trapezoid (version 1.0-0)

Trapezoid: The Trapezoidal Distribution

Description

Density function, distribution function, quantile function, and random generation for the trapezoidal distribution with minimum equal to min, lower mode equal to mode1, upper mode equal to mode2, and maximum equal to max. For the generalized trapezoidal distribution, n1, n3, and alpha may optionally be specified.

Usage

dtrapezoid(x, min=0, mode1=1/3, mode2=2/3, max=1, n1=2, n3=2, alpha=1, log=FALSE)
ptrapezoid(q, min=0, mode1=1/3, mode2=2/3, max=1, n1=2, n3=2, alpha=1, lower.tail=TRUE, log.p=FALSE)
qtrapezoid(p, min=0, mode1=1/3, mode2=2/3, max=1, n1=2, n3=2, alpha=1, lower.tail=TRUE, log.p=FALSE)
rtrapezoid(n, min=0, mode1=1/3, mode2=2/3, max=1, n1=2, n3=2, alpha=1)

Arguments

x,q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) $>$ 1, the length is taken to be the number required.

min

vector of minima.

mode1

vector of lower modes.

mode2

vector of upper modes.

max

vector of maxima.

n1

vector of growth parameters.

n3

vector of decay parameters.

alpha

vector of boundary ratio parameters.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X <= x]<="" samp="">, otherwise, P[X > x].

Value

• dtrapezoid gives the density function, ptrapezoid gives the distribution function, qtrapezoid gives the quantile function, and rtrapezoid generates random deviates.

Details

The generalized trapezoidal distribution is described by van Dorp and Kotz (2003) and van Dorp and colleagues (2007). With n1, n3, and alpha equal to the default values of 2, 2, and 1, respectively, the distribution shape is of a quadrilateral trapezoid. Altering n1, n3, or alpha changes the growth, decay, and boundary ratio parameters, respectively, as demonstrated in the examples below.

References

van Dorp, J. R. and Kotz, S. (2003) Generalized trapezoidal distributions. Metrika. 58(1):85--97. Preprint available: http://www.seas.gwu.edu/~dorpjr/Publications/JournalPapers/Metrika2003VanDorp.pdf van Dorp, J. R., Rambaud, S.C., Perez, J. G., and Pleguezuelo, R. H. (2007) An elicitation procedure for the generalized trapezoidal distribution with a uniform central stage. Decision Analysis Journal. 4:156--166. Preprint available: http://www.seas.gwu.edu/~dorpjr/Publications/JournalPapers/DA2007.pdf

See Also

Distributions for standard distributions.

Examples

## Plot default trapezoid distribution
curve(dtrapezoid(x, min=0, mode1=1/3, mode2=2/3, max=1, n1=2, n3=2, alpha=1),
	from=0, to=1)


## Plot triangular trapezoid distribution
curve(dtrapezoid(x, min=0, mode1=1/2, mode2=1/2, max=1, n1=2, n3=2, alpha=1),
	from=0, to=1)


## Explore effects of n1, n3, and alpha parameters
# ggplot2 required for this example
require(ggplot2)

x <- seq(from = 0, to = 1, by = 0.01)	

# Create a lists of arguments, varying n1, n3, and alpha
arguments <- list()
arguments[['A']] <- list(x = x, n1 = 2, n3 = 2, alpha = 0.8)
arguments[['B']] <- list(x = x, n1 = 1.5, n3 = 1.5, alpha = 1)
arguments[['C']] <- list(x = x, n1 = 2.5, n3 = 2.5, alpha = 1.5)
arguments[['D']] <- list(x = x, n1 = 1.5, n3 = 2.5, alpha = 0.5)
arguments[['E']] <- list(x = x, n1 = 2.5, n3 = 1.5, alpha = 1)
arguments[['F']] <- list(x = x, n1 = 0.5, n3 = 0.5, alpha = 1.5)
arguments[['G']] <- list(x = x, n1 = 1.5, n3 = 0.5, alpha = 0.5)
arguments[['H']] <- list(x = x, n1 = 2.5, n3 = 0.5, alpha = 1)
arguments[['I']] <- list(x = x, n1 = 0.5, n3 = 1.5, alpha = 1.5)
arguments[['J']] <- list(x = x, n1 = 0.5, n3 = 2.5, alpha = 0.5)

# Calculate the distributions
plot.data <- ldply(arguments, function(z)
	{
		x <- z$x
		density <- dtrapezoid(x = z$x, min = 0, mode1 = 0.2, mode2 = 0.8,
			max = 1, n1 = z$n1, n3 = z$n3, alpha = z$alpha)
		args <- paste("n1 = ", z$n1, ", n3 = ", z$n3, ", alpha = ", z$alpha, 
			sep="", collapse="")
		out <- data.frame(x, density, args)
	})

# Create labels for later use in displaying the arguments on the plots 
plot.data$label <- paste(plot.data$.id, ": ", plot.data$args, sep="")

# Create plots
generalizedTrapezoids <- ggplot(data = plot.data, aes(x = x, y = density)) + 
		geom_line() + theme_bw() + 
		facet_wrap(~label, ncol = 2, scales = "free_y")
print(generalizedTrapezoids)