Some Current Problems in Point Process Research: 1. Prototype point processes 2. Nonsimple point pro

1
Some Current Problems in Point Process
Research 1. Prototype point processes 2.
Non-simple point processes3. Voronoi diagrams
2

• Global Earthquake Data
• Local e.q. catalogs tend to have problems, esp.
  missing data.
• 1977 Harvard (global) catalog created.
• Considered the most complete. Errors best
  understood.
• A collection of aftershock sequences
• Harvard Catalog, 1/1/77 to 3/1/03
• Shallow events only (depth lt 70km)
• Mw 7.5 to 8.0
• Aftershocks Mw gt 5.5, within 100km, 0.133 days
  - 2 yrs.
• No Mw 7.5 within 200km in previous 2 yrs.
• No Mw 8.0 w/in 400km within 4 yrs (Molchan et
  al., 1997)
• 49 mainshocks, avg. 5.47 aftershocks, SD 4.3.
  

3
(No Transcript)
4

• 1. Prototypes.
• Some motivating questions
• A) What does a typical aftershock sequence look
  like?
• B) How can we tell if a particular sequence is an
  outlier?
• C) How can we group aftershock sequences into
  clusters based on the similarity of their
  features?

5
A) What does the typical aftershock sequence look
like? e.g. What is typically observed after an
eq of Mw 7.5 - 8.0?

• Modified Omori K/(tc)p

6
A) What does the typical aftershock sequence look
like? e.g. What is typically observed after an
eq of Mw 7.5 - 8.0?

• Modified Omori K/(tc)p
• May desire a prototype
• a point pattern of min. distance to those
  observed.
• Requires distance between point patterns.

7
(No Transcript)
8
Victor-Purpura (1997) distance

• Given two point patterns
• Match each point in A to the nearest point in B
  and record the horizontal distance moved (penalty
  pm1 per unit moved)
• Delete excess points (with penalty pa)

9
Considerations
10
Calculating the distance between two point
patterns

• Reduces to which points are kept and which are
  removed.

• A point gt 2pa /pm from its nearest neighbor is
  automatically removed.

• Mutual nearest neighbors within 2pa/pm are
  automatically kept.

11
Prototype Point Pattern

• Defined such that the sum of distances from the
  prototype to all observed point patterns in the
  data set is minimized.
• Represents a typical observation.

12
Some properties of the prototype

• Prototype is not necessarily unique.
• There exists a prototype pattern composed
  entirely of points in the dataset.
• In fact, a prototype can be found such that each
  point it contains is the median of its associated
  points in distance calculations.

13
Uses Data summary, outlier identification,
clustering,
14
(No Transcript)
15
(No Transcript)
16
(No Transcript)
17
Clusters of aftershock sequences

• Distance of each aftershock sequence to the
  prototypes for time and magnitude

18
Cluster Map
19

• With multidimensional point processes (time, mw,
  location)
• No simple sequential pairing.
• Mutual nearest neighbors are kept.
• There exists a prototype consisting only of
  points whose coordinates are medians of
  coordinates of associated pts.

20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
2. Non-simple point processes. Simple
point processes are characterized by the
conditional intensity, l(t). But what about
non-simple point processes?
24
Poisson process with intensity l 2
Poisson process with intensity l 1, but with
each point doubled
Both have the same conditional intensity! (l 2)

• Two types of simplicity, for multi-dimensional
  point processes
• 1) Completely simple
• No two points overlap exactly the same triple
  (t,x,a).
• 2) Simple ground process
• No two points at exactly the same time.
• Multi-dimensional marked point processes are
  only uniquely characterized
• by the conditional intensity l(t,x,m) if they
  have simple ground process.

25
Multi-dimensional marked point processes are
only uniquely characterized by the conditional
intensity l(t,x,m) if they have simple ground
process.
m1 m2
m1 m2
t
t
2 independent Poisson processes A
Poisson process with l 1, each with intensity l
1. and an exact copy.
l(t, m1) l(t, m2) 1. l(t, m1) l(t, m2)
1.
How can one model non-simple point processes?
26
How can one model non-simple marked point
processes?
m1 m2 m3
m1m2m3m12m13m23
t
t

• Consider an extended mark space, consisting of
  pairs (and triplets, quadruplets, etc.) of marks
• Z m1, m2, m3, m12m1, m2, m13, m23.
• The resulting point process will have simple
  ground process. (Daley Vere-Jones 1988, p208)
• The conditional intensity l(t, m) of the
  resulting process can be written in terms of the
  original conditional intensity l(t, m)
• For instance, l(t, m2) l(t,m2) l(t,
  m12) l(t, m23).
• Can have models where l(t,mij) l(t,mi)
  l(t,mj). (Schoenberg 2005)

27
3. Voronoi Tessellations.
Given a collection of points p1, p2, , divide
the space into cells C1 , C2 , , such that cell
Ci consists of all locations closer to pi than
to any of the other points pj. Ci x x -
pi lt x - pj for all j.
28
(No Transcript)
29
(No Transcript)
30
(No Transcript)
31
(No Transcript)
32
(No Transcript)
33
Southern California Earthquake Center (SCEC) data

• Lat 32-37 (733 km)
• Lon (-114, -112) (556 km)
• Time (1/1/1984 - 17/6/2004)
• Mag Mo gt 2.
• n 6796.
• Errors in the catalog
• Missing earthquakes, esp in clusters.
• Discrimination problems.
• Location projection errors.

34
Many models for the cell characteristics were
fitted Frechet, gamma, lognormal, exponential,
Pareto, tapered Pareto. Pareto F(x) 1 -
(a/x)b. Tapered Pareto F(x) 1 - (a/x)b
e(a-x)/q.
35
Q-Q plots for the tapered Pareto F(x) 1 -
(a/x)b e(a-x)/q.
Cell area
Cell perimeter
36

• Summary and Open Questions
• Prototypes may be useful data summaries for point
  
• processes, and for clustering, identifying
  outliers, etc.
• Prototypes for particular models? Applications?
• Non-simple point processes can be viewed as
  simple
• on an extended mark space. More non-simple
  models?
• Applications?
• Cell sizes in Voronoi tessellations of earthquake
  data
• seem to be tapered Pareto distribution (like many
  other
• features of earthquakes). Why? What is the
  theoretical
• cell size distribution for a particular model?
  Other
• applications?