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
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