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

Final canonical block form of : = .

Matrix Algebra

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

TEM Model

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

. . .

Ellipsoid Games Interval Games

.cooperative

. . .

Ellipsoid Games Interval Games

.cooperative

. . .

Ellipsoid Games Interval Games

.cooperative

. . .

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

Ellipsoid Games cooperative

rer

rr

r

Ellipsoid Games cooperative

rer

rr

r

Farkas Lemma

( +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!

gweber@metu.edu.tr

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