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Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Fourier Analysis of Stationary and Non-StationaryTime Series
Jo Evans, Tim Park
September 6, 2012
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
A time series is a stochastic process indexed at discrete points intime i.e Xt for t = 0, 1, 2, 3, ... The mean is defined as
µt = E[Xt ]
and the autocovariance function is defined as
γ(t, s) = E[(Xt − µt)(Xs − µs)].
A weakly-stationary (or just stationary) process is one in whichµt ≡ µ where µ is constant in time and γ(t, s) only depends onh = |t − s| called lag.Stationary time series describe processes whose behavior does notchange over time. White noise series are stationary. Often seriescan be transformed into roughly stationary series.
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
If autocovariance function is absolutely summable
Xt =
∫ 1/2
−1/2A(ω)e2πiωtdZ (ω).
Here dZ (ω) is an independent mean zero stochastic process withE[|dZ (ω)|2] = 1 and A(ω) is called the transfer function and
|A(ω)|2 = f (ω).
f (ω) or spectral density is the proportion of the varianceconcentrated at a component oscillating with frequency ω. This iscalled the Cramer model.
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Why estimate f (ω):
Some series (e.g. the sum of an AR(0.9) and AR(-0.9)) maybe difficult to show simply in the time domain but easy in theFourier domain. The example series will give two broad peaksat low and high frequencies.
Sometimes the underlying process is periodic in nature (e.gspeech) and f (ω) might reveal this.
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
I have been trying to estimate f (ω). We do this using aperiodogram. If we have T realizations of a stationary time series.We can fit the model
Xt =T−1∑k=0
(ak cos (2πωkt) + bk sin (2πωkt)).
This model has no error terms as we are using T values toestimate T parameters. The ak , bk can be calculated using a fastfourier transform. We then define the periodogram by
I (ωk) = (a2k + b2k)/4πT .
Where ωk = kT and k = 0, 1, . . . ,T − 1. It can be shown that if
ωk → ω then as T →∞ then I (ωk) is an asymptotically unbiasedestimator of f (ω).
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Timew
hite
noi
se
0 200 400 600 800 1000
−3
−1
13
Time
nois
y si
ne s
erie
s
0 200 400 600 800 1000
−1.
50.
01.
0
Time
AR
(0.9
)
0 200 400 600 800 1000
−5
05
Time
AR
(−0.
9)
0 200 400 600 800 1000
−6
−2
26
Figure: 4 simulated series all of length 1000Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
0.0 0.1 0.2 0.3 0.4 0.5
0.00
00.
002
0.00
4
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.02
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.10
0.20
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.50.
000.
100.
200.
30
frequency
perio
dogr
am
Figure: Periodograms of Series 1-4
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
However as a point estimator of f (ωk) the periodogram is notconsistent (i.e. the variance does not tend to zero as T isincreased).To deal with this we have to smooth the periodogram.
Box Kernel Smoother
P(ωk) =1
2m + 1
k+m∑j=k−m
I (ωj)
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
I have data showing the activity of the FTSE100 for 7101 dailytime steps. This time series is clearly neither mean zero notstationary. The standard transform to improve this is to takelogarithmic returns.
Time
0 1000 2000 3000 4000 5000 6000 7000
1000
3000
5000
7000
−0.
100.
000.
100.
20
Figure: FTSE100 time series with logarithmic returns shown below
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Figure: Periodogram of FTSE100 ts with bandwidth=0.168.
0.0 0.1 0.2 0.3 0.4 0.5
5.5e
−08
6.5e
−08
7.5e
−08
frequency
perio
dogr
am
This periodograms appear to show that most of the variance isconcentrated at periodic components with high frequencies withlarger peaks at 0.27 and 0.48.
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Many time series that people want to study are non-stationary.EEGs will exhibit different behavior before, after and during aseizure. Earthquakes and Explosion detection may show verydifferent behavior.
