Gerhard-Wilhelm Weber 1*
Sırma Zeynep Alparslan Gök 2, Erik Kropat 3, Özlem Defterli 4, Fatma Yelikaya-Özkurt 1, Armin Fügenschuh 5
1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey 2 Department of Mathematics, Süleyman Demirel University, Isparta, Turkey 3 Department of Computer Science, Universität der Bundeswehr München, Munich, Germany 4 Department of Mathematics and Computer Science, Cankaya University, Ankara, Turkey 5 Optimierung, Zuse Institut Berlin, Germany
* Faculty of Economics, Management and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal Universiti Teknologi Malaysia, Skudai, Malaysia
New Insights and Applications of Eco-Finance Networks
and Collaborative Games
6th International Summer School
National University of Technology of the UkraineKiev, Ukraine, August 8-20, 2011
Bio- and Financial Systems
Genetic , Gene-Environment and Eco-Finance Networks
Time-Continuous and Time-Discrete Models
Optimization Problems
Numerical Example and Results
Networks under Uncertainty
Ellipsoidal Model
Optimization of the Ellipsoidal Model
Kyoto Game
Ellipsoidal Game Theory
Related Aspects from Finance
Hybrid Stochastic Control
Conclusion
Outline
Bio-Systems
medicine
food
education health caredevelopment
sustainability
bio materials bio energy
environment
Stock Markets
Regulatory Networks: Examples
Target variables Environmental items
Genetic Networks
Gene expression Transscription factors,toxins, radiation
Eco-Finance Networks
CO2-emissions Financial means, technical means
Further examples:Socio-econo-networks, stock markets, portfolio optimization, immune system, epidemiological processes …
DNA microarray chip experiments
prediction of gene patterns based on
with
M.U. Akhmet, H. Öktem
S.W. Pickl, E. Quek Ming Poh
T. Ergenç, B. Karasözen
J. Gebert, N. Radde
Ö. Uğur, R. Wünschiers
M. Taştan, A. Tezel, P. Taylan
F.B. Yilmaz, B. Akteke-Öztürk
S. Özöğür, Z. Alparslan-Gök
A. Soyler, B. Soyler, M. Çetin
S. Özöğür-Akyüz, Ö. Defterli
N. Gökgöz, E. Kropat
... Finance
Environment
Health Care
MedicineBio-Systems
Ex.: yeast data
GENE / time 0 9.5 11.5 13.5 15.5 18.5 20.5
'YHR007C' 0.224 0.367 0.312 0.014 -0.003 -1.357 -0.811
'YAL051W' 0.002 0.634 0.31 0.441 0.458 -0.136 0.275
'YAL054C' -1.07 -0.51 -0.22 -0.012 -0.215 1.741 4.239
'YAL056W' 0.09 0.884 0.165 0.199 0.034 0.148 0.935
'PRS316' -0.046 0.635 0.194 0.291 0.271 0.488 0.533
'KAN-MX' 0.162 0.159 0.609 0.481 0.447 1.541 1.449
'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192
'E. COLI #33' -0.405 0.853 -0.259 -0.124 -1.181 0.095 0.027
http://genome-www5.stanford.edu/
DNA experiments
Analysis of DNA experiments
E0 : metabolic state of a cell at t0 (:=gene expression pattern),ith element of the vector E0 :=expression level of gene i,
Mk := I + hkM(Ek) , Ek (k є IN0) is recursively defined as Ek+1 := MkEk.
Metabolic ShiftGebert et al. (2006)
Modeling & Prediction
( ): nE
0(0) =( ) ,
E EE M E E
( ): n nM
prediction, anticipation least squares – max likelihood
expression data
matrix-valued function – metabolic reaction
Tn tetetetEE ))(,...,)(,)(()( 21
Expression
data
kkk EE M1
1 ( ( )) ,k k k kE I h M E E 2
21 2( )k
k k khE I h M M E
Ex.:
)(Μ jik em M
We analyze the influence of em -parameters on the dynamics (expression-metabolic).
