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Point processes on the line . Nerve firing.

Point processes on the line . Nerve firing. Stochastic point process . Building blocks Process on R {N(t)}, t in R, with consistent set of distributions Pr{N(I 1 )=k 1 ,..., N(I n )=k n } k 1 ,...,k n integers  0 I's Borel sets of R.

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Point processes on the line . Nerve firing.

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  1. Point processes on the line. Nerve firing.

  2. Stochastic point process. Building blocks Process on R {N(t)}, t in R, with consistent set of distributions Pr{N(I1)=k1 ,..., N(In)=kn } k1 ,...,kn integers  0 I's Borel sets of R. Consistentency example. If I1 , I2 disjoint Pr{N(I1)= k1 , N(I2)=k2 , N(I1 or I2)=k3 } =1 if k1 + k2 =k3 = 0 otherwise Guttorp book, Chapter 5

  3. Points: ... -1  0  1  ... discontinuities of {N} N(t) = #{0 < j  t} Simple: j k if j  k points are isolated dN(t) = 0 or 1 Surprise. A simple point process is determined by its void probabilities Pr{N(I) = 0} I compact

  4. Conditional intensity. Simple case History Ht = {j  t} Pr{dN(t)=1 | Ht } = (t:)dt  r.v. Has all the information Probability points in [0,T) are t1 ,...,tN Pr{dN(t1)=1,..., dN(tN)=1} = (t1)...(tN)exp{- (t)dt}dt1 ... dtN [1-(h)h][1-(2h)h] ... (t1)(t2) ...

  5. Parameters. Suppose points are isolated dN(t) = 1 if point in (t,t+dt] = 0 otherwise 1. (Mean) rate/intensity. E{dN(t)} = pN(t)dt = Pr{dN(t) = 1} j g(j) =  g(s)dN(s) E{j g(j)} =  g(s)pN(s)ds Trend: pN(t) = exp{+t} Cycle:  + cos(t+)  0

  6. Product density of order 2. Pr{dN(s)=1 and dN(t)=1} = E{dN(s)dN(t)} = [(s-t)pN(t) + pNN (s,t)]dsdt Factorial moment

  7. Autointensity. Pr{dN(t)=1|dN(s)=1} = (pNN (s,t)/pN (s))dt s  t = hNN(s,t)dt = pN (t)dt if increments uncorrelated

  8. Covariance density/cumulant density of order 2. cov{dN(s),dN(t)} = qNN(s,t)dsdt st = [(s-t)pN(s)+qNN(s,t)]dsdt generally qNN(s,t) = pNN(s,t) - pN(s) pN(t) st

  9. Identities. 1. j,k g(j ,k ) =  g(s,t)dN(s)dN(t) Expected value. E{ g(s,t)dN(s)dN(t)} =  g(s,t)[(s-t)pN(t)+pNN (s,t)]dsdt =  g(t,t)pN(t)dt +  g(s,t)pNN(s,t)dsdt

  10. 2. cov{ g(j ),  h(k )} = cov{ g(s)dN(s),  h(t)dN(t)} =  g(s) h(t)[(s-t)pN(s)+qNN(s,t)]dsdt =  g(t)h(t)pN(t)dt +  g(s)h(t)qNN(s,t)dsdt

  11. Product density of order k. t1,...,tk all distinct Prob{dN(t1)=1,...,dN(tk)=1} =E{dN(t1)...dN(tk)} = pN...N (t1,...,tk)dt1 ...dtk

  12. Proof of Central Limit Theorem via cumulants in i.i.d. case. Normal distribution facts. 1. Determined by its moments 2. Cumulants of order  2 identically 0 Y1, Y2, ... i.i.d. mean 0, variance 2, all moments, E{Yk} k=1,2,3,4,... existing Sn = Y1 + Y2 + ... + Yn E{Sn } = 0 var{ Sn} = n 2 cumr Sn = n r cumr Y = cum{Y,...,Y} cumr {Sn / n} = n r / nr/2  0 for r = 3, 4, ...  2 r = 2 as n  

