6th International Summer School National University of Technology of the Ukraine Kiev, Ukraine, August 8-20, 2011 Let Us Predict Dynamicsunder Various Assumptions on Time and Uncertainty Gerhard-Wilhelm Weber 1*, Özlem Defterli 2, Linet Özdamar3, Zeynep Alparslan-Gök 4, Chandra Sekhar Pedamallu5, Büşra Temoçin1, 6,Azer Kerimov 1, Ceren Eda Can7, Efsun Kürüm1, 1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey 2 Department of Mathematics and Computer Science, Cankaya Uniiversity, Ankara, Turkey 3 Department of Systems Engineering, Yeditepe University, Istanbul, Turkey 4 Department of Mathematics, Süleyman Demirel University, Isparta, Turkey 5 Department of Medical Oncology, Dana-Farber Cancer Institute, Boston, MA, USA, The Broad Institute of MIT and Harvard, Cambridge, MA, USA 6 Department of Statistics, Ankara University, Ankara, Turkey 7 Department of Statistics, Hacettepe University, Ankara, Turkey * 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 Advisor to EURO Conferences
Outline Introduction and Motivation System Dynamics Primary Education (PE) Simulation in PE in Turkey and Its Results Simulation in PE in India and Its Results Transmission of HIV in Developing Countries Time-Continuous and Time-Discrete Models Gene-Environment Networks Numerical Example and Results Regulatory Networks under Uncertainty Ellipsoidal Model Stochastic Model Stochastic Hybrid Systems Portfolio Optimization subject to SHSs Lévy Processes, Simulation and Asset Price Dynamics Conclusion
System Dynamics Primary Education Education Index (according to 2007/2008, Human Development Report) 101 million children of primary school age are out of school Number of primary-school-agechildren not in school, by region (2007) http://www.childinfo.org/education.html
To design a good policy, issues to be considered / understood are:
- reasons behind enrollment, drop-outs and repeaters in school, - perceived quality of teaching by students and parents, - educational level and income level of parents, - expectations from school by parents, - perceived quality of teaching by the District EducationalOfficer(DEO), - needs of different infrastructure facilities(space, ventilation, sanitation, etc.) in the school, - and much more….
The choice of relevant attributes has to be made carefully, keeping in mind both
short-termandlong-termconsequences of solutions (decisions).
All attributes can be associated with given levels that may indicate
quantitativeor qualitative possession.
When entities interact through their attributes, the levelsof the attributes might change, i.e., the system behaves in certain directions.
Some changes in attribute levels may be desirablewhile others may not be so.
Each attribute influences several others, thus creating a web of complex interactions that eventually determine system behaviour. In other terms, attributes are variables that vary from time to time.
They can vary in an unsupervisedway in the system.
However, variables can be controlleddirectly or indirectly, and partially by introducing new intervention policies.
However, interrelationsamong variables should be analyzedcarefully beforeintroducing new policies.
System Dynamics 4 Steps of Implementation Step 1.Set the initial values to identified attributes obtained from published sources and surveys conducted. Step 2.Build a cross-impact matrix with the identified relevant attributes. a. Summing the effectsof column attributes on rows indicates the effect of each attribute in the matrix. b. The parameters ijcan be determined by creating a pairwise correlation matrixafter collecting the data,and adjusted by subjective assessment. c. Qualitative impacts are quantifiedsubjectively as shown in the table. Qualitative impacts can be extracted from a published sources and survey data set.
