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Spectral Density Estimation (Chapter 13)

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  • Spectral Density Estimation (Chapter 13)

  • Nonparametric Spectral Estimation Sunspot Numbers

    Outline

    1 Nonparametric Spectral Estimation

    2 Sunspot Numbers

    Arthur Berg Spectral Density Estimation (Chapter 13) 2/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    Outline

    1 Nonparametric Spectral Estimation

    2 Sunspot Numbers

    Arthur Berg Spectral Density Estimation (Chapter 13) 3/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    Arthur Berg Spectral Density Estimation (Chapter 13) 4/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    Asymptotic Properties of the Periodogram

    Under general conditions of the time series:

    Bias

    bias (I(j)) = E(I(j)) f (j) = O(

    1n

    )Bias is very small!

    Variancevar (I(j)) = O(1)

    Variance is very large!

    Lets strike a compromise!

    Increase the bias Decrease the variance

    Arthur Berg Spectral Density Estimation (Chapter 13) 5/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    Averaging the Periodogram

    The Periodogram estimates at different Fourier frequencies are approximatelyindependent. So averaging neighboring estimates is the key to improving theestimate of the spectral density.

    Arthur Berg Spectral Density Estimation (Chapter 13) 6/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    A Closer Look at the Periodogram

    This is where we are:

    I(j) =n1

    h=(n1)

    (h)e2ijh

    This is where we want to be:

    f () =

    h=(h)e2ih

    One way of looking at the problem:

    (h) is no good for values of h close to n!!!

    Arthur Berg Spectral Density Estimation (Chapter 13) 7/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    The Solution

    Reduce the influence of (h) at extreme values of h.Consider the following estimator:

    f () =n1

    h=(n1)

    (h)(h)e2ijh

    where (h) starts out at 1 when h 0, but then decreases as h increases.

    Arthur Berg Spectral Density Estimation (Chapter 13) 8/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    Examples of Lag Windows

    Arthur Berg Spectral Density Estimation (Chapter 13) 9/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    Outline

    1 Nonparametric Spectral Estimation

    2 Sunspot Numbers

    Arthur Berg Spectral Density Estimation (Chapter 13) 10/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    Arthur Berg Spectral Density Estimation (Chapter 13) 11/ 13

  • Nonparametric Spectral Estimation Sunspot Numbers

    Sir (Franz) Arthur (Friedrich ) Schuster FRS (1851 1934)

    Schuster credited with the formulation of the periodogram.Arthur Schuster, On Lunar and Solar Periodicities of Earthquakes,Proceedings of the Royal Society of London, Vol. 61 (1897), pp.455-465.Available Online!

    Arthur Berg Spectral Density Estimation (Chapter 13) 12/ 13

    http://visualiseur.bnf.fr/CadresFenetre?O=NUMM-56154&I=481&M=tdm
  • Nonparametric Spectral Estimation Sunspot Numbers

    Smoothing the Sunsport Periodogram

    "Mr. A Schuster of Owens College has ingeniously pointed out that theperiods of good vintage in Western Europe have occurred at intervalssomewhat approximating to eleven years, the average length of theprincipal sun-spot cycle." William Stanley Jevons

    Arthur Berg Spectral Density Estimation (Chapter 13) 13/ 13

    http://web.cecs.pdx.edu/~ssp/Reports/2005/Olvera.pdfNonparametric Spectral EstimationSunspot Numbers