4.4 Non-homogeneous Poisson processes

Non-homogeneous Poisson processes (NHPPs) extend the classic Poisson process by letting the event rate change over time. Instead of a fixed rate $\lambda$, you work with a time-dependent intensity function $\lambda(t)$, which opens the door to modeling situations where events speed up, slow down, or follow cyclical patterns.

Definition of non-homogeneous Poisson processes

A counting process $\{N(t), t \geq 0\}$ is a non-homogeneous Poisson process if it satisfies three conditions: it starts at zero ($N(0) = 0$), it has independent increments, and the number of events in any interval $[a, b]$ follows a Poisson distribution whose mean depends on the integral of $\lambda(t)$ over that interval.

The key difference from a homogeneous Poisson process is that the rate parameter is no longer constant. The count in an interval still follows a Poisson distribution, but the parameter of that distribution is computed from the intensity function rather than being a simple product of rate and length.

Intensity function λ(t)

The intensity function $\lambda(t) \geq 0$ gives the instantaneous rate of event occurrence at time $t$. In a small interval $[t, t + dt)$, the expected number of events is approximately $\lambda(t)\, dt$, and the probability of more than one event in that interval is $o(dt)$.

$\lambda(t)$ can take many forms depending on the application:

• Linear: $\lambda(t) = at + b$ (steadily increasing or decreasing rate)
• Exponential: $\lambda(t) = ae^{bt}$ (accelerating growth or decay in the rate)
• Periodic: $\lambda(t) = a + b\sin(ct)$ (cyclical patterns, like daily rush-hour traffic)

The only constraint is that $\lambda(t) \geq 0$ for all $t$.

Mean value function Λ(t)

The mean value function (also called the cumulative intensity) accumulates the intensity over time:

$\Lambda(t) = \int_0^t \lambda(s)\, ds$

This gives the expected number of events in $[0, t]$. More generally, the expected count in any interval $[a, b]$ is $\Lambda(b) - \Lambda(a)$.

The distribution of the count in $[0, t]$ is:

$P(N(t) = n) = \frac{[\Lambda(t)]^n \, e^{-\Lambda(t)}}{n!}$

Think of $\Lambda(t)$ as playing the role that $\lambda t$ plays in the homogeneous case. Whenever you see $\lambda t$ in a homogeneous formula, you can often replace it with $\Lambda(t)$ (or $\Lambda(b) - \Lambda(a)$ for a general interval) to get the NHPP version.

Properties of non-homogeneous Poisson processes

NHPPs retain some core Poisson process properties but lose others. Understanding which properties carry over is important.

Independent increments

The number of events in disjoint time intervals are independent random variables. If $[a, b]$ and $[c, d]$ don't overlap, then $N(b) - N(a)$ and $N(d) - N(c)$ are independent. This property holds exactly as in the homogeneous case.

Poisson distribution of increments

The count in any interval $[a, b]$ follows a Poisson distribution:

$P(N(b) - N(a) = n) = \frac{[\Lambda(b) - \Lambda(a)]^n \, e^{-(\Lambda(b) - \Lambda(a))}}{n!}$

Notice the mean depends on the integral of $\lambda(t)$ over the interval, not just the interval's length. Two intervals of equal length can have very different expected counts if $\lambda(t)$ differs across them.

Stationary increments are lost

Unlike the homogeneous case, an NHPP does not have stationary increments. The distribution of $N(t+h) - N(t)$ depends on where you are in time (on $t$), not just on the length $h$.

Conditional uniformity

Given that exactly $n$ events occurred in $[0, T]$, the event times are independent with a common density proportional to $\lambda(t)$:

$f(t) = \frac{\lambda(t)}{\Lambda(T)}, \quad 0 \leq t \leq T$

In the homogeneous case this reduces to a uniform distribution on $[0, T]$. For an NHPP, events cluster where $\lambda(t)$ is large. This property is useful both for understanding the process and for simulation.

