Prediction Interval for a Poisson Distribution

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Description

Estimate the mean of a Poisson distribution, and construct a prediction interval for the next k observations or next set of k sums.

Usage

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  predIntPois(x, k = 1, n.sum = 1, method = "conditional", 
    pi.type = "two-sided", conf.level = 0.95, round.limits = TRUE)

Arguments

x

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.

k

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.

n.sum

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.

method

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".

pi.type

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".

conf.level

a scalar between 0 and 1 indicating the confidence level of the prediction interval. The default value is conf.level=0.95.

round.limits

logical scalar indicating whether to round the computed prediction limits to the nearest integer. The default value is round.limits=TRUE.

Details

A prediction interval for some population is an interval on the real line constructed so that it will contain k future observations or averages from that population with some specified probability (1-α)100\%, where 0 < α < 1 and k is some pre-specified positive integer. The quantity (1-α)100\% is call the confidence coefficient or confidence level associated with the prediction interval.

In the case of a Poisson distribution, we have modified the usual meaning of a prediction interval and instead construct an interval that will contain k future observations or k future sums with a certain confidence level.

A prediction interval is a random interval; that is, the lower and/or upper bounds are random variables computed based on sample statistics in the baseline sample. Prior to taking one specific baseline sample, the probability that the prediction interval will contain the next k averages is (1-α)100\%. Once a specific baseline sample is taken and the prediction interval based on that sample is computed, the probability that that prediction interval will contain the next k averages is not necessarily (1-a)100\%, but it should be close.

If an experiment is repeated N times, and for each experiment:

  1. A sample is taken and a (1-a)100\% prediction interval for k=1 future observation is computed, and

  2. One future observation is generated and compared to the prediction interval,

then the number of prediction intervals that actually contain the future observation generated in step 2 above is a binomial random variable with parameters size=N and prob=(1-α)100\% (see Binomial).

If, on the other hand, only one baseline sample is taken and only one prediction interval for k=1 future observation is computed, then the number of future observations out of a total of N future observations that will be contained in that one prediction interval is a binomial random variable with parameters size=N and prob=(1-α^*)100\%, where α^* depends on the true population parameters and the computed bounds of the prediction interval.

Because of the discrete nature of the Poisson distribution, even if the true mean of the distribution λ were known exactly, the actual confidence level associated with a prediction limit will usually not be exactly equal to (1-α)100\%. For example, for the 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, no interval can contain exactly 95% of this distribution, so it is impossible to construct an exact 95% prediction interval for the next k=1 observation for a Poisson distribution with parameter lambda=2.

The Form of a Poisson Prediction Interval
Let \underline{x} = x_1, x_2, …, x_n denote a vector of n observations from a Poisson distribution with parameter lambda=λ. Also, let X denote the sum of these n random variables, i.e.,

X = ∑_{i=1}^n x_i \;\;\;\;\;\; (1)

Finally, let m denote the sample size associated with the k future sums (i.e., n.sum=m). When m=1, each sum is really just a single observation, so in the rest of this help file the term “sums” replaces the phrase “observations or sums”.

Let \underline{y} = y_1, y_2, …, y_m denote a vector of m future observations from a Poisson distribution with parameter lambda=λ^{*}, and set Y equal to the sum of these m random variables, i.e.,

Y = ∑_{i=1}^m y_i \;\;\;\;\;\; (2)

Then Y has a Poisson distribution with parameter lambda=mλ^{*} (Johnson et al., 1992, p.160). We are interested in constructing a prediction limit for the next value of Y, or else the next k sums of m Poisson random variables, based on the observed value of X and assuming λ^{*} = λ.

For a Poisson distribution, the form of a two-sided prediction interval is:

[m\bar{x} - K, m\bar{x} + K] = [cX - K, cX + K] \;\;\;\;\;\; (3)

where

\bar{x} = \frac{X}{n} = ∑_{i=1}^n x_i \;\;\;\;\;\; (4)

c = \frac{m}{n} \;\;\;\;\;\; (5)

and K is a constant that depends on the sample size n, the confidence level (1-α)100\%, the number of future sums k, and the sample size associated with the future sums m. Do not confuse the constant K (uppercase K) with the number of future sums k (lowercase k). The symbol K is used here to be consistent with the notation used for prediction intervals for the normal distribution (see predIntNorm).