System Dynamics 4 Steps of Implementation d. The impact of infrastructural facilities on primary school enrollments and progression become visible by running the simulation model. e. An exemplary partial cross-impact matrix with the attributes and their hypothetical values above is illustrated as follows:
System Dynamics Problems with Migrant Primary School Students in Turkey Problems with Migrant Primary School Students in Turkey
Not able to assimilate in urban life and society
System Dynamics Problems with Migrant Primary School Students in Turkey Entities in the System Dynamics Model student, teacher, family, school environment, Ministry of Education. Entity Relationshipdiagram
System Dynamics Problems with Migrant Primary School Students in Turkey Attributes under each entity Entity 1: Student: F1.1Not believing in education F1.2Dislike of school books F1.3Lack of good Turkish language skills F1.4Dislike of school F1.5Weak academic self confidence F1.6Frequent disciplinary problem Entity 2: Teacher: F2.1Relating to guidance teacher Entity 3: Family: F3.1Economic difficulty F3.2Child labor F3.3Large families F3.4Malnutrition F3.5Homes with poor infrastructure (e.g., lack of heating, lack of room) F3.6Different home language F3.7Parent interest Entity 4: School environment: F4.1 Violence at school F4.2Alien school environment F4.3Lecture not meeting student needs
System Dynamics Problems with Migrant Primary School Students in Turkey The Proposed Policies to remediate the system P1. Daily distribution of milk P2. Distribution of coal, food, etc. to Turkish green card holders P3. The three-children per family aspiration by government P4. School infrastructure - renovation P5. Providing adult education for poor parents in migrant communities P6. Providing one year of Turkish language class before migrant student attends urban school P7.Combined policy: P2 + P4 + P5 + P6 While the mere basics of survival in the city are maintained by P2, policy P5would enable the parents to find better employment and improve the migrant family’s economic conditions. Policies P2and P5support parentsand, hence, their impacts on their offspring are indirect. On the other hand, policies P4and P6target the academic performance of students directly. The combined policy P7would naturally produce the best overall impact on the system if the budget of the Ministry of Education can afford it.
System Dynamics Problems with Migrant Primary School Students in Turkey Implementation Results
System Dynamics Problems with Primary Education in India The following simulationis based on data from Gujarat (India).
System Dynamics Problems with Primary Education in India
Level of enrollment(loe)
Beforeimplementation of policy variables (red line): - sharp increaseat the beginning phase of the simulation (first 12 iterations), and then there is steady decrease after a certain period of time (first 12 iterations). After implementation of policy variables (blue line):
steadyincreasefrom initial value (0.71) to unity.
increase in level of enrollment in first 12 iterations happens
because of the response lag in population to bad quality school system,
instant impact on the level enrollment after implementation
of policy variables because students and parents are more eager to have the children attend a nice looking healthy school.
System Dynamics Problems with Primary Education in India
Level of repeaters (lr)
Before implementation of policy variables (red line): - steady increasefrom 0.05 to unity in 50 iterations. After implementation of policy variables (blue line):
steadyincreasefrom initial value 0.05 to 0.12 in first 14 iterations and then declinedto zero in 50 iterations.
level of repeaters has instant impact on level of repeaters
ifthe infrastructural facilities are bad (assumption: teachers are doing their best in teaching – this variable is static -sovariable impacted here is teaching aids),
increase in level of repeaters in first 14 iterations is because improvement in the infrastructure doesnot have an instant impact on the level of repeaters.
Bio-Systems Medicine Environment ... Finance Health Care prediction of gene patterns based on DNA microarraychip experiments 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
DNA experiments Ex.:yeastdata http://genome-www5.stanford.edu/
Gene-Environment Networks if gene j regulates gene i otherwise
The Model Class is the firstly introduced time-autonomous form, where d-vector of concentration levels of proteins and of certain levels of environmental factors continuous change in the gene-expression data in time nonlinearities initial values of the gene-exprssion 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 the d-vector of genetic and environmental states. 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) is the set of genes.
The Model Class (i)is an constant (nxn)-matrix is an (nx1)-vector of gene-expression levels 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) 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) environmental effects (*) n genes , m environmental effects are (n+m)-vector and (n+m)x(n+m)-matrix, respectively.
