Binary and Ordinal Distributions
marginal <- list(c(0.3, 0.75), c(0.2, 0.5, 0.9))
support <- list() # default support will be generated inside simulation

# Poisson Distributions
lam <- c(1, 5, 10)

# Negative Binomial Distributions
size <- c(3, 6)
prob <- c(0.2, 0.8)

ncat <- length(marginal)
ncont <- ncol(M)
npois <- length(lam)
nnb <- length(size)

# Create correlation matrix from a uniform distribution (0.2, 0.7)
set.seed(seed)
Rey <- diag(1, nrow = (ncat + ncont + npois + nnb))
for (i in 1:nrow(Rey)) {
  for (j in 1:ncol(Rey)) {
    if (i > j) Rey[i, j] <- runif(1, 0.2, 0.7)
    Rey[j, i] <- Rey[i, j]
  }
}

# Check to see if Rey is positive-definite
min(eigen(Rey, symmetric = TRUE)$values) < 0

## [1] FALSE

Lower <- list()

# list of standardized kurtosis values to add in case only invalid power 
#     method pdfs are produced
Skurt <- list(seq(0.5, 2, 0.5), seq(0.02, 0.05, 0.01), seq(0.02, 0.05, 0.01))

start.time <- Sys.time()
for (i in 1:ncol(M)) {
  Lower[[i]] <- calc_lower_skurt(method = "Polynomial", skews = M[3, i], 
                                 fifths = M[5, i], sixths = M[6, i], 
                                 Skurt = Skurt[[i]], seed = 104)
}

## Only invalid pdf constants could be found.
##                Try more starting values (increase n)
##                or a different seed or Skurt vector.
##                Also verify standardized cumulant values.

stop.time <- Sys.time()
Time <- round(difftime(stop.time, start.time, units = "min"), 3)
cat("Total computation time:", Time, "minutes \n")

## Total computation time: 0.703 minutes

# Note the message given for Distribution 1 (Normal).

as.matrix(Lower[[1]]$Min[1, c("skew", "fifth", "sixth", "valid.pdf", 
                              "skurtosis")], 
          nrow = 1, ncol = 5, byrow = TRUE) 

as.matrix(Lower[[2]]$Min[1, c("skew", "fifth", "sixth", "valid.pdf", 
                              "skurtosis")], 
          nrow = 1, ncol = 5, byrow = TRUE) 

Lower[[2]]$SkurtCorr1

## [1] 0.03

as.matrix(Lower[[3]]$Min[1, c("skew", "fifth", "sixth", "valid.pdf", 
                              "skurtosis")], 
          nrow = 1, ncol = 5, byrow = TRUE) 

Lower[[3]]$SkurtCorr1

## [1] 0.02

# Make sure Rey is within upper and lower correlation limits
valid <- valid_corr(k_cat = ncat, k_cont = ncont, k_pois = npois,
                    k_nb = nnb, method = "Polynomial", means =  M[1, ],
                    vars =  (M[2, ])^2, skews = M[3, ], skurts = M[4, ],
                    fifths = M[5, ], sixths = M[6, ], marginal = marginal, 
                    lam = lam, size = size, prob = prob, rho = Rey, 
                    seed = seed)

## 
##  Constants: Distribution  1  
## 
##  Constants: Distribution  2  
## 
##  Constants: Distribution  3  
## All correlations are in feasible range!

A <- rcorrvar(n = 10000, k_cont = ncont, k_cat = ncat, k_pois = npois,
              k_nb = nnb, method = "Polynomial", means =  M[1, ], 
              vars =  (M[2, ])^2, skews = M[3, ], skurts = M[4, ], 
              fifths = M[5, ], sixths = M[6, ], marginal = marginal,
              lam = lam, size = size, prob = prob, rho = Rey, seed = seed)

## 
##  Constants: Distribution  1  
## 
##  Constants: Distribution  2  
## 
##  Constants: Distribution  3  
## 
## Constants calculation time: 0 minutes 
## Intercorrelation calculation time: 0.114 minutes 
## Error loop calculation time: 0 minutes 
## Total Simulation time: 0.115 minutes