Similarly, the form of a one-sided lower prediction interval is:

[m\bar{x} - K, ∞] = [cX - K, ∞] \;\;\;\;\;\; (6)

and the form of a one-sided upper prediction interval is:

[0, m\bar{x} + K] = [0, cX + K] \;\;\;\;\;\; (7)

The derivation of the constant K is explained below.

Conditional Distribution (method="conditional")
Nelson (1970) derives a prediction interval for the case k=1 based on the conditional distribution of Y given X+Y. He notes that the conditional distribution of Y given the quantity X+Y=w is binomial with parameters size=w and prob=[mλ^{*} / (mλ^{*} + nλ)] (Johnson et al., 1992, p.161). When k=1, the prediction limits are computed as those most extreme values of Y that still yield a non-significant test of the hypothesis H_0: λ^{*} = λ, which for the conditional distribution of Y is equivalent to the hypothesis H_0: prob=[m /(m + n)].

Using the relationship between the binomial and F-distribution (see the explanation of exact confidence intervals in the help file for ebinom), Nelson (1982, p. 203) states that exact two-sided (1-α)100\% prediction limits [LPL, UPL] are the closest integer solutions to the following equations:

\frac{m}{LPL + 1} = \frac{n}{X} F(2 LPL + 2, 2X, 1 - α/2) \;\;\;\;\;\; (8)

\frac{UPL}{n} = \frac{X+1}{n} F(2X + 2, 2 UPL, 1 - α/2) \;\;\;\;\;\; (9)

where F(ν_1, ν_2, p) denotes the p'th quantile of the F-distribution with ν_1 and ν_2 degrees of freedom.

If ci.type="lower", α/2 is replaced with α in Equation (8) above for LPL, and UPL is set to .

If ci.type="upper", α/2 is replaced with α in Equation (9) above for UPL, and LPL is set to 0.

NOTE: This method is not extended to the case k > 1.

Conditional Distribution Approximation Based on Normal Distribution
(method="conditional.approx.normal")
Cox and Hinkley (1974, p.245) derive an approximate prediction interval for the case k=1. Like Nelson (1970), they note that the conditional distribution of Y given the quantity X+Y=w is binomial with parameters size=w and prob=[mλ^{*} / (mλ^{*} + nλ)], and that the hypothesis H_0: λ^{*} = λ is equivalent to the hypothesis H_0: prob=[m /(m + n)].

Cox and Hinkley (1974, p.245) suggest using the normal approximation to the binomial distribution (in this case, without the continuity correction; see Zar, 2010, pp.534-536 for information on the continuity correction associated with the normal approximation to the binomial distribution). Under the null hypothesis H_0: λ^{*} = λ, the quantity

z = [Y - \frac{c(X+Y)}{1+c}] / \{ [\frac{c(X+Y)}{(1+c)^2}]^{1/2} \} \;\;\;\;\;\; (10)

is approximately distributed as a standard normal random variable.

The Case When k = 1
When k = 1 and pi.type="two-sided", the prediction limits are computed by solving the equation

z^2 ≤ z_{1 - α/2}^2 \;\;\;\;\;\; (11)

where z_p denotes the p'th quantile of the standard normal distribution. In this case, Gibbons (1987b) notes that the quantity K in Equation (3) above is given by:

K = \frac{t^2c}{2} tc[X (1 + \frac{1}{c}) + \frac{t^2}{4}]^{1/2} \;\;\;\;\;\; (12)

where t = z_{1-α/2}.

When pi.type="lower" or pi.type="upper", K is computed exactly as above, except t is set to t = z_{1-α}.

