sasp slides
TRANSCRIPT
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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
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8/11/2019 SASP Slides
2/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
ParameterEstimation
Model Selection
E2C2 Summer Cup
Appendix
Outline
Stochastic Processes
Spectral Analysis
Some Stationary Stochastic Processes
Parameter Estimation
Model Selection
The FFF E2C2 Summer Cup
Lecture notes in shared folder Spectral Analysis
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8/11/2019 SASP Slides
3/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
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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8/11/2019 SASP Slides
4/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
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
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8/11/2019 SASP Slides
5/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
White Noise
AR 1
AR 2 ,Oscillating
AR 2, Relaxating
Fractional Differenced
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
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8/11/2019 SASP Slides
6/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
White Noise
AR 1
AR 2 ,Oscillating
AR 2, Relaxating
Fractional Differenced
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
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8/11/2019 SASP Slides
7/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
White Noise
AR 1
AR 2 ,Oscillating
AR 2, Relaxating
Fractional Differenced
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
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8/11/2019 SASP Slides
8/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral AnalysisStationary StochasticProcesses
White Noise
AR 1
AR 2 ,Oscillating
AR 2, Relaxating
Fractional Differenced
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
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8/11/2019 SASP Slides
9/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral AnalysisStationary StochasticProcesses
White Noise
AR 1
AR 2 ,Oscillating
AR 2, Relaxating
Fractional Differenced
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
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8/11/2019 SASP Slides
10/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral AnalysisStationary StochasticProcesses
White Noise
AR 1
AR 2 ,Oscillating
AR 2, Relaxating
Fractional Differenced
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
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8/11/2019 SASP Slides
11/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral AnalysisStationary StochasticProcesses
White Noise
AR 1
AR 2 ,Oscillating
AR 2, Relaxating
Fractional Differenced
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
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8/11/2019 SASP Slides
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
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8/11/2019 SASP Slides
13/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral AnalysisStationary StochasticProcesses
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
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8/11/2019 SASP Slides
14/23
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8/11/2019 SASP Slides
15/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
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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8/11/2019 SASP Slides
16/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
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 t l A l i f
-
8/11/2019 SASP Slides
17/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
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
Spectral Analysis ofC S C
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8/11/2019 SASP Slides
18/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
ParameterEstimation
Model Selection
E2C2 Summer Cup
Appendix
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
Spectral Analysis of
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8/11/2019 SASP Slides
19/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
ParameterEstimation
Model Selection
E2C2 Summer Cup
Appendix
Definition LRD
FARIMA Processes
Appendix
13/12
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8/11/2019 SASP Slides
20/23
Spectral Analysis ofL R D d
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8/11/2019 SASP Slides
21/23
Spectral Analysis ofStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
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 ofL R D d
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8/11/2019 SASP Slides
22/23
Sp yStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
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
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8/11/2019 SASP Slides
23/23
p yStochastic Processes
Hening Rust
Stochastic Processes
Spectral Analysis
Stationary StochasticProcesses
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
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