6th International Summer School<br />                            National Universi...
Outline<br />Introduction and Motivation<br />System Dynamics <br />Primary Education  (PE)<br />Simulation in PE in Turke...
Introduction and Motivation <br />
System Dynamics       Primary Education<br />Education Index (according to 2007/2008, Human Development Report)<br />101 m...
System Dynamics       Primary Education<br /><ul><li>The first stage of compulsory education is primary or elementary educ...
In most countries, it is compulsory for children to receive primary education, </li></ul>though in many jurisdictions it i...
System Dynamics       Primary Education<br /><ul><li>Because of the importance of primary education, there are several mod...
logistic regression models (Admassu(2008)),
Poisson regression models(Admassu(2008)),
system dynamics models(Altamirano and van Daalen  (2004),Karadeli et al.  (2001), Pedamallu (2001), Terlouet al.  (1999)),
behavioral models (Benson (1995), Hanusheket al. (2008)),in the contexts of different countries.
Several factors have been identified which influence </li></ul>the school enrollment and drop outs.<br />
System Dynamics       Primary Education<br /><ul><li>To design a good policy, issues to be considered / understood are:</l...
System Dynamics       Primary Education<br /><ul><li>to analyzeimportance of infrastructural facilities on the quality of ...
System Dynamics       Primary Education<br /><ul><li>Modeling and forecasting the behaviour of complex systems are necessa...
System variables are bounded:
A variable increases or decreases according to whether the net impact </li></ul>of the other variables is positive or nega...
System Dynamics       Primary Education<br /><ul><li>A variable’s response to a given impact decreases to 0as </li></ul>th...
System Dynamics<br /><ul><li>The choice of relevant attributes has to be made carefully, keeping in mind both</li></ul>sho...
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 beh...
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.</li></li></ul><li>Syste...
System Dynamics       4 Steps of Implementation<br />d. The impact of infrastructural facilities on primary school enrollm...
System Dynamics     Problems with Migrant Primary School  Students in Turkey<br />Problems with Migrant Primary School Stu...
Lack of parental interest
Lack of adequate shelter, food
Not able to assimilate in urban life and society</li></li></ul><li>System Dynamics     Problems with Migrant Primary Schoo...
System Dynamics     Problems with Migrant Primary School  Students in Turkey<br />Attributes under each entity<br />Entity...
System Dynamics     Problems with Migrant Primary School  Students in Turkey<br />The Proposed Policies to remediate the s...
System Dynamics     Problems with Migrant Primary School  Students in Turkey<br />Implementation Results<br />
System Dynamics     Problems with Primary Education in India<br />The following simulationis based on data from Gujarat (I...
System Dynamics     Problems with Primary Education in India<br /><ul><li>Level of enrollment(loe)</li></ul>Beforeimplemen...
System Dynamics     Problems with Primary Education in India<br /><ul><li>Level of repeaters (lr)</li></ul>Before implemen...
DNA experiments<br />Ex.:yeastdata<br />http://genome-www5.stanford.edu/<br />
Analysis of DNA experiments<br />
Modeling & Prediction<br />      least squares  –  max likelihood<br />statistical learning<br />prediction,   anticipatio...
Modeling & Prediction<br />Ex.:<br />Ex.:   Euler,   Runge-Kutta<br />M<br />We analyze the influence of em-parameters  <b...
Genetic Network<br />Ex. :<br />
Genetic Network<br />0.4x1<br />gene2<br />gene1<br />0.2 x2<br />1 x1<br />gene3<br />gene4<br />
Gene-Environment Networks<br />             if  gene j regulates gene i<br />             otherwise<br />
The Model Class<br />is the firstly introduced time-autonomous form, where<br />d-vector                                  ...
The Model Class<br />(i)is an  constant (nxn)-matrix<br />                                                                ...
The Model Class<br />In general, in the d-dimensional extended space,<br />                                               ...
The Time-Discretized Model<br />-  Euler’s method, <br />-  Runge-Kutta methods, e.g., 2nd-order Heun's method<br />3rd-or...
The Time-Discretized Model  <br />(**)<br />in the extended spacedenotes the                                              ...
Matrix Algebra<br />are  (nxn)- and  (nxm)-matrices, respectively<br /> (n+m)x(n+m) -matrix<br />are  (n+m)-vectors<br />A...
Matrix Algebra<br />Final canonical block form of :                     =  .<br />
Optimization Problem<br />mixed-integer least-squares optimization problem:<br />Boolean variables<br />subject to<br />Ug...
The Mixed-Integer Problem<br />:  constant (nxn)-matrix with entries            representing the effect<br />    which the...
Numerical Example<br />Consider our MINLP for the following data: <br />Gebert et al. (2004a)<br />Apply 3rd-order Heun me...
Numerical Example<br />Apply 3rd-order Heun’s time discretization :<br />
____    gene A<br />........   gene B<br />_ . _ .   gene C<br />- - - -     gene D<br />Results of Euler Method for all g...
____    gene A<br />........   gene B<br />_ . _ .   gene C<br />- - - -   gene D<br />Results of 3rd-order Heun Method fo...
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
Regulatory NetworksunderUncertainty<br />Interaction of Target & Environmental Clusters<br />Determine the degree of conne...
Regulatory NetworksunderUncertainty<br />Clusters and Ellipsoids:<br />Target clusters: 	             C1,C2,…,CREnvironmen...
Regulatory NetworksunderUncertainty<br />Time-Discrete Model<br />Target  Target<br />Environment  Target<br />(<br />R<...
Regulatory NetworksunderUncertainty<br />
Regulatory NetworksunderUncertainty<br />
Regulatory NetworksunderUncertainty<br />Time-Discrete Model<br />controlfactor<br />(<br />(<br />)<br />)<br />TT<br />(...
Regulatory NetworksunderUncertainty<br />The Mixed-Integer Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<b...
Regulatory NetworksunderUncertainty<br />The Continuous Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br /...
Stock Markets<br />
Financial Dynamics<br />drift   and     diffusion       term<br />Ex.:       price,          wealth,        interest rate,...
Financial Dynamics<br />Milstein Scheme :<br />and, based on our finitely many data:<br />
HybridStochastic Control<br /><ul><li>standard Brownian motion
continuous state    Solves an SDE whose jumps are governed by the discrete state.
discrete state   Continuous time Markov chain.
control</li></ul>Chess Review, Nov. 21, 2005<br />Control of Stochastic Hybrid Systems, Robin Raffard<br />
Applications<br />hybrid<br /><ul><li>Engineering:Maintain dynamical system in safe domain for maximum time.
