Point processes Some special cases 1 a Homogeneous

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)

Debris. Env. gif

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 d. N(t) = d. M(Q(t))q(t)dt Random translation i ( j + uj)

Probability generating functional. G[ ] = E{ exp(log (t)) d. N(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) = a(t-u)d. N(u) for some suitable a(. ) - < t <

(Stationary) interval functions. Y(I), I an interval {Y(I 1+u), . . . , Y(IK+u)} ~ {Y(I 1), . . . , Y(IK)} Y(I) = I [(exp{it }-1)/i ]d. ZY( ) cov{d. ZY( ), d. ZY( )| = ( - )f. YY( )d d Cov(Y(I), Y(J)} = I J cov{d. Y(s), d. Y(t)} e. g. m. p. p.