20110706 d2 spectral density

7/28/2019 20110706 D2 Spectral Density

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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/


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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

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2

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3

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4

Frequency,

f(w

)

w


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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


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Frequency,

f(w

)

w


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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


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4

Frequency,

f(w

)

w


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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


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Frequency,

f(w

)

w


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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)


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Frequency,

f(w

)

w


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Outline

1. Motivation

2. Spectral Density

3. Filtering

4. Conclusion


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Frequency,

f(w

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w


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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)|


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Periodogram

Sample-based spectral density, j=j/n

f(j) =(n1)

h=(n1)

(h)e2ijh, j = 0, 1, ..., n 1 (5)


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4

Frequency,

f(w

)

w


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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.


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4

Frequency,

f(w

)

w


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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.


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4

Frequency,

f(w

)

w


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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.


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3

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4

Frequency,

f(w

)

w


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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.


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4

Frequency,

f(w

)

w


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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.


1

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3

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4

Frequency,

f(w

)

w


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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.


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Frequency,

f(w

)

w


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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


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Frequency,

f(w

)

w


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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|


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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


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Frequency,

f(w

)

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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


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Frequency,

f(w

)

w


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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


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Frequency,

f(w

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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/


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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


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Frequency,

f(w

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20110706 d2 spectral density

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