An Introduction to Point Processes

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• An Introduction to Point Processes

• Definitions examples
• Conditional intensity
  Papangelou intensity
• Models
• a) Renewal processes
• b) Poisson processes
• c) Cluster models
• d) Inhibition models

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Point pattern a collection of points in some
space.
Point process a random point pattern.
Centroids of Los Angeles County wildfires,
1960-2000
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Aftershocks from global large earthquakes
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Epicenters times of microearthquakes in
Parkfield, CA
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Marked point process a random variable (mark)
with each point.
Hollister, CA earthquakes locations, times,
magnitudes
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Los Angeles Wildfires dates and sizes
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Time series
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Time series Palermo football rank vs. time
Marked point process Hollister earthquake times
magnitudes
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Antiquated definition a point process N(t) is a
right-continuous, Z-valued stochastic process
--x-------x--------------x-------------------
----x---x-x--------------- 0 t
T N(t) Number of points with times lt t.
Problem does not extend readily to higher
dimensions.
Modern definition A point process N is a
Z-valued random measure N(a,b) Number of
points with times between a b. N(A) Number
of points in the set A.
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• More Definitions
• s-finite finite number of pts in any bounded
  set.
• Simple N(x) 0 or 1 for all x, almost
  surely. (No overlapping pts.)
• Orderly N(t, t D)/D ----gtp 0, for each t.
• Stationary The joint distribution of N(A1u),
  , N(Aku) does not
• depend on u.

• Notation Calculus
• ?A f(x) dN ?f(xi ), for xi in A.
• ?A dN N(A) of points in A.

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• Intensities (rates) and Compensators
• -------------x-x-----------x-----------
  ----------x---x--------------x------
• 0 t- t t T
• Consider the case where the points are observed
  in time only.
• Nt,u of pts between times t and u.
• Overall rate m(t) limDt -gt 0 ENt, tDt) /
  Dt.
• Conditional intensity l(t) limDt -gt 0 ENt,
  tDt) Ht / Dt,
• where Ht history of N for all times before t.
• If N is orderly, then l(t) limDt -gt 0 PNt,
  tDt) gt 0 Ht / Dt.
• Compensator predictable process C(t) such that
  N-C is a martingale.
• If l(x) exists, then ?ot l(u) du C(t).
• Papangelou intensity lp(t) limDt -gt 0 ENt,
  tDt) Pt / Dt,
• where Pt information on N for all times
  before and after t.

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Intensities (rates) and Compensators ----------
---x-x-----------x----------- ----------x---x----
----------x------ 0 t- t
t T These definitions extend to space and
space-time Conditional intensity l(t,x)
limDt,Dx -gt 0 ENt, tDt) x Bx,Dx Ht / DtDx,
where Ht history of N for all times before
t, and Bx,Dx is a ball around x of size
Dx. Compensator ?A l(t,x) dt dx
C(A). Papangelou intensity lp(t,x)
limDt,Dx -gt 0 ENt, tDt) x Bx,Dx Pt,x /
DtDx, where Pt,x information on N for all
times and locations except (t,x).
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• Some Basic Properties of Intensities
• Fact 1 (Uniqueness). If l exists, then it
  determines the distribution of N.
• (Daley and Vere-Jones, 1988).
• Fact 2 (Existence). For any simple point process
  N, the compensator C
• exists and is unique. (Jacod, 1975)
• Typically we assume that l exists, and use it
  to model N.
• Fact 3 (Kurtz Theorem). The avoidance
  probabilities, PN(A)0 for all measurable sets
  A, also uniquely determine the distribution of N.
  
• Fact 4 (Martingale Theorem). For any predictable
  process f(t),
• E ? f(t) dN E ? f(t) l(t) dt.
• Fact 5 (Georgii-Zessin-Nguyen Theorem). For any
  ex-visible process f(x),
• E ? f(x) dN E ? f(x) lp(x) dx.

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• Some Important Point Process Models
• Renewal process.
• The inter-event times t2 - t1, t3 - t2, t4 -
  t3, etc. are independent and identically
  distributed random variables.
• (Classical density estimation.)
• Ex. Normal, exponential, power-law, Weibull,
  gamma, log-normal.

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• 2) Poisson process.
• Fact 6 If N is orderly and l does not depend on
  the history of the process, then N is a Poisson
  process
• N(A1), N(A2), , N(Ak) are independent, and
  each has the Poisson dist.
• PN(A) j C(A)j exp-C(A) / j!.
• Recall C(A) ?A l(x) dx.

• Stationary (homogeneous) Poisson process l(x)
  m.
• Fact 7 Equivalent to a renewal process with
  exponential inter-event times.
• Inhomogeneous Poisson process l(x) f(x),
• where f(x) is some fixed, deterministic function.

