Package 'triangle'

Title: Distribution Functions and Parameter Estimates for the Triangle Distribution
Description: Provides the "r, q, p, and d" distribution functions for the triangle distribution. Also includes maximum likelihood estimation of parameters.
Authors: Rob Carnell [aut, cre]
Maintainer: Rob Carnell <[email protected]>
License: GPL (>= 2)
Version: 1.0.1
Built: 2024-11-10 05:45:43 UTC
Source: https://github.com/bertcarnell/triangle

Help Index

Compare multiple triangle distributions fits

Description

Compare multiple triangle distributions fits

Usage

compare_triangle_fit(
  y,
  cols = c("red", "blue", "green"),
  main = "Triangle Fit Comparison",
  ...
)

Arguments

y

the triangle distributed sample
cols

the colors of the CDF-based estimates, the maximum likelihood estimates, and the method of moments estimates
main

the plot title
...

other parameters passed to plot.ecdf

Examples

set.seed(10304)
xtest <- rtriangle(100, 1, 5, 2)
compare_triangle_fit(xtest)

The Log-Triangle Distribution

Description

These functions provide information about the triangle distribution on the logarithmic interval from a to b with a maximum at c. dltriangle gives the density, pltriangle gives the distribution function, qltriangle gives the quantile function, and rltriangle generates n random deviates.

Usage

rltriangle(
  n = 1,
  a = 1,
  b = 100,
  c = 10^((log10(a) + log10(b))/2),
  logbase = 10
)

dltriangle(x, a = 1, b = 100, c = 10^((log10(a) + log10(b))/2), logbase = 10)

pltriangle(q, a = 1, b = 100, c = 10^((log10(a) + log10(b))/2), logbase = 10)

qltriangle(p, a = 1, b = 100, c = 10^((log10(a) + log10(b))/2), logbase = 10)

Arguments

n

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

lower limit of the distribution.
b

upper limit of the distribution.
c

mode of the distribution.
logbase

the base of the logarithmic scale to use (default to 10)
x, q

vector of quantiles.
p

vector of probabilities.

Details

All probabilities are lower tailed probabilties. a, b, and c may be appropriate length vectors except in the case of rtriangle.

Value

dltriangle gives the density, pltriangle gives the distribution function, qltriangle gives the quantile function, and rltraingle generates random deviates. Invalid arguments will result in return value NaN or NA.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

See Also

.Random.seed about random number generation, runif, etc for other distributions.

Examples

tri <- rltriangle(100000, 1, 100, 10)
hist(log10(tri), breaks=100, main="Triangle Distribution", xlab="x")
dltriangle(10, 1, 100, 10) # 2/(log10(b)-log10(a)) = 1
qltriangle(pltriangle(10)) # 10

Quantile-Quantile Plot for Triangle Distributed Data

Description

Quantile-Quantile Plot for Triangle Distributed Data

Usage

qqtriangle(
  y,
  a,
  b,
  c,
  main = "Triangle Q-Q Plot",
  xlab = "Theoretical Quantiles",
  ylab = "Sample Quantiles",
  ...
)

Arguments

y

the triangle distributed sample
a

the theoretical distribution triangle minimum parameter
b

the theoretical distribution triangle maximum parameter
c

the theoretical distribution triangle mode parameter
main

the plot title
xlab

the x-axis label
ylab

the y-axis label
...

other parameters passed to qqplot

Value

a list of x-y coordinates on the plot

Examples

set.seed(10304)
xtest <- rtriangle(100, 1, 5, 2)
theta <- coef(triangle_mle(xtest))
qqtriangle(xtest, theta[1], theta[2], theta[3])

Maximum likelihood estimate of the standard triangle distribution mode

Description

Maximum likelihood estimate of the standard triangle distribution mode

Usage

standard_triangle_mle(x, debug = FALSE)

Arguments

x

sample from a triangle distribution
debug

if TRUE then the function will check the input parameters and print calculation information

Value

an object of S3 class triangle_mle containing a list with the call, coefficients, variance co-variance matrix, minimum negative log likelihood, number of observations, and the sample

References

Samuel Kotz and Johan Rene van Dorp. Beyond Beta doi:10.1142/5720

Examples

xtest <- c(0.1, 0.25, 0.3, 0.4, 0.45, 0.6, 0.75, 0.8)
standard_triangle_mle(xtest)

xtest <- rtriangle(20, 0, 1, 0.63)
standard_triangle_mle(xtest)

Utility Methods for S3 class triangle_mle

Description

Utility Methods for S3 class triangle_mle

Usage

## S3 method for class 'triangle_mle'
summary(object, ...)

## S3 method for class 'triangle_mle'
print(x, ...)

## S3 method for class 'triangle_mle'
coef(object, ...)

## S3 method for class 'triangle_mle'
logLik(object, ...)

## S3 method for class 'triangle_mle'
AIC(object, ..., k = 2)

## S3 method for class 'triangle_mle'
BIC(object, ...)

## S3 method for class 'triangle_mle'
vcov(object, ...)

## S3 method for class 'triangle_mle'
profile(fitted, ...)

## S3 method for class 'triangle_mle'
confint(object, parm, level = 0.95, ...)

Arguments

object

class triangle_mle from a call to triangle_mle()
...

