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+) 0
t
N 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 ), 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
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
)(
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
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
"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
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)]()([)(
/)}]()({
)}()({)}()({)}()({[
/)}()({)()(
uvpuvpvppuvuvvp
tusp
utsptspptusutstsp
dsdtutdMusdME
tdMusdMEutdMsdMEtdMsdME
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 ()
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
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
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
Dirac delta.
H() () d = H(0)
exp {it}() d = 1
inverse
() = (2)-1 exp {-it}dt
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
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
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)]()([)(
/)}]()({
)}()({)}()({)}()({[
/)}()({)()(
uvpuvpvppuvuvvp
tusp
utsptspptusutstsp
dsdtutdMusdME
tdMusdMEutdMsdMEtdMsdME
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 ()