Systems biology: Parameter identification.
Finance: Optimal portfolio selection.</li></li></ul><li>Method:1st step<br />hybrid<br />Derive a PDE satisfied by the obj...
Method:2ndand 3rd step<br />hybrid<br />Rewrite original problem as deterministic PDE optimization program<br />Solve PDE ...
References<br />http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf<br />Thank you very much for your attention!<br /...
Appendix<br />System Dynamics     Epidemics of HIV<br /><ul><li>The epidemic of AIDS has been steadily spreading for the p...
Well-known modes of transmission of HIV are sexual contact, direct contact with HIV-infectedblood or fluids and perinatal ...
Transmission can be classified into intentional and unintentional transmission</li></ul>   -  intentional if the infected ...
Appendix<br />System Dynamics     Epidemics of HIV<br />Different issues to consider:<br /><ul><li>To design good policy, ...
Appendix<br />System Dynamics     Epidemics of HIV<br />Goals:<br /><ul><li>to analyze the phenomenon of intentional trans...
to predict the effects of intentional transmission on the spread of HIV so that a discussion can be started on how to deve...
to develop intervention policies using societal problem handling methods such asCOMPRAM (DeTombe 1994, 2003) once the outp...
system dynamics,
mathematical modeling,
etc..</li></li></ul><li>Appendix<br />System Dynamics     Epidemics of HIV<br />Proposed approach:<br />A cross impact ana...
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 ...
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.</li><...
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:</li></ul>-  The first could be investing funds in ...
riskyassets with prices                     following:</li></li></ul><li>Stochastic Control of Hybrid Systems<br />Appendi...
Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />We assume:<br />
Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />Be<br />
Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />Be<br />
Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />The investor is faced with the problem of find...
Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />
Stock Prices and Indices<br />Appendix<br />d-fine, 2002<br />Ö. Önalan, 2006<br />
Root Monte Carlo Simulation<br />Appendix<br />B. Dupire, 2007<br />
Lévy Processes<br />Appendix<br />Definition: A cadlag adapted processes <br />defined on a probability space (Ώ,F,P) is s...
Lévy Processes<br />Appendix<br />ί )<br />ίί )For   is independent of       , <br />i.e., has independentincrements.<br /...
Lévy Processes<br />Appendix<br />There is strong interplay between Lévy processes and infinitely divisibledistributions.<...
Lévy Processes<br />Appendix<br />Moreover, for alland all            we define<br />hence, for rational       :<br />    ...
Lévy Processes<br />Appendix<br />For every Lévy process, the following property holds:<br />where  is the characteristic ...
Lévy Processes<br />Appendix<br />The triplet is called the Lévyor characteristic triplet;<br />is called the Lévyor chara...
Lévy Processes<br />Appendix<br />The Lévy measure is a measure on  which satisfies<br />                                 ...
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Dynamics under Various Assumptions on Time and Uncertainty

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AACIMP 2011 Summer School. Operational Research stream. Lecture by Gerhard-Wilhelm Weber.

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Dynamics under Various Assumptions on Time and Uncertainty

  1. 1. 6th International Summer School<br /> National University of Technology of the Ukraine<br /> Kiev, Ukraine, August 8-20, 2011<br />Let Us Predict Dynamicsunder <br />Various Assumptions on Time and Uncertainty<br />Gerhard-Wilhelm Weber 1*,<br /> Özlem Defterli 2, Linet Özdamar3, Zeynep Alparslan-Gök 4, Chandra Sekhar Pedamallu5,<br />Büşra Temoçin1, 6,Azer Kerimov 1, Ceren Eda Can7, Efsun Kürüm1, <br /> 1 Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey<br /> 2 Department of Mathematics and Computer Science, Cankaya Uniiversity, Ankara, Turkey 3 Department of Systems Engineering, Yeditepe University, Istanbul, Turkey<br /> 4 Department of Mathematics, Süleyman Demirel University, Isparta, Turkey<br /> 5 Department of Medical Oncology, Dana-Farber Cancer Institute, Boston, MA, USA,<br />The Broad Institute of MIT and Harvard, Cambridge, MA, USA<br /> 6 Department of Statistics, Ankara University, Ankara, Turkey<br /> 7 Department of Statistics, Hacettepe University, Ankara, Turkey <br /> * Faculty of Economics, Management and Law, University of Siegen, Germany<br />Center for Research on Optimization and Control, University of Aveiro, Portugal<br />Universiti Teknologi Malaysia, Skudai, Malaysia<br /> Advisor to EURO Conferences<br />
  2. 2. Outline<br />Introduction and Motivation<br />System Dynamics <br />Primary Education (PE)<br />Simulation in PE in Turkey and Its Results <br />Simulation in PE in India and Its Results<br />Transmission of HIV in Developing Countries<br />Time-Continuous and Time-Discrete Models<br />Gene-Environment Networks<br />Numerical Example and Results<br />Regulatory Networks under Uncertainty<br />Ellipsoidal Model<br />Stochastic Model <br />Stochastic Hybrid Systems<br />Portfolio Optimization subject to SHSs <br />Lévy Processes, Simulation and Asset Price Dynamics<br />Conclusion<br />
  3. 3. Introduction and Motivation <br />
  4. 4. System Dynamics Primary Education<br />Education Index (according to 2007/2008, Human Development Report)<br />101 million children of primary school age are out of school                                                   <br />Number of primary-school-agechildren not in school, by region (2007)<br />http://www.childinfo.org/education.html<br />
  5. 5. System Dynamics Primary Education<br /><ul><li>The first stage of compulsory education is primary or elementary education.
  6. 6. In most countries, it is compulsory for children to receive primary education, </li></ul>though in many jurisdictions it is permissible for parents to provide it.<br /><ul><li>The transition from elementary school to secondary school or high school is </li></ul>somewhat arbitrary, but it generally occurs at about eleven or twelve yearsof age. <br /><ul><li>The major goals of primary education are achieving basic literacy and </li></ul>numeracy amongst all pupils, as well as establishing foundations <br />in science,geography, history and other social sciences.<br /><ul><li>The relative priority of various fields, and the methods used to teach them, </li></ul>are an area of considerable political debate.<br /><ul><li>Some of the expected benefits from primary education are the reduction of </li></ul>the infant mortality rate, the population growth rate, of the <br />crude birth and death rate, and so on.<br />
  7. 7. System Dynamics Primary Education<br /><ul><li>Because of the importance of primary education, there are several models </li></ul>proposed to study the factors influencing the primary school enrollment and <br />progressions. <br /><ul><li>Various models developed to analyze issues in basic education are as follows:
  8. 8. logistic regression models (Admassu(2008)),
  9. 9. Poisson regression models(Admassu(2008)),
  10. 10. system dynamics models(Altamirano and van Daalen (2004),Karadeli et al. (2001), Pedamallu (2001), Terlouet al. (1999)),
  11. 11. behavioral models (Benson (1995), Hanusheket al. (2008)),in the contexts of different countries.
