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.