Point processes. Some special cases.

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Presentation transcript:

Point processes. Some special cases. 1. a). (Homogeneous) Poisson. Rate  N(t), t in R Approaches.

Can use to check homogenious Poisson assumption Examples Brillinger, Bryant, Segundo Biological Cybernetics 22,213-228 (1976)

1. b) Inhomogeneous Poisson.

Time substitution. N(t) Poisson rate (t)

Zero probability function. Characterizes simple N (I) = Pr{N(I;)=0}, for all bounded Baire For Poisson, (t) (I) =exp{-(I)} capital  M(I) = I (t)dt Risk analysis. Poisson events, rate , period T Prob{event in T} = 1 - exp{-|T|}  |T|

2. Doubly stochastic (stationary) Poisson.

3. Self-exciting process.

4. Renewal process.

Waiting time paradox. Homogeneous Poisson, Y's i.i.d. exponentials Forward recurrence time - waiting time Backward recurrence time Time between events all exponential parameter 

5. Cluster process.

Operations on point processes. To produce other p.p.'s Superposition M(t) + N(t) Thinning i Ij (t-j), Ij=0 or 1 Time substitution N(t) = M(Q(t)) Q monotonic nondecreasing dN(t) = dM(Q(t))q(t)dt Random translation i (j + uj)

Probability generating functional. G[] = E{exp(log (t)) dN(t)} For Poisson G[] = exp{((t)-1)(t)dt}

Limiting cases. Poisson 1. Superpose p.p. 2. Rare events 3. Deletion

Can make a p.p. into a t.s. Y(t) =  a(t - j) - < t <  =  a(t-u)dN(u) for some suitable a(.)

(Stationary) interval functions. Y(I), I an interval {Y(I1+u),...,Y(IK+u)} ~ {Y(I1),...,Y(IK)} Y(I) = I [(exp{it}-1)/i]dZY() cov{dZY(),dZY()| = (-)fYY()d d Cov(Y(I),Y(J)} = I J cov{dY(s),dY(t)} e.g. m.p.p.