1. Point processes rates are a point process concern

2. Point process data points along the line radioactive emissions, nerve cell firings, … 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) Y 0 =  1, Y 1 =  2 -  1,..., Y N-1 =  N -  N-1 intervals  0 e) Y(t) =  j  (t-  j ) = dN(t)/dt  (.): Dirac delta function

3. Data displays

4. Point process data can arise from crossings

5. empirical rate: N(T)/T slope empirical running rate: [N(t+  )-N(t-  )]/2  change?

6. Stacking

7. Clustering

8. Properties of the Dirac delta,  (.). a generalized function, Schwartz distribution  (0) =   (t) = 0, t  0 density function of a r.v., Ƭ, that = 0 with probability 1 cdf H(t) = 0, t<0 H(t) =1, t  0 for suitable g(.), E(g( Ƭ )) = g(.): test function

9. 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 Y k =  k+1 -  k, non-negative

10. Vector-valued point process points of several types N(t) Y(t) = dN(t)/dt

12. Marked point process. {  j, M j } mark M j is associated with time  j examples: earthquakes, insurance If marks real-valued: jump or cumulative process point process if M j = 1

13. Y(t) =  j M j  (t-  j ) = dJ(t)/dt

14. stacking

15. Sampled time series, hybrid. X(  j ) Point processes are very, very basic in science particle vs. wave theory of light Computing. can replace {  j } by t.s. Y k = dN(t) with k = [  j /dt] [.]: integral part