Acorr_error = round(A$correlations - Rey, 6)
summary(as.numeric(Acorr_error))

B <- rcorrvar(n = 10000, k_cont = ncont, k_cat = ncat, k_pois = npois,
              k_nb = nnb, method = "Polynomial", means =  M[1, ], 
              vars =  (M[2, ])^2, skews = M[3, ], skurts = M[4, ], 
              fifths = M[5, ], sixths = M[6, ], marginal = marginal,
              lam = lam, size = size, prob = prob, rho = Rey, seed = seed, 
              errorloop = TRUE)

## 
##  Constants: Distribution  1  
## 
##  Constants: Distribution  2  
## 
##  Constants: Distribution  3  
## 
## Constants calculation time: 0 minutes 
## Intercorrelation calculation time: 0.109 minutes 
## Error loop calculation time: 0.265 minutes 
## Total Simulation time: 0.375 minutes

Bcorr_error = round(B$correlations - Rey, 6)
summary(as.numeric(Bcorr_error))

knitr::kable(B$summary_ordinal[[1]], caption = "Variable 1")

knitr::kable(B$summary_ordinal[[2]], caption = "Variable 2")

as.matrix(B$summary_Poisson[, c(1, 3:6, 8:9)], nrow = 3, ncol = 7, 
          byrow = TRUE)

as.matrix(B$summary_Neg_Bin[, c(1, 3:7, 9:10)], nrow = 2, ncol = 8, 
          byrow = TRUE)

as.matrix(round(B$constants, 6), nrow = 3, ncol = 6, byrow = TRUE)

as.matrix(round(B$summary_targetcont, 5), nrow = 3, ncol = 7, byrow = TRUE)

as.matrix(round(B$summary_continuous[, c("Distribution", "mean", "sd", 
                                         "skew", "skurtosis", "fifth", 
                                         "sixth")], 5), nrow = 3, ncol = 7, 
          byrow = TRUE)

B$valid.pdf

## [1] "TRUE" "TRUE" "TRUE"

as.matrix(t(round(stats_pdf(c = B$constants[1, ], method = "Polynomial", 
                            alpha = 0.025), 4)))

as.matrix(t(round(stats_pdf(c = B$constants[2, ], method = "Polynomial", 
                            alpha = 0.025), 4)))

as.matrix(t(round(stats_pdf(c = B$constants[3, ], method = "Polynomial", 
                            alpha = 0.025), 4)))

plot_sim_cdf(B$continuous_variables[, 2], calc_cprob = TRUE, delta = 10)

plot_sim_pdf_theory(B$continuous_variables[, 2], Dist = "Chisq", params = 4)

plot_sim_cdf(B$ordinal_variables[, 2])

plot_sim_theory(B$Poisson_variables[, 2], cont_var = FALSE, Dist = "Poisson", 
                params = 5)

plot_sim_pdf_theory(B$Poisson_variables[, 2], cont_var = FALSE, 
                    Dist = "Poisson", params = 5)

plot_sim_theory(B$Neg_Bin_variables[, 1], cont_var = FALSE, 
                Dist = "Negative_Binomial", params = c(3, 0.2))

plot_sim_pdf_theory(B$Neg_Bin_variables[, 1], cont_var = FALSE, 
                Dist = "Negative_Binomial", params = c(3, 0.2))

pois_eps <- rep(0.0001, npois)
nb_eps <- rep(0.0001, nnb)

# Make sure Rey is within upper and lower correlation limits
valid2 <- valid_corr2(k_cat = ncat, k_cont = ncont, k_pois = npois,
                      k_nb = nnb, method = "Polynomial", means =  M[1, ],
                      vars =  (M[2, ])^2, skews = M[3, ], skurts = M[4, ],
                      fifths = M[5, ], sixths = M[6, ], marginal = marginal, 
                      lam = lam, pois_eps = pois_eps, size = size, 
                      prob = prob, nb_eps = nb_eps, rho = Rey, seed = seed)

## 
##  Constants: Distribution  1  
## 
##  Constants: Distribution  2  
## 
##  Constants: Distribution  3  
## All correlations are in feasible range!

