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(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