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Time series power spectral density. frequency-side, , vs. time-side, t X(t), t= 0, ±1, ±2,… Suppose stationary c XX (u) = cov{X(t+u),X(t)} u = 0, ±1, ±2, … lag f XX ()= (1/2π) Σ exp {-iu} c XX (u) - < period 2π non-negative /2 : frequency (cycles/unit time)

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Time series power spectral density . frequency-side,  , vs. time-side, t X(t), t= 0, ±1, ±2,… Suppose stationary c XX (u) = cov{X(t+u),X(t)} u = 0, ±1, ±2, … lag f XX (  )= (1/2 π ) Σ exp {-i  u} c XX (u) -  <  ≤  period 2 π non-negative - PowerPoint PPT Presentation

### Transcript of Time series power spectral density . frequency-side, , vs. time-side, t Time series power spectral density.

frequency-side, , vs. time-side, t

X(t), t= 0, ±1, ±2,… Suppose stationary

cXX(u) = cov{X(t+u),X(t)} u = 0, ±1, ±2, … lag

f XX()= (1/2π) Σ exp {-iu} cXX(u) - < ≤

period 2π non-negative

/2 : frequency (cycles/unit time)

cXX(u) = ∫ -pi pi exp{iuλ} f XX() d  The Empirical FT.

Data X(0), …, X(T-1)

dXT()= Σ exp {-iu} X(u)

dXT(0)= Σ X(u)

What is the large sample distribution of the EFT? The complex normal.

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Also get asymp independence if consider separate stretches

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independent exponentials

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var log|AT| [|R|-2 -1]/L

var argAT [|R|-2 -1]/L Berlin andVienna monthlytemperatures   RecifeSOI Furnace data  RXZ|Y =

(R XZ – R XZ R ZY )/[(1- |R XZ|2 )(1- |RZY | 2 )]

Partial coherence/coherency. Mississipi dams  techniques for many stationary processes look the same

approximate i.i.d sample values

assessing models (character of departure)

time varying variant... Cleveland, RB, Cleveland, WS, McRae, JE & Terpenning, I (1990), ‘STL: a seasonal-trend decomposition procedurebased on loess’, Journal of Official Statistics

Y(t) = S(t) + T(t) + E(t)

Seasonal, trend. error

London water usage   Dynamic spectrum, spectrogram: IT (t,). London water Earthquake? Explosion?   Lucilia cuprina nobs = length(EXP6) # number of observations wsize = 256 # window size overlap = 128 # overlap ovr = wsize-overlap nseg = floor(nobs/ovr)-1; # number of segments krnl = kernel("daniell", c(1,1)) # kernel ex.spec = matrix(0, wsize/2, nseg)for (k in 1:nseg){ a = ovr*(k-1)+1 b = wsize+ovr*(k-1) ex.spec[,k] = mvspec(EXP6[a:b], krnl, taper=.5, plot=FALSE)\$spec } x = seq(0, 10, len = nrow(ex.spec)/2) y = seq(0, ovr*nseg, len = ncol(ex.spec)) z = ex.spec[1:(nrow(ex.spec)/2),] # below is text version filled.contour(x,y,log(z),ylab="time",xlab="frequency (Hz)",nlevels=12 ,col=gray(11:0/11),main="Explosion") dev.new() # a nicer version with color filled.contour(x, y, log(z), ylab="time", xlab="frequency(Hz)", main="Explosion") dev.new() # below not shown in text persp(x,y,z,zlab="Power",xlab="frequency(Hz)",ylab="time",ticktype="detailed",theta=25,d=2,main="Explosion")