segue from time series to point processes. y = 0,1 e(y) = prob{y = 1}
DESCRIPTION
Segue from time series to point processes. Y = 0,1 E(Y) = Prob{Y = 1} (Y 1 , Y 2 ) E(Y 1 Y 2 } = Prob{( Y 1 ,Y 2 ) = (1,1)} {Y(t)} case. mean level: c Y (t) = Prob{Y(t) = 1} - PowerPoint PPT PresentationTRANSCRIPT
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 ()