  12. 12. Several factors have been identified which influence </li></ul>the school enrollment and drop outs.<br />
  13. 13. System Dynamics Primary Education<br /><ul><li>To design a good policy, issues to be considered / understood are:</li></ul> - reasons behind enrollment, drop-outs and repeaters in school,<br /> - perceived quality of teaching by students and parents,<br /> - educational level and income level of parents,<br /> - expectations from school by parents,<br /> - perceived quality of teaching by the District EducationalOfficer(DEO),<br /> - needs of different infrastructure facilities(space, ventilation, sanitation, etc.)<br />in the school,<br /> - and much more….<br />
  14. 14. System Dynamics Primary Education<br /><ul><li>to analyzeimportance of infrastructural facilities on the quality of the </li></ul>primary education system and its correlation to attributes related to the <br />primary education system and environment including parents, teachers, <br />infrastructure and so on (cross impact matrix),<br /><ul><li>to predict the effects of infrastructural facilities on the quality of primary </li></ul>education system so that a discussion can be started on <br />how to developpolicies,<br /><ul><li>to developintervention policies using societal problem handling methods </li></ul>such as COMPRAM(DeTombe1994, 2003),<br />once the outputs of the model become known with available data.<br />
  15. 15. System Dynamics Primary Education<br /><ul><li>Modeling and forecasting the behaviour of complex systems are necessary</li></ul>if we are to exert some degree of control over them.<br /><ul><li>Propertiesof variables and interactions in our large-scale system:
  16. 16. System variables are bounded:
  17. 17. A variable increases or decreases according to whether the net impact </li></ul>of the other variables is positive or negative.<br />Julius (2002)<br />
  18. 18. System Dynamics Primary Education<br /><ul><li>A variable’s response to a given impact decreases to 0as </li></ul>that variable approachesits upper or lower bound. <br /><ul><li>It is generally found that bounded growth and decay processes exhibit </li></ul>this sigmoidal character.<br /><ul><li>All other issues being constant, a variable (attribute) produces </li></ul>agreater impact onthe system as it grows larger(ceteris paribus).<br /><ul><li>Complex interactions are described by a looped network of </li></ul>binary interactions<br />(this is the basis of cross-impact analysis).<br />
  19. 19. System Dynamics<br /><ul><li>The choice of relevant attributes has to be made carefully, keeping in mind both</li></ul>short-termandlong-termconsequences of solutions (decisions). <br /><ul><li>All attributes can be associated with given levels that may indicate </li></ul>quantitativeor qualitative possession.<br /><ul><li>When entities interact through their attributes, the levelsof the attributes might change, i.e., the system behaves in certain directions.
  20. 20. Some changes in attribute levels may be desirablewhile others may not be so.
  21. 21. 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.
  22. 22. They can vary in an unsupervisedway in the system.
  23. 23. However, variables can be controlleddirectly or indirectly, and partially by introducing new intervention policies.
  24. 24. However, interrelationsamong variables should be analyzedcarefully beforeintroducing new policies.</li></li></ul><li>System Dynamics 4 Steps of Implementation<br />Step 1.Set the initial values to identified attributes obtained from published sources and surveys conducted.<br />Step 2.Build a cross-impact matrix with the identified relevant attributes. <br />a. Summing the effectsof column attributes on rows indicates the effect of each attribute in the matrix. <br />b. The parameters ijcan be determined by creating a pairwise correlation matrixafter collecting the data,and adjusted by subjective assessment. <br />c. Qualitative impacts <br /> are quantifiedsubjectively as <br /> shown in the table. <br /> Qualitative impacts can be <br /> extracted from a published <br /> sources and survey data set.<br />
  25. 25. System Dynamics 4 Steps of Implementation<br />d. The impact of infrastructural facilities on primary school enrollments and progression become visible by running the simulation model. <br />e. An exemplary partial cross-impact matrix with the attributes and their hypothetical values above is illustrated as follows:<br />
  26. 26. System Dynamics Problems with Migrant Primary School Students in Turkey<br />Problems with Migrant Primary School Students in Turkey<br /><ul><li>Lack of Turkish language
  27. 27. Lack of parental interest
  28. 28. Lack of adequate shelter, food
  29. 29. Not able to assimilate in urban life and society</li></li></ul><li>System Dynamics Problems with Migrant Primary School Students in Turkey<br />Entities in the System Dynamics Model<br />student,<br />teacher,<br />family,<br />school environment, <br />Ministry of Education.<br />Entity Relationshipdiagram<br />
  30. 30. System Dynamics Problems with Migrant Primary School Students in Turkey<br />Attributes under each entity<br />Entity 1: Student:<br />F1.1Not believing in education <br />F1.2Dislike of school books <br />F1.3Lack of good Turkish language skills<br />F1.4Dislike of school <br />F1.5Weak academic self confidence <br />F1.6Frequent disciplinary problem<br />Entity 2: Teacher:<br />F2.1Relating to guidance teacher<br />Entity 3: Family:<br />F3.1Economic difficulty<br />F3.2Child labor<br />F3.3Large families <br />F3.4Malnutrition <br />F3.5Homes with poor infrastructure <br />(e.g., lack of heating, lack of room)<br />F3.6Different home language <br />F3.7Parent interest <br />Entity 4: School environment:<br />F4.1 Violence at school<br />F4.2Alien school environment <br />F4.3Lecture not meeting student needs<br />
  31. 31. System Dynamics Problems with Migrant Primary School Students in Turkey<br />The Proposed Policies to remediate the system<br />P1. Daily distribution of milk <br />P2. Distribution of coal, food, etc. to Turkish green card holders <br />P3. The three-children per family aspiration by government <br />P4. School infrastructure - renovation<br />P5. Providing adult education for poor parents in migrant communities <br />P6. Providing one year of Turkish language class <br />before migrant student attends urban school <br />P7.Combined policy: P2 + P4 + P5 + P6<br />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. <br />Policies P2and P5support parentsand, hence, their impacts on their offspring are indirect. <br />On the other hand, policies P4and P6target the academic performance of students directly. <br />The combined policy P7would naturally produce the best overall impact on the system if the budget of the Ministry of Education can afford it. <br />
  32. 32. System Dynamics Problems with Migrant Primary School Students in Turkey<br />Implementation Results<br />
  33. 33. System Dynamics Problems with Primary Education in India<br />The following simulationis based on data from Gujarat (India).<br />
