time series analysis in scientific computing (24.11.2008 … · 2010-02-04 · 1) numerical methods...
TRANSCRIPT
![Page 1: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/1.jpg)
Institute of MathematicsFreie
Universität
Berlin (FU)
Time Series Analysis in Scientific Computing
(24.11.2008-28.11.2008)I. Horenko, R. Klein, Ch. Schütte
DFG Research Center MATHEON „Mathematics
in key
technologies“
![Page 2: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/2.jpg)
Scientific Computing: Complex Systems
Meteorology/Climate
Fluid MechanicsMarket Phase
1
Market
Phase 2
Biophysics/Drug Design
Computational Finance
Deterministic description “from first principles“isfrequently Unavailable or Unfeasible!
Time Series
Analysis
![Page 3: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/3.jpg)
Complex
Systems
Properties:
1)
non-stationarity
2)
a lot of d.o.f.s are
involved
(multidimensionality)
3)
stochasticity
4)
presence
of hidden phases/regimes
Aim
of the
Seminar:
mathematical concepts and methods of multidimensional stochastic
time series analysys and identification of hidden phases
![Page 4: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/4.jpg)
Plan of the
Seminar (mornings):
Monday (Illia Horenko):Deterministic view on stochastic processes (direct and inverse numerical problems)
Tuesday (Christof Schütte):Identification of hidden phases: introduction (K-Means), subspace iteration methods, Expectation-Maximisation algorithm,Gaussian Mixture Models (GMM)
Wednesday (Christof Schütte):Hidden Markov models (HMM), HMM in multiple dimensions (HMM-VAR)
Thursday (Illia Horenko):Variational approach to time series analysis, finite element methods (FEM) in data analysis (FEM-Clustering)
Friday (Illia Horenko): Methods of non-stationary time series analysis
![Page 5: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/5.jpg)
Numerics
of Direct and InverseProblems in Stochastics: deterministic
viewpoint
![Page 6: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/6.jpg)
Complex
Systems
Properties:
1)
non-stationarity
2)
a lot of d.o.f.s are
involved
(multidimensionality)
3)
stochasticity
4)
presence
of hidden phases/regimes
Today
we
look
at:
1) stochastic processes and their
deterministic
interpretation
2)
dynamical
systems
viewpoint
2) inverse
problems
in stochastics: functional minimization
![Page 7: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/7.jpg)
Complex
Systems
Properties:
1)
non-stationarity
2)
a lot of d.o.f.s are
involved
(multidimensionality)
3)
stochasticity
4)
presence
of hidden phases/regimes
Today
we
look
at:
1) stochastic processes and their
deterministic
interpretation
2)
dynamical
systems
viewpoint
2) inverse
problems
in stochastics: functional minimization
![Page 8: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/8.jpg)
Memo I: Probability
![Page 9: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/9.jpg)
Memo I: Probability
![Page 10: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/10.jpg)
Memo II: Stochastic
Process
Probability Density Function:
Expectation Value:
Variance:
White Noise:
![Page 11: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/11.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
![Page 12: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/12.jpg)
Direct
Stochastic
Problems
![Page 13: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/13.jpg)
Markov
Chains
Realizations of the process:
Markov-Property:
Example:
![Page 14: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/14.jpg)
Realizations of the process:
Markov-Property:
State Probabilities:
Discrete
Markov
Process…
![Page 15: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/15.jpg)
Realizations of the process:
Markov-Property:
State Probabilities:
Discrete
Markov
Process…
![Page 16: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/16.jpg)
Realizations of the process:
Markov-Property:
State Probabilities:
Discrete
Markov
Process…
![Page 17: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/17.jpg)
Realizations of the process:
Markov-Property:
State Probabilities:
Discrete
Markov
Process…
![Page 18: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/18.jpg)
Discrete
Markov
Process…
Realizations of the process:
Markov-Property:
State Probabilities:
This Equation is Deterministic
![Page 19: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/19.jpg)
Continuous
Markov
Process…
![Page 20: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/20.jpg)
Continuous
Markov
Process…
![Page 21: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/21.jpg)
Continuous
Markov
Process…
![Page 22: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/22.jpg)
Continuous
Markov
Process…
infenitisimalgenerator
![Page 23: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/23.jpg)
Continuous
Markov
Process…
infenitisimalgenerator
This Equation is Deterministic: ODE
![Page 24: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/24.jpg)
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
![Page 25: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/25.jpg)
Markov
Process
in R
Realizations of the process:
Process:
![Page 26: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/26.jpg)
Markov
Process
in R
Realizations of the process:
Process:
If then
![Page 27: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/27.jpg)
Markov
Process
in R
Realizations of the process:
Process:
If then
Representing the arbitrary intial probability density with Dirac-
testfunctions results in:
This Equation is Deterministic
(follows
also fromIto‘s
formula)
![Page 28: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/28.jpg)
Infenitisimal Generator:
Markov Process Dynamics:
This Equation is Deterministic: PDE
Markov
Process
in R
![Page 29: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/29.jpg)
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
![Page 30: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/30.jpg)
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
![Page 31: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/31.jpg)
Intermediate
Conclusions…1) Numerical Methods from ODEs and (multidimensional) PDEs like
Runge-Kutta-Methods, FEM and (adaptive) Rothe particle methodsare applicable to stochatic 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 (Whitneyand Takens theorems, model reduction by identification of attractors)
Adaptive PDE particle methods in multiple dimensions:H./Weiser, JCC 24(15), 2003 H./Weiser/Schmidt/Schütte, JCP 120(19), 2004H./Lorenz/Schütte/Huisinga, JCC 26(9), 2005Weiße/H./Huisinga, LNCS 4216, 2006
![Page 32: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/32.jpg)
Short Trip to the World ofDynamical Systems
![Page 33: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/33.jpg)
Essential Dynamics of many
dynamical
systems
is
defined
by attractors.
