SASP Slides

8/11/2019 SASP Slides

1/23

Spectral Analysis ofStochastic Processes

Hening Rust

Stochastic Processes

Spectral Analysis

Stationary StochasticProcesses

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Spectral Analysis of Stochastic ProcessesTheory

Henning [email protected]

University of Potsdam and Potsdam Institut for Climate ImpactResearch, Germany

E2C2 Summer School, Comorova, September 2007

1 / 1 2


2/23


Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Outline


Spectral Analysis

Some Stationary Stochastic Processes

Parameter Estimation

Model Selection

The FFF E2C2 Summer Cup

Lecture notes in shared folder Spectral Analysis

2 / 1 2


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Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Spectral Analysis

0.0 0.1 0.2 0.3 0.4 0.5

1e0

3

1e01

1e+01

frequency

spectrum

periodogrammod. Daniell (8,25)

AR[2]

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4/23


Hening Rust


Spectral Analysis


White Noise

AR 1

AR 2 ,Oscillating

AR 2, Relaxating

Fractional Differenced

FARIMA 1,d,0

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

White Noise

Time

dat

.wn

0 50 100 150 200

2

2

4

0.0 0.1 0.2 0.3 0.4 0.5

0.0

02

0.5

00

frequency

spe

ctrum

4 / 1 2


5/23


Hening Rust


Spectral Analysis


White Noise

AR 1

AR 2 ,Oscillating

AR 2, Relaxating


FARIMA 1,d,0

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

AR[1]

Time

dat

.ar1

0 50 100 150 200

4

0

4

0.0 0.1 0.2 0.3 0.4 0.51e04

1e+00

frequency

spe

ctrum

5 / 1 2


6/23


Hening Rust


Spectral Analysis


White Noise

AR 1

AR 2 ,Oscillating

AR 2, Relaxating


FARIMA 1,d,0

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

AR[2], Oscillating

Time

dat.a

r2.

osc

0 50 100 150 200

4

0

4

0.0 0.1 0.2 0.3 0.4 0.51e03

1e+01

frequency

spe

ctrum

6 / 1 2


7/23


Hening Rust


Spectral Analysis


White Noise

AR 1

AR 2 ,Oscillating

AR 2, Relaxating


FARIMA 1,d,0

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

AR[2], Relaxating

Time

dat.a

r2.

rel

0 50 100 150 200

10

0

5

0.0 0.1 0.2 0.3 0.4 0.51e03

1e+03

frequency

spe

ctrum

7 / 1 2


8/23


Hening Rust


Spectral AnalysisStationary StochasticProcesses

White Noise

AR 1

AR 2 ,Oscillating

AR 2, Relaxating


FARIMA 1,d,0

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Fractional Differenced (FD)

Time

dat.

fd

0 50 100 150 200

4

0

2

4

0.0 0.1 0.2 0.3 0.4 0.5

1e03

1e+01

frequency

spec

trum

8 / 1 2


9/23


Hening Rust



White Noise

AR 1

AR 2 ,Oscillating

AR 2, Relaxating


FARIMA 1,d,0

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Fractional Differenced (FD)

Time

dat.

fd

0 50 100 150 200

4

0

2

4

5e04 2e03 1e02 5e02 2e01

1e03

1e+01

frequency

spec

trum

8 / 1 2


10/23


Hening Rust



White Noise

AR 1

AR 2 ,Oscillating

AR 2, Relaxating


FARIMA 1,d,0

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

FARIMA[1,d,0]

Time

dat.

far

0 50 100 150 200

6

0

4

0.0 0.1 0.2 0.3 0.4 0.51e04

1

e+00

frequency

spec

trum

9 / 1 2


11/23


Hening Rust



White Noise

AR 1

AR 2 ,Oscillating

AR 2, Relaxating


FARIMA 1,d,0

ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

FARIMA[1,d,0]

Time

dat.

far

0 50 100 150 200

6

0

4

5e04 2e03 1e02 5e02 2e011e041

e+00

frequency

spec

trum

9 / 1 2


12/23

Non-Nested Model Selection, Scheme

model f() model g()

data x

Ensemble

xf,r

Ensemble

xg,r

lf(f,r|xf,r) lg(f,r|xf,r) lg(g,r|xg,r)lf(g,r|xg,r)

lrf,r lrg,r


13/23


Hening Rust



ParameterEstimation

Model Selection

Non-Nested Model

Selection

E2C2 Summer Cup

Appendix

Non-Nested Model Selection, Outcome

5 0 5

0.0

0.1

0.2

0.3

0.4

lr

density

a)b)

5 0 5

0.0

0.4

0.8

lr

ECDF

a)b)

11/12


14/23


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Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection

E2C2 Summer CupAppendix

E2C2 Summer Cup Round II

Prices one good answer one good drink 1

Questions

What is the difference between the spectral densityand the periodogram?

What is the difference between SRD and LRD?

Which theorem links the ACF and the spectral density?

1can be substituted with organic bittersweet chocolate12/12

S l A l i f


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Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection




Questions





S t l A l i f


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Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection




Questions





Spectral Analysis ofC S C


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Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix



Questions





Spectral Analysis of


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Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Definition LRD

FARIMA Processes

Appendix

13/12


20/23

Spectral Analysis ofL R D d


21/23


Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Definition LRD

FARIMA Processes

Long-Range Dependence

Definition

A stationary stochastic process Xt is calledlong-rangedependent(LRD) if its autocorrelation function () is notsummable, i.e.

() = .

Examples

algebraically decaying ACF for large lags

lim () = c

pole in the spectrum at = 0lim0 S() = cS||

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Spectral Analysis ofL R D d


22/23

Sp yStochastic Processes

Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Definition LRD

FARIMA Processes

Long-Range Dependence

Definition

A stationary stochastic process Xt is calledlong-rangedependent(LRD) if its autocorrelation function () is notsummable, i.e.

() = .

Examples

algebraically decaying ACF for large lags

lim () = c

pole in the spectrum at = 0lim0 S() = cS||

14/12

Spectral Analysis ofFARIMA P ocesses


23/23

p yStochastic Processes

Hening Rust


Spectral Analysis


ParameterEstimation

Model Selection

E2C2 Summer Cup

Appendix

Definition LRD

FARIMA Processes

FARIMA Processes

Example

5e04 1e02 5e01

0.2

2.0

50.0

frequency

sdf

FAR[1]

AR[1]

Spectral Density

S() =

2

2

|(ei)|2

|(ei

)|2|1 ei|2d

0 cS||

2d

15/12

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