20110706 d2 spectral density
TRANSCRIPT
-
7/28/2019 20110706 D2 Spectral Density
1/22
Spectral Density and Filtering
Dedy Dwi Prastyo
Ladislaus von Bortkiewicz Chair of StatisticsC.A.S.E. Center for Applied Statisticsand EconomicsHumboldtUniversitt zu Berlinhttp://lvb.wiwi.hu-berlin.de
http://www.case.hu-berlin.de
http://lvb.wiwi.hu-berlin.de/http://www.case.hu-berlin.de/http://www.case.hu-berlin.de/http://lvb.wiwi.hu-berlin.de/ -
7/28/2019 20110706 D2 Spectral Density
2/22
Motivation 1-1
Stochastic Process
1950 1960 1970 19801.5
1
0.5
0
0.5
1
1.5
Time, t
SOI
Figure 1: Monthly Southern Oscillation Index (SOI)
Stochastic process {Xt}
t=0 in the frequency domain (Shumway andStoffer (2006) and Hamilton (1994))
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.51
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
3/22
Motivation 1-2
Time Domain
0 5 10 15 20 25
0.4
0.2
0.6
1.0
Lag Time
ACF
0 5 10 15 20 25
0.2
0.2
0.6
Lag Time
PACF
Figure 2: ACF and PACF of SOI series
Time dependence
Order of dependence, stationarity, seasonality, long memory
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.51
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
4/22
Motivation 1-3
Frequency Domain
0.0 0.1 0.2 0.3 0.4 0.5
0
4
8
12
Frequency
spectrum
Figure 3: Periodogram of SOI series
Frequency of the (business) cycle
Repeating pattern, dominant frequency
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.51
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
5/22
Motivation 1-4
Stochastic Process Representation
Signal St and noise t
Xt = St + t
Xt = a {cos(2t + b)} + t (1)
Parameters
a = amplitude
= frequencyb = shift
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.51
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
6/22
Motivation 1-5
Example
0 100 200 300 400 500
2
0
2
St
0 100 200 300 400 5005
0
5
S
t
+
t1
0 100 200 300 400 500
20
0
20
Time t
St
+
t2
Figure 4: Simulated signal St = 2cos{2(t/50) + 0.6}, with
t1 N(0, 1) and t2 N(0, 25)
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
7/22
Outline
1. Motivation
2. Spectral Density
3. Filtering
4. Conclusion
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
8/22
Spectral Density 2-1
Spectral Density
Stationary process Xt with autocovariance (h Z )
(h) = E[(xt+h E[xt+h])(xt E[xt])],
h=
|(h)|
-
7/28/2019 20110706 D2 Spectral Density
9/22
Spectral Density 2-2
Periodogram
Sample-based spectral density, j=j/n
f(j) =(n1)
h=(n1)
(h)e2ijh, j = 0, 1, ..., n 1 (5)
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
10/22
Spectral Density 2-3
AR(1)
0.0 0.1 0.2 0.3 0.4 0.5
0
40
80
Frequency
SpectralDensity
0.0 0.1 0.2 0.3 0.4 0.5
0
40
80
Frequency
SpectralDensity
Figure 5: Spectral density ofXt = 0.9Xt1 + t and Xt = 0.9Xt1 + twith t N(0, 1) i.i.d.
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
11/22
Spectral Density 2-4
AR(2)
0.0 0.1 0.2 0.3 0.4 0.5
0
20
40
60
Frequency
Spe
ctralDensity
0.0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
4
Frequency
Spe
ctralDensity
Figure 6: Spectral density ofXt = 0.5Xt1 0.75Xt2 + t and
Xt = 0.1Xt1 + 0.4Xt2 + t with t N(0, 1) i.i.d.
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
12/22
Spectral Density 2-5
MA(1)
0.0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
Frequency
Sp
ectralDensity
0.0 0.1 0.2 0.3 0.4 0.5
0
1
2
3
Frequency
Sp
ectralDensity
Figure 7: Spectral density of Xt = t + 0.9t1 and Xt = t 0.9t1with t N(0, 1) i.i.d.
