4.4 Non-homogeneous Poisson processes

Non-homogeneous Poisson processes (NHPPs) extend the classic Poisson process by allowing event rates to vary over time. This generalization enables more realistic modeling of real-world phenomena where event frequencies change throughout the day or with external factors.

NHPPs are characterized by a time-dependent intensity function λ(t), which represents the instantaneous event rate. This flexibility allows for modeling complex patterns in various fields, from customer arrivals at stores to earthquake occurrences, providing valuable insights into time-varying random events.

Definition of non-homogeneous Poisson processes

• Non-homogeneous Poisson processes (NHPPs) are a generalization of homogeneous Poisson processes where the rate parameter varies over time
• NHPPs are used to model events that occur randomly over time with a time-dependent intensity function $\lambda(t)$
• The number of events in any interval follows a Poisson distribution, but the rate parameter changes according to the intensity function

Intensity function λ(t)

• The intensity function $\lambda(t)$ represents the instantaneous rate of event occurrence at time $t$
• $\lambda(t)$ can be any non-negative function of time, allowing for modeling of time-varying event rates
  • Examples: linear function $\lambda(t) = at + b$, exponential function $\lambda(t) = ae^{bt}$, or periodic function $\lambda(t) = a + b\sin(ct)$
• The expected number of events in a small interval $[t, t+dt)$ is approximately $\lambda(t)dt$

Mean value function Λ(t)

• The mean value function $\Lambda(t)$ represents the expected number of events up to time $t$
• $\Lambda(t)$ is the integral of the intensity function from 0 to $t$: $\Lambda(t) = \int_0^t \lambda(s) ds$
• The probability of observing $n$ events in the interval $[0, t]$ follows a Poisson distribution with mean $\Lambda(t)$: $P(N(t) = n) = \frac{(\Lambda(t))^n e^{-\Lambda(t)}}{n!}$

Properties of non-homogeneous Poisson processes

• NHPPs share some properties with homogeneous Poisson processes but differ in others due to the time-varying intensity function

Independent increments

• The number of events in disjoint time intervals are independent random variables
• Knowledge of the number of events in one interval does not provide information about the number of events in another non-overlapping interval

Poisson distribution of events

• The number of events in any interval $[a, b]$ follows a Poisson distribution with mean $\Lambda(b) - \Lambda(a)$
• $P(N(b) - N(a) = n) = \frac{(\Lambda(b) - \Lambda(a))^n e^{-(\Lambda(b) - \Lambda(a))}}{n!}$

Memoryless property

• The waiting time until the next event, given the history of the process up to the current time, depends only on the current time and not on the past events
• The distribution of the waiting time until the next event at time $t$ is exponential with rate $\lambda(t)$

Examples of non-homogeneous Poisson processes

• NHPPs can model various real-world phenomena where the event rate varies over time

Time-varying arrival rates

• Customer arrivals at a store or call center with different rates depending on the time of day (higher during peak hours, lower during off-peak hours)
• Website traffic with varying rates based on the time of day or day of the week

Modeling non-stationary processes

• Earthquake occurrences, where the rate of earthquakes may change over time due to changes in geophysical conditions
• Machine failures, where the failure rate may increase as the machine ages or wears out

Simulation of non-homogeneous Poisson processes

• Simulating NHPPs is more complex than homogeneous Poisson processes due to the time-varying intensity function

Thinning method

• Generate events from a homogeneous Poisson process with a rate $\lambda^*$ greater than or equal to the maximum value of $\lambda(t)$ over the simulation interval
• For each generated event at time $t$, accept it with probability $\lambda(t) / \lambda^*$, otherwise reject it
• The accepted events form a NHPP with intensity function $\lambda(t)$

Inversion method

• Generate the waiting times between events by inverting the cumulative distribution function (CDF) of the exponential distribution with rate $\lambda(t)$
• The waiting time until the next event at time $t$ is given by $W_t = F_t^{-1}(U)$, where $F_t$ is the CDF of the exponential distribution with rate $\lambda(t)$ and $U$ is a uniform random variable on $[0, 1]$
• Repeat this process, updating the current time by adding the generated waiting times, until the desired simulation interval is covered

Inference for non-homogeneous Poisson processes

• Estimating the intensity function $\lambda(t)$ from observed event data is a key task in NHPP inference

