point processes on the line . nerve firing
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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(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
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
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) ...
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 )()}({
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)()[({
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
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
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
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
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
)(
Cumulant density of order k.
t1,...,tk distinct
cum{dN(t1),...,dN(tk)}
= qN...N (t1 ,...,tk)dt1 ...dtk
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
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
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
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
Examples.
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
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
)()()(2)]()([)()(
)()(2)]()([)(
/)}]()({ )}()({)}()({)}()({[
/)}()({)()(
uvpuvpvppuvuvvptusp
utsptspptusutstsp
dsdtutdMusdMEtdMusdMEutdMsdMEtdMsdME
dsdttdNsdNEtsppts
MMMMMMMNN
MM
MMMMMNN
NNN
Taking "E" again,
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
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 ()
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
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)
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.
Empirical examples.
sea hare
Muscle spindle
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()
Partial coherency. Trivariate process {M,N,O}
]}||1][||1{[/][ 22| ONMOONMOMNOMN ffffff
“Removes” the linear time invariant effects of O from M and N
