SimMultiCorrData (version 0.2.2)

plot_cdf: Plot Theoretical Power Method Cumulative Distribution Function for Continuous Variables

Description

This plots the theoretical power method cumulative distribution function: $$F_p(Z)(p(z)) = F_p(Z)(p(z), F_Z(z)),$$ as given in Headrick & Kowalchuk (2007, 10.1080/10629360600605065). It is a parametric plot with \(sigma * y + mu\), where \(y = p(z)\), on the x-axis and \(F_Z(z)\) on the y-axis, where \(z\) is vector of \(n\) random standard normal numbers (generated with a seed set by user). Given a vector of polynomial transformation constants, the function generates \(sigma * y + mu\) and calculates the theoretical cumulative probabilities using \(F_p(Z)(p(z), F_Z(z))\). If calc_cprob = TRUE, the cumulative probability up to \(delta = sigma * y + mu\) is calculated (see cdf_prob) and the region on the plot is filled with a dashed horizontal line drawn at \(F_p(Z)(delta)\). The cumulative probability is stated on top of the line. It returns a ggplot2-package object so the user can modify as necessary. The graph parameters (i.e. title, color, fill, hline) are ggplot2-package parameters. It works for valid or invalid power method pdfs.

Usage

plot_cdf(c = NULL, method = c("Fleishman", "Polynomial"), mu = 0,
  sigma = 1, title = "Cumulative Distribution Function", ylower = NULL,
  yupper = NULL, calc_cprob = FALSE, delta = 5, color = "dark blue",
  fill = "blue", hline = "dark green", n = 10000, seed = 1234,
  text.size = 11, title.text.size = 15, axis.text.size = 10,
  axis.title.size = 13, lower = -1000000, upper = 1000000)

Arguments

c

a vector of constants c0, c1, c2, c3 (if method = "Fleishman") or c0, c1, c2, c3, c4, c5 (if method = "Polynomial"), like that returned by find_constants

method

the method used to generate the continuous variable \(y = p(z)\). "Fleishman" uses Fleishman's third-order polynomial transformation and "Polynomial" uses Headrick's fifth-order transformation.

mu

mean for the continuous variable (default = 0)

sigma

standard deviation for the continuous variable (default = 1)

title

the title for the graph (default = "Cumulative Distribution Function")

ylower

the lower y value to use in the plot (default = NULL, uses minimum simulated y value)

yupper

the upper y value (default = NULL, uses maximum simulated y value)

calc_cprob

if TRUE (default = FALSE), cdf_prob is used to find the cumulative probability up to \(delta = sigma * y + mu\) and the region on the plot is filled with a dashed horizontal line drawn at \(F_p(Z)(delta)\)

delta

the value \(sigma * y + mu\), where \(y = p(z)\), at which to evaluate the cumulative probability

color

the line color for the cdf (default = "dark blue")

fill

the fill color if calc_cprob = TRUE (default = "blue)

hline

the dashed horizontal line color drawn at delta if calc_cprob = TRUE (default = "dark green")

n

the number of random standard normal numbers to use in generating \(y = p(z)\) (default = 10000)

seed

the seed value for random number generation (default = 1234)

text.size

the size of the text displaying the cumulative probability up to delta if calc_cprob = TRUE

title.text.size

the size of the plot title

axis.text.size

the size of the axes text (tick labels)

axis.title.size

the size of the axes titles

lower

lower bound for cdf_prob

upper

upper bound for cdf_prob

Value

A ggplot2-package object.

References

Fleishman AI (1978). A Method for Simulating Non-normal Distributions. Psychometrika, 43, 521-532. 10.1007/BF02293811.

Headrick TC (2002). Fast Fifth-order Polynomial Transforms for Generating Univariate and Multivariate Non-normal Distributions. Computational Statistics & Data Analysis, 40(4):685-711. 10.1016/S0167-9473(02)00072-5. (ScienceDirect)

Headrick TC (2004). On Polynomial Transformations for Simulating Multivariate Nonnormal Distributions. Journal of Modern Applied Statistical Methods, 3(1), 65-71. 10.22237/jmasm/1083370080.

Headrick TC, Kowalchuk RK (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, 229-249. 10.1080/10629360600605065.

Headrick TC, Sawilowsky SS (1999). Simulating Correlated Non-normal Distributions: Extending the Fleishman Power Method. Psychometrika, 64, 25-35. 10.1007/BF02294317.

Headrick TC, Sheng Y, & Hodis FA (2007). Numerical Computing and Graphics for the Power Method Transformation Using Mathematica. Journal of Statistical Software, 19(3), 1 - 17. 10.18637/jss.v019.i03.

Wickham H. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2009.

See Also

find_constants, cdf_prob, ggplot2-package, geom_path, geom_abline, geom_ribbon

Examples

# NOT RUN {
# Logistic Distribution: mean = 0, sigma = 1

# Find standardized cumulants
stcum <- calc_theory(Dist = "Logistic", params = c(0, 1))

# Find constants without the sixth cumulant correction
# (invalid power method pdf)
con1 <- find_constants(method = "Polynomial", skews = stcum[3],
                      skurts = stcum[4], fifths = stcum[5],
                      sixths = stcum[6], n = 25, seed = 1234)

# Plot cdf with cumulative probability calculated up to delta = 5
plot_cdf(c = con1$constants, method = "Polynomial",
         title = "Invalid Logistic CDF", calc_cprob = TRUE, delta = 5)

# Find constants with the sixth cumulant correction
# (valid power method pdf)
con2 <- find_constants(method = "Polynomial", skews = stcum[3],
                      skurts = stcum[4], fifths = stcum[5],
                      sixths = stcum[6], Six = seq(1.5, 2, 0.05))

# Plot cdf with cumulative probability calculated up to delta = 5
plot_cdf(c = con2$constants, method = "Polynomial",
         title = "Valid Logistic CDF", calc_cprob = TRUE, delta = 5)
# }
# NOT RUN {
# }