Segue from time series to point processes.

Y = 0,1 E(Y) = Prob{Y = 1} (Y1, Y2 ) E(Y1 Y2} = Prob{( Y1 ,Y2) = (1,1)}

{Y(t)} case.

mean level: c Y(t) = Prob{Y(t) = 1}

product moment: Prob{Y(t1)=1, Y(t2) = 1} = E{Y(t 1)Y(t2)} Naïve interpretations

Stationary case : cYY(t1 – t 2)

Can use acf, ccf, … i,e, stationary series R functions

Can approximate point process data {τj , j=1,…,J } by a 0-1 tt.s. data

points isolated, pick small Δt

time series Y(t/ Δt) T = J/Δt can be large

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) ...

Dirac delta.

Picture a r.v. , U, = 0 with probability 1

then E{g(U)} = g(0)

Picture a r.v. , V, with distribution N(0, σ 2), σ small

then E{g(V)}approaches g(0) as σ decreases, g cts at 0

Picture that U has a density δ(u), a generalized function

then E{g(U)} = ∫ g(u) δ(u) du

Properties: ∫ δ(u) du = 1, δ(u) = 0 for u ≠ 0

dH(u) = δ(u) du for H the Heavyside function

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: exp{cos(t+)}

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 ), 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

= Prob{dN(t1)=1,...,dN(tk)=1}

E{N(t) (k)} = ∫0t… ∫0

t pN...N (t1,...,tk)dt1 ...dtk

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 <

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 ()