computational time series analysis (ipam, part i...
TRANSCRIPT
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Computational Time Series Analysis
(IPAM, Part I, stationary methods) Illia Horenko
Research Group “Computational Time Series Analysis “ Institute of Mathematics Freie Universität Berlin Germany
and
Institute of Computational Science Universita della Swizzera Italiana, Lugano Switzerland
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Motivation (I): Data
Inverse problem: data-based identification of model parameters
Challenges:
VERY LARGE (~Tb size) , non-Markovian (memory), non-stationary (trends), multidimensional, ext. influence
time series of vectors (continous, finite dim.)
time series of regimes (discrete, finite dim.)
anticyclonic
other
cyclonic
time series of functions (continuous, infin. dim.)
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Motivation (II): Models
Inverse problem: data-based identification of model parameters
Challenges:
existence and uniqueness, robustness, conditioning
Models
Dynamical (f.e., deterministic, stochastic, ...)
Geometrical (f.e, essential manifolds, ...)
Mixed (f.e., MTV, PIP/POP, ...)
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Plan for Today (stationary case)
• “eagle eye“ perspective on stochastic processes from the viewpoint of
deterministic dynamical systems
• geometric model inference: EOF/SSA
• multivariate dynamical model inference: VARX
• handling the ill-posed problem
• motivation for tomorrow : examples where stationarity assumption
does not work
![Page 5: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/5.jpg)
Plan for Today (stationary case)
• “eagle eye“ perspective on stochastic processes from the viewpoint of
deterministic dynamical systems
• geometric model inference: EOF/SSA
• multivariate dynamical model inference: VARX
• handling the ill-posed problem
• motivation for tomorrow : examples where stationarity assumption
does not work
![Page 6: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/6.jpg)
Classification of Stochastic Process
Stochastic Processes
Discrete State Space, Discrete Time: Markov Chain
Discrete State Space, Continuous Time: Markov Process
Continuous State Space, Discrete Time: Autoregressive Process
Continuous State Space, Continuous Time: Stochastic Differential Equation
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Realizations of the process:
Markov-Property:
Example:
Discrete Markov Process…
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Realizations of the process:
Markov-Property:
State Probabilities:
Discrete Markov Process…
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Realizations of the process:
Markov-Property:
State Probabilities:
Discrete Markov Process…
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Discrete Markov Process…
Realizations of the process:
Markov-Property:
State Probabilities:
This Equation is Deterministic
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Continuous Markov Process…
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Continuous Markov Process…
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Continuous Markov Process…
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Continuous Markov Process…
infenitisimal generator
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Continuous Markov Process…
infenitisimal generator
This Equation is Deterministic: ODE
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Continuous State Space
Stochastic Processes
Discrete State Space, Discrete Time: Markov Chain
Discrete State Space, Continuous Time: Markov Process
Continuous State Space, Discrete Time: AR(1)
Continuous State Space, Continuous Time: Stochastic Differential Equation
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Deterministic Markov Process in R
Realizations of the process:
Process: Flow operator:
Properties:
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Deterministic Markov Process in R
Realizations of the process:
Process: Flow operator:
Properties:
Example: ODE with
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Deterministic Markov Process in R
Realizations of the process:
Liouville Theorem: let be the flow ( ), given as a solution of
If is defined as
then the following equation is fulfilled:
Process: Flow operator:
Properties:
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Deterministic Markov Process in R
Realizations of the process:
Liouville Theorem: let be the flow ( ), given as a solution of
If is defined as
then the following equation is fulfilled:
Process: Flow operator:
Properties:
Excercise 1: think of the proof (hint: use the partial integration ), think about the Hilbert space H and appropriate boundary conditions
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Realizations of the process:
Process:
This Equation is Deterministic
(follows from Ito‘s lemma)
Stochastic Markov Process in R
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Infenitisimal Generator:
Markov Process Dynamics:
This Equation is Deterministic: PDE
Stochastic Markov Process in R
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Numerics of Stochastic Processes
Discrete State Space, Discrete Time: Markov Chain
Discrete State Space, Continuous Time: Markov Process
Continuous State Space, Discrete Time: Autoregressive Process
Continuous State Space, Continuous Time: Stochastic Differential Equation
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Numerics of Stochastic Processes
Deterministic Numerics of ODEs and PDEs!
