# Generate 20 observations from a gamma distribution with parameters
# shape=3 and scale=2, then create a prediciton interval for the
# next observation.
# (Note: the call to set.seed simply allows you to reproduce this
# example.)
set.seed(250)
dat <- rgamma(20, shape = 3, scale = 2)
predIntGamma(dat)
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): shape = 2.203862
# scale = 2.174928
#
#Estimation Method: mle
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: exact using
# Kulkarni & Powar (2010)
# transformation to Normality
# based on mle of 'shape'
#
#Normal Transform Power: 0.246
#
#Prediction Interval Type: two-sided
#
#Confidence Level: 95%
#
#Number of Future Observations: 1
#
#Prediction Interval: LPL = 0.5371569
# UPL = 15.5313783
#--------------------------------------------------------------------
# Using the same data as in the previous example, create an upper
# one-sided prediction interval for the next three averages of
# order 2 (i.e., each mean is based on 2 future observations), and
# use the bias-corrected estimate of shape.
pred.list <- predIntGamma(dat, n.transmean = 2, k = 3,
pi.type = "upper", est.method = "bcmle")
pred.list
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): shape = 1.906616
# scale = 2.514005
#
#Estimation Method: bcmle
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: Bonferroni using
# Kulkarni & Powar (2010)
# transformation to Normality
# based on bcmle of 'shape'
#
#Normal Transform Power: 0.246
#
#Prediction Interval Type: upper
#
#Confidence Level: 95%
#
#Number of Future
#Transformed Means: 3
#
#Sample Size for
#Transformed Means: 2
#
#Prediction Interval: LPL = 0.00000
# UPL = 12.17404
#--------------------------------------------------------------------
# Continuing with the above example, assume the distribution shifts
# in the future to a gamma distribution with shape = 5 and scale = 2.
# Create 6 future observations from this distribution, and create 3
# means by pairing the observations sequentially. Note we must first
# transform these observations using the power 0.246, then compute the
# means based on the transformed data, and then transform the means
# back to the original scale and compare them to the upper prediction
# limit of 12.17
set.seed(427)
new.dat <- rgamma(6, shape = 5, scale = 2)
p <- pred.list$interval$normal.transform.power
p
#[1] 0.246
new.dat.trans <- new.dat^p
means.trans <- c(mean(new.dat.trans[1:2]), mean(new.dat.trans[3:4]),
mean(new.dat.trans[5:6]))
means <- means.trans^(1/p)
means
#[1] 11.74214 17.05299 11.65272
any(means > pred.list$interval$limits["UPL"])
#[1] TRUE
#----------
# Clean up
rm(dat, pred.list, new.dat, p, new.dat.trans, means.trans, means)
#--------------------------------------------------------------------
# Reproduce part of the example on page 73 of
# Krishnamoorthy et al. (2008), which uses alkalinity concentrations
# reported in Gibbons (1994) and Gibbons et al. (2009) to construct a
# one-sided upper 90% prediction limit.
predIntGamma(Gibbons.et.al.09.Alkilinity.vec, pi.type = "upper",
conf.level = 0.9, normal.approx.transform = "cube.root")
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Gamma
#
#Estimated Parameter(s): shape = 9.375013
# scale = 6.202461
#
#Estimation Method: mle
#
#Data: Gibbons.et.al.09.Alkilinity.vec
#
#Sample Size: 27
#
#Prediction Interval Method: exact using
# Wilson & Hilferty (1931) cube-root
# transformation to Normality
#
#Normal Transform Power: 0.3333333
#
#Prediction Interval Type: upper
#
#Confidence Level: 90%
#
#Number of Future Observations: 1
#
#Prediction Interval: LPL = 0.0000
# UPL = 85.3495Run the code above in your browser using DataLab