Stat 153 - 19 Oct 2008 D. R. Brillinger Chapter 8 - Bivariate processes 8.1 Cross-covariance and cross correlation time-side 8.2 Cross-covariance frequency-side Chapter 9 - Linear systems regression system - fixed input, stochastic output
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Bivariate time series. random process: (Xt , Yt ) , t = 0, ±1, ±2, ... data: (x1 , y1 ), ..., (xN , yN ) Typically leads to more specific conclusions
"Ordinary" statistics correlation. (X,Y): μX , μY , σX , σY σXY = E{(X - μX)(Y - μY)} = σYX joint distribution -1 ρXY 1
MSE linear prediction min E{(Y - βX)2} = σY2 (1- ρ2) β = σYX σXX-1 min E{(X - αY)2} = σX2 (1- ρ2) α = σXY σYY-1 ρ2 measures goodness of prediction
Cross-covariance function, stationary case γXY (k) = cov{Xt , Yt+k } = γYX (-k) Cross-correlation function ρXY (k) = corr{Xt , Yt+k} |ρXY (k)| 1 Example. Is there a common signal present? Xt = Σ au Zt-u + Mt Yt = Σ bv Zt-v + Nt γXY (k) = σZ2 Σ au bk+u
Estimates rXY (k) = cXY (k)/{cXX (0)cYY(0)} If {Xt } and {Yt} uncorrelated at all lags and {Xt } noise E[rXY (k)] 0 Var[rXY (k)] 1/N
Examples. Berlin-Vienna Temperatures
Seasonally adjusted
Mississippi River discharge
Binary data X(t), Y(t) = 0,1 Two neurons from Aplysia californica
The cross-spectrum f XY(ω) = [Σ γXY(k) exp{-iωk}]/π, 0 < ω < π cospectrum: c(ω ) = Re{f XY(ω)} quadspectrum: q(ω ) = - Im{f XY(ω)}
Estimation of crosspectrum Cross-periodogram Smooth
|f XY(ω)|2 f X(ω)f Y(ω) Squared coherency/coherence C(ω ) = | f XY(ω)|2 / f X(ω)f Y(ω) 0 C(ω ) 1 Xt = Σ au Zt-u + Mt Yt = Σ bv Zt-v + Nt A(ω) = Σ ak exp{-iωk} B(ω) = Σ bk exp{-iωk} f X(ω) = |A(ω)|2 σZ2/π + f M(ω) f Y(ω)=|B(ω)|2 σZ2/π + f N(ω) f XY(ω) = A(ω)B(ω)* σZ2 /π
Coherence is a measure of how well one can predict Yt from {Xt} at frequency ω by Σ hk Xt-k
Berlin-Vienna monthlt data
Mississippi
Aplysia