The Frequency Domain Time domain methods: regress present on past; - - PowerPoint PPT Presentation

The Frequency Domain

• Time domain methods:

– regress present on past; – capture dynamics in terms of velocity (first order), accel- eration (second order), inertia, etc.

• Frequency domain methods:

– regress present on periodic sines and cosines; – capture dynamics in terms of resonant frequencies, etc.

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E.g. AR(2):

plot(ts(arima.sim(list(order = c(2,0,0), ar = c(1.5,-.95)), n = 144)))

• Strong periodicity, around 16 peaks ⇒ period of around 9

samples.

• Fitting an AR model doesn’t describe this:

xt = 1.50xt−1 − 0.95xt−2 + wt.

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Cyclical Behavior

• Simplest case is the periodic process

xt = A cos(2πωt + φ) = U1 cos(2πωt) + U2 sin(2πωt). where: – A is amplitude; – ω is frequency, in cycles per sample; – φ is phase, in radians;

• and U1 = A cos(φ), U2 = −A sin(φ).

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Folding Frequency; Aliasing

• If ω = 0, xt = A cos(φ), constant.
• If ω = 1? At t = 0, 1, 2, . . . , same thing!
• ω = 0 is an alias of ω = 1.
• All frequencies higher than ω = 1/2 have an alias in 0 ≤ ω ≤

1/2: cos[2π(k ± ω)t + φ] = cos(2πωt ± φ), t = 0, 1, 2, . . .

• ω = 1/2 is the folding frequency.

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• For example, ω = 0.8:
• mega = 0.8;

phi = pi / 6; plot(function(x) cos(2 * pi * omega * x + phi), from = 0, to = 10); plot(function(x) cos(2 * pi * (1 - omega) * x - phi), from = 0, to = 10, add = TRUE, col = "red"); abline(v = 0:10, lty = 2, col = "blue");

• Note:

– ω = 0.8 = 0.5 + 0.3, and 1 − ω = 0.2 = 0.5 − 0.3; – 1 − ω is ω “folded” around 0.5.

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Stationarity

• If

xt = A cos(2πωt + φ) = U1 cos(2πωt) + U2 sin(2πωt). and φ is random, uniformly distributed on [0, 2π), then: E(xt) = 0, E

• xt+hxt
• = 1

2A2 cos(2πωh).

• So xt is weakly stationary.

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• Also

E(U1) = E(U2) = 0, E

• U2

1

• = E
• U2

2

• = 1

2A2, and E(U1U2) = 0.

• Alternatively, if the U’s have these properties, xt is stationary

with the same mean and autocovariances: E(xt) = 0, E

• xt+hxt
• = 1

2A2 cos(2πωh).

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• More generally, if

xt =

q

• k=1
• Uk,1 cos(2πωkt) + Uk,2 sin(2πωkt)
• ,

where: – the U’s are uncorrelated with zero mean; – var

• Uk,1
• = var
• Uk,2
• = σ2

k;

then xt is stationary with zero mean and autocovariances γ(h) =

q

• k=1

σ2

k cos(2πωkh).

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Harmonic Analysis

• Any time series sample x1, x2, . . . , xn can be written

xt = a0 +

(n−1)/2

• j=1
• aj cos(2πjt/n) + bj sin(2πjt/n)
• if n is odd; if n is even, an extra term is needed.
• The periodogram is

P(j/n) = a2

j + b2 j .

• The R function spectrum can calculate and plot the peri-
• dogram.

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• R examples:

par(mfcol = c(2, 1)); # one frequency: x = cos(2*pi*(0.123)*(1:144)) plot.ts(x); spectrum(x, log = "no") # and a second frequency: x = x + 2 * cos(2*pi*(0.234)*(1:144)) plot.ts(x); spectrum(x, log = "no") # and added noise: x = x + rnorm(144) plot.ts(x); spectrum(x, log = "no") # the AR(2) series: x = ts(arima.sim(list(order = c(2,0,0), ar = c(1.5,-.95)), n = 144)) plot(x); spectrum(x, log = "no")

• Using SAS: proc spectra program and output.

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