Point processes rates are a point process concern

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) Y 0 = 1, Y 1 = 2 - 1 , . . . , YN-1 = N - N-1 intervals 0 e) Y(t) = j (t- j) = d. N(t)/dt (. ): Dirac delta function

Data displays

Point process data can arise from crossings

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

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) = d. N(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) = d. N(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 ) = d. J(t)/dt

stacking

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