The document discusses robust conic generalized partial linear models (RCGPLMs) and their application in forecasting default probabilities in emerging markets. It provides background on regression techniques like MARS and CMARS. It then introduces generalized linear models (GLMs) and generalized partial linear models (GPLMs), describing how conic GPLMs combine linear and nonlinear regression methods. The document outlines the development of robust conic GPLMs to account for uncertainty in input data, and provides examples of applying RCGPLMs to real-world default prediction problems.

1. Gerhard-W ilhelm W EBER * Ayşe ÖZMEN Zehra Çavuşoğlu Özlem Defterli Institute of Applied Mathematics, METU, Ankara, Turkey * Faculty of Economics, Management Science and Law, University of Siegen, Germany Center for Research on Optimization and Control, U niversity of Aveiro, Portuga l Universiti Teknologi Malaysia, Skudai, Malaysia Forecasting Default Probabilities in Emerging Markets and Dynamical Regulatory Networks through New Robust Conic GPLMs and Optimization 6th International Summer School Achievements and Applications of Contemporary Informatics, Mathematics and Physics National University of Technology of the Ukraine Kiev, Ukraine, August 8-20, 2011

10. L is an matrix. CQP and Tikhonov Regularization for MARS

15. Robust Optimization Laurent El Ghaoui . Robust Optimization and Applications , IMA Tutorial, March 11, 2003

16. d ual of conic program , . . . Robust Optimization

17. r obust conic programming . . . . Robust Optimization

18. p olytopic uncertainty . . . Robust Optimization

19. r obust C Q P CQP , . , . Robust Optimization

21. General model on the relation between input and response : error term mean noisy input data value is random variable, and we assume that it is normally distributed. Robustification of CMARS The Idea of Robust CMARS ,

24. polyhedral uncertainty sets Cartesian product . . . . vertex vertices . . . . . . . . . . . . Polyhedral Uncertainty

26. is the p olytope with vertices w here is the convex hull . Polyhedral Uncertainty and Robust Counterpart for CMARS Model :

27. Robust CQP with the Polytopic Uncertainity Robust conic quadratic program ming o f our CMARS: where L ice-cream (or second-order, or Lorentz) cones. equivalently ( Standard ) C onic Q uadratic P rogram ming

32. The Least-Squares Estimation with Tikhonov Regularization The procedure is as follows: The vector is found by the application of the linear least squares on the given data: (1) Then, parametric part has the form: To estimate the regression coefficients the method of least squares is employed: in to minimize the residual sum of squares (RSS). Conic GPLM (CGPLM)

35. General model on the relation between input and response : error term mean is random variable, and we assume that it is normally distributed. Robustification of Conic GPLM The Idea of Robust Conic GPLM is a link function that connect the mean of the response variable, to the predictor varaibles. Then, additive semiparametric model :

36. Variables of Robust Conic GPLM Robustification of Conic GPLM

37. Linear Part of Robust Conic GPLM A linear estimator is found as: As a Tikhonov Regularization form it can be written as: Finally, it can be written as a standard CQP problem: Robustification of Conic GPLM

38. Nonlinear Part of Robust Conic GPLM By the help of the smooth function found by RCMARS the PRSS form is obtained: It can be converted into: Robustification of Conic GPLM Finally, it can be written as a standard CQP problem:

39. Numerical Experience To employ the robust optimization technique on the linear part of CGPLM model, we include perturbation s (uncertainty) into the real input data in each dimension, and into the output data ( i =1,2, … ,24). For this purpose, the uncertainty matrices and vectors are elements in polyhedral uncertainty set s for the linear part. Then, uncertainty is evaluated for all input and output values which are represented by CIs. Afterward s , we transform the variables into the standard normal distribution , the CI is obtained to be [ -3, 3 ] . Robustification of Conic GPLM

40. For nonlinear part, we constructed model functions for these data using MARS Software , where we selected the maximum number of basis elements: Then, the large model becomes Robustification of Conic GPLM Numerical Experience

47. Real-word Application for RCGPLM Results and Comparision Training Sample Validation Sample D-D ND-ND Correct Classification Rate D-D ND-ND Correct Classification Rate CGPLM 90.09% 93.24% 91.81% 86.27% 90.05% 89.31% RCGPLM 87.80% 96.20% 93.33% 96.88% 89.71% 92%

48. Process Version of RCGPLM Bio-Systems medicine food education health care development sustainability bio materials bio energy environment

49. 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 Medicine Process Version of RCGPLM Bio-Systems

50. Financial Systems Process Version of RCGPLM

51. Regulatory Networks: Examples Further examples: Socio-econo-networks, stock markets, portfolio optimization, immune system, epidemiological processes … Process Version of RCGPLM Target variables Environmental items Genetic Networks Gene expression Transscription factors, toxins, radiation Eco-Finance Networks CO 2 -emissions Financial means, technical means