The 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)
The Time-Discretized Model - Euler’s method, - Runge-Kutta methods, e.g., 2nd-order Heun's method 3rd-order Heun's method is introduced byDefterli et al. (2009) we rewrite it as where Ergenc and Weber (2004), Tastan (2005), Tastan et al. (2006), Tastan et al. 2005)
The Time-Discretized Model (**) in the extended spacedenotes the DNA microarray experimental data and the dataof environmental items obtained at the time-level the approximations obtainedby the iterative formula above initial values kthapproximation or prediction is calculated as:
Matrix Algebra are (nxn)- and (nxm)-matrices, respectively (n+m)x(n+m) -matrix are (n+m)-vectors Applying the 3rd-order Heun’s method to the eqn. (*) gives the iterative formula (**), where
Matrix Algebra Final canonical block form of : = .
Optimization Problem mixed-integer least-squares optimization problem: Boolean variables 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). , : th : the numbers of genes regulated by gene (its outdegree), by environmental item , or by the cumulative environment, resp..
The 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 vectorrepresenting the lower bounds for the decrease of the transcript concentration. Binary variables :
Numerical Example Consider our MINLP for the following data: Gebert et al. (2004a) Apply 3rd-order Heun method: Take using the modeling language Zimpl 3.0, we solve by SCIP 1.2 as a branch-and-cut framework, together with SOPLEX 1.4.1 as our LP-solver
Numerical Example Apply 3rd-order Heun’s time discretization :
____ gene A ........ gene B _ . _ . gene C - - - - gene D Results of Euler Method for all genes:
____ gene A ........ gene B _ . _ . gene C - - - - gene D Results of 3rd-order Heun Method for all genes:
Regulatory NetworksunderUncertainty Time-Discrete Model Target Target Environment Target ( R ) ( S ) Targetcluster TT (k) (k+1) ET (k) X ξ X A + + = E A j r r j0 j s j s r =1 s =1 ( R ) ( S ) Environmental cluster TE (k) (k+1) EE (k) X ζ E A + + = E A i r r i0 is i s r =1 s =1 Target Environment Environment Environment Determine system matrices and intercepts.
Regulatory NetworksunderUncertainty Time-Discrete Model controlfactor ( ( ) ) TT (k) (k+1) ET (k) Emissionclusters X ξ X A + + = E A + 0 j r r j0 j s j s r =1 s =1 ( R ) ( S ) Environ-mental cluster (k) TE (k) (k+1) EE (k) u + X ζ E A + + = E A i r r i0 is i s r =1 s =1 Predictionof CO2-emissions
Regulatory NetworksunderUncertainty The Mixed-Integer Regression Problem: R S T Σ Σ Σ − − ^ (k) (k) (k) (k) + ^ E E X Maximize X ∩ ∩ s r r s r = 1 s = 1 k= 1 α TT ≤ deg(C )TT s.t. j j α TE ≤ deg(C )TE j j boundson outdegrees α ET ≤ deg(D )ET i i α EE ≤ deg(D )EE i i
Regulatory NetworksunderUncertainty The Continuous Regression Problem: R S T Σ Σ Σ − − ^ (k) (k) (k) (k) + ^ E E X Maximize X ∩ ∩ s r r s r = 1 s = 1 k= 1 R Σ α TT TT ≤ P TT ( )TT s.t. , ξ A j j r jr j0 r =1 R α TE Σ ≤ TE P TE ( )TE , ξ A j j r j0 jr r =1 boundson outdegrees R ET Σ α ET P ET ( )ET ≤ , ζ A i i s i0 is ContinuousConstraints /Probabilities s =1 R Σ EE α EE P EE ( )EE ≤ , ζ A i i s is i0 s =1
The epidemic of AIDS has been steadily spreading for the past two decades, and now affects every country in the world.
Estimated prevalence of HIV among young adults (15-49) per country at the end of 2005 (source: http://en.wikipedia.org/wiki/AIDS)
Each year, more people die, and the number of HIV+ people continues to rise despite national and international HIV prevention policies and dedicated public healthcare strategies.
Well-known modes of transmission of HIV are sexual contact, direct contact with HIV-infectedblood or fluids and perinatal transmission from mother to child.
