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Large Sample Distribution of EFT and Frequency Domain Techniques for Stationary Processes

This article explores the large sample distribution of the Empirical Fourier Transform (EFT) and its application in analyzing stationary processes. It covers topics such as complex normal theorem, mixing processes, cumulants evaluation, power spectrum estimation, bias correction, and parametric estimation. The text delves into concepts like spectral density matrix, periodogram, crossperiodogram, and prediction models. It also discusses the importance of frequency domain approaches for assessing models and analyzing time-varying variants. Case studies on temperature data, seismic signals, water usage, and spectral analysis techniques are presented.

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Large Sample Distribution of EFT and Frequency Domain Techniques for Stationary Processes

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  1. The Empirical FT. What is the large sample distribution of the EFT?

  2. The complex normal.

  3. Theorem. Suppose X is stationary mixing, then

  4. Proof. Write Evaluate first and second-order cumulants Bound higher cumulants Normal is determined by its moments

  5. Consider

  6. 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( )

  7. Estimation of the (power) spectrum. An estimate whose limit is a random variable

  8. Some moments. The estimate is asymptotically unbiased Final term drops out if  = 2r/T  0 Best to correct for mean, work with

  9. Periodogram values are asymptotically independent since dT values are - independent exponentials Use to form estimates

  10. Approximate marginal confidence intervals

  11. More on choice of L

  12. Approximation to bias

  13. Indirect estimate

  14. Estimation of finite dimensional . approximate likelihood (assuming IT values independent exponentials)

  15. Bivariate case.

  16. Crossperiodogram. Smoothed periodogram.

  17. Complex Wishart

  18. Predicting Y via X

  19. Plug in estimates.

  20. Large sample distributions. var log|AT|  [|R|-2 -1]/L var argAT [|R|-2 -1]/L

  21. Berlin and Vienna monthly temperatures

  22. Recife SOI

  23. Furnace data

  24. Partial coherence/coherency. Mississipi dams RXZ|Y = (R XZ – R XZ R ZY )/[(1- |R XZ|2 )(1- |RZY |2 )]

  25. 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

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

  27. Earthquake? Explosion?

  28. Lucilia cuprina

  29. 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")

  30. 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...

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