not used except for print (other arguments passed to printCoefmat)
x

the triangle_mle object
k

the penalty per parameter to be used; the default k = 2
fitted

an object of class triangle_mle
parm

parameters to be given confidence intervals passed to stats4::confint
level

confidence interval level passed to stats4::confint

Value

an object of class summary.mle

print.triangle_mle: x invisibly

coef.triangle_mle: a vector of coefficients

logLik.triangle_mle: an object of class logLik

AIC.triangle_mle: the AIC

BIC.triangle_mle: the BIC

vcov.triangle_mle: the variance co-variance matrix

profile.triangle_mle: an object of class profile.mle

confint.triangle_mle: a matrix of parameter confidence intervals

Examples

set.seed(1234)
x <- rtriangle(100, 0, 1, 0.5)
mle1 <- triangle_mle(x)
summary(mle1)
print(mle1)
coef(mle1)
logLik(mle1)
AIC(mle1)
BIC(mle1)
vcov(mle1)
## Not run: 
  prof <- profile(mle1)
  stats4::plot(prof)
  confint(mle1, 1:3, level = 0.95)

## End(Not run)

The Triangle Distribution

Description

These functions provide information about the triangle distribution on the interval from a to b with a maximum at c. dtriangle gives the density, ptriangle gives the distribution function, qtriangle gives the quantile function, and rtriangle generates n random deviates.

Usage

dtriangle(x, a = 0, b = 1, c = (a + b)/2)

ptriangle(q, a = 0, b = 1, c = (a + b)/2)

qtriangle(p, a = 0, b = 1, c = (a + b)/2)

rtriangle(n = 1, a = 0, b = 1, c = (a + b)/2)

Arguments

x, q

vector of quantiles.
a

lower limit of the distribution.
b

upper limit of the distribution.
c

mode of the distribution.
p

vector of probabilities.
n

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

Details

All probabilities are lower tailed probabilities. a, b, and c may be appropriate length vectors except in the case of rtriangle. rtriangle is derived from a draw from runif. The triangle distribution has density:

f(x)=2(x−a)(b−a)(c−a)f(x) = \frac{2(x-a)}{(b-a)(c-a)}

for a≤x<ca \le x < c.

f(x)=2(b−x)(b−a)(b−c)f(x) = \frac{2(b-x)}{(b-a)(b-c)}

for c≤x≤bc \le x \le b. f(x)=0f(x) = 0 elsewhere. The mean and variance are:

E(x)=(a+b+c)3E(x) = \frac{(a + b + c)}{3}

V(x)=118(a2+b2+c2−ab−ac−bc)V(x) = \frac{1}{18}(a^2 + b^2 + c^2 - ab - ac - bc)

Value

dtriangle gives the density, ptriangle gives the distribution function, qtriangle gives the quantile function, and rtriangle generates random deviates. Invalid arguments will result in return value NaN or NA.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

See Also

.Random.seed about random number generation, runif, etc for other distributions.

Examples

## view the distribution
tri <- rtriangle(100000, 1, 5, 3)
hist(tri, breaks=100, main="Triangle Distribution", xlab="x")
mean(tri) # 1/3*(1 + 5 + 3) = 3
var(tri)  # 1/18*(1^2 + 3^2 + 5^2 - 1*5 - 1*3 - 5*3) = 0.666667
dtriangle(0.5, 0, 1, 0.5) # 2/(b-a) = 2
qtriangle(ptriangle(0.7)) # 0.7

Triangle parameter estimates using a non-linear fit of the empirical CDF

Description

Triangle parameter estimates using a non-linear fit of the empirical CDF

Usage

triangle_cdfe(x, control = stats::nls.control(maxiter = 100, warnOnly = TRUE))

Arguments

x

the triangle distributed sample
control

an object created by stats::nls.control

Value

an object of class nls

Examples

set.seed(10304)
xtest <- rtriangle(100, 1, 5, 2)
cdfe <- triangle_cdfe(xtest)
print(cdfe)
summary(cdfe)
coef(cdfe)
## Not run: 
  confint(cdfe)

## End(Not run)

Maximum likelihood estimate of the triangle distribution parameters

Description

Maximum likelihood estimate of the triangle distribution parameters

Usage

triangle_mle(x, debug = FALSE, maxiter = 100)

Arguments

x

sample from a triangle distribution
debug

if TRUE then the function will check the input parameters and print calculation information
maxiter

the maximum number of cycles of optimization between maximizing a and b given c and maximizing c given a and b

Value

an object of S3 class triangle_mle containing a list with the call, coefficients, variance co-variance matrix, minimum negative log likelihood, details of the optimization number of observations, and the sample

References

Samuel Kotz and Johan Rene van Dorp. Beyond Beta doi:10.1142/5720

Examples

xtest <- c(0.1, 0.25, 0.3, 0.4, 0.45, 0.6, 0.75, 0.8)
triangle_mle(xtest)

xtest <- rtriangle(20, 1, 5, 3.5)
triangle_mle(xtest)

Triangle distribution method of moments estimate

Description

Triangle distribution method of moments estimate

Usage

triangle_mom(x)

Arguments

x

triangle distribution sample

Value

a named vector of the parameter estimates

Examples

set.seed(1204)
x <- rtriangle(20, 0, 2, 1.5)
triangle_mom(x)