Maximum likelihood estimation

• The likelihood function for a NHPP with observed event times $t_1, \ldots, t_n$ in the interval $[0, T]$ is $L(\lambda) = \exp\left(-\int_0^T \lambda(t) dt\right) \prod_{i=1}^n \lambda(t_i)$
• The maximum likelihood estimate (MLE) of $\lambda(t)$ is obtained by maximizing the log-likelihood function, which may require numerical optimization techniques
  • Example: for a linear intensity function $\lambda(t) = at + b$, the MLEs of $a$ and $b$ can be obtained analytically

Bayesian inference

• Bayesian inference combines prior knowledge about the intensity function with observed data to obtain a posterior distribution for $\lambda(t)$
• Specify a prior distribution for the parameters of the intensity function, such as a Gamma prior for the rate parameter of an exponential intensity function
• The posterior distribution is proportional to the product of the prior and the likelihood, and can be sampled using Markov chain Monte Carlo (MCMC) methods

Applications of non-homogeneous Poisson processes

• NHPPs find applications in various fields where the event rate varies over time

Queueing systems with time-varying arrival rates

• Modeling customer arrivals at a service system (bank, hospital, call center) where the arrival rate depends on the time of day
• Analyzing performance measures such as waiting times, queue lengths, and server utilization in non-stationary queueing systems

Reliability analysis

• Modeling the failure process of a system or component where the failure rate changes over time due to aging, wear, or environmental factors
• Estimating reliability metrics such as the mean time to failure (MTTF) and the probability of failure-free operation over a given time interval

Earthquake modeling

• Describing the occurrence of earthquakes in a region where the seismic activity rate may vary over time due to changes in geophysical conditions or after a major earthquake (aftershocks)
• Assessing seismic hazard and estimating the probability of earthquakes exceeding a certain magnitude in a given time window

Financial modeling

• Modeling the arrival of trades or price changes in financial markets, where the activity level may vary depending on market conditions, news events, or time of day
• Estimating volatility, value-at-risk, and other risk measures for financial assets or portfolios

Relationship to other processes

• NHPPs are related to and can be compared with other stochastic processes

Comparison with homogeneous Poisson processes

• Homogeneous Poisson processes are a special case of NHPPs where the intensity function is constant over time, i.e., $\lambda(t) = \lambda$
• NHPPs allow for more flexible modeling of time-varying event rates, while homogeneous Poisson processes are suitable for modeling events that occur at a constant average rate

Connection to renewal processes

• A renewal process is a generalization of a Poisson process where the inter-event times are independent and identically distributed (i.i.d.) random variables from an arbitrary distribution
• If the inter-event times in a renewal process follow an exponential distribution with a time-varying rate parameter $\lambda(t)$, the resulting process is a NHPP

Doubly stochastic Poisson processes

• Doubly stochastic Poisson processes, also known as Cox processes, are a further generalization of NHPPs where the intensity function itself is a stochastic process
• The intensity function $\lambda(t)$ is modeled as a realization of a random process, allowing for additional randomness in the event rate
• Example: the intensity function may be modeled as a Gaussian process, leading to a Gaussian Cox process

Extensions and generalizations

• NHPPs can be extended and generalized to model more complex event processes

Marked non-homogeneous Poisson processes

• A marked NHPP associates a random mark (attribute) with each event, in addition to its occurrence time
• The marks can represent additional information about the events, such as the magnitude of an earthquake or the size of an insurance claim
• The intensity function can depend on both time and the mark value, allowing for modeling of time-varying event rates and mark distributions

Spatial non-homogeneous Poisson processes

• Spatial NHPPs extend the concept of NHPPs to multiple dimensions, modeling events that occur in a spatial region with a location-dependent intensity function $\lambda(\mathbf{x}, t)$
• The intensity function depends on both the spatial location $\mathbf{x}$ and time $t$, allowing for modeling of spatiotemporal event patterns
• Examples: modeling the occurrence of crime incidents in a city, or the spread of a disease in a population

Cox processes

• Cox processes, also known as doubly stochastic Poisson processes, are a generalization of NHPPs where the intensity function is itself a stochastic process
• The intensity function $\lambda(t)$ is modeled as a realization of a random process, such as a Gaussian process or a Markov process
• Cox processes allow for additional randomness and uncertainty in the event rate, beyond the time-varying nature of NHPPs
• Examples: modeling the occurrence of insurance claims, where the claim rate may depend on unobserved risk factors that evolve over time