Los Angeles polution mortality study Shumway at al (1988) Environ. Res. 45, Los Angeles County: average daily cardiovascular mortality particulate polution (six day smoothed averages) n = 508, Bivariate time series

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

Stat Oct 2008 D. R. Brillinger Chapter 9 - Linear Systems Λ[λ 1 x 1 + λ 2 x 2 ](t) = λ 1 Λ[x 1 ](t) + λ 2 Λ[x 2 ](t) Time invariant. Λ[B τ x](t) = B τ Λ[x](t) Λ[B τ x](t) = y(t- τ) if Λ[x](t) = y(t) Linear. Example. How to describe ? Common in nature

System identification Y t =  h k X t-k + N t Is there a relationship? Estimate h, H given {x(t),y(t)}, t=0,...,N Predict Y from X Control Studying causality Studying delay …

{h k }: impulse response Define δ k = 1 if k = 0 and = 0 otherwise, then h k = 0 k<0: causal/physically realizable One way

Transfer / frequency response function 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. 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 (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

Ideal band-pass filter H(ω ) = 1 |ω-ω 0 |  Δ = 0 otherwise

Construction of general filter Inverse Filtered series via fft( )

Bandpass filtering of Vienna monthly temperatures, Bank of bandpass filters Taper, form g(t/(N+1)x t, t=1,...,N e.g. g(u) = (1 + cos πu)/2

Pure lag filter. y = x t-τ h k = 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

The effect of filtering on second-order parameters γ YY (k) = Σ Σ h i h j γ XX (k-j+i) Proof. Cov{y t+k,x t } =

f YY (ω) = |H(ω)| 2 f XX (ω) Proof. f YY (ω)= Σ γ YY (k) exp{-iωk}]/π = ΣΣ Σ h i h j γ XX (k-j+i) exp{-iωk}]/π

Interpretation of f XX (ω 0 ) γ YY (0) = var Y t =  f YY (ω)dω =  |H(ω)| 2 f XX (ω)dω  f(ω 0 ) if H(.) narrow bandpass centered at ω 0 Provides an estimate of f(ω 0 ) ave{x t (ω 0 ) 2 }

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

The coherence may be estimated via estimate of corr{x t (ω),y t (ω)} 2 and … Hilbert transform

f YY (ω) = |H(ω)| 2 f XX (ω) suggests e.g. fit AR(p), p large Another estimate

Spectral density of an MA(q)

Spectral density of an AR(p) Proof f YY (ω) = |H(ω)| 2 f XX (ω) φ(B)Y t = Z t

Spectral density of an ARMA(p,q) Proof. f YY (ω) = |H(ω)| 2 f XX (ω) φ(B)Y t = θ(B)Z t

System identification Y t =  h k X t-k + N t Black box What is inside?

Estimating the frequency response function Y t =  h k X t-k + N t γ XY (τ) = Σ h k γ XX (τ-k) f XY (ω) = H(ω)f XX (ω) Proof Y t = h(B) X t + N t

Estimate Form estimates by smoothing m periodograms Coherence/squared coherency 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 CO 2 in outlet gas

acf ccf

Box-Jenkins approach h(B) = δ(B) -1 ω(B)B b Parametric model Y t = h(B) X t-k + N t Y t = ω 0 X t +...+ω 11 X t-11 +β 0 Z t +...+β 9 Z t-9 impulse response {ω k } E.g. furnace data

Get uncertainty by bootstrapping

Discussion time side vs. frequency side quantities parametric vs. nonparametric models acf, ccf f YY, f XY, C {h k } H(ω) ARMAX(p,q) Y t =  h k X t-k + N t Plot the data (x t, y t ), t=1,...,N

Actuarial example

Science Oct

AR(p): Y t = α 1 Y t-1 + α 2 Y t α p Y t-p + X t ARMA(p,q): Y t = α 1 Y t-1 + α 2 Y t α p Y t-p + X t + β 1 X t β q X t-q φ(B)Y t = θ(B)X t Y t = φ(B) -1 θ(B)X t MA(q): Y t = X t + β 1 X t β q X t-q

AR(p): Y t = α 1 Y t-1 + α 2 Y t α p Y t-p + X t ARMA(p,q): Y t = α 1 Y t-1 + α 2 Y t α p Y t-p + X t + β 1 X t β q X t-q MA(q): Y t = X t + β 1 X t β q X t-q Y t = φ(B) -1 θ(B)X t

ARMA(p,q): Y t = α 1 Y t-1 + α 2 Y t α p Y t-p + X t + β 1 X t β q X t-q Y t = φ(B) -1 θ(B)X t