The Case When k > 1
When k > 1, Gibbons (1987b) suggests using the Bonferroni inequality. That is, the value of K is computed exactly as for the case k=1 described above, except that the Bonferroni value of t is used in place of the usual value of t:

When pi.type="two-side", t = z_{1 - (α/k)/2}.

When pi.type="lower" or pi.type="upper", t = z_{1 - α/k}.

Conditional Distribution Approximation Based on Student's t-Distribution
(method="conditional.approx.t")
When method="conditional.approx.t", the exact same procedure is used as when
method="conditional.approx.normal", except that the quantity in Equation (10) is assumed to follow a Student's t-distribution with n-1 degrees of freedom. Thus, all occurrences of z_p are replaced with t_{n-1, p}, where t_{ν, p} denotes the p'th quantile of Student's t-distribution with ν degrees of freedom.

Normal Approximation (method="normal.approx")
The normal approximation for Poisson prediction limits was given by Nelson (1970; 1982, p.203) and is based on the fact that the mean and variance of a Poisson distribution are the same (Johnson et al, 1992, p.157), and for “large” values of n and m, both X and Y are approximately normally distributed.

The Case When k = 1
The quantity Y - cX is approximately normally distributed with expectation and variance given by:

E(Y - cX) = E(Y) - cE(X) = mλ - cnλ = 0 \;\;\;\;\;\; (13)

Var(Y - cX) = Var(Y) + c^2 Var(X) = mλ + c^2 nλ = mλ (1 + \frac{m}{n}) \;\;\;\;\;\; (14)

so the quantity

z = \frac{Y-cX}{√{m\hat{λ}(1 + \frac{m}{n})}} = \frac{Y-cX}{√{m\bar{x}(1 + \frac{m}{n})}} \;\;\;\;\;\; (15)

is approximately distributed as a standard normal random variable. The function predIntPois, however, assumes this quantity is distributed as approximately a Student's t-distribution with n-1 degrees of freedom.

Thus, following the idea of prediction intervals for a normal distribution (see predIntNorm), when pi.type="two-sided", the constant K for a (1-α)100\% prediction interval for the next k=1 sum of m observations is computed as:

K = t_{n-1, 1-α/2} √{m\bar{x} (1 + \frac{m}{n})} \;\;\;\;\;\; (16)

where t_{ν, p} denotes the p'th quantile of a Student's t-distribution with ν degrees of freedom.

Similarly, when pi.type="lower" or pi.type="upper", the constant K is computed as:

K = t_{n-1, 1-α} √{m\bar{x} (1 + \frac{m}{n})} \;\;\;\;\;\; (17)

The Case When k > 1
When k > 1, the value of K is computed exactly as for the case k = 1 described above, except that the Bonferroni value of t is used in place of the usual value of t:

When pi.type="two-sided",

K = t_{n-1, 1-(α/k)/2} √{m\bar{x} (1 + \frac{m}{n})} \;\;\;\;\;\; (18)

When pi.type="lower" or pi.type="upper",

K = t_{n-1, 1-(α/k)} √{m\bar{x} (1 + \frac{m}{n})} \;\;\;\;\;\; (19)

Hahn and Nelson (1973, p.182) discuss another method of computing K when k > 1, but this method is not implemented here.

Value

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.

Note

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.

Author(s)

Steven P. Millard (EnvStats@ProbStatInfo.com)

References

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.

See Also

Poisson, epois, estimate.object, Prediction Intervals, tolIntPois, Estimating Distribution Parameters.

Examples

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  # 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

  #----------

  # Compare results above with the other approximation methods:

  predIntPois(dat, method = "conditional.approx.normal", 
    pi.type = "upper")$interval$limits
  #LPL UPL 
  #  0   4  
 

  predIntPois(dat, method = "conditional.approx.t", 
    pi.type = "upper")$interval$limits 
  #LPL UPL 
  #  0   4 


  predIntPois(dat, method = "normal.approx", 
    pi.type = "upper")$interval$limits 
  #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

  
  # 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 

  #==========

  # 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

  #==========

  # 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

  #==========

  # Cleanup
  #--------
  rm(dat)

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