Ex.: Euler, Runge-Kutta
Modeling & Prediction
M stable
unstable metabolic reaction
feasible
unfeasible
Stability
goodness-of-fit (model) test
Def.: M is stable : B : (complex) bounded neighbourhood of
0 1 1, M ,M ,..., Mkk ΙΝ M :1 2 0(M M ... M ) .k k
• For which parameters, i.e., for which set M (hence, dynamics), is stability guaranteed ?
• For which parameters, i.e., for which set M (hence, dynamics), is stability guaranteed ?
M stable
unstable metabolic reaction
feasible
unfeasible
Stability
combinatorial algorithm
Akhmet, Gebert, Öktem, Pickl, Weber (2005), Gebert, Laetsch, Pickl, Weber, Wünschiers (2006), Weber, Ugur, Taylan, Tezel (2009), Ugur, Pickl, Weber, Wünschiers (2009)
Genetic Network
, 1
E M E h
)()()()(
)()()()(
)()()()(
)()()()(
34333231
24232221
14131211
04030201
tEtEtEtE
tEtEtEtE
tEtEtEtE
tEtEtEtE
080170255
25570180255
050200255
2550250255
2001
039.02.00
0061.04.0
0000
MĖ 4Ė 2
Ė 0
Ė 5
Ė 1
Ė 3
0123456789
0 2 4 6 8
Time, t
Ex
pre
ss
ion
lev
el,
Ė
Ex. :
gene2
gene3
gene1
gene4
0.4 E1
0.2 E2 1 E1
Genetic Network
Gene-Environment Networks
1:
0i j
if gene j regulates gene i
otherwise
Model Class
: d-vector of concentration levels of proteins and
of certain levels of environmental factors
: change in the gene-expression data in time
: time-autonomous form, where
: initial values of the gene-expression levels
: experimental data vectors obtained from microarray experiments
and environmental measurements : the gene-expression level (concentration rate) of the i th gene at time t
denotes anyone of the first n coordinates in thed-vector of genetic and environmental states.
: the set of genes.
Weber et al. (2008c), Chen et al. (1999), Gebert et al. (2004a),Gebert et al. (2006), Gebert et al. (2007), Tastan (2005), Yilmaz (2004), Yilmaz et al. (2005),Sakamoto and Iba (2001), Tastan et al. (2005)
(i) : a constant (nxn)-matrix : an (nx1)-vector of gene-expression levels(ii) represents and t the dynamical system of the n genes and their interaction alone. : : (nxn)-matrix with entries as functions of polynomials, exponential, trigonometric, splines or wavelets, containing some parameters to be optimized.
(iii)
environmental effects
n genes , m environmental effects
: (n+m)-vector and (n+m)x(n+m)-matrix, respectively.
Weber et al. (2008c), Tastan (2005), Tastan et al. (2006),Ugur et al. (2009), Tastan et al. (2005), Yilmaz (2004), Yilmaz et al. (2005),Weber et al. (2008b), Weber et al. (2009b)
(*)
Model Class
Model Class
In general, in the d-dimensional extended space,
with
: : (dxd)-matrix,
: (dx1)-vectors.
Ugur and Weber (2007), Weber et al. (2008c),Weber et al. (2008b), Weber et al. (2009b)
Time-Discretized Model
- Euler’s method, - Runge-Kutta methods, e.g., 2nd-order Heun's method
3rd-order Heun's method is introduced by Defterli et al. (2009)
we rewrite it as
where
Ergenc and Weber (2004), Tastan (2005), Tastan et al. (2006), Tastan et al. 2005)
Time-Discretized Model
: in the extended space denotes the DNA microarray experimental data and the data of environmental items obtained at the time-level
: approximations obtained by the iterative formula above
: initial values
k th approximation (prediction):
(**)
Matrix Algebra
: (nxn)- and (nxm)-matrices, respectively
: (n+m)x(n+m) -matrix
: (n+m)-vectors
Applying the 3rd-order Heun’s method to (*) gives the iterative formula (**), where
Optimization Problem
mixed-integer least-squares optimization problem:
subject to
Ugur and Weber (2007),Weber et al.(2008c),Weber et al. (2008b),Weber et al. (2009b), Gebert et al. (2004a), Gebert et al. (2006), Gebert et al. (2007)
Boolean variables
, , : th : the numbers of genes regulated by gene (its outdegree), by environmental item , or by the cumulative environment, resp..
