Time series power spectral density
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Time series power spectral density . frequency-side,  , vs. time-side, t PowerPoint PPT Presentation


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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

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Time series power spectral density . frequency-side,  , vs. time-side, t

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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 {-iu} cXX(u) - < ≤

period 2π non-negative

/2 : frequency (cycles/unit time)

cXX(u) = ∫ -pi pi exp{iuλ} f XX() d 


Time series power spectral density frequency side vs time side t

The Empirical FT.

Data X(0), …, X(T-1)

dXT()= Σ exp {-iu} X(u)

dXT(0)= Σ X(u)

What is the large sample distribution of the EFT?


Time series power spectral density frequency side vs time side t

The complex normal.


Time series power spectral density frequency side vs time side t

Theorem. Suppose X is stationary mixing, then


Time series power spectral density frequency side vs time side t

Proof. Write

Evaluate first and second-order cumulants

Bound higher cumulants

Normal is determined by its moments


Time series power spectral density frequency side vs time side t

Consider


Time series power spectral density frequency side vs time side t

Comments.

Already used to study mean estimate

Tapering, h(t)X(t). makes

Get asymp independence for different frequencies

The frequencies 2r/T are special, e.g. T(2r/T)=0, r 0

Also get asymp independence if consider separate stretches

p-vector version involves p by p spectral density matrix fXX( )


Time series power spectral density frequency side vs time side t

Estimation of the (power) spectrum.

An estimate whose limit is a random variable


Time series power spectral density frequency side vs time side t

Some moments.

The estimate is asymptotically unbiased

Final term drops out if  = 2r/T  0

Best to correct for mean, work with


Time series power spectral density frequency side vs time side t

Periodogram values are asymptotically independent since dT values are -

independent exponentials

Use to form estimates


Time series power spectral density frequency side vs time side t

Approximate marginal confidence intervals


Time series power spectral density frequency side vs time side t

More on choice of L


Time series power spectral density frequency side vs time side t

Approximation to bias


Time series power spectral density frequency side vs time side t

Indirect estimate


Time series power spectral density frequency side vs time side t

Estimation of finite dimensional .

approximate likelihood (assuming IT values independent exponentials)


Time series power spectral density frequency side vs time side t

Bivariate case.


Time series power spectral density frequency side vs time side t

Crossperiodogram.

Smoothed periodogram.


Time series power spectral density frequency side vs time side t

Complex Wishart


Time series power spectral density frequency side vs time side t

Predicting Y via X


Time series power spectral density frequency side vs time side t

Plug in estimates.


Time series power spectral density frequency side vs time side t

Large sample distributions.

var log|AT|  [|R|-2 -1]/L

var argAT [|R|-2 -1]/L


Time series power spectral density frequency side vs time side t

Berlin and

Vienna monthly

temperatures


Time series power spectral density frequency side vs time side t

Recife

SOI


Time series power spectral density frequency side vs time side t

Furnace data


Time series power spectral density frequency side vs time side t

Partial coherence/coherency. Mississipi dams

RXZ|Y =

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


Time series power spectral density frequency side vs time side t

Advantages of frequency domain approach.

techniques for many stationary processes look the same

approximate i.i.d sample values

assessing models (character of departure)

time varying variant...


Time series power spectral density frequency side vs time side t

London water usage

Cleveland, RB, Cleveland, WS, McRae, JE & Terpenning, I (1990), ‘STL: a seasonal-trend decomposition procedure

based on loess’, Journal of Official Statistics

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

Seasonal, trend. error


Time series power spectral density frequency side vs time side t

Dynamic spectrum, spectrogram: IT (t,). London water


Time series power spectral density frequency side vs time side t

Earthquake? Explosion?


Time series power spectral density frequency side vs time side t

Lucilia cuprina


Time series power spectral density frequency side vs time side t

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")


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