Estimate the mean of a Poisson distribution, and
construct a prediction interval for the next \(k\) observations or
next set of \(k\) sums.
predIntPois(x, k = 1, n.sum = 1, method = "conditional",
pi.type = "two-sided", conf.level = 0.95, round.limits = TRUE)Extended methods and derivations are available in
Prediction intervals for environmental monitoring: predIntPois.
Open the vignette locally with
vignette("prediction-intervals", package = "EnvStats").
numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a Poisson distribution
(i.e., epois or epoisCensored).
If x is a numeric vector,
missing (NA), undefined (NaN), and
infinite (Inf, -Inf) values are allowed but will be removed.
positive integer specifying the number of future observations or sums the
prediction interval should contain with confidence level conf.level.
The default value is k=1.
positive integer specifying the sample size associated with the \(k\) future
sums. The default value is n.sum=1 (i.e., individual observations).
Note that all future sums must be based on the same sample size.
character string specifying the method to use. The possible values are: "conditional" (based on a conditional distribution; the default), "conditional.approx.normal" (method based on approximating a conditional
distribution with the standard normal distribution), "conditional.approx.t" (method based on approximating a conditional
distribution with Student's t-distribution), and "normal.approx" (approximate method based on the fact that the
mean and varaince of a Poisson distribution are the same).
See the DETAILS section for more information on these methods. The "conditional" method
is only implemented for k=1; when k is bigger than 1, the value of
method cannot be "conditional".
character string indicating what kind of prediction interval to compute.
The possible values are pi.type="two-sided" (the default),
pi.type="lower", and pi.type="upper".
a scalar between 0 and 1 indicating the confidence level of the prediction interval.
The default value is conf.level=0.95.
logical scalar indicating whether to round the computed prediction limits to the
nearest integer. The default value is round.limits=TRUE.
If x is a numeric vector, predIntPois returns a list of class
"estimate" containing the estimated parameter, the prediction interval,
and other information. See the help file for estimate.object for details.
If x is the result of calling an estimation function,
predIntPois returns a list whose class is the same as x.
The list contains the same components as x, as well as a component called
interval containing the prediction interval information.
If x already has a component called interval, this component is
replaced with the prediction interval information.
Cox, D.R., and D.V. Hinkley. (1974). Theoretical Statistics. Chapman and Hall, New York, pp.242–245.
Gibbons, R.D. (1987b). Statistical Models for the Analysis of Volatile Organic Compounds in Waste Disposal Sites. Ground Water 25, 572–580.
Gibbons, R.D., D.K. Bhaumik, and S. Aryal. (2009). Statistical Methods for Groundwater Monitoring, Second Edition. John Wiley & Sons, Hoboken, pp. 72–76.
Hahn, G.J., and W.Q. Meeker. (1991). Statistical Intervals: A Guide for Practitioners. John Wiley and Sons, New York.
Hahn, G., and W. Nelson. (1973). A Survey of Prediction Intervals and Their Applications. Journal of Quality Technology 5, 178–188.
Johnson, N. L., S. Kotz, and A. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, Chapter 4.
Millard, S.P., and N.K. Neerchal. (2001). Environmental Statistics with S-PLUS. CRC Press, Boca Raton.
Miller, R.G. (1981a). Simultaneous Statistical Inference. McGraw-Hill, New York, pp.8, 76–81.
Nelson, W.R. (1970). Confidence Intervals for the Ratio of Two Poisson Means and Poisson Predictor Intervals. IEEE Transactions of Reliability R-19, 42–49.
Nelson, W.R. (1982). Applied Life Data Analysis. John Wiley and Sons, New York, pp.200–204.
Zar, J.H. (2010). Biostatistical Analysis. Fifth Edition. Prentice-Hall, Upper Saddle River, NJ, pp. 585–586.
Prediction and tolerance intervals have long been applied to quality control and life testing problems. Nelson (1970) notes that his development of confidence and prediction limits for the Poisson distribution is based on well-known results dating back to the 1950's. Hahn and Nelson (1973) review predicion intervals for several distributions, including Poisson prediction intervals. The mongraph by Hahn and Meeker (1991) includes a discussion of Poisson prediction intervals.
Gibbons (1987b) uses the Poisson distribution to model the number of detected
compounds per scan of the 32 volatile organic priority pollutants (VOC), and also
to model the distribution of chemical concentration (in ppb), and presents formulas
for prediction and tolerance intervals. The formulas for prediction intervals are
based on Cox and Hinkley (1974, p.245). Gibbons (1987b) only deals with
the case where n.sum=1.
Gibbons et al. (2009, pp. 72–76) discuss methods for Poisson prediction limits.