Mixed-Integer Problem
: constant (nxn)-matrix with entries representing the effect which the expression level of gene has on the change of expression of gene
genetic regulation network
mixed-integer nonlinear optimization problem (MINLP):
subject to
: constant vector representing the lower bounds
for the decrease of the transcript concentration.
Binary variables :
Numerical Example MINLP for data:
Gebert et al. (2004a)
Apply 3rd-order Heun method:
Take
using modeling language Zimpl 3.0, we solveby SCIP 1.2 as a branch-and-cutframework, together with SOPLEX 1.4.1 as LP solver
Numerical Example
Apply 3rd-order Heun’s time discretization :
Results of Euler Method for all genes:
____ gene A........ gene B_ . _ . gene C- - - - gene D
Results of 3rd-order Heun Method for all genes:
____ gene A........ gene B_ . _ . gene C- - - - gene D
Errors uncorrelated Errors correlated Fuzzy values
Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics
θ1
θ2
Regulatory Networks under Uncertainty
Errors uncorrelated Errors correlated Stochastics
Interval arithmetics Ellipsoidal calculus Lévy processes
θ1
θ2
Regulatory Networks under Uncertainty
Errors uncorrelated Errors correlated Stochastics
Interval arithmetics Ellipsoidal calculus Lévy processes
θ1
θ2
Regulatory Networks under Uncertainty
( ), ΜkM E
E
( )NI R
Model Class under Interval Uncertainty
( ) ( )( 1) M ( )s k s kE k E k C
θ1,1 θ1,2
θ2,1
θ2,2
( ) : ( ( 1))
1 if ( )( ( )) :
0 else
B
i ii
s k F Q E k
E kQ E k
Model Class under Interval Uncertainty
hybrid
min( ), ( ), ( )ij i im c d
( ( , ))y Y C D
21
0
l
M E C E D E
1
1
1
,min
( 1, ..., )
( 1, ..., )
( , ) ( )
( , ) ( )
( , ) ( )
&
n
ij ij ji
n
i ii
n
i ii
ii i
j n
m
p m y y
q c y y
d y y
m
overall box constraints
( ( , ))y Y C D
( 1,..., )i n
subject to
Model Class under Interval Uncertainty
I, K, L finite
2C
Generalized Semi-Infinite Programming
: structurally stable
global local global
)(
nIR
asymptotic
effect
)(
homeom.
:),(),( 0C
),(
Generalized Semi-Infinite Programming
Jongen, Weber, Guddat et al.
Generalized Semi-Infinite Programming
Thm. (W. 1999/2003, 2006):
Fulfilled!
Errors uncorrelated Errors correlated Stochastics
Interval arithmetics Ellipsoidal calculus Lévy processes
θ1
θ2
Regulatory Networks under Uncertainty
Errors uncorrelated Errors correlated Stochastics
Interval arithmetics Ellipsoidal calculus Lévy processes
θ1
θ2
Coalitions under uncertainty
Regulatory Networks under Uncertainty
Determine the degree of connectivity.
Regulatory Networks: Interactions
Clusters and Ellipsoids:
Target clusters: C1 , C2 ,…,CR Environmental clusters: D1, D2,…, DS
Target ellipsoids: X1, X2,…, XR Xi = E (μi , Σi) Environmental ellipsoids: E1 , E2,…, ES Ej = E (ρj , Πj)
Center Covariance matrix
Time-Discrete Model
Time-Discrete Model:
Targetcluster
Environmental cluster
Target Target Environment Target
Target Environment Environment Environment
R
r = 1A
TTj r X r
(k)ξ j0 +( ) +
S
s = 1A
ETj s E s
(k)( )=X j(k + 1)
R
r = 1A
TEi r X r
(k)ζ i0 +( ) +
S
s = 1A
EEis E s
(k)( )=E i(k + 1)
Determine system matrices and intercepts.