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
13/22
Spectral Density 2-6
MA(2)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
1.0
2.0
3.0
Frequency
SpectralDensity
0.0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
Frequency
SpectralDensity
Figure 8: Spectral density ofXt = t + 0.9t1 0.65t2 andXt = t 0.9t1 + 0.65t2 with t N(0, 1) i.i.d.
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
14/22
Spectral Density 2-7
ARMA(1, 1)
0.0 0.1 0.2 0.3 0.4 0.5
0
4
8
12
Frequency
SpectralDensity
0.0 0.1 0.2 0.3 0.4 0.5
0
4
8
12
Frequency
SpectralDensity
Figure 9: Spectral density ofXt = 0.5Xt1 + t + 0.8t1 andXt = 0.5Xt1 + t 0.8t1 with t N(0, 1) i.i.d.
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
15/22
Spectral Density 2-8
SAR(1) and SMA(1)
0.0 0.1 0.2 0.3 0.4 0.5
0
1
00
300
Frequency
Sp
ectralDensity
0.0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
Frequency
Sp
ectralDensity
Figure 10: Spectral density Xt = 0.5Xt1+0.9Xt12+(0.5)(0.9)Xt13+tand Xt = t + 0.4t1 + 0.9t12 + (0.4)(0.9)t13 with t N(0, 1) i.i.d.
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
16/22
Spectral Density 2-9
Periodogram of SOI
0.0 0.1 0.2 0.3 0.4 0.5
0
4
8
12
Frequency
spectrum
Figure 11: Periodogram of SOI series, dominant frequency at 1/0.021=48
and 1/0.083=12 months
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
17/22
Filtering 3-1
Linear Filtering
Extract signal St from data Xt contaminated by noise t
Linear filter, cr is impulse response
St =
r=
crxtr,
r=
|cr|
-
7/28/2019 20110706 D2 Spectral Density
18/22
Filtering 3-2
Example
Time
St
0 100 200 300 400
1.0
0.0
0.5
Time
St
0 100 200 300 400
0.4
0.0
0.4
Figure 12: First differencing St = xtxt1 and 12-month centered movingaverage St =
1
24(xt6 + xt+6) +
1
12
5
r=5 xtr of SOI series
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
19/22
Filtering 3-3
Example
0.0 0.1 0.2 0.3 0.4 0.5
0
2
4
6
8
1
2
frequency
spectrum
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
frequency
spectrum
Figure 13: Periodogram ofSOI series and 12-month centered moving aver-
age filter with dominant frequency 1/0.02 = 52 (El Nino) and 1/0.083 =12 months
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
20/22
Conclusion 4-1
Conclusion
Spectral density
Express the information in cycle term Identify the dominant frequency in the series
Filtering: Extract the signal from data
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w
-
7/28/2019 20110706 D2 Spectral Density
21/22
Spectral Density and Filtering
Dedy Dwi Prastyo
Ladislaus von Bortkiewicz Chair of StatisticsC.A.S.E. Center for Applied Statisticsand EconomicsHumboldtUniversitt zu Berlinhttp://lvb.wiwi.hu-berlin.de
http://www.case.hu-berlin.de
http://lvb.wiwi.hu-berlin.de/http://www.case.hu-berlin.de/http://www.case.hu-berlin.de/http://lvb.wiwi.hu-berlin.de/ -
7/28/2019 20110706 D2 Spectral Density
22/22
References 5-1
Bibliography
Hamilton, J.D., 1994Time Series Analysis
Princeton University Press
Shumway, R. and Stoffer, D.S., 2006Time Series Analysis and Its Application, with R examples, 2nd
Edition
Springer Science and Business Media
Spectral Analysis 0 0 . 05 0 . 1 0 . 15 0 . 2 0 . 2 5 0 . 3 0 . 3 5 0 . 4 0 . 4 5 0 . 500.5
1
1.5
2
2.5
3
3.5
4
Frequency,
f(w
)
w