Nonparametric Spectral Estimation Sunspot Numbers
Outline
1 Nonparametric Spectral Estimation
2 Sunspot Numbers
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Nonparametric Spectral Estimation Sunspot Numbers
Outline
1 Nonparametric Spectral Estimation
2 Sunspot Numbers
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Nonparametric Spectral Estimation Sunspot Numbers
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Nonparametric Spectral Estimation Sunspot Numbers
Asymptotic Properties of the Periodogram
Under general conditions of the time series:
Bias
bias (I(ωj)) = E(I(ωj))− f (ωj) = O(
1n
)Bias is very small!
Variancevar (I(ωj)) = O(1)
Variance is very large!
Let’s strike a compromise!
Increase the bias←→ Decrease the variance
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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.
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Nonparametric Spectral Estimation Sunspot Numbers
A Closer Look at the Periodogram
This is where we are:
I(ωj) =n−1∑
h=−(n−1)
γ̂(h)e−2πiωjh
This is where we want to be:
f (ω) =∞∑
h=−∞γ(h)e−2πiωh
One way of looking at the problem:
γ̂(h) is no good for values of h close to n!!!
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Nonparametric Spectral Estimation Sunspot Numbers
The Solution
Reduce the influence of γ̂(h) at extreme values of h.Consider the following estimator:
f̂ (ω) =n−1∑
h=−(n−1)
λ(h)γ̂(h)e−2πiωjh
where λ(h) starts out at 1 when h ≈ 0, but then decreases as h increases.
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Nonparametric Spectral Estimation Sunspot Numbers
Examples of Lag Windows
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Nonparametric Spectral Estimation Sunspot Numbers
Outline
1 Nonparametric Spectral Estimation
2 Sunspot Numbers
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Nonparametric Spectral Estimation Sunspot Numbers
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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!
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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
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