plpoisson (version 0.3.1)

poisBayes: Bayesian Prediction Limits for Poisson Distribution (Gamma Prior)

Description

The function provides the Bayesian prediction limits of a Poisson random variable derived based on a gamma prior. The resulting prediction bounds quantify the uncertainty associated with the predicted future number of occurences in a time window of size \(t\).

Usage

poisBayes(xobs, n, s, t, a, b, alpha = 0.05)

Value

A list containing the following components:

lower

An integer value representing the lower bound of the prediction limit.

upper

An integer value representing the upper bound of the prediction limit.

Arguments

xobs

a numeric value denoting the number of the observed occurrencies.

n

a numeric value representing the total number of the time windows s in the past (observed time windows).

s

a numeric value corresponding to the fixed size (or average size) of the observed time windows.

t

a numeric value indicating the size of the future time window.

a

a poisitive real number denoting the shape hyperparameter of a gamma prior distribution.

b

a poisitive real number representing the rate hyperparameter of a gamma prior distribution.

alpha

a numeric value associated to the credible probability. By default alpha = 0.05, thus an prediction interval at 95% will be returned.

Author

Valbona Bejleri, Luca Sartore and Balgobin Nandram

Details

When the argument b = Inf, one can obtain prediction limits with uniform prior by setting the argument a = 1. Similarly, one can get the limits with a Jeffreys prior by setting the argument a = 0.

References

Bejleri, V., & Nandram, B. (2018). Bayesian and frequentist prediction limits for the Poisson distribution. Communications in Statistics-Theory and Methods, 47(17), 4254-4271.

Bejleri, V. (2005). Bayesian Prediction Intervals for the Poisson Model, Noninformative Priors, Ph.D. Dissertation, American University, Washington, DC.

See Also

poiss, poisJEFF, poisUNIF

Examples

# Loading the package
library(plpoisson)
set.seed(2020L)

# Number of observed time windows
n <- 555L

# Simulating a dataset
data <- cbind.data.frame(
    occ_obs = rpois(n, rgamma(n, 5.5, .5)),
    win_siz = rgamma(n, 1.44, .777)
) 

## Bayesian prediction limits 
##  (with gamma prior)
poisBayes(sum(data$occ_obs), # Past occurrencies 
    nrow(data), # Total past time windows
    mean(data$win_siz), # Window size
    333, # Size of future window
    2, 2.22) # Hyper-parameters for gamma prior