Los Angeles polution mortality study

Shumway at al (1988) Environ. Res. 45, 224-241

Los Angeles County:

average daily cardiovascular mortality

particulate polution

(six day smoothed averages)

n = 508, 1970-1979

Bivariate time series

Correlation is largest at 8 weeks lag, but ...

acf

ccf

Stat 153 - 29 Oct 2008 D. R. Brillinger

Chapter 9 - Linear Systems

Λ[λ1x1 + λ2x2](t) = λ1Λ[x1](t) + λ2 Λ[x2](t)

Time invariant.

Λ[Bτx](t) = BτΛ[x](t)

Λ[Bτx](t) = y(t- τ) if Λ[x](t) = y(t)

Linear.

Example. How to describe ?

-kkh ktt xy

Common in nature

System identification

Yt = hk Xt-k + Nt

Is there a relationship?

Estimate h, H given {x(t),y(t)}, t=0,...,N

Predict Y from X

Control

Studying causality

Studying delay

…

-kkh ktt xy

{hk}: impulse response

Define δk= 1 if k = 0 and = 0 otherwise, then

-k ktkt hh

hk = 0 k<0: causal/physically realizable

One way

Transfer / frequency response function

k

ki

kehH )(

G(ω ) = |H(ω)|: gain

φ(ω) = arg{H(ω)}: phase

complex-valued

H(ω+2π) = H(ω)

H(-ω) = H(ω)* complex conjugate

Another way

Example.

11 31

31

31 tttt xxxy

H(ω) = (1+2cos ω)/3

If input x(t) = exp{-iωt}, then output

Fundamental property of a linear time invariant system

cosinusoids are carried into cosinusoids of the same frequency

frequencies are not mixed up

If

then

j

ti

jjjeAtx )( )(

j

ti

jjjjeHAty )()( )(

(useful approximation)

y(t) = H(ω)exp{-iωt}

60 Hz can creep into lab measurements

Ideal low-pass filter

H(ω ) = 1 |ω| Ω

= 0 otherwise, |ω| π

||

)( )(j

jti

jj eAty

Ideal band-pass filter

H(ω ) = 1 |ω-ω0| Δ

= 0 otherwise

||

0

)( )(

j

jti

jj eAty

Construction of general filter

t

tp ptN

pix

Nc ,...2,1,0 },

2exp{

1

Inverse

p

pt tN

picx N1,..., t},2

exp{

Filtered series

via fft( )

}2

exp{)2

( tN

pi

Np

Hcyp

pt

Bandpass filtering of Vienna monthly temperatures, 1775-1950

Bank of bandpass filters

Taper, form g(t/(N+1)xt, t=1,...,N

e.g. g(u) = (1 + cos πu)/2

Pure lag filter.

y = xt-τ

hk = 1 if k = τ

= 0 otherwise

H(ω ) = exp{-iωτ}

φ(ω) = -ωτ mod(2π)

G(ω) = 1

Product sales and a leading indicator series

Box and Jenkins

BJsales

acf

ccf

Work with differences

-prewhitening

-kkh ktt xy

The effect of filtering on second-order parameters

γYY (k) = Σ Σ hi hj γXX (k-j+i)

Proof.

Cov{yt+k ,xt } =

fYY(ω) = |H(ω)|2 fXX(ω)

Proof.

fYY (ω)= Σ γYY(k) exp{-iωk}]/π

= ΣΣ Σ hi hj γXX (k-j+i) exp{-iωk}]/π

Interpretation of fXX(ω0)

γYY (0) = var Yt = fYY(ω)dω

= |H(ω)|2 fXX(ω)dω

f(ω0) if H(.) narrow bandpass centered at ω0

0ki )(e )( dfk YYYY

Provides an estimate of f(ω0)

ave{xt (ω0)2}

Remember

The narrower the filter the less biased the estimate, generally.

The coherence may be estimated via estimate of

corr{xt (ω),yt(ω)}2

and …

Hilbert transform

fYY(ω) = |H(ω)|2 fXX(ω)

suggests

)(ˆ|)(|)(ˆ 2 YYXX fHf

e.g. fit AR(p), p large

Another estimate

Spectral density of an MA(q)

/|)(| )( 22

Z

i

YY ef

Spectral density of an AR(p)

Proof

fYY(ω) = |H(ω)|2 fXX(ω)

φ(B)Yt = Zt

/|)(| )( 22

Z

i

YY ef

/)(|)(| 22

ZYY

i fe

Spectral density of an ARMA(p,q)

Proof.

fYY(ω) = |H(ω)|2 fXX(ω)

φ(B)Yt = θ(B)Zt

/||)(|)(| 222

Z

i

YY

i efe

/|)(||)(| )( 222

Z

ii

YY eef

System identification

Yt = hk Xt-k + Nt

Black box

What is inside?

Estimating the frequency response function

Yt = hk Xt-k + Nt

γXY (τ) = Σ hk γXX (τ-k)

fXY(ω) = H(ω)fXX(ω)

Proof

k

ki

XX

ki

k ekeh

/)( )(

Yt = h(B) Xt + Nt

Estimate

)(ˆ)(ˆ)(ˆ 1 XYXX ffH

Form estimates by smoothing m periodograms

Coherence/squared coherency

)}(ˆ)(ˆ/{|)(ˆ|)(ˆ 2 YYXXXY fffC

Expected value m/M in case C(ω)=0

Upper 100α% null point 1-(1-α)1/(m-1)

BJsales

Lag of about 3 days

Took m = 5

Gas furnace data

Cross-spectral analysis

nonparametric model

Input: (.6 - methane feed)/.04

Output: percent CO2 in outlet gas

acf

ccf

Box-Jenkins approach

h(B) = δ(B)-1ω(B)Bb

Parametric model

Yt = h(B) Xt-k + Nt

Yt = ω0Xt+...+ω11Xt-11+β0Zt+...+β9Zt-9

impulse response {ωk}

E.g. furnace data

Get uncertainty by bootstrapping

Discussion

time side vs. frequency side quantities

parametric vs. nonparametric models

acf, ccf fYY , fXY, C

{hk} H(ω)

ARMAX(p,q) Yt = hk Xt-k + Nt

Plot the data (xt , yt ), t=1,...,N

Actuarial example

Science Oct 24 2008

AR(p):

Yt = α1Yt-1 + α2Yt-2 + ... + αpYt-p + Xt

ARMA(p,q):

Yt = α1Yt-1 + α2Yt-2 + ... + αpYt-p + Xt + β1 Xt-1 + ... + βq Xt-q

φ(B)Yt = θ(B)Xt

Yt = φ(B)-1θ(B)Xt

MA(q):

Yt = Xt + β1 Xt-1 + ... + βq Xt-q

AR(p): Yt = α1Yt-1 + α2Yt-2 + ... + αpYt-p + Xt

ARMA(p,q):

Yt = α1Yt-1 + α2Yt-2 + ... + αpYt-p + Xt + β1 Xt-1 + ... + βq Xt-q

)( )( ieH

MA(q): Yt = Xt + β1 Xt-1 + ... + βq Xt-q

Yt = φ(B)-1θ(B)Xt

)( )( 1 ieH

ARMA(p,q):

Yt = α1Yt-1 + α2Yt-2 + ... + αpYt-p + Xt + β1 Xt-1 + ... + βq Xt-q

Yt = φ(B)-1θ(B)Xt