Point processes

definitions, displays, examples, representations, algebra, linear quantities

Point process data

points along the line

radioactive emissions, nerve cell firings, wildfires, accidents, …

Describe by: a) 0 1 < 2 < ... < N < T in [0,T)

b) N(t) = #{ j | 0 j <t}, a step function

c) counting measure N(I) =

d) Y0 = 1, Y1 = 2 - 1 , ..., YN-1 = N - N-1

intervals 0

e) Y(t) = j (t-j) = dN(t)/dt

(.): Dirac delta function definition later

I tdN )(

Data displays

cat data

Manaus. Confluence of Rio Negro and the Solimoes

Point process data can arise from crossings

empirical rate: (N(T) - N(0))/T

slope

empirical running rate: [N(t+)-N(t- )]/2 , > 0, small

change?

Clustering

Stacking

Refractory period

•1812 December 8 Wrightwood, San Juan Capistrano

•1812 December 21 Lompoc

•1852 November 29 Volcano Lake

•1857 January 9 Fort Tejon

•1872 March 26 Owens Valley

•1892 February 23 Imperial Valley

•1899 December 25 San Jacinto

•1906 April 18 San Francisco

•1918 April 21 San Jacinto

•1925 June 29 Santa Barbara

•1927 November 24 Lompoc

•1933 March 10 Long Beach

•1940 May 18 Imperial Valley

•1942 October 21 Fish Creek Mountains

•1947 April 10 Manix

•1952 July 21 Kern County

•1952 August 22 Bakersfield (aftershock)

•1968 April 8 Borrego Mountain

•1971 February 9 San Fernando

•1979 October 15 Imperial Valley

•1989 October 17 Loma Prieta

•1992 April 22 Joshua Tree

•1992 June 28 Landers

•1994 January 17 Northridge

California earthquakes

magnitude >= 7

1 D trajectories.

El rapido going north on San Pablo Avenue, Berkeley

weekdays October 2008 6:10 to 19:30, approx every 20 min

velocity in seconds

distance = velocity time, dx = vdt

Where to situate holding points? D. Singham

Properties of the Dirac delta, (.).

a generalized function, Schwartz distribution

(0) = (t) = 0, t 0 1)( dtt

density function of a r.v.,Ƭ, that = 0 with probability 1

cdf (Heavyside function) H(t) = 0, t<0 H(t) =1, t 0

for suitable g(.), E(g(Ƭ)) = )0()()( gdtttg

g(.): test function

A calculus.

Y(t) = j (t-j) = dN(t)/dt

T

j j tdNtggdttYtg0

)()()()()( = N(g)

Can treat a point process as an "ordinary" time series using

orderly: points are isolated no twins

In survival analysis just 1 point

Might work with interval series Yk = k+1 - k , non-negative

Vector-valued point process

points of several types

N(t)

Y(t) = dN(t)/dt instantaneous vector rate

Aplysia, sea slug, sea hare

Marked point process.

{j , Mj }

mark Mj is associated with time j

examples: earthquakes, insurance

If marks real-valued: jump or cumulative process

point process if Mj = 1

Y(t) = j Mj (t-j ) = dJ(t)/dt

I inI jjMtdJIJ )()(

t sdJtJ 0 )()(

stacking pattern? no pattern?

Observation times

-3,-1,1,4,8,10,12,

15,17,19,24,26,28,30,

35,38,40,41,43,45,48,

49,50,52,53,54,55

Argentina vs. Serbia Montenegro

Locations of the soocer ball

Sampled time series, hybrid. X(j )

Computing.

can replace orderly {j} by t.s.

Yt = 1 if t = [j/], small

= 0 otherwise, t = 0, 1, 2, ...

[u]: integral part of u

0-1 process, Bernoulli

Point processes are very, very basic to science and engineering

particle vs. wave theory of light

particle physics

molecular structure of matter

neuron firing - calcium ions

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