Time-Discrete Model
Ellipsoidal Calculus:
• Affine-linear transformations
• Sums of ellipsoids
• Intersections / fusions of ellipsoidsinner / outer
approximations
Kurzhanski, Varaiya (2008)Parameterized family of ellipsoidal approximations
Ros et al. (2002)
E1 + E2
E1 ∩ E2
AE + b
Time-Discrete Model
The Regression Problem:
Maximize (overlap of ellipsoids)
measurement
prediction
ATTj r , A
ETj s , A
TEi r , A
EEis
ξ j0 , ζ i0
Determine
matrices
vectors
and
Ellipsoidal Calculus
X r(k)^ X r
(k)−∩ E s
(k)^ E s(k)−
∩+ ΣR
r = 1ΣS
s = 1ΣT
k = 1
Set-Theoretic Regression Problem
Measures for the size of intersection:
• Volume → ellipsoid matrix determinant
• Sum of squares of semiaxes → trace of covariance matrix
• Length of largest semiaxes → eigenvalues of covariance matrix
rr ,E
r
semidefinite programming interior point methods
Set-Theoretic Regression Problem
0
0
1
Tj
Ci
Cj
1
1
Curse of Dimensionality
χij =1, if Cj Ci
0, if Cj \ Ci
Mixed-Integer Regression Problem:
maximize
such that ≤j jαTT
deg(C )TE ≤j jαTE
deg(D )ET ≤i iαET
deg(D )EE ≤i iαEE
bounds on outdegreesdeg(C )TT
X r(k)^ X r
(k)−∩ E s
(k)^ E s(k)−
∩+ ΣR
r = 1ΣS
s = 1ΣT
k = 1
Curse of Dimensionality
Scale free networks(metabolic networks, world wide web, …)
• High error tolerance
• High attack vulnerability (removal of important nodes)
Curse of Dimensionality
Continuous Regression Problem:
X r(k)^ X r
(k)−∩ E s
(k)^ E s(k)−
∩+ maximize
such that P TT ( TT ≤j r jαTT
≤ jαTE
≤ iαET
≤ iαEE
bounds on outdegrees
ΣR
r = 1ΣS
s = 1ΣT
k = 1
ATTj r , ξ
j0
P TE ( TE j r
ATE
j r , ξ j0
P ET ( ET AETi s , ζ
i0
P EE ( EE AEEi s
ΣR
r = 1
, ζ i0
ΣR
r = 1
ΣR
s = 1
ΣR
s = 1
Continuous Constraints /Probabilities
is
is
)
)
)
)
Curse of Dimensionality
Continuous Regression Problem:
X r(k)^ X r
(k)−∩ E s
(k)^ E s(k)−
∩+ maximize
such that P TT ( TT ≤j r jαTT
≤ jαTE
≤ iαET
≤ iαEE
ΣR
r = 1ΣS
s = 1ΣT
k = 1
ATTj r , ξ
j0
P TE ( TE j r
ATE
j r , ξ j0
P ET ( ET AETi s , ζ
i0
P EE ( EE AEEi s
ΣR
r = 1
, ζ i0
ΣR
r = 1
ΣR
s = 1
ΣR
s = 1
is
is
)
)
)
)
Curse of Dimensionality
Ex.:Robust Optimization
Cost Games
Cost games are very important in the practice of OR.
Ex.: airport game, unanimity game, production economy with landowners and peasants, bankrupcy game, etc..
There is also a cost game in environmental protection (TEM model):
The aim is to reach a state which is mentioned in Kyoto Protocol by choosing control parameters such that the emissions of each player become minimized.
For example, the value is taken as a control parameter.
The central problem in cooperative game theory is how to allocate the gain among the individual players in a “fair” way.
There are various notions of fairness and corresponding allocation rules (solution concepts).
Any with is an allocation.
So, a core allocation guarantees each coalition to be satisfied in the sense that it gets at least what it could get on its own.
