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Point processes rates are a point process concern

Explore the concept of point processes, their properties, and their applications in various fields. Understand the formation, analysis, and interpretation of point process data points and their relevance in scientific research.

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Point processes rates are a point process concern

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  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) Y0 = 1, Y1 = 2 - 1 , ..., YN-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 Yk = k+1 - k , non-negative

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

  11. 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

  12. Y(t) = j Mj (t-j ) = dJ(t)/dt

  13. stacking

  14. 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

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