periodogram analysis

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Stat 565

Charlotte Wickham stat565.cwick.co.nz

The Periodogram

May 9 2012

Wednesday, May 9, 12



Help sessions

This week both

Today 2-3.30pm &

Tomorrow 2-3pm in

Library 6420




Moving to the frequency domain

Time DomainFrequency

Domain

xt linear combination of past xt linear combination of periodic components

population ACF Spectral Density

sample acf Periodogram

Object of interest

Data analysis tool

Identify dominantfreqenc(y/ies)Wednesday, May 9, 12



Where are we going?

Today: Periodogram

motivation, examples and issues

Then: Spectral Density

improving the periodogram as an estimate of the spectral

density

Later: Linear filters




Can you guess the frequencies in these (simulated) series?




A quick trig review

Imagine a series,

xt = A cos ( 2π t ω + ɸ) Amplitude Frequency

(cycles per unit time)

Phase

(nothing random yet)




x t = A cos ( 2π t ω + ɸ)

Frequency, ω

ω = 1/2, cycle/unit time

ω = 1, cycle/unit time

ω = 1/5, cycle/unit time

Period = 1/ ω

time to complete one

cycle





Amplitude, A

A = 5

A = 1

A = 10





ɸ = π /2 ɸ = 0 ɸ = -π /2

Phase, ɸ




xt = A cos ( 2π t ω + ɸ)

Generally we rewrite this as:

xt = a cos (2π t ω) + b sin(2π t ω)

where a = A cos(ɸ), b = - A sin(ɸ)

and a2 + b2 = A 2

(using the identity: cos(u + v) = cos(u)cos(v) - sin(u)sin(v) )

we saw on Monday if we know the frequency we can

estimate a and b using ordinary least regression




Extend to

xt = ∑k ak cos (2π t ωk ) + bk sin(2π t ωk )

a sum of k periodic components

xt = 2 cos(2πt 6/n) + 3 sin(2π 6/n) +4 cos(2πt 10/n) +5 sin(2πt 10/n) +

6 cos(2πt 40/n) + 7 sin(2πt 40/n)

Deterministic example

k=3








Finding ak and bk

We can estimate the ak and bk by regressing xt

on sin(2π t ωk ) and cos(2π t ωk ).

But what frequencies will we use?

Numerically it is very efficient to use, k/n, (the

Fourier frequencies).

k/n = k complete cycles over our record (the kth

harmonic).

We don't actually do the regression, we can use

the fast fourier transform (fft in R).




Assume n is odd

xt = a0 + a1 cos (2π t 1/n) + b1 sin(2π t 1/n) +

a2 cos (2π t 2/n) + b2 sin(2π t 2/n) +

... +

a(n-1)/2 cos (2π t ((n-1)/2)/ n) + b2 sin(2π t ((n-1)/2)/n)

= ∑k ak cos (2π t k/n) + bk sin(2π t k/n) k = 1, ..., (n - 1)/2

How many parameters (ak & bk ) are there?

We can perfectly fit any data with this expansion.

The Discrete Fourier transform




Scaled Periodogram

xt = 2 cos(2πt 6/n) + 3 sin(2π 6/n) +

4 cos(2πt 10/n) + 5 sin(2πt 10/n) +

6 cos(2πt 40/n) + 7 sin(2πt 40/n)

Deterministic example

A plot of k/n versus A k 2

Remember A k 2 = ak

2 + bk 2




What about when there is noise?

Scaled periodogram of the first example, x

0.0390625 = 10/256Wednesday, May 9, 12



Your turn

the 2nd example, x2

the 3rd example, x3





Look for peaks, they occur at dominant

frequencies.

There are some issues:

1. The frequencies we find depend on

our record length.

2. The periodogram doesn't get

smoother with longer series.

The periodogram




Periodogram for white noise

n=

Ideally, with a very

long series we

would estimate a

flat spectrum.

Documents

periodogram analysis