1. Point processes definitions, displays, examples, representations, algebra, linear quantities

2. 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) 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 definition later

3. Data displays cat data

4. Manaus. Confluence of Rio Negro and the Solimoes

5. Point process data can arise from crossings

6. empirical rate: (N(T) - N(0))/T slope empirical running rate: [N(t+  )-N(t-  )]/2 ,  > 0, small change?

7. Clustering

8. Stacking Refractory period

9. 1812 December 8 Wrightwood, San Juan CapistranoWrightwood, San Juan Capistrano 1812 December 21 LompocLompoc 1852 November 29 Volcano LakeVolcano Lake 1857 January 9 Fort TejonFort Tejon 1872 March 26 Owens ValleyOwens Valley 1892 February 23 Imperial ValleyImperial Valley 1899 December 25 San JacintoSan Jacinto 1906 April 18 San FranciscoSan Francisco 1918 April 21 San JacintoSan Jacinto 1925 June 29 Santa BarbaraSanta Barbara 1927 November 24 LompocLompoc 1933 March 10 Long BeachLong Beach 1940 May 18 Imperial ValleyImperial Valley 1942 October 21 Fish Creek MountainsFish Creek Mountains 1947 April 10 ManixManix 1952 July 21 Kern CountyKern County 1952 August 22 Bakersfield (aftershock)Bakersfield 1968 April 8 Borrego MountainBorrego Mountain 1971 February 9 San FernandoSan Fernando 1979 October 15 Imperial ValleyImperial Valley 1989 October 17 Loma PrietaLoma Prieta 1992 April 22 Joshua TreeJoshua Tree 1992 June 28 LandersLanders 1994 January 17 NorthridgeNorthridge California earthquakes magnitude >= 7

10. 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

11. Where to situate holding points?D. Singham

12. 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 (Heavyside function) H(t) = 0, t<0 H(t) =1, t  0 for suitable g(.), E(g( Ƭ )) = g(.): test function

13. A calculus. 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 work with interval series Y k =  k+1 -  k, non-negative

14. Vector-valued point process points of several types N(t) Y(t) = dN(t)/dt instantaneous vector rate

16. Aplysia, sea slug, sea hare

17. 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

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

19. stacking pattern? no pattern?

20. 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 )

21. Computing. can replace orderly {  j } by t.s. Y t = 1 if t = [  j /  ],  small = 0 otherwise, t = 0,  1,  2,... [u]: integral part of u 0-1 process, Bernoulli

22. Point processes are very, very basic to science and engineering particle vs. wave theory of light particle physics molecular structure of matter neuron firing - calcium ions.....