Point processes on the line . Nerve firing.
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![Page 1: Point processes on the line . Nerve firing.](https://reader035.fdocuments.in/reader035/viewer/2022070502/56813650550346895d9dd320/html5/thumbnails/1.jpg)
Point processes on the line. Nerve firing.
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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
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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
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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) ...
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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+)
tN dssptNE 0 )()}({
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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
tvu
NN dudvvuptNtNE,0
),(]}1)()[({
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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
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Covariance density/cumulant density of order 2.
cov{dN(s),dN(t)} = qNN(s,t)dsdt st
= [(s-t)pN(s)+qNN(s,t)]dsdt generally
qNN(s,t) = pNN(s,t) - pN(s) pN(t) st
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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
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2. cov{ g(j ), g(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
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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
kkkttk dtdtttptNE ...),...,(})({ 1100
)(
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Cumulant density of order k.
t1,...,tk distinct
cum{dN(t1),...,dN(tk)}
= qN...N (t1 ,...,tk)dt1 ...dtk
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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
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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
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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
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Power spectral density. frequency-side, , vs. time-side, t
/2 : frequency (cycles/unit time)
|| largefor 21~
)(}exp{21
21
)]()(}[exp{21)(
N
NNN
NNNNN
p
duuquip
duuqpuuif
Non-negative
Unifies analyses of processes of widely varying types
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Examples.
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Spectral representation. stationary increments - Kolmogorov
)(}exp{/)(
)(1}exp{)(
N
N
dZitdttdN
dZiittN
})(){(},cov{ increments orthogonal
)()()}(),(cov{order of spectrumcumulant
...),...,()...()}(),...,({)()}({
)()(dZ valued,-complex random, :
111...11
N
YX
NNNN
KKNNKKNN
N
NN
YXEYX
ddfdZdZK
ddfdZdZcumddZE
dZZ
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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
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)()()(2)]()([)()(
)()(2)]()([)(
/)}]()({ )}()({)}()({)}()({[
/)}()({)()(
uvpuvpvppuvuvvptusp
utsptspptusutstsp
dsdtutdMusdMEtdMusdMEutdMsdMEtdMsdME
dsdttdNsdNEtsppts
MMMMMMMNN
MM
MMMMMNN
NNN
Taking "E" again,
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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
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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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Frequency domain approach. Coherency, coherence
Cross-spectrum.
duuquif MNMN )(}exp{21)(
Coherency.
R MN() = f MN()/{f MM() f NN()}
complex-valued, 0 if denominator 0
Coherence
|R MN()|2 = |f MN()| 2 /{f MM() f NN()|
|R MN()|2 1, c.p. multiple R2
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where
A() = exp{-iu}a(u)du
fOO () is a minimum at A() = fNM()fMM()-1
Minimum: (1 - |RMN()|2 )fNN()
0 |R MN()|2 1
AAfAfAfff MMNMMNNNOO
Proof. Filtering. M = {j }
a(t-v)dM(v) = a(t-j )
Consider
dO(t) = dN(t) - a(t-v)dM(v)dt, (stationary increments)
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Proof.
0 Take
0
sderivative second andfirst Consider
1
1
MNMMNMNN
MMNM
OO
MMNMMNNNOO
ffffffA
f
AAfAfAfff
Coherence, measure of the linear time invariant association of the components of a stationary bivariate process.
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Empirical examples.
sea hare
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Muscle spindle
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Spectral representation approach.
b.v. of ,)()()}(),(cov{
)(}exp{/)(
)(}exp{/)(
NMMNNM
N
M
FddFdZdZ
dZitdttdN
dZitdttdM
Filtering.
dO(t)/dt = a(t-v)dM(v) = a(t-j )
= exp{it}dZM()
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Partial coherency. Trivariate process {M,N,O}
]}||1][||1{[/][ 22| ONMOONMOMNOMN ffffff
“Removes” the linear time invariant effects of O from M and N
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