52. Modeling & Prediction prediction, anticipation least squares – max likelihood statistical learning expression data m atrix - valued function – metabolic reaction E xpression Process Version of RCGPLM

53. Process Version of RCGPLM Ex.: M We analyze the influence of em -parameters on the dynamics ( e xpression- m etabolic). Ex.: Euler, Runge-Kutta , Heun Modeling & Prediction

54. Process Version of RCGPLM g en e 2 g en e 3 g en e 1 g en e 4 0.4 E 1 0.2 E 2 1 E 1 Genetic Networks

55. Process Version of RCGPLM Gene-Environment Networks

56. Process Version of RCGPLM The Model Class d- vector of concentration levels of proteins and of certain levels of environmental factors d = m + n continuous change in the gene-expression data in time is the firstly introduced time-autonomous form, where 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. is 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)

58. θ 1 θ 2 Regulatory Networks under Uncertainty Process Version of RCGPLM Errors uncorrelated Errors correlated Fuzzy values Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics

59. Process Version of RCGPLM Robustification of GPLM Approach for Regulatory, Dynamical Systems W e can represent their generalized multiplicative form with our GPLM approach as follows : represents the expression levels of targets, consists of environmental factors which affect the targets in the network , is called as network matrix , which can be identified by solving the following least-squares (or maximum likelihood) estimation problem: : some vector of unknowns

60. Process Version of RCGPLM Robustification of GPLM Approach for Regulatory, Dynamical Systems W e represent the process version of the GPLM formulation in the following way: corresponding to the parameters of The unknown parameters appearing inside of ( nonlinear part ). can be collected separately vector s ( linear part ), corresponding to the parameters of Hence,

61. When the entiries of the matrix are splines, to solve the problem given in ( ** ) , CGPLM can be used for target-environment networks. Process Version of RCGPLM Robustification of GPLM Approach for Regulatory, Dynamical Systems Furthermore , in the case of the existence of uncertainty in the expression data, then the presented RCGPLM technique can be applied with RCMARS in order to study a robustification of our target-environment networks. Then, for each row of the matrix equation in ( * ), we represent the process version of the RCGPLM model in the subsequent manner:

62. Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004. Ben-Tal, A., Nemirovski, A., Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, MPR-SIAM Series on Optimization, SIAM, Philadelphia, 2001 . Chen, T., He, H.L., and Church, G.M., Modeling gene expression with differential equations, Proceedings of Pacific Symposium on Biocomputing 1999, 29-40. Defterli, O., Fügenschuh, A, and Weber, G-W., New discretization and optimization techniques with results in the dynamics of gene- environment networks. In: Proceedings of the 3rd Global Conference on Power Control & Optimization (PCO 2010), Editors: N. Barsoum, P. Vasant, R. Habash, ISBN: 978-983-44483-1-8. Defterli, O., Fügenschuh, A., and Weber, G.-W., Modern Tools For The Tıme-dıscrete Dynamıcs and Optımızatıon Of Gene-envıronment Networks, Communications in Nonlinear Science and Numerical Simulation, in press, 2011. El Ghaoui, L., Robust Optimization and Applications, IMA Tutorial, 2003. 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. Friedman, J.H., Multivariate adaptive regression splines, The Annals of Statistics 19, 1 (1991) 1-141. Hansen, P.C., Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1998. Hastie, T., Tibshirani, R., and Friedman, J.H., The Element of Statistical Learning, Springer Verlag, NY, 2001. Hoon, M.D., Imoto, S., Kobayashi, K., Ogasawara, N ., and Miyano, 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. 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. Kropat, E., Weber, G.-W. , Robust regression analysis for gene-environment and eco-finance networks under polyhedral and ellipsoidal uncertainty. preprint_2 (2010) at Institute of Applied Mathematics, METU . Myers, R.H., and Montgomery, D.C., Response Surface Methodology: Process and Product Optimization Using Designed Experiments,New York: Wiley (2002). Nemirovski, A., Lectures on modern convex optimization, Israel Institute Technology (2002), http://iew3.technion.ac.il/Labs/Opt/LN/Final.pdf . Nesterov, Y.E., and Nemirovskii, A.S., Interior Point Methods in Convex Programming, SIAM, 1993. References

63. Özmen, A., Weber, G.-W., Batmaz, I. and Kropat E ., RCMARS: Robustification of CMARS with Different Scenarios under Polyhedral Uncertainty Set. To appear in Communications in Nonlinear Science and Numerical Simulation (CNSNS), Special Issue Nonlinear, Fractional and Complex Systems with Discontinuity and Chaos, D. Baleanu and J.A. Tenreiro Machado (guest editors) , 2010 . Özmen, A., Weber, G.-W. , and Kerimov , A . , RCMARS: A New Optimization Supported Tool - Applied on Financial Market Data -under Polyhedral