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
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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+) 0
t
N 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 ), 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
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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
kkk
ttk dtdtttptNE ...),...,(})({ 1100
)(
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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
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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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"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
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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)]()([)(
/)}]()({
)}()({)}()({)}()({[
/)}()({)()(
uvpuvpvppuvuvvp
tusp
utsptspptusutstsp
dsdtutdMusdME
tdMusdMEutdMsdMEtdMsdME
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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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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The Fourier transform. regularity conditions
Functions, A(), - < <
|A()|d finite
FT. a(t) = exp{it)A()d
Inverse A() =(2)-1 exp{-it} a(t) dt
unique
C()= A() + B()
c(t) = c(t) + b(t)
2 1
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Convolution (filtering).
d(t) = b(t-s) c( s)ds
D() = B()C()
Discrete FT.
a(t) = exp{-i2ts/T} A(2s/T) s, t = 0,1,...,T-1
A(2s/T) =T-1 exp {i2st/T) a(t)
FFTs exist
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Dirac delta.
H() () d = H(0)
exp {it}() d = 1
inverse
() = (2)-1 exp {-it}dt
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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
dZiit
tN
})(){(},cov{
increments orthogonal
)()()}(),(cov{
order of spectrumcumulant
...),...,()...()}(),...,({
)()}({
)()(dZ valued,-complex random, :
111...11
N
YX
NNNN
KKNNKKNN
N
NN
YXEYX
ddfdZdZ
K
ddfdZdZcum
ddZE
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)]()([)(
/)}]()({
)}()({)}()({)}()({[
/)}()({)()(
uvpuvpvppuvuvvp
tusp
utsptspptusutstsp
dsdtutdMusdME
tdMusdMEutdMsdMEtdMsdME
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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