Discrete State Space, Discrete Time: Markov Chain
Discrete State Space, Continuous Time: Markov Process
Continuous State Space, Discrete Time: Autoregressive Process
Continuous State Space, Continuous Time: Stochastic Differential Equation
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Conclusions 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable to stochastic processes
2) Monte-Carlo-Sampling of resulting p.d.f.‘s
Stochastic Numerics = ODE/PDE numerics + Rand. Numb. Generator
3) Concepts from the Theory of Dynamical Systems are Applicable (Whitney and Takens theorems, model reduction by identification of attractors)
H./Weiser, JCC 24(15), 2003
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Plan for Today (stationary case)
• “eagle eye“ perspective on stochastic processes from the viewpoint of
deterministic dynamical systems
• geometric model inference: EOF/SSA
• multivariate dynamical model inference: VARX
• handling the ill-posed problem
• motivation for tomorrow : examples where stationarity assumption
does not work
![Page 27: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/27.jpg)
EOF/PCA/SSA/...: Geometrical methods
For a given time series we look for a
minimum of the reconstruction error
µ
(X-µ)
TT (X-µ) T
T x t
t
t
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EOF/PCA/SSA/...: Geometrical methods
For a given time series we look for a
minimum of the reconstruction error
µ
(X-µ)
TT (X-µ) T
T x t
t
t
![Page 29: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/29.jpg)
EOF/PCA/SSA/...: Geometrical methods
For a given time series we look for a
minimum of the reconstruction error
µ
(X-µ)
TT (X-µ) T
T x t
t
t
![Page 30: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/30.jpg)
EOF/PCA/SSA/...: Geometrical methods
For a given time series we look for a
minimum of the reconstruction error
µ
(X-µ)
TT (X-µ) T
T x t
t
t
Excercise 2: think of the proof (hint: use the Lagrange multipliers ), think of the estimate of projection error
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Plan for Today (stationary case)
• “eagle eye“ perspective on stochastic processes from the viewpoint of
deterministic dynamical systems
• geometric model inference: EOF/SSA
• multivariate dynamical model inference: VARX
• handling the ill-posed problem
• motivation for tomorrow : examples where stationarity assumption
does not work
![Page 32: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/32.jpg)
Predicting Stochastic Processes
Deterministic Numerics of ODEs and PDEs!
Discrete State Space, Discrete Time: Markov Chain
Discrete State Space, Continuous Time: Markov Process
Continuous State Space, Discrete Time: Autoregressive Process
Continuous State Space, Continuous Time: Stochastic Differential Equation
![Page 33: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/33.jpg)
Predicting the Dynamics: VAR(p)
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Estimating VAR(p) : least squares
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Estimating VAR(p) : least squares
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Estimating VAR(p) : least squares
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Estimating VAR(p) : least squares
Excercise 3: think of the proof (hint: use the matrix function derivatives from the Matrix Coockbook: www2.imm.dtu.dk/pubdb/views/edoc_download.../imm3274.pdf))
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Plan for Today (stationary case)
• “eagle eye“ perspective on stochastic processes from the viewpoint of
deterministic dynamical systems
• geometric model inference: EOF/SSA
• multivariate dynamical model inference: VARX
• handling the ill-posed problem
• motivation for tomorrow : examples where stationarity assumption
does not work
![Page 39: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/39.jpg)
Estimating VAR(p) : regularization
A. N. Tikhonov (http://en.wikipedia.org)
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Estimating VAR(p) : regularization
A. N. Tikhonov (http://en.wikipedia.org)
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Plan for Today (stationary case)
• “eagle eye“ perspective on stochastic processes from the viewpoint of
deterministic dynamical systems
• geometric model inference: EOF/SSA
• multivariate dynamical model inference: VARX
• handling the ill-posed problem
• motivation for tomorrow : examples where stationarity assumption
does not work
![Page 42: Computational Time Series Analysis (IPAM, Part I ...helper.ipam.ucla.edu/publications/cltut/cltut_9008.pdfRunge-Kutta-Methods, FEM and (adaptive) Rothe particle methods are applicable](https://reader034.vdocument.in/reader034/viewer/2022050219/5f6534fde79d624a401bc443/html5/thumbnails/42.jpg)
Lorenz96: “order zero“ atmosph. model W
ilks,
Qua
rt.J
.Roy
al M
et.S
oc.,
2005
“slow“
“fast“
• Lorenz, Proc. Of. Sem. On Predict., 1996 • Majda/Timoffev/V.-Eijnden, PNAS, 1999 • Orell, JAS, 2003 • Wilks, Quart.J.Royal Met.Soc., 2005 • Crommelin/V.-Eijnden, Journal of Atmos. Sci., 2008
VARX (standard stationary stochastic Model)
stationary model (time-independent parameters)
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Lorenz96: “order zero“ atmosph. model
???
+ = non-stationary model
(time-dependent parameters)
VARX (standard stationary stochastic Model)
• Lorenz, Proc. Of. Sem. On Predict., 1996 • Majda/Timoffev/V.-Eijnden, PNAS, 1999 • Orell, JAS, 2003 • Wilks, Quart.J.Royal Met.Soc., 2005 • Crommelin/V.-Eijnden, Journal of Atmos. Sci., 2008
Wilk
s, Q
uart
.J.R
oyal
Met
.Soc
., 20
05
“slow“
“fast“
+ F(t)
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Lorenz96: “order zero“ atmosph. model
???
+ = non-stationary model
(time-dependent parameters)
Predictors of Lorenz‘96-Model
VARX (standard stationary stochastic Model)
• Lorenz, Proc. Of. Sem. On Predict., 1996 • Orell, JAS, 2003 • Wilks, Quart.J.Royal Met.Soc., 2005 • Crommelin/V.-Eijnden, Journal of Atmos. Sci., 2008
Wilk
s, Q
uart
.J.R
oyal
Met
.Soc
., 20
05
“slow“
“fast“
+ F(t)
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Tomorrow: Non-stationarity in TSA
Direct problem:
Inverse problem:
Example: