Tapering and prewhitening

fFT

taper, h(u)

Need for prewhitening/prefiltering

periodogram is generally biased

Smoothed periodogram

Expected value generally biased, but not for white noise

Improve by prefiltering

Y filtered version of X with transfer function A

dZY(λ) = A(λ)dZX(λ)

fYY(λ) = |A (λ)|-2 fXX(λ)

fXX(λ) = |A (λ)|-2 fYY(λ)

For example: detrend, “remove” peaks, fit autoregressive

The importance of prefiltering.

The estimates are generally biased

E{ fT()} = WT (-) f() d

Seismic noise

horizontal components recorded at UCB

Estimated coherence can be 1,0 when no relation

Estimated coherence can be 0.0 when linearly time invariantly related

postscript(file="seismic.eps")

junkx<-scan("ts_drb.16.dat")

junky<-scan("ts_drb.17.dat")

par(mfrow=c(2,1))

xaxis<-c(1:2500)*.08

plot(xaxis,junkx,type="l",main="Vertical component seismic noise at Berkeley",xlab="time (sec)",ylab="",las=1)

plot(xaxis,junky,type="l",main="West component",xlab="time(sec)",ylab="",las=1)

par(mfrow=c(2,3))

junk<-spec.pgram(cbind(junkx,junky),spans=15,taper=0,detrend=F,demean=T,plot=F)

junk$freq<-junk$freq/.08

plot(junk$freq,junk$spec[,1],type="l",las=1,main="Vertical noise",xlab="frequency (hertz)",log="y")

plot(junk$freq,junk$spec[,2],type="l",las=1,main="West noise",log="y")

plot(junk$freq,junk$coh,type="l",ylim=c(0,1),main="Raw data",las=1)

abline(h=1-(1-.95)^(1/(.5*(junk$df-2))))

junkxx<-ar(junkx,order.max=2)

junkyy<-ar(junky,order.max=2)

Junkx<-junkxx$resid

Junky<-junkyy$resid

Junkx<-Junkx[3:length(Junkx)];Junky<-Junky[3:length(Junky)]

Junk<-spec.pgram(cbind(Junkx,Junky),spans=15,taper=0,detrend=F,demean=T,plot=F)

Junk$freq<-Junk$freq/.08

plot(Junk$freq,Junk$spec[,1],type="l",las=1,log="y")

plot(Junk$freq,Junk$spec[,2],type="l",las=1,log="y")

plot(Junk$freq,Junk$coh,type="l",ylim=c(0,1),main="AR(2) residuals",las=1)

abline(h=1-(1-.95)^(1/(.5*(junk$df-2))))

graphics.off()