Attractor stays
invariant under
the
flow operator
Separation of informative (attractor) and non-informative (rest)parts of the space leads to dimension reduction
Dynamical
systems
viewpoint
Figuresfrom
http:\\ww
w.w
ikipedia.org
![Page 34: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/34.jpg)
Problem: Euclidean
distance
cannot
beused
as a measure
for
the
relative neighbourghood
of attractor
elements.
How
to localize
the
attractor?
Attractors can have very complex, even fractal geometry
Strategy: the
data
have
to be
“embedded“
into
Eucllidean
space
Whitney embedding theorem (Whitney, 1936) : sufficiently
smooth connected -dimensional manifolds can be smoothly
embedded in -dimensional Euclidean space.
![Page 35: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/35.jpg)
Takens‘ Embedding (Takens, 1981)
and is a smooth map.
Assume that has an attractor with dimension
. Let , be a “proper“
measurement process. Then
is a -dimensional embedding in Euclidean space
How
to construct
the
embedding?
![Page 36: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/36.jpg)
Illustration of Takens‘
EmbeddingThe
image of is
unfolded
in
Problems:-
how to “filter out“
the attractive subspace?
-
how
to connect
the
changes
in attractive
subspace
with
hiddenphase?
Strategy:
Let
. Define
a new
variable
. Let
the
attractor
for
each
of the
hidden
phases
be
contained
in a distinct
linear
manifold defined
via
![Page 37: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/37.jpg)
Topological
dimension
reduction
(H. 07): for
a given
time series
we
look
for
a
minimum
of the
reconstruction error
μ
T
![Page 38: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/38.jpg)
Topological
dimension
reduction
(H. 07): for
a given
time series
we
look
for
a
minimum
of the
reconstruction error
μ
(X-μ)
TT (X-μ)T
Txt
t
t
![Page 39: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/39.jpg)
Topological
dimension
reduction
(H. 07): for
a given
time series
we
look
for
a
minimum
of the
reconstruction error
μ
(X-μ)
TT (X-μ)T
Txt
t
t
![Page 40: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/40.jpg)
Analysis of embedded
data
PCA + Takens‘ Embedding (Broomhead&King, 1986)
Let . Define a new variable
. Let the attractor
from Takens‘
Theorem be in a linear manifold defined via
. Then
Reconstruction from m
essential coordinates:
![Page 41: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/41.jpg)
PCA+Takens: data
compression/reconstruction
1.
2.
3.
4.
5.
![Page 42: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/42.jpg)
0
10
20
30
40
50
−20−10
010
2030
−40
−20
0
20
40
Lorenz-Oszillator with measurement noise
Example: Lorenz
![Page 43: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/43.jpg)
Example: Lorenz
Reconstructed trajectory(red, for m=2)
![Page 44: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/44.jpg)
Inverse Stochastic Problems
![Page 45: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/45.jpg)
And Again: Memo I
![Page 46: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/46.jpg)
Markov Processes: Log-Likelihood
Observed Time Series:
Markov-Property:
State-Discrete (Markov Chains) State-Discrete (AR(1))
![Page 47: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/47.jpg)
Markov Processes: Log-Likelihood
Observed Time Series:
Markov-Property:
State-Discrete (Markov Chains) State-Discrete (AR(1))
![Page 48: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/48.jpg)
Markov Chains: Log-Likelihood
Observed Time Series: ,
Markov-Property:
Probability of Observed Time Series (Likelihood):
![Page 49: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/49.jpg)
Markov Chains: Log-Likelihood
Observed Time Series: ,
Markov-Property:
Log-Likelihood Functional:
Maximization problem is ill-posed
![Page 50: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/50.jpg)
Memo III: Optimization with Constraints
Lagrange Principle
![Page 51: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/51.jpg)
Take-Home Messages:
1.
Numerics
of ODEs and PDEs is applicable to
Stochastic Processes
2.
Numerical inverse problems in stochastics can be understood
as deterministic minimization problems with constraints
3. Minimization problems can be ill-posed: additional
information/assumptions may become necessary
![Page 52: Time Series Analysis in Scientific Computing (24.11.2008 … · 2010-02-04 · 1) Numerical Methods from ODEs and (multidimensional) PDEs like Runge-Kutta-Methods, FEM and (adaptive)](https://reader033.vdocument.in/reader033/viewer/2022060314/5f0b8dc57e708231d431160a/html5/thumbnails/52.jpg)
Thank you for attention!