Point Process Modeling Foundations

Definitions and Notation

Counting Process

A process that generates well-defined events ocurring, or not, over an interval (often in space and or time).

What is an event?

All that matters is that it has a fixed definition and can be modeled to 'occur' at a point in time.

\(N(t)_{i,m} \)

The \(i\)'th observation of the number of events at time \(t\) on unit \(m\)

Writing a counting process

Since we only need to count events, we can write a counting process as \( \{N(s,t): t \gt s\}\)

The beginning

Since the definition is relative, we can assume \(N(0) = 0\)

How many events do we expect?

The 'intensity' of the counting process, \(\lambda_m(s,t)\), defines the rate of event occurrence after time \(s\) up to time \(t\) (inclusive).

Why does this sound like a Poisson model?

When \(\lambda_m(s,t)\) is time-constant, \(N(s,t)_{i,m} \sim Poisson(\lambda_m(s,t)(t-s))\)

What happens when \(\lambda_m(s,t)\) is not time-constant?

That's an Non-Homogeneous Poisson Process (NHPP) and shares only some of the properties of a Poisson Process.

When \(lambda_m(s,t)\) is not time-constant, how do you incorporate time with the rate?

\(\Lambda_m(s_1,s_2) = \int_{s_1}^{s_2} \lambda_m(t)dt\)

What's helpful about this?

The inter-arrival time density is still exponential although the distribution of inter-arrival times can not be simulated from an exponential. Simulation can however be done using a thinning process from a Poisson process (possibly piece-wise) that has a higher rate at all points. Since model-fitting relies on calculating the density rather than direct simulation the model-fitting is not restricted by this issue.