* ( )w w N i N
( ) : { | ( ) *, ( ) ( ) ( )}, NCore w x x N w x S w S S NR
( ) : ( : ).
ii S
x S x S N coalition
Nx R ( ) *x N w
( )x Core wS N
Cost Games
Influence of memory parameter on the emissions reduced and financial means expended
TEM Model
( 1) ( ) ( )
( )
k k k
kM
E E E
M M M
( +1) ( ) ( ) IE IM IEk k k
( 1) ( ) ( )
( )( )
0k k k
kkM
u
E E E
M M M
TEM Model
Gamescooperative
. . . .
cooperative Interval Games
. . .
Ellipsoid Games Interval Games
.cooperative
Robust Optimization
Interval Gamescooperative
: ( ) | ( ), ( ) ( )
Ni j
i N
w I I I w N I w j j NRI
: ( ) | ( ) ( , )
ii S
Core w I w I w S S N SI
1 2 1 2
1 2
1 2
, :
: ( )( ) ( ) ( )
: ( )( ) ( )
N
< N, >
< N, >
S S S
S S
w w
w w IG
w
w w
w
w w
w
:= 1,2,
...
, , : 2 ( ), ( ) [0,
0]
< , >
N
NnN I ww
N IGw
R
:= 1,2,
...
, , : 2 ( ), ( ) [0,
0]
< , >
N
NnN I ww
N IGw
R
Interval Glove Game
: , :
: 0 €, : 0 €
: 10 - 20 €.
(1,3) (2,3) (1,2,3) [10,20]
( ) [0,0], else.
( ) = {([0,0],[0,0],[10,20])}.
= 1,2,3
= 1,2 :
L R
w w w
w S
Co
N
L
re w
L R
Interval : core
Interval Gamescooperative
:= 1, 2,..., , : 2
)
, ( 0
:
< , >
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games cooperative
2
1
1/ 2
( , ) = | ( ) ( ) 1 ,
( , ) = + | || || 1
E
E
Tpc x x c x c
c u c u
R
( , ) ( , ) E E TA c +b = Ac+b A A
1 2
11
2 2
2
11 : ( , ( )
( ) : (1 )
)
) (1
E EE E E+ = c +c
s s
s
s
:= 1, 2,..., , : 2
)
, ( 0
:
< , >
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Glove Game
: , :
: 0 €, : 0 €
: ( , ) €.
(1,3) (2,3) (1,2,3) ( , )
( ) 0 , else.
= 1,2,3
= 1,2 :
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore, value, . . . .
E
E
Ellipsoid Games cooperative
:= 1, 2,..., , : 2
)
, ( 0
:
< , >
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Kyoto Game
, : (individual roles in TEM Model)
: (individual role in TEM Model)
Ellipsoid Games cooperative
: , :
: 0 €, : 0 €
: ( , ) €.
(1,3) (2,3) (1,2,3) ( , )
( ) 0 , else.
= 1,2,3
= 1,2 :
L R
c
w w w c
w
R
S
N L
L
Ellipsoid ellipsoidcore, value, . . . .
E
E
Ellipsoid Glove Game
:= 1, 2,..., , : 2
)
, ( 0
:
< , >
N
N
wn wN
EGN w
set of all ellipsoids
Ellipsoid Games cooperative
Ellipsoid Malacca Police Game
R
( +1) ( ) ( ) IE IM IEk k k
( ) ( ) ( ) 2( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( )1E (E , ) (E , ) ( )(E , ) 1
2
k k kk k k k k ki i
i i i i ik k
W Wa t b t b'b t
h h
( ) ( ) ( ) ( )E (E , ) (E , )t t t ti i i id a t dt b t dW
.
Finance Networks
( +1) ( ) ( ) IE IM IEk k k
( ) ( ) ( ) 2( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( )1E (E , ) (E , ) ( )(E , ) 1
2
k k kk k k k k ki i
i i i i ik k
W Wa t b t b'b t
h h
( ) ( ) ( ) ( )E (E , ) (E , )t t t ti i i id a t dt b t dW
.