Transmission can be classified into intentional and unintentional transmission
- intentional if the infected person knows that he/she is HIV+ and he/she does not disclose it when there is a risk of transmission, - otherwise, transmission is not intentional.
Appendix System Dynamics Epidemics of HIV Different issues to consider:
To design good policy, issues to be considered / understood are:
- psyche and behavioral pattern of HIV+, - interactions between the society and HIV+, - socio and economic situation in which HIV+ are living, - awareness level of society about HIV and its transmission channels, - circumstances that make HIV people to do intentional transmission, - and much more.
Appendix System Dynamics Epidemics of HIV Goals:
to analyze the phenomenon of intentional transmission of HIV/AIDS and its correlation to attributes related to the virus carrier and his/her community including family members, work colleagues and healthcare infrastructure (cross impact matrix),
to predict the effects of intentional transmission on the spread of HIV so that a discussion can be started on how to develop policies that induce openness about the HIV+ status of individuals,
to develop intervention policies using societal problem handling methods such asCOMPRAM (DeTombe 1994, 2003) once the outputs of the model become known with available data.
Appendix System Dynamics Epidemics of HIV Proposed approach: A cross impact analysis method has been proposed here to study the behaviour of HIV+ persons with respect to the disclosure of their status:
basic entities and their relationships with the behavior of HIV+ persons are described, and a list of attributes are provided,
the data requirements for the model are summarized along with a questionnaire to collect the data,
the cross impact among attributes can be measured by pairwise correlation analysis,
partial cross impact matrix that conveys information on the influence of one variable over the other is illustrated using qualitative judgement,
parameters are then fed into mathematical equations that change the level of variables throughout simulation iterations,
significance of intentional infection transmission rates due to various sources can then be identified and analyzed.
Appendix System Dynamics Epidemics of HIV Conclusions:
A cross-impact model is developed here to study the intentional transmission of HIV bynon-disclosure of status in various risky situations.
Here, we identify certain entities and attributes that might affect an HIV+ patient’s attitude towards disclosure.
We describe the simulation method and the data to be collected if such a model is to be executed.
Two policy variables may be proposed as intervention to non-disclosure:
- The first could be investing funds in improving hygiene and preventative measures in healthcare institutions. - However, such a policy should be accompanied by supporting the HIV+ individuals with economic aid if they are unemployed, free access to special AIDS clinics and access to housing units where they will not be subjected to any harassment.
The proposed cross impact model enables the identification of important factors that result in non-disclosure and could invoke new intervention policies and regulations to prevent the intentional transmission of HIV/AIDS in different countries and societal environments.
Stochastic Control of Hybrid Systems Appendix We consider a financial market consisting of:
Stochastic Control of Hybrid Systems Appendix We assume:
Stochastic Control of Hybrid Systems Appendix Wealth Process We assume:
Stochastic Control of Hybrid Systems Appendix Wealth Process Be
Stochastic Control of Hybrid Systems Appendix Wealth Process Be
Stochastic Control of Hybrid Systems Appendix Wealth Process The investor is faced with the problem of finding strategies that maximize the utilityof (i) his consumption for all, and (ii) his terminal wealth at time T.
Stochastic Control of Hybrid Systems Appendix Wealth Process
Stock Prices and Indices Appendix d-fine, 2002 Ö. Önalan, 2006
Root Monte Carlo Simulation Appendix B. Dupire, 2007
Lévy Processes Appendix Definition: A cadlag adapted processes defined on a probability space (Ώ,F,P) is said to be a Lévy processes,if it possesses the following properties:
Lévy Processes Appendix ί ) ίί )For is independent of , i.e., has independentincrements. ίίί)For any is equal in distribution to (the distribution of does not depend on t ); has stationary increments. ίv) For every, , i.e.,is stochastically continuous. In the presence of ( ί ), ( ίί ), ( ίίί ), this is equivalent to the condition
Lévy Processes Appendix There is strong interplay between Lévy processes and infinitely divisibledistributions. Proposition:If is a Lévy processes, thenis infinitely divisible foreach. Proof : For any and any : Together with the stationarity and independence of increments we concludethat the random variableis infinitely divisible.