  34. 34. System Dynamics Problems with Primary Education in India<br /><ul><li>Level of enrollment(loe)</li></ul>Beforeimplementation of policy variables (red line):<br /> - sharp increaseat the beginning phase of the simulation <br />(first 12 iterations), and then there is steady decrease<br />after a certain period of time (first 12 iterations).<br />After implementation of policy variables (blue line):<br /><ul><li>steadyincreasefrom initial value (0.71) to unity.</li></ul>Analysis: <br /><ul><li>increase in level of enrollment in first 12 iterations happens </li></ul>because of the response lag in population <br />to bad quality school system,<br /><ul><li>instant impact on the level enrollment after implementation </li></ul>of policy variables because students and parents are <br />more eager to have the children attend a nice looking <br />healthy school.<br />
  35. 35. System Dynamics Problems with Primary Education in India<br /><ul><li>Level of repeaters (lr)</li></ul>Before implementation of policy variables (red line):<br />- steady increasefrom 0.05 to unity in 50 iterations.<br />After implementation of policy variables (blue line):<br /><ul><li>steadyincreasefrom initial value 0.05 to 0.12 in first 14 iterations and then declinedto zero in 50 iterations.</li></ul>Analysis: <br /><ul><li>level of repeaters has instant impact on level of repeaters </li></ul>ifthe infrastructural facilities are bad<br />(assumption: teachers are doing their best in teaching – <br />this variable is static -sovariable impacted here is teaching aids),<br /><ul><li>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.</li></li></ul><li>Bio-Systems <br /> Medicine<br />Environment<br />... Finance<br /> Health Care<br />prediction of gene patterns based on<br />DNA microarraychip experiments<br />with<br />M.U. Akhmet, H. Öktem <br />S.W. Pickl, E. Quek Ming Poh<br />T. Ergenç, B. Karasözen <br />J. Gebert, N. Radde <br />Ö. Uğur, R. Wünschiers<br />M. Taştan, A. Tezel, P. Taylan <br />F.B. Yilmaz, B. Akteke-Öztürk<br />S. Özöğür, Z. Alparslan-Gök <br /> A. Soyler, B. Soyler, M. Çetin<br />S. Özöğür-Akyüz, Ö. Defterli<br /> N. Gökgöz, E. Kropat<br />
  36. 36. DNA experiments<br />Ex.:yeastdata<br />http://genome-www5.stanford.edu/<br />
  37. 37. Analysis of DNA experiments<br />
  38. 38. Modeling & Prediction<br /> least squares – max likelihood<br />statistical learning<br />prediction, anticipation<br />Expression<br />expression data<br />matrix-valued function – metabolic reaction<br />
  39. 39. Modeling & Prediction<br />Ex.:<br />Ex.: Euler, Runge-Kutta<br />M<br />We analyze the influence of em-parameters <br />on the dynamics (expression-metabolic).<br />
  40. 40. Genetic Network<br />Ex. :<br />
  41. 41. Genetic Network<br />0.4x1<br />gene2<br />gene1<br />0.2 x2<br />1 x1<br />gene3<br />gene4<br />
  42. 42. Gene-Environment Networks<br /> if gene j regulates gene i<br /> otherwise<br />
  43. 43. The Model Class<br />is the firstly introduced time-autonomous form, where<br />d-vector of concentration levels of proteins and of certain levels of environmental factors <br />continuous change in the gene-expression data in time<br />nonlinearities<br />initial values of the gene-exprssion levels<br />: experimental data vectors obtained from microarray experiments <br /> and environmental measurements <br />: the gene-expression level (concentration rate) of the i th gene at time t<br />denotes anyone of the first n coordinates in the<br />d-vector of genetic and environmental states.<br />Weber et al. (2008c), Chen et al. (1999), <br />Gebert et al. (2004a),<br />Gebert et al. (2006), Gebert et al. (2007), <br />Tastan (2005), Yilmaz (2004), Yilmaz et al. (2005),<br />Sakamoto and Iba (2001), Tastan et al. (2005)<br />is the set of genes.<br />
  44. 44. The Model Class<br />(i)is an constant (nxn)-matrix<br /> is an (nx1)-vector of gene-expression levels<br />represents and t the dynamical system of the n genes <br /> and their interaction alone.<br /> : (nxn)-matrix with entries as functions of polynomials, exponential, trigonometric, <br /> splines or wavelets containing some parameters to be optimized.<br />(iii)<br />Weber et al. (2008c), Tastan (2005), <br />Tastan et al. (2006),<br />Ugur et al. (2009), Tastan et al. (2005), <br />Yilmaz (2004), Yilmaz et al. (2005),<br />Weber et al. (2008b), Weber et al. (2009b)<br />environmental effects<br />(*)<br />n genes , m environmental effects<br />are (n+m)-vector and <br /> (n+m)x(n+m)-matrix, respectively.<br />
  45. 45. The Model Class<br />In general, in the d-dimensional extended space,<br /> with <br /> : (dxd)-matrix<br /> : (dx1)-vectors<br />Ugur and Weber (2007), <br />Weber et al. (2008c),<br />Weber et al. (2008b), <br />Weber et al. (2009b)<br />
  46. 46. The Time-Discretized Model<br />- Euler’s method, <br />- Runge-Kutta methods, e.g., 2nd-order Heun's method<br />3rd-order Heun's method is introduced byDefterli et al. (2009)<br />we rewrite it as <br />where<br />Ergenc and Weber (2004), <br />Tastan (2005), Tastan et al. (2006), Tastan et al. 2005)<br />
  47. 47. The Time-Discretized Model <br />(**)<br />in the extended spacedenotes the DNA microarray experimental data and the dataof environmental items obtained at the time-level<br />the approximations obtainedby the iterative formula above<br />initial values<br />kthapproximation or prediction is calculated as:<br />
  48. 48. Matrix Algebra<br />are (nxn)- and (nxm)-matrices, respectively<br /> (n+m)x(n+m) -matrix<br />are (n+m)-vectors<br />Applying the 3rd-order Heun’s method to the eqn. (*) gives the iterative formula (**), where <br />
  49. 49. Matrix Algebra<br />Final canonical block form of : = .<br />
  50. 50. Optimization Problem<br />mixed-integer least-squares optimization problem:<br />Boolean variables<br />subject to<br />Ugur and Weber (2007),<br />Weber et al.(2008c),<br />Weber et al. (2008b),<br />Weber et al. (2009b), <br />Gebert et al. (2004a), <br />Gebert et al. (2006), <br />Gebert et al. (2007).<br />, : th : the numbers of genes regulated by gene (its outdegree), <br /> by environmental item , or by the cumulative environment, resp..<br />
  51. 51. The Mixed-Integer Problem<br />: constant (nxn)-matrix with entries representing the effect<br /> which the expression level of gene has on the change of expression of gene<br />Genetic regulation network<br />mixed-integer nonlinear optimization problem (MINLP):<br />subject to <br />: constant vectorrepresenting the lower bounds <br /> for the decrease of the transcript concentration.<br />Binary variables : <br />
  52. 52. Numerical Example<br />Consider our MINLP for the following data: <br />Gebert et al. (2004a)<br />Apply 3rd-order Heun method:<br />Take<br />using the modeling language Zimpl 3.0, we solve<br />by SCIP 1.2 as a branch-and-cut framework, together with SOPLEX 1.4.1 as our LP-solver<br />
  53. 53. Numerical Example<br />Apply 3rd-order Heun’s time discretization :<br />
  54. 54. ____ gene A<br />........ gene B<br />_ . _ . gene C<br />- - - - gene D<br />Results of Euler Method for all genes:<br />
  55. 55. ____ gene A<br />........ gene B<br />_ . _ . gene C<br />- - - - gene D<br />Results of 3rd-order Heun Method for all genes:<br />
  56. 56.