  13. Cumulant density of order k. t1,...,tk distinct cum{dN(t1),...,dN(tk)} = qN...N (t1 ,...,tk)dt1 ...dtk

  14. Stationarity. Joint distributions, Pr{N(I1+t)=k1 ,..., N(In+t)=kn} k1 ,...,kn integers  0 do not depend on t for n=1,2,... Rate. E{dN(t)=pNdt Product density of order 2. Pr{dN(t+u)=1 and dN(t)=1} = [(u)pN + pNN (u)]dtdu

  15. Autointensity. Pr{dN(t+u)=1|dN(t)=1} = (pNN (u)/pN)du u  0 = hN(u)du Covariance density. cov{dN(t+u),dN(t)} = [(u)pN + qNN (u)]dtdu

  16. "Estimation of the second-order intensities of a bivariate stationary point process," Journal of the Royal Statistical Society B Vol. 38 (1976), pp. 60-66

  17. Algebra/calculus of point processes. Consider process {j, j+u}. Stationary case dN(t) = dM(t) + dM(t+u) Taking "E", pNdt = pMdt+ pMdt pN = 2 pM

  18. Taking "E" again,

  19. Association. Measuring? Due to chance? Are two processes associated? Eg. t.s. and p.p. How strongly? Can one predict one from the other? Some characteristics of dependence: E(XY)  E(X) E(Y) E(Y|X) = g(X) X = g (), Y = h(),  r.v. f (x,y)  f (x) f(y) corr(X,Y)  0

  20. Bivariate point process case. Two types of points (j ,k) Crossintensity. Prob{dN(t)=1|dM(s)=1} =(pMN(t,s)/pM(s))dt Cross-covariance density. cov{dM(s),dN(t)} = qMN(s,t)dsdt no ()

  21. Mixing. cov{dN(t+u),dN(t)} small for large |u| |pNN(u) - pNpN| small for large |u| hNN(u) = pNN(u)/pN ~ pN for large |u|  |qNN(u)|du <  See preceding examples

  22. The Fourier transform.  regularity conditions Functions, A(), - <  <   |A()|d finite FT. a(t) =  exp{it)A()d Inverse A() =(2)-1 exp{-it} a(t) dt unique C()=  A() +  B() c(t) =  c(t) +  b(t) 2  1

  23. Convolution (filtering). d(t) =  b(t-s) c( s)ds D() = B()C() Discrete FT. a(t) =  exp{-i2ts/T} A(2s/T) s, t = 0,1,...,T-1 A(2s/T) =T-1 exp {i2st/T) a(t) FFTs exist

  24. Dirac delta.  H() () d = H(0)  exp {it}() d  = 1 inverse () = (2)-1 exp {-it}dt

  25. Power spectral density. frequency-side, , vs. time-side, t /2 : frequency (cycles/unit time) Non-negative Unifies analyses of processes of widely varying types

  26. Examples.

  27. Spectral representation. stationary increments - Kolmogorov

  28. Algebra/calculus of point processes. Consider process {j, j+u}. Stationary case dN(t) = dM(t) + dM(t+u) Taking "E", pNdt = pMdt+ pMdt pN = 2 pM

  29. Taking "E" again,

  30. Association. Measuring? Due to chance? Are two processes associated? Eg. t.s. and p.p. How strongly? Can one predict one from the other? Some characteristics of dependence: E(XY)  E(X) E(Y) E(Y|X) = g(X) X = g (), Y = h(),  r.v. f (x,y)  f (x) f(y) corr(X,Y)  0

  31. Bivariate point process case. Two types of points (j ,k) Crossintensity. Prob{dN(t)=1|dM(s)=1} =(pMN(t,s)/pM(s))dt Cross-covariance density. cov{dM(s),dN(t)} = qMN(s,t)dsdt no ()

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