Point processes rates are a point process concern

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) Y0 = 1, Y1 = 2 - 1 , ..., YN-1 = N - N-1 intervals  0 e) Y(t) = j (t-j) = dN(t)/dt (.): Dirac delta function

Data displays

Point process data can arise from crossings

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

Stacking

Clustering

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

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 = k+1 - k , non-negative

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

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

Y(t) = j Mj (t-j ) = dJ(t)/dt

stacking

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