Nonhomogeneous Poisson Process

Introduction

The model that represents the number of failures experienced up to time t {N(t), t ≥ 0} is a nonhomogeneous Poisson process model (NHPP). 

For example,
 

Let N(t) be the number of cars arriving at a mall by time t. We assume that the vehicles arrive somewhat randomly, so we want to model N(t) as a Poisson process. However, we noticed that this process does not have stationary increments. For example, we noted that the arrival rate of vehicles is larger during lunchtime than, say, 4 p.m. We model N(t) as a nonhomogeneous Poisson process in such cases.   

The nonhomogeneous Poisson process has all the properties of a Poisson process, except that its rate is a function of time, i.e., λ=λ(t).

The main problem with the NHPP model is deciding on an appropriate mean value function to represent the predicted number of failures up to a specific point in time.

The model will produce multiple functional forms of the mean value function depending on the following assumptions:

1. The failure process has an independent increment, which means that the number of failures during the time period (t, t + s) is determined by the present time t and the length of time intervals s, rather than the process's previous history.
2. The process failure rate is P {exactly one failure in (t, t + ∆t) = PN(t, t + ∆t) – N(t) = 1} = λ(t)∆t + o(t), where λ(t) is the intensity function.
3. The probability of more than one failure during a small interval ∆t is negligible, that is, P{two or more failures in (t, t+∆t) = o(∆t).
4. The base condition is N(0) = 0.

In other words, we can say that a nonhomogeneous Poisson Process is:

Let λ(t):[0,∞)↦[0,∞) be an integrable function. The counting process {N(t),t∈[0,∞)} is called a nonhomogeneous Poisson process with rate λ(t) if all the following conditions are met:

1. N(0)=0
2. N(t) has independent increments;
3. for any t∈[0,∞), we have
   1. P(N(t+Δt)−N(t)=0)=1−λ(t)Δt+o(Δt),
   2. P(N(t+Δt)−N(t)=1)=λ(t)Δt+o(Δt),
   3. P(N(t+Δt)−N(t)≥2)=o(Δt).

So, for the above example, the number of arrivals in any interval is a Poisson random variable for a nonhomogeneous Poisson process with the rate λ(t); however, its parameter can vary on the location of the interval. We can write

FAQs

1. What are the properties of the Poisson process?
A Poisson distribution is a discrete probability distribution for an event. Poisson distribution events are independent. The events are timed to occur at a specific interval. The Poisson distribution has a lambda value that is always greater than 0.

2. What is an inhomogeneous Poisson process?
A Poisson process with a time-varying rate is called an inhomogeneous Poisson process. It can be used to simulate customers' arrival timings at a store, traffic incidents, and damage positions along a road. The process's probability density function is Poisson distributed at any time slice t.