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
I generated a series using the R code:
Code
y1 = arima.sim(n=1024,model=list(ar=0.9))y2 = arima.sim(n=1024,model=list(ar=-0.9))y = c(y1,y2)
Time
y
0 500 1000 1500 2000
−5
05
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Here is the resulting stationary periodogram:
0.0 0.1 0.2 0.3 0.4 0.5
0.00
00.
005
0.01
00.
015
0.02
00.
025
0.03
0
frequency
perio
dogr
am
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
In ... Priestley introduced the idea of a smoothly time varyingspectral density function. This changed the Cramer model to oneof the form
X (t) =
∫ 1/2
−1/2At(ω)e2πiωtdZ (ω)
where At(ω) is the time varying transfer function.
γ(s, t) =
∫ 1/2
−1/2A∗s (ω)At(ω)e2πiω(t−s)dµ(ω)
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
I decided to assume that my spectrum was changing slowly withtime and that the series would be approximately stationary in smalltime blocks. With the FTSE series I analyzed the first 4096 timesteps by splitting it into 16 blocks and doing stationary fourieranalysis on each of these blocks.
Figure: A Slowly Changing Sloth
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
0.0 0.2 0.4
frequency
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4
frequency
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4
frequency
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4
frequency
perio
dogr
am
0.0 0.2 0.4 0.0 0.2 0.4
perio
dogr
am
0.0 0.2 0.4
perio
dogr
am
0.0 0.2 0.4
perio
dogr
am
Figure: Periodograms for each of the smaller time series
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Here are 8 peridograms produced by splitting the half and half ARnon-stationary series:
0.0 0.1 0.2 0.3 0.4 0.5
frequency
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Issues with the Priestley-Dahlhause method.
I know no way of selecting or justifying the blocks in which toregard the series as stationary.
Choosing blocks which are all the same size can missinformation when ft(ω) is changing quickly or use a lot ofcomputational power and produce greater error bands whenft(ω) is changing slowly.
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
The last model I looked at was SLEX. This model also incorporatesthe idea of a time variable spectrum which is constant in timeblocks. SLEX fits the time series in each block to a set of SLEX(Smooth Localized EXponential) vectors which allows the blocksto overlap without producing a bias.
Figure: A SLEX vector
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
The SLEX algorithm uses a Fourier transform so can becomputed using a fast fourier transform making it as efficientas earlier methods.
The SLEX procedure uses the Best Basis Algorithm developedby Coifman and Wickerhauser to automatically select theblocks.
The SLEX procedure minimizes a generalized cross validationdeviance function to calculate the smoothing bandwidth
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
The SLEX algorithm:
Split the series dyadically into 2j blocks for j = 1, . . . , J.
Compute the raw SLEXgrams in each of these blocks
Select the optimal bandwidth and smooth the SLEXgrams ineach block
Select the best blocks to model the series using BBA
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
1.000000 7.888889 14.777778 21.666667 28.555556
65
43
21
Segmentation
Position
Leve
l
Figure: Segmentation chosen by the BBA for the FTSE100 time series
The BBA computes the cost associated with selecting each block.It then compares the cost of each bigger block to the two halfblocks beneath it and will reject the bigger block if the cost isgreater.
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Time
logr
etur
ns[1
:409
6]
0 1000 2000 3000 4000
−0.
10−
0.05
0.00
0.05
0.10
Figure: SLEX blocks shown on the log returns time series
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
0.0 0.1 0.2 0.3 0.4 0.5
frequency
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
perio
dogr
am
Figure: SLEXgrams
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Performing SLEX on the half and half series gives thesegmentation:
1.000000 4.333333 7.666667 11.000000 14.333333
54
32
1
Segmentation
Position
Leve
l
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Here are the resulting SLEX periodograms:
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
0.0 0.1 0.2 0.3 0.4 0.5
frequency
perio
dogr
am
Jo Evans, Tim Park Fourier Domain Analysis
Time Series BasicsEstimating Spectral Density
Stationary Analysis of FTSE100 dataEvolutionary Spectra
SLEX
Figure: Any Questions?
Jo Evans, Tim Park Fourier Domain Analysis