  57. 57. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
  58. 58. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
  59. 59. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
  60. 60. Regulatory NetworksunderUncertainty<br />θ2<br />θ1<br />
  61. 61. Regulatory NetworksunderUncertainty<br />Interaction of Target & Environmental Clusters<br />Determine the degree of connectivity.<br />
  62. 62. Regulatory NetworksunderUncertainty<br />Clusters and Ellipsoids:<br />Target clusters: C1,C2,…,CREnvironmental clusters: D1,D2,…,DS<br />Target ellipsoids: X1,X2,…,XRXi = E(μi , Σi) Environmental ellipsoids: E1,E2,…,ESEj = E(ρj ,Πj) <br />Center<br />Covariancematrix<br />
  63. 63. Regulatory NetworksunderUncertainty<br />Time-Discrete Model<br />Target  Target<br />Environment  Target<br />(<br />R<br />)<br />(<br />S<br />)<br />Targetcluster<br />TT<br />(k)<br />(k+1)<br />ET<br />(k)<br />X<br />ξ<br />X<br />A<br />+<br />+<br />=<br />E<br />A<br />j r<br /> r<br /> j0<br />j s<br /> j<br /> s<br />r =1<br />s =1<br />(<br />R<br />)<br />(<br />S<br />)<br />Environmental cluster<br />TE<br />(k)<br />(k+1)<br />EE<br />(k)<br />X<br />ζ<br />E<br />A<br />+<br />+<br />=<br />E<br />A<br />i r<br /> r<br /> i0<br />is<br /> i<br /> s<br />r =1<br />s =1<br />Target  Environment<br />Environment  Environment<br /> Determine system matrices and intercepts.<br />
  64. 64. Regulatory NetworksunderUncertainty<br />
  65. 65. Regulatory NetworksunderUncertainty<br />
  66. 66. Regulatory NetworksunderUncertainty<br />Time-Discrete Model<br />controlfactor<br />(<br />(<br />)<br />)<br />TT<br />(k)<br />(k+1)<br />ET<br />(k)<br />Emissionclusters<br />X<br />ξ<br />X<br />A<br />+<br />+<br />=<br />E<br />A<br />+<br />0<br />j r<br /> r<br /> j0<br />j s<br /> j<br /> s<br />r =1<br />s =1<br />(<br />R<br />)<br />(<br />S<br />)<br />Environ-mental cluster<br />(k)<br />TE<br />(k)<br />(k+1)<br />EE<br />(k)<br />u<br />+<br />X<br />ζ<br />E<br />A<br />+<br />+<br />=<br />E<br />A<br />i r<br /> r<br /> i0<br />is<br /> i<br /> s<br />r =1<br />s =1<br /><br />Predictionof CO2-emissions<br />
  67. 67. Regulatory NetworksunderUncertainty<br />The Mixed-Integer Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br />−<br />−<br />^<br />(k)<br />(k)<br />(k)<br /><br />(k)<br />+<br /><br />^<br />E<br />E<br />X<br />Maximize<br />X<br />∩<br />∩<br /> s<br /> r<br /> r<br /> s<br />r = 1<br />s = 1<br />k= 1<br />α<br />TT<br />≤<br />deg(C )TT <br />s.t.<br /><br /><br /><br /><br /><br /><br />j<br />j<br />α<br />TE<br />≤<br />deg(C )TE <br />j<br />j<br />boundson outdegrees<br />α<br />ET<br />≤<br />deg(D )ET <br />i<br />i<br />α<br />EE<br />≤<br />deg(D )EE <br />i<br />i<br />
  68. 68. Regulatory NetworksunderUncertainty<br />The Continuous Regression Problem:<br />R<br />S<br />T<br />Σ<br />Σ<br />Σ<br />−<br />−<br />^<br />(k)<br />(k)<br />(k)<br /><br />(k)<br />+<br /><br />^<br />E<br />E<br />X<br />Maximize<br />X<br />∩<br />∩<br /> s<br /> r<br /> r<br /> s<br />r = 1<br />s = 1<br />k= 1<br />R<br />Σ<br />α<br />TT<br />TT<br />≤<br />P TT ( )TT <br />s.t.<br />, ξ<br />A<br /><br /><br /><br /><br /><br /><br />j<br />j r<br />jr<br /> j0<br />r =1<br />R<br />α<br />TE<br />Σ<br />≤<br />TE<br />P TE ( )TE <br />, ξ<br />A<br />j<br />j r<br /> j0<br />jr<br />r =1<br />boundson outdegrees<br />R<br />ET<br />Σ<br />α<br />ET<br />P ET ( )ET <br />≤<br />, ζ<br />A<br />i<br />i s<br /> i0<br />is<br />ContinuousConstraints /Probabilities<br />s =1<br />R<br />Σ<br />EE<br />α<br />EE<br />P EE ( )EE <br />≤<br />, ζ<br />A<br />i<br />i s<br />is<br /> i0<br />s =1<br />
  69. 69. Stock Markets<br />
  70. 70. Financial Dynamics<br />drift and diffusion term<br />Ex.: price, wealth, interest rate, volatility <br /> processes<br />
  71. 71. Financial Dynamics<br />Milstein Scheme :<br />and, based on our finitely many data:<br />
  72. 72. HybridStochastic Control<br /><ul><li>standard Brownian motion
  73. 73. continuous state Solves an SDE whose jumps are governed by the discrete state.
