Point processes

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Point processes rates are a point process concern
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Point process data
points along the line radioactive
emissions, nerve cell firings,
Describe by a) 0? ?1 lt ?2 lt ... lt ?N lt T in
0,T) b) N(t) j 0?
?j ltT, 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
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Data displays
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Point process data can arise from crossings
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empirical rate N(T)/T slope
empirical running rate N(t?)-N(t- ?)/2
? change?
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Stacking
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Clustering
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Properties of the Dirac delta, ?(.). a
generalized function, Schwartz distribution
?(0) ? ?(t) 0, t ? 0
density function of a r.v.,?, that 0 with
probability 1 cdf H(t) 0, tlt0
H(t) 1, t ? 0 for suitable g(.), E(g(?))

g(.) test function
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Y(t) ?j ?(t-?j) dN(t)/dt
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 analyze
interval series Yk ?k1 - ?k , non-negative
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Vector-valued point process points of
several types N(t) Y(t) dN(t)/dt
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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
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Y(t) ?j Mj ?(t-?j ) dJ(t)/dt
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stacking
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Sampled time series, hybrid. X(?j )
Computing. can replace ?j by t.s. Yk
dN(t) with k ?j/dt . integral
part
Point processes are very, very basic in science
particle vs. wave theory of light