Post on 28-Dec-2015
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 )(
•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
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
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
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