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The Poisson process is the limiting distribution
in many important results Fact 8 (thinning
Westcott 1976) Suppose N is simple, stationary,
ergodic.
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Fact 9 (superposition Palm 1943) Suppose N is
simple stationary.
Then Mk --gt stationary Poisson.
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Fact 10 (translation Vere-Jones 1968 Stone
1968) Suppose N is stationary.
For each point xi in N, move it to xi yi, where
yi are iid. Let Mk be the result of k such
translations.
Then Mk --gt stationary Poisson.
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Fact 11 (rescaling Meyer 1971) Suppose N is
simple and has at most one point on any vertical
line. Rescale the y-coordinates move each point
(xi, yi) to (xi , ?oyi l(xi,y) dy).
Then the resulting process is stationary Poisson.
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• 3) Some cluster models.
• Neyman-Scott process clusters of points whose
  centers are formed from a stationary Poisson
  process. Typically each cluster consists of a
  fixed integer k of points which are placed
  uniformly and independently within a ball of
  radius r around each clusters center.
• Cox-Matern process cluster sizes are random
  independent and identically distributed Poisson
  random variables.
• Thomas process cluster sizes are Poisson, and
  the points in each cluster are distributed
  independently and isotropically according to a
  Gaussian distribution.
• Hawkes (self-exciting) process mothers are
  formed from a stationary Poisson process, and
  each produces a cluster of daughter points, and
  each of them produces a cluster of further
  daughter points, etc. l(t, x) m ?
  g(t-ti, x-xi).
• ti lt t

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• 4) Some inhibition models.
• Matern (I) process first generate points from a
  stationary Poisson process, and then if there are
  any pairs of points within distance d of each
  other, delete both of them.
• Matern (II) process generate a stationary
  Poisson process, then index the points j
  1,2,,n at random, and then successively delete
  any point j if it is within distance d from any
  retained point with smaller index.
• c) Simple Sequential Inhibition (SSI) Keep
  simulating points from a stationary Poisson
  process, deleting any if it is within distance d
  from any retained point, until exactly k points
  are kept.
• Self-correcting process Hawkes process where g
  can be negative l(t, x) m ?
  g(t-ti, x-xi).
• ti lt t

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Poisson (100) Poisson (5050x50y)
Neyman-Scott(10,5,0.05) Cox-Matern(10,5,0.05)
Thomas (10,5,0.05) Matern I (200, 0.05)
Matern II (200, 0.05)
SSI (200, 0.05)
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• In modeling a space-time marked point process,
  usually directly model l(t,x,a).
• For example, for Los Angeles County wildfires
• Windspeed. Relative Humidity, Temperature,
  Precipitation,
• Tapered Pareto size distribution f, smooth
  spatial background m.
• l(t,x,a)
• b1expb2R(t) b3W(t) b4P(t) b5A(t60)
• b6T(t) b7b8 - D(t)2 m(x) g(a).
• Could also include fuel age, wind direction,
  interactions

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r 0.16
(sq m)
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(sq m)
(F)
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• In modeling a space-time marked point process,
  usually directly model l(t,x,a).
• For example, for Los Angeles County wildfires
• Windspeed. Relative Humidity, Temperature,
  Precipitation,
• Tapered Pareto size distribution f, smooth
  spatial background m.
• l(t,x,a)
• b1expb2R(t) b3W(t) b4P(t) b5A(t60)
• b6T(t) b7b8 - D(t)2 m(x) g(a).
• Could also include fuel age, wind direction,
  interactions

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• In modeling a space-time marked point process,
  usually directly model l(t,x,a).
• For example, for Los Angeles County wildfires
• Relative Humidity, Windspeed, Precipitation,
  Aggregated rainfall over previous 60 days,
  Temperature, Date
• Tapered Pareto size distribution f, smooth
  spatial background m.
• l(t,x,a) b1expb2R(t) b3W(t) b4P(t)
  b5A(t60)
• b6T(t) b7b8 - D(t)2 m(x) g(a).
• Could also include fuel age, wind direction,
  interactions

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(Ogata 1998)
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• Simulation.
• Sequential.
• a) Renewal processes are easy to simulate
  generate iid random variables z1, z2, from the
  renewal distribution, and let t1z1,
• t2 z1 z2, t3 z1z2z3, etc.
• b) Reverse Rescaling. In general, can simulate
  a Poisson process with rate 1, and move each
  point (ti, xi) to (ti , yi),
• where xi ?oyi l(ti,x) dx.
• Thinning.
• If m sup l(t, x),
• first generate a Poisson process with rate m,
• and then keep each point (ti, xi) with
  probability l(ti, xi)/m.

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• Summary
• Point processes are random measures
• N(A) of points in A.
• l(t,x) Expected rate around x, given history lt
  time t.
• Classical models are renewal Poisson
  processes.
• For Poisson processes, l(t,x) is deterministic.
• Poisson processes are limits in thinning,
  superposition, translation, and rescaling
  theorems.
• Non-Poisson processes may have clustering
  (Neyman-Scott, Cox-Matern, Thomas, Hawkes) or
  inhibition (MaternI, MaternII, SSI,
  self-correcting).
• Next time How to estimate the parameters in
  these models, and how to tell how well a model
  fits.