  74. 74. discrete state Continuous time Markov chain.
  75. 75. control</li></ul>Chess Review, Nov. 21, 2005<br />Control of Stochastic Hybrid Systems, Robin Raffard<br />
  76. 76. Applications<br />hybrid<br /><ul><li>Engineering:Maintain dynamical system in safe domain for maximum time.
  77. 77. Systems biology: Parameter identification.
  78. 78. Finance: Optimal portfolio selection.</li></li></ul><li>Method:1st step<br />hybrid<br />Derive a PDE satisfied by the objective function in terms of the generator:<br /><ul><li>Example 1: </li></ul> If <br /> then<br /><ul><li>Example 2:</li></ul> If<br /> then<br />
  79. 79. Method:2ndand 3rd step<br />hybrid<br />Rewrite original problem as deterministic PDE optimization program<br />Solve PDE optimization program using adjoint method<br /> Simple and robust…<br />
  80. 80. References<br />http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf<br />Thank you very much for your attention!<br />gweber@metu.edu.tr<br />
  81. 81. Appendix<br />System Dynamics Epidemics of HIV<br /><ul><li>The epidemic of AIDS has been steadily spreading for the past two decades, and now affects every country in the world.</li></ul>Estimated prevalence of HIV among young adults (15-49) per country at the end of 2005 (source: http://en.wikipedia.org/wiki/AIDS)<br /><ul><li>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.
  82. 82. Well-known modes of transmission of HIV are sexual contact, direct contact with HIV-infectedblood or fluids and perinatal transmission from mother to child.
  83. 83. Transmission can be classified into intentional and unintentional transmission</li></ul> - 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, <br /> - otherwise, transmission is not intentional.<br />
  84. 84. Appendix<br />System Dynamics Epidemics of HIV<br />Different issues to consider:<br /><ul><li>To design good policy, issues to be considered / understood are:</li></ul>- psyche and behavioral pattern of HIV+,<br />- interactions between the society and HIV+,<br />- socio and economic situation in which HIV+ are living,<br />- awareness level of society about HIV and its transmission channels,<br />- circumstances that make HIV people to do intentional transmission,<br />- and much more.<br />
  85. 85. Appendix<br />System Dynamics Epidemics of HIV<br />Goals:<br /><ul><li>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),
  86. 86. 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,
  87. 87. 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.</li></ul>Possible Solutions:<br />Some popular approaches are:<br /><ul><li>system thinking approach (cross impact analysis),
  88. 88. system dynamics,
  89. 89. mathematical modeling,
  90. 90. etc..</li></li></ul><li>Appendix<br />System Dynamics Epidemics of HIV<br />Proposed approach:<br />A cross impact analysis method has been proposed here to study the behaviour of HIV+ persons with respect to the disclosure of their status:<br /><ul><li>basic entities and their relationships with the behavior of HIV+ persons are described, and a list of attributes are provided,
  91. 91. the data requirements for the model are summarized along with a questionnaire to collect the data,
  92. 92. the cross impact among attributes can be measured by pairwise correlation analysis,
  93. 93. partial cross impact matrix that conveys information on the influence of one variable over the other is illustrated using qualitative judgement,
  94. 94. parameters are then fed into mathematical equations that change the level of variables throughout simulation iterations,
  95. 95. significance of intentional infection transmission rates due to various sources can then be identified and analyzed.</li></li></ul><li>Appendix<br />System Dynamics Epidemics of HIV<br />Conclusions:<br /><ul><li>A cross-impact model is developed here to study the intentional transmission of HIV bynon-disclosure of status in various risky situations.
  96. 96. Here, we identify certain entities and attributes that might affect an HIV+ patient’s attitude towards disclosure.
  97. 97. We describe the simulation method and the data to be collected if such a model is to be executed.
  98. 98. Two policy variables may be proposed as intervention to non-disclosure:</li></ul>- The first could be investing funds in improving hygiene and preventative measures in healthcare institutions. <br />- 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. <br /><ul><li>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. </li></li></ul><li>Stochastic Control of Hybrid Systems<br />Appendix<br />We consider a financial market consisting of:<br /><ul><li>one risk-free asset with price satisfying:
  99. 99. riskyassets with prices following:</li></li></ul><li>Stochastic Control of Hybrid Systems<br />Appendix<br />We assume:<br />
  100. 100. Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />We assume:<br />
  101. 101. Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />Be<br />
  102. 102. Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />Be<br />
  103. 103. Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />The investor is faced with the problem of finding strategies that maximize the utilityof<br />(i) his consumption for all, and<br />(ii) his terminal wealth at time T.<br />
  104. 104. Stochastic Control of Hybrid Systems<br />Appendix<br />Wealth Process<br />
  105. 105. Stock Prices and Indices<br />Appendix<br />d-fine, 2002<br />Ö. Önalan, 2006<br />
  106. 106. Root Monte Carlo Simulation<br />Appendix<br />B. Dupire, 2007<br />
  107. 107. Lévy Processes<br />Appendix<br />Definition: A cadlag adapted processes <br />defined on a probability space (Ώ,F,P) is said to be a <br />Lévy processes,if it possesses the following properties:<br />
  108. 108. Lévy Processes<br />Appendix<br />ί )<br />ίί )For is independent of , <br />i.e., has independentincrements.<br />ίίί)For any is equal in distribution to<br />(the distribution of does not depend on t );<br />has stationary increments.<br />ίv) For every, <br /> , <br />i.e.,is stochastically continuous.<br />In the presence of ( ί ), ( ίί ), ( ίίί ), this is equivalent to the condition<br />
  109. 109. Lévy Processes<br />Appendix<br />There is strong interplay between Lévy processes and infinitely divisibledistributions.<br />Proposition:If is a Lévy processes, thenis infinitely divisible foreach.<br />Proof : For any and any : <br />Together with the stationarity and independence of increments we concludethat the <br />random variableis infinitely divisible. <br />