C <- rcorrvar2(n = 10000, k_cont = ncont, k_cat = ncat, k_pois = npois,
               k_nb = nnb, method = "Polynomial", means =  M[1, ], 
               vars =  (M[2, ])^2, skews = M[3, ], skurts = M[4, ], 
               fifths = M[5, ], sixths = M[6, ], marginal = marginal, 
               lam = lam, pois_eps = pois_eps, size = size, prob = prob, 
               nb_eps = nb_eps, rho = Rey, seed = seed)

## 
##  Constants: Distribution  1  
## 
##  Constants: Distribution  2  
## 
##  Constants: Distribution  3  
## 
## Constants calculation time: 0 minutes 
## Intercorrelation calculation time: 0.044 minutes 
## Error loop calculation time: 0 minutes 
## Total Simulation time: 0.046 minutes

Ccorr_error = round(C$correlations - Rey, 6)
summary(as.numeric(Ccorr_error))

D <- rcorrvar2(n = 10000, k_cont = ncont, k_cat = ncat, k_pois = npois,
               k_nb = nnb, method = "Polynomial", means =  M[1, ], 
               vars =  (M[2, ])^2, skews = M[3, ], skurts = M[4, ], 
               fifths = M[5, ], sixths = M[6, ], marginal = marginal, 
               lam = lam, pois_eps = pois_eps, size = size, prob = prob, 
               nb_eps = nb_eps, rho = Rey, seed = seed, errorloop = TRUE)

## 
##  Constants: Distribution  1  
## 
##  Constants: Distribution  2  
## 
##  Constants: Distribution  3  
## 
## Constants calculation time: 0 minutes 
## Intercorrelation calculation time: 0.045 minutes 
## Error loop calculation time: 0.145 minutes 
## Total Simulation time: 0.191 minutes

Dcorr_error = round(D$correlations - Rey, 6)
summary(as.numeric(Dcorr_error))

knitr::kable(D$summary_ordinal[[1]], caption = "Variable 1")

knitr::kable(D$summary_ordinal[[2]], caption = "Variable 2")

as.matrix(D$summary_Poisson[, c(1, 3:6, 8:9)], nrow = 3, ncol = 7, 
          byrow = TRUE)

as.matrix(D$summary_Neg_Bin[, c(1, 3:7, 9:10)], nrow = 2, ncol = 8, 
          byrow = TRUE)

as.matrix(round(D$summary_targetcont, 5), nrow = 3, ncol = 7, byrow = TRUE)

as.matrix(round(D$summary_continuous[, c("Distribution", "mean", "sd", 
                                         "skew", "skurtosis", "fifth", 
                                         "sixth")], 5), nrow = 3, ncol = 7, 
          byrow = TRUE)

D$valid.pdf

## [1] "TRUE" "TRUE" "TRUE"

as.matrix(t(round(stats_pdf(c = D$constants[1, ], method = "Polynomial", 
                            alpha = 0.025), 4)))

as.matrix(t(round(stats_pdf(c = B$constants[2, ], method = "Polynomial", 
                            alpha = 0.025), 4)))

as.matrix(t(round(stats_pdf(c = B$constants[3, ], method = "Polynomial", 
                            alpha = 0.025), 4)))

plot_sim_theory(D$Poisson_variables[, 2], cont_var = FALSE, Dist = "Poisson", 
                params = 5)

plot_sim_pdf_theory(D$Poisson_variables[, 2], cont_var = FALSE, 
                    Dist = "Poisson", params = 5)

plot_sim_theory(D$Neg_Bin_variables[, 1], cont_var = FALSE, 
                Dist = "Negative_Binomial", params = c(3, 0.2))

plot_sim_pdf_theory(D$Neg_Bin_variables[, 1], cont_var = FALSE, 
                    Dist = "Negative_Binomial", params = c(3, 0.2))

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