Finance Networks with Bubbles
( +1) ( ) ( ) IE IM IEk k k
( ) ( ) ( ) 2( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
( )1E (E , ) (E , ) ( )(E , ) 1
2
k k kk k k k k ki i
i i i i ik k
W Wa t b t b'b t
h h
( ) ( ) ( ) ( )E (E , ) (E , )t t t ti i i id a t dt b t dW
.
Finance Networks with Bubbles
hybrid
( , ) ( , ) t t t tdX a X t dt b X t dW
( [0, ])(0, ) t t TW N t
Ex.: price, wealth, interest rate, volatility
processes
drift diffusion
Financial Dynamics
Milstein Scheme:
and, based on finitely many data:
21 1 1 1 1
1ˆ ˆ ˆ ˆ ˆ( , )( ) ( , )( ) ( )( , ) ( ) ( )2 j j j j j j j j j j j j j j j jX X a X t t t b X t W W b b X t W W t t
2( )( , ) ( , ) 1 2( )( , ) 1 .
j jj j j j j j j
j j
W WX a X t b X t b b X t
h h
Financial Dynamics
Financial Dynamics Identified
2 2
22min μX A L
Tikhonov regularization
,
2
2
subject to
min ,
,
t
t
t
M
X A
L
conic quadratic programming
Interior Point Methods
Important new class of (Generalized) Partial Linear Models: Important new class of (Generalized) Partial Linear Models:
, ,
( , ) GPLM ( ) ( )LM MARS
= +
TE Y X T G X T
X T X T
e.g.,
x
y
+( , )=[ ( )]c x x ( , )=[ ( )]-c x x
CMARS
Özmen, Weber, Batmaz
Financial Dynamics Identified
Robust CMARS:
RCMARS
Financial Dynamics Identified
semi-length of confidence interval
.. ..outlier outlier
confidence interval
. ......( ) jT
... . .. ... .. .... .. . . . ..
Özmen, Weber, Batmaz
Robust CMARS:
RCMARS
semi-length of confidence interval
.. ..outlier outlier
confidence interval
. ......( ) jT
... . .. ... .. .... .. . . . ..RCGPLM
Financial Dynamics Identified
Özmen, Weber, Batmaz
Portfolio Optimization Identified
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic control
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic control
Parameter Estimation
Portfolio Optimization Identified
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic controlParameter Estimation
Portfolio Optimization Identified
max utility ! or
min costs ! or
min risk!
martingale method:
Optimization Problem
Representation Problem
or stochastic controlParameter Estimation
Portfolio Optimization Identified
Hybrid Stochastic Control
• standard Brownian motion
• continuous state
Solves an SDE whose jumps are governed by the discrete state.
• discrete state Continuous time Markov chain.
• control
Control of Stochastic Hybrid Systems, R.Raffard
Applications
• Engineering: Maintain dynamical system in safe domain for maximum time.
• Systems biology: Parameter identification.
• Finance: Optimal portfolio selection.
hybrid
Method: 1st step
1. Derive a PDE satisfied by the objective function in terms of the generator:
• Example 1: If
then
• Example 2:If
then
hybrid
Method: 2nd and 3rd step
2. Rewrite original problem as deterministic PDE optimization program:
3. Solve PDE optimization program using adjoint method.
Simple and robust …
hybrid
References
Thank you very much for your attention!
http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf
References Part 1
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Sequence Data(cDNA, Genome,Genbank, etc.)
Selection or Design andSynthesis of the Probes
Array Production
Laser Scan of the Array
Picture Analysis
Test Material Control Material
mRNA-Isolation
cDNA-Synthesisand Labeling
Hybridization
Array Preparation Sample Preparation Data Analysis
DNA experimentsAppendix
Application
Evaluation of the models based on performance values: • CMARS performs better than Tikhonov regularization with respect to all the measures for both data sets. • On the other hand, GLM with CMARS (GPLM) performs better than both Tikhonov regularization and CMARS with
respect to all the measures for both data sets.
F. Yerlikaya Özkurt, G.-W. Weber, P. Taylan
Identifying Stochastic Differential EquationsAppendix