  110. 110. Lévy Processes<br />Appendix<br />Moreover, for alland all we define<br />hence, for rational :<br /> . <br />
  111. 111. Lévy Processes<br />Appendix<br />For every Lévy process, the following property holds:<br />where is the characteristic exponent of . <br />
  112. 112. Lévy Processes<br />Appendix<br />The triplet is called the Lévyor characteristic triplet;<br />is called the Lévyor characteristic exponent.<br /> Here,: drift term,<br />: diffusion coefficient, and<br />:Lévy measure.<br />
  113. 113. Lévy Processes<br />Appendix<br />The Lévy measure is a measure on which satisfies<br /> .<br />This means that a Lévy measure has no mass at the origin, <br />but infinitely many jumps can occur around the origin.<br />The Lévy measure describes <br />the expected number of jumps <br />of a certain height in a time in interval length 1. <br />
  114. 114. Lévy Processes<br />Appendix<br /> The sum of all jumps smaller than some does not converge. <br />However, the sum of the jumps compensated by their mean does converge. <br />This pecularity leads to the necessity of the compensator term . <br />If the Lévy measure is of the form , <br />then f (x) is called the Lévy density. <br /> In the same way as the instant volatility describes the local uncertainty of a diffusion, <br />the Lévy density describes the local uncertainity of a pure jump process.<br />The Lévy-Khintchine Formulaallows us to study the distributional properties <br />of a Lévy process. <br />Another key concept, the Lévy-Ito Decomposition Theorem, <br />allows one to describe the structure of a Lévy process sample path.<br />
  115. 115. Lévy Processes<br />Appendix<br /> Every Lévy process is a combination of a Brownian motion with drift and <br />a possibly infinite sum of independent compound Poisson processes. <br />This also means that every Lévy process can be approximated <br />with arbitrary precision by a jump-diffusion process. <br />In particular, the Lévy measure v describes arrival rates for jumps of every size <br />for each component of Lt .<br />Jumps of sizes in the set A occur according to a Poisson processwith intensity parameter , A being a any interval bounded away from 0.<br />The Lévy measure of the process L may also be defined as <br /> 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).<br />
  116. 116. Lévy Processes<br />Appendix<br /> 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.<br /> Source: Barndorff Nielsen and N.Shephard (2002) , pp. 22.<br />
  117. 117. NormalInverse Gaussian Distribution<br />Appendix<br />The normal inverse Gaussian (NIG) distributions were introduced by <br />Barndorff-Nielsen (1995) as a subclass of generalized hyperbolic laws with ,<br />so that<br /> .<br />The normal inverse Gaussian (NIG) distribution is defined by the following probability density function: <br />,<br />
  118. 118. NormalInverse Gaussian Distribution<br />Appendix<br />where,0 ≤│β│≤ α<br />and K1is the modified Bessel function of third kind with index 1:<br />The tail behavior of the NIG density is characterized by the following asymptotic<br />relation:<br />
  119. 119. NIG-density for different values of ; here, and . <br />NormalInverse Gaussian Distribution<br />Appendix<br />
  120. 120. NIG-density for different values of ; here, and . <br />NormalInverse Gaussian Distribution<br />Appendix<br />
  121. 121. NIG Lévy Asset Price Model<br />Appendix<br />Under the probability measureP, we consider the modeling of the asset price<br />process as the exponential of anLévy process<br /> whereL is an NIG Lévy process. <br />The NIG distribution is infinitely divisible and, hence, it generates a Lévy process. <br />The log-returns of the model have independent and stationary increments.<br />
  122. 122. NIG Lévy Asset Price Model<br />Appendix<br />Any infinitely divisible distributionX generates a Lévy process. According to the construction, increments of length 1 have distributionX, i.e., .<br />But, in general, none of the increments of length different from 1 has a distribution of the same class.<br />The price process is the solution of the stochastic differential equation<br />where is an NIG Lévy process and <br />is the jump of L at time t.<br />The solution of above SDE is given by<br />
  123. 123. NIG Lévy Asset Price Model<br />Appendix<br />is the unique solution of the following equation:<br /> .<br />Under the corresponding probability , the process is again a Lévy process <br />which is called the Esscher transform.<br />This Esscher equivalent martingale measureis given by . <br />
  124. 124. Simulating NIG Distributed Random Variables<br />Appendix<br /> We consider the simulation algorithm for sampling from an<br />NIG (α, β, μ, δ) distributed random variableL. <br /><ul><li>Sample Z from .
  125. 125. Sample Y from N (0,1) .
  126. 126. Return</li></ul>An IGvariateZcan be sampled as follows, <br />firstly drawing a random variable ,which is distributed, defining a random variable:<br />T.H. Rydberg (1997)<br />
  127. 127. Simulating NIG Distributed Random Variables<br />Appendix<br />and then letting<br />being uniform distributed and. <br />We see that to sample,<br />a standard normal Y, a distributed and a uniform appear.<br />
  128. 128. Simulating NIG Distributed Random Variables<br />Appendix<br />The probability density function of the inverse Gaussian distribution:<br />The parameters aand bare functions of α and β :<br /> , .<br />
  129. 129. NIG Lévy Asset Price Model<br />Appendix<br />
  130. 130. Asset Price Dynamics<br />Appendix<br />Dynamics of Jump Models:<br /><ul><li>Jump-DiffusionModels,
  131. 131. Infinite Activity Lévy Models.</li></li></ul><li>Jump Diffusion Models<br />Appendix<br /><ul><li>Special cases of exponential-Lévy models.
  132. 132. Frequency of jumps is finite.
  133. 133. Stock price follows a geometric Brownian motion between jumps.</li></li></ul><li>Jump Diffusion Models<br />Appendix<br /><ul><li>Merton’s Model
  134. 134. Tries to capture skewnessand excess kurtosisof the log return density.
  135. 135. Stock price has the following SDE:</li></ul> with : a Brownian motion, and <br /> : a compound Poisson process<br /> with i.i.d. lognormally distributed jump sizes.<br />
  136. 136. Jump Diffusion Models<br />Appendix<br /><ul><li>Kou’s Model
  137. 137. Employed to produce analytical solutions for path-dependent options, (barrier, lookback) because of the memoryless property of exponential density.