# Generate 20 observations from a Poisson distribution with parameter
# lambda=2. The interval [0, 4] contains 94.7% of this distribution and
# the interval [0,5] contains 98.3% of this distribution. Thus, because
# of the discrete nature of the Poisson distribution, no interval contains
# exactly 95% of this distribution. Use predIntPois to estimate the mean
# parameter of the true distribution, and construct a one-sided upper
# 95% prediction interval for the next single observation from this distribution.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat <- rpois(20, lambda = 2)
predIntPois(dat, pi.type = "upper")
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Poisson
#>
#> Estimated Parameter(s): lambda = 1.8
#>
#> Estimation Method: mle/mme/mvue
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#> Prediction Interval Method: conditional
#>
#> Prediction Interval Type: upper
#>
#> Confidence Level: 95%
#>
#> Number of Future Observations: 1
#>
#> Prediction Interval: LPL = 0
#> UPL = 5
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Poisson
#
#Estimated Parameter(s): lambda = 1.8
#
#Estimation Method: mle/mme/mvue
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: conditional
#
#Prediction Interval Type: upper
#
#Confidence Level: 95%
#
#Number of Future Observations: 1
#
#Prediction Interval: LPL = 0
# UPL = 5
#----------
# Compare results above with the other approximation methods:
predIntPois(dat, method = "conditional.approx.normal",
pi.type = "upper")$interval$limits
#> LPL UPL
#> 0 4
#LPL UPL
# 0 4
predIntPois(dat, method = "conditional.approx.t",
pi.type = "upper")$interval$limits
#> LPL UPL
#> 0 4
#LPL UPL
# 0 4
predIntPois(dat, method = "normal.approx",
pi.type = "upper")$interval$limits
#> Warning: Estimated value of 'lambda' and/or number of future observations is/are probably too small for the normal approximation to work well.
#> LPL UPL
#> 0 4
#LPL UPL
# 0 4
#Warning message:
#In predIntPois(dat, method = "normal.approx", pi.type = "upper") :
# Estimated value of 'lambda' and/or number of future observations
# is/are probably too small for the normal approximation to work well.
#==========
# Using the same data as in the previous example, compute a one-sided
# upper 95% prediction limit for k=10 future observations.
# Using conditional approximation method based on the normal distribution.
predIntPois(dat, k = 10, method = "conditional.approx.normal",
pi.type = "upper")
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Poisson
#>
#> Estimated Parameter(s): lambda = 1.8
#>
#> Estimation Method: mle/mme/mvue
#>
#> Data: dat
#>
#> Sample Size: 20
#>
#> Prediction Interval Method: conditional.approx.normal
#>
#> Prediction Interval Type: upper
#>
#> Confidence Level: 95%
#>
#> Number of Future Observations: 10
#>
#> Prediction Interval: LPL = 0
#> UPL = 6
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Poisson
#
#Estimated Parameter(s): lambda = 1.8
#
#Estimation Method: mle/mme/mvue
#
#Data: dat
#
#Sample Size: 20
#
#Prediction Interval Method: conditional.approx.normal
#
#Prediction Interval Type: upper
#
#Confidence Level: 95%
#
#Number of Future Observations: 10
#
#Prediction Interval: LPL = 0
# UPL = 6
# Using method based on approximating conditional distribution with
# Student's t-distribution
predIntPois(dat, k = 10, method = "conditional.approx.t",
pi.type = "upper")$interval$limits
#> LPL UPL
#> 0 6
#LPL UPL
# 0 6
#==========
# Repeat the above example, but set k=5 and n.sum=3. Thus, we want a
# 95% upper prediction limit for the next 5 sets of sums of 3 observations.
predIntPois(dat, k = 5, n.sum = 3, method = "conditional.approx.t",
pi.type = "upper")$interval$limits
#> LPL UPL
#> 0 12
#LPL UPL
# 0 12
#==========
# Reproduce Example 3.6 in Gibbons et al. (2009, p. 75)
# A 32-constituent VOC scan was performed for n=16 upgradient
# samples and there were 5 detections out of these 16. We
# want to construct a one-sided upper 95% prediction limit
# for 20 monitoring wells (so k=20 future observations) based
# on these data.
# First we need to create a data set that will yield a mean
# of 5/16 based on a sample size of 16. Any number of data
# sets will do. Here are two possible ones:
dat <- c(rep(1, 5), rep(0, 11))
dat <- c(2, rep(1, 3), rep(0, 12))
# Now call predIntPois. Don't round the limits so we can
# compare to the example in Gibbons et al. (2009).
predIntPois(dat, k = 20, method = "conditional.approx.t",
pi.type = "upper", round.limits = FALSE)
#>
#> Results of Distribution Parameter Estimation
#> --------------------------------------------
#>
#> Assumed Distribution: Poisson
#>
#> Estimated Parameter(s): lambda = 0.3125
#>
#> Estimation Method: mle/mme/mvue
#>
#> Data: dat
#>
#> Sample Size: 16
#>
#> Prediction Interval Method: conditional.approx.t
#>
#> Prediction Interval Type: upper
#>
#> Confidence Level: 95%
#>
#> Number of Future Observations: 20
#>
#> Prediction Interval: LPL = 0.000000
#> UPL = 2.573258
#>
#Results of Distribution Parameter Estimation
#--------------------------------------------
#
#Assumed Distribution: Poisson
#
#Estimated Parameter(s): lambda = 0.3125
#
#Estimation Method: mle/mme/mvue
#
#Data: dat
#
#Sample Size: 16
#
#Prediction Interval Method: conditional.approx.t
#
#Prediction Interval Type: upper
#
#Confidence Level: 95%
#
#Number of Future Observations: 20
#
#Prediction Interval: LPL = 0.000000
# UPL = 2.573258
#==========
# Cleanup
#--------
rm(dat)