Lévy Processes Appendix Moreover, for alland all we define hence, for rational : .
Lévy Processes Appendix For every Lévy process, the following property holds: where is the characteristic exponent of .
Lévy Processes Appendix The triplet is called the Lévyor characteristic triplet; is called the Lévyor characteristic exponent. Here,: drift term, : diffusion coefficient, and :Lévy measure.
Lévy Processes Appendix The Lévy measure is a measure on which satisfies . This means that a Lévy measure has no mass at the origin, but infinitely many jumps can occur around the origin. The Lévy measure describes the expected number of jumps of a certain height in a time in interval length 1.
Lévy Processes Appendix The sum of all jumps smaller than some does not converge. However, the sum of the jumps compensated by their mean does converge. This pecularity leads to the necessity of the compensator term . If the Lévy measure is of the form , then f (x) is called the Lévy density. In the same way as the instant volatility describes the local uncertainty of a diffusion, the Lévy density describes the local uncertainity of a pure jump process. The Lévy-Khintchine Formulaallows us to study the distributional properties of a Lévy process. Another key concept, the Lévy-Ito Decomposition Theorem, allows one to describe the structure of a Lévy process sample path.
Lévy Processes Appendix Every Lévy process is a combination of a Brownian motion with drift and a possibly infinite sum of independent compound Poisson processes. This also means that every Lévy process can be approximated with arbitrary precision by a jump-diffusion process. In particular, the Lévy measure v describes arrival rates for jumps of every size for each component of Lt . Jumps of sizes in the set A occur according to a Poisson processwith intensity parameter , A being a any interval bounded away from 0. The Lévy measure of the process L may also be defined as A pure jump Lévy process can have a finite activity (aggregate jump arrival rate: finite) or infinite activity (infinitely many jumps possibly occuring in any finite time interval).
Lévy Processes Appendix A sample path of an NIG Lévy process is drawn with jumps being of irregular size which implies the very jagged shape of the picture. Source: Barndorff Nielsen and N.Shephard (2002) , pp. 22.
NormalInverse Gaussian Distribution Appendix The normal inverse Gaussian (NIG) distributions were introduced by Barndorff-Nielsen (1995) as a subclass of generalized hyperbolic laws with , so that . The normal inverse Gaussian (NIG) distribution is defined by the following probability density function: ,
NormalInverse Gaussian Distribution Appendix where,0 ≤│β│≤ α and K1is the modified Bessel function of third kind with index 1: The tail behavior of the NIG density is characterized by the following asymptotic relation:
NIG-density for different values of ; here, and . NormalInverse Gaussian Distribution Appendix
NIG-density for different values of ; here, and . NormalInverse Gaussian Distribution Appendix
NIG Lévy Asset Price Model Appendix Under the probability measureP, we consider the modeling of the asset price process as the exponential of anLévy process whereL is an NIG Lévy process. The NIG distribution is infinitely divisible and, hence, it generates a Lévy process. The log-returns of the model have independent and stationary increments.
NIG Lévy Asset Price Model Appendix Any infinitely divisible distributionX generates a Lévy process. According to the construction, increments of length 1 have distributionX, i.e., . But, in general, none of the increments of length different from 1 has a distribution of the same class. The price process is the solution of the stochastic differential equation where is an NIG Lévy process and is the jump of L at time t. The solution of above SDE is given by
NIG Lévy Asset Price Model Appendix is the unique solution of the following equation: . Under the corresponding probability , the process is again a Lévy process which is called the Esscher transform. This Esscher equivalent martingale measureis given by .
Simulating NIG Distributed Random Variables Appendix We consider the simulation algorithm for sampling from an NIG (α, β, μ, δ) distributed random variableL.
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