  138. 138. Stock price has the following SDE:</li></ul> with : a Brownian motion, and<br /> : a Poisson process with <br /> asymmetric double exponentially <br /> distributed logarithmic jump sizes.<br />
  139. 139. Jump Diffusion Models<br />Appendix<br /><ul><li>Bates’s Model
  140. 140. Combines compound Poisson jumps and stochastic volatility.
  141. 141. Easy to simulate and efficient MC methods can be employedfor pricing path-dependent options.
  142. 142. Dynamics of the stock price is given by the following SDEs:</li></ul> with , : Brownian motionswith non-zero correlation, and <br /> : a compound Poisson process.<br /><ul><li>Can also be considered as a SV extension of the Merton’s model.
  143. 143. Jumps of the log-price process do not have to be Gaussian.</li></li></ul><li>Infinite Activity Lévy Models<br />Appendix<br /><ul><li>Infinitely many jumps in any finite time interval.
  144. 144. Empirical performance of these models generally is not improved by adding a diffusion component.
  145. 145. Easier to calibrate than JD models since
  146. 146. the globalerror declines rapidly,
  147. 147. all the parameters vary simultaneously from the initial ones during the calibration.
  148. 148. High activityis accounted for by a large/infinite number of small jumps.</li></li></ul><li>Variance Gamma Process<br />Appendix<br /><ul><li>The characteristic function is given by
  149. 149. Infinitely divisibledistribution.
  150. 150. Can also be defined by considering as time-changed Brownian motionwith drift</li></li></ul><li>Normal Inverse Gaussian Process<br />Appendix<br /><ul><li>The characteristic function is given by
  151. 151. Can also be defined by Inverse Gaussian time-changed Brownian motion.
  152. 152. Negative and positive values of result in negative and positive skewness, respectively.</li></li></ul><li>Meixner Process<br />Appendix<br /><ul><li>The characteristic function is given by
  153. 153. Special case of the Generalized z(GZ) distributions.
  154. 154. Moments of all orders exist.
  155. 155. Has semi-heavy tails. </li></li></ul><li>References Part 1<br />Akar, H., Poverty, and Schooling in Turkey: a Needs Assessment Study, presentation at EURO ORD Workshop on Complex Societal Problems, Sustainable Living and Development, May 13-16, 2008, IAM, METU, Ankara.<br />Admassu, K., Primary School Enrollment and Progression in Ethiopia: Family and School Factors,American Sociological Association Annual Meeting, July 31, 2008, Boston, MA.<br />Hanushek, E.A.,Lavy, V., and Kohtaro, H., Do studentscare about schoolquality? Determinants of dropoutbehaviorin developingcountries, Journal of Human Capital 2:1, 2008, pp. 69-105.<br />Julius, The Delphi Method: Techniques and Applications, eds.: Harold,A.L., and Murray, T. , Addison-Wesley, 2002.<br />Leopold-Wildburger, U., Weber, G.-W., and Zachariasen, M.,OR for Better Management of Sustainable Development, European Journal of Operational Research 193:3 (March 2009); special issue at the occasion of EURO XXI 2006, Reykjavik, Iceland, 2006.<br />Lane, D.C., Social theory and system dynamics practice, European Journal of Operational Research113:3 (1999), pp. 501-527. <br />Pedamallu, C.S.,Externally AidedConstructionof SchoolRoomsfor PrimaryClasses- Preparation of ProjectReport. Master’s Dissertation, Indian Institute of Technology Madras, 2001.<br />
  156. 156. References Part 1<br />Pedamallu, C.S.,Özdamar, L.,Akar, H., Weber, G.-W., and Özsoy, A., Investigating academic <br />performanceof migrant students: a system dynamics perspectivewith an application to Turkey, to <br />appear in International Journal ofProduction Economics, the special issue on Models for <br />CompassionateOperations,Sarkis, J., guest editor.<br /> Pedamallu, C.S., Ozdamar, L., Kropat , E., and Weber, G.-W., A system dynamics model for intentional<br /> transmission of HIV/AIDS using cross impact analysis, to appear in CEJOR (Central European Journal <br /> of Operations Research).<br />Pedamallu, C.S.,Özdamar, L., Weber, G.-W., and Kropat,E., A system dynamics model to studythe importance of infrastructure facilities on quality of primary education system in developing countries, in the proceedings of PCO 2010, 3rd Global Conference on Power Control and <br />Optimization,February 2-4, 2010, Gold Coast, Queensland, Australia ISBN: 978-0-7354-<br />0785-5 (AIP: American Institute of Physics) 321-325.<br />Weimer-Jehle, W., Cross-impact balances: A system-theoretical approach to cross-impact analysis, Technological Forecasting and Social Change, 73:4, May 2006, pp. 334-361.<br />
  157. 157. References Part 2<br />Achterberg, T., Constraint integer programming, PhD. Thesis, Technische Universitat Berlin, Berlin, 2007.<br />Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems. Academic Press, San Diego; 2004.<br />Chen, T., He, H.L., and Church, G.M., Modeling gene expression with differential equations, Proceedings of Pacific Symposium on Biocomputing 1999, 29-40.<br />Ergenc, T., and Weber, G.-W., Modeling and prediction of gene-expression patterns reconsidered with Runge-Kutta discretization, Journal of Computational Technologies 9, 6 (2004) 40-48.<br />Gebert, J., Laetsch, M., Pickl, S.W., Weber, G.-W., and Wünschiers ,R., Genetic networks and anticipation of gene expression patterns, Computing Anticipatory Systems: CASYS(92)03 - Sixth International Conference,AIP Conference Proceedings 718 (2004) 474-485.<br />Hoon, M.D., Imoto, S., Kobayashi, K., Ogasawara, N., andMiyano, S., Inferring gene regulatory networks from time-ordered gene expression data of Bacillus subtilis using dierential equations, Proceedings of Pacific Symposium on Biocomputing (2003) 17-28.<br />Pickl, S.W., and Weber, G.-W., Optimization of a time-discrete nonlinear dynamical system from a problem of ecology - an analytical and numerical approach, Journal of Computational Technologies 6, 1 (2001) 43-52.<br />Sakamoto, E., and Iba, H., Inferring a system of differential equations for a gene regulatory network by using genetic programming, Proc. Congress on Evolutionary Computation 2001, 720-726.<br />Tastan, M., Analysis and Prediction of Gene Expression Patterns by Dynamical Systems, and by a Combinatorial Algorithm, MSc Thesis, Institute of Applied Mathematics, METU, Turkey, 2005.<br />
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