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