A note on inter-event times

The "memoryless property" of the homogeneous Poisson process does not carry over in the usual sense. Inter-event times in an NHPP are not exponentially distributed with a fixed rate, and they are not identically distributed. The time until the next event, starting from time $t$, depends on the shape of $\lambda(s)$ for $s > t$. Specifically, if the last event occurred at time $t$, the next event time $T_{\text{next}}$ satisfies:

$P(T_{\text{next}} > t + x \mid T_{\text{next}} > t) = \exp\!\left(-\int_t^{t+x} \lambda(s)\, ds\right)$

Examples of non-homogeneous Poisson processes

Time-varying arrival rates

• Call centers: Call volume peaks mid-morning and mid-afternoon, with a dip around lunch. A periodic intensity like $\lambda(t) = 50 + 30\sin(\pi t / 12)$ (calls per hour) could capture this pattern.
• Website traffic: A news site might see a baseline of 200 visits/min that spikes to 1,000 visits/min after a major story breaks, then decays exponentially back to baseline.

Reliability and aging

• Machine failures: A power-law intensity $\lambda(t) = \alpha \beta t^{\beta - 1}$ (the Weibull process) models equipment whose failure rate increases with age when $\beta > 1$, or decreases (early "burn-in" failures) when $\beta < 1$.

Natural events

• Earthquake aftershocks: After a major earthquake, aftershock rates are high and decay over time. The modified Omori law gives an intensity of the form $\lambda(t) = K(t + c)^{-p}$ for the aftershock sequence.

Simulation of non-homogeneous Poisson processes

Simulating an NHPP is trickier than the homogeneous case because you can't just generate exponential inter-arrival times with a fixed rate. Two standard methods handle this.

Thinning method (Lewis–Shedler)

This is the most widely used approach when $\lambda(t)$ is bounded on the simulation interval $[0, T]$.

1. Find an upper bound $\lambda^* \geq \max_{t \in [0,T]} \lambda(t)$.
2. Generate events from a homogeneous Poisson process with rate $\lambda^*$.
3. For each generated event at time $t_i$, independently accept it with probability $\lambda(t_i) / \lambda^*$. Otherwise, discard it.
4. The accepted events form a realization of the NHPP with intensity $\lambda(t)$.

The tighter your bound $\lambda^*$, the fewer events you waste. If $\lambda(t)$ varies a lot, you can use a piecewise constant upper bound to improve efficiency.

Inversion method (time-transformation)

This method exploits the fact that if you "warp" time using $\Lambda(t)$, an NHPP becomes a homogeneous Poisson process with rate 1.

1. Generate arrival times $s_1 < s_2 < \cdots$ from a homogeneous Poisson process with rate 1 (equivalently, set $s_k = s_{k-1} + E_k$ where $E_k \sim \text{Exp}(1)$).
2. Transform each arrival time back to the original time scale by solving $\Lambda(t_k) = s_k$ for $t_k$, i.e., $t_k = \Lambda^{-1}(s_k)$.
3. Stop when $t_k > T$.

This method requires you to invert $\Lambda(t)$, which is straightforward when $\lambda(t)$ has a simple closed-form integral (e.g., linear or exponential intensity) but may need numerical root-finding otherwise.

Inference for non-homogeneous Poisson processes

Maximum likelihood estimation

Suppose you observe event times $t_1, \ldots, t_n$ in $[0, T]$. The likelihood of an NHPP is:

$L(\theta) = \exp\!\left(-\int_0^T \lambda(t;\theta)\, dt\right) \prod_{i=1}^n \lambda(t_i;\theta)$

where $\theta$ collects the parameters of the intensity model. The log-likelihood is:

$\ell(\theta) = -\Lambda(T;\theta) + \sum_{i=1}^n \log \lambda(t_i;\theta)$

For simple parametric forms, you can sometimes solve $\partial \ell / \partial \theta = 0$ analytically. For example, with a linear intensity $\lambda(t) = at + b$, you get a system of two equations in $a$ and $b$. For more complex models, numerical optimization (Newton-Raphson, EM, etc.) is needed.

Bayesian inference

Bayesian methods place a prior distribution on the intensity parameters $\theta$ and combine it with the likelihood above to get a posterior:

$\pi(\theta \mid \text{data}) \propto \pi(\theta) \cdot L(\theta)$

• For parametric models, conjugate or weakly informative priors on $\theta$ are common. For instance, a Gamma prior on a rate parameter.
• For nonparametric models (e.g., modeling $\lambda(t)$ as a Gaussian process on the log scale), MCMC sampling or variational inference is typically required.

Bayesian approaches are especially useful when data are sparse and you want to incorporate domain knowledge into the intensity estimate.

Applications of non-homogeneous Poisson processes

Queueing systems with time-varying arrival rates

Service systems like hospitals, banks, and call centers experience demand that fluctuates throughout the day. Modeling arrivals as an NHPP lets you estimate time-dependent performance measures (expected wait times, staffing needs) rather than relying on a single average rate that may be misleading during peak or off-peak periods.

Reliability analysis

The non-homogeneous Poisson process is a standard model for repairable systems. Under "minimal repair" assumptions (each repair returns the system to the state it was in just before failure), successive failures follow an NHPP. The power-law process $\lambda(t) = \alpha \beta t^{\beta - 1}$ is widely used, with $\beta > 1$ indicating a deteriorating system and $\beta < 1$ indicating reliability growth.

Key reliability metrics derived from the NHPP include:

• Expected number of failures in $[0, t]$: $\Lambda(t)$
• Probability of zero failures in $[t_1, t_2]$: $\exp(-[\Lambda(t_2) - \Lambda(t_1)])$

Earthquake modeling

Seismologists use NHPPs (and extensions like the ETAS model) to describe earthquake catalogs. The intensity captures background seismicity plus triggered aftershock sequences. This feeds into probabilistic seismic hazard analysis, which estimates the probability of ground shaking exceeding a threshold at a given site over a given time window.

Financial modeling

Trade arrivals and order-book events in financial markets are not uniformly spaced. Activity clusters around market open/close and news releases. NHPP models (and their extensions to self-exciting processes) help estimate intraday volatility patterns and inform risk measures like value-at-risk.

Relationship to other processes

Comparison with homogeneous Poisson processes

A homogeneous Poisson process is simply an NHPP with $\lambda(t) = \lambda$ (constant). Every formula in this guide reduces to the familiar homogeneous version when you set $\Lambda(t) = \lambda t$. The homogeneous process has stationary increments; the NHPP does not.

Connection to renewal processes

A renewal process has i.i.d. inter-arrival times from some distribution. A homogeneous Poisson process is the special case where that distribution is exponential. An NHPP is not a renewal process in general, because its inter-arrival times are neither identically distributed nor independent of the current time. The two generalizations go in different directions: renewal processes generalize the inter-arrival distribution, while NHPPs generalize the rate structure.

Doubly stochastic Poisson processes (Cox processes)

A Cox process takes the NHPP one step further: the intensity function $\lambda(t)$ is itself random, modeled as a realization of some stochastic process. Conditional on a particular realization of $\lambda(t)$, the event process is an NHPP.

• If $\log \lambda(t)$ is a Gaussian process, you get a log-Gaussian Cox process.
• If $\lambda(t)$ is driven by a latent Markov chain, you get a Markov-modulated Poisson process.

Cox processes are useful when the rate variation has both a systematic component (time of day, trend) and an unpredictable component (unobserved risk factors, random environmental changes).

Extensions and generalizations

Marked non-homogeneous Poisson processes

A marked NHPP attaches a random mark $m_i$ to each event, representing some attribute: earthquake magnitude, insurance claim size, trade volume, etc. The process is characterized by a joint intensity:

$\lambda(t, m) = \lambda_g(t) \cdot f(m \mid t)$

where $\lambda_g(t)$ is the ground intensity (overall event rate) and $f(m \mid t)$ is the conditional mark distribution at time $t$. If the mark distribution doesn't depend on $t$, the marks and the timing are independent, which simplifies analysis considerably.

Spatial and spatiotemporal NHPPs

Extending to spatial dimensions, the intensity becomes $\lambda(\mathbf{x}, t)$, where $\mathbf{x}$ is a location. The expected number of events in a region $A$ during $[a, b]$ is:

$\int_a^b \int_A \lambda(\mathbf{x}, t)\, d\mathbf{x}\, dt$

Applications include mapping crime hotspots, tracking disease spread, and modeling wildfire ignitions. Estimation often uses kernel smoothing or log-Gaussian Cox process models.

Self-exciting processes (Hawkes processes)

While not strictly NHPPs (because they violate independent increments), Hawkes processes are a closely related generalization worth knowing. The intensity depends on the process's own history:

$\lambda(t) = \mu(t) + \sum_{t_i < t} g(t - t_i)$

where $\mu(t)$ is a baseline intensity and $g(\cdot)$ is a triggering kernel. Each past event temporarily boosts the rate, creating clustering. The ETAS model for earthquakes and many financial microstructure models are Hawkes processes.