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,..., Y N-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 Y k = k+1 - k, non-negative
Vector-valued point process points of several types N(t) Y(t) = dN(t)/dt
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
Y(t) = j M j (t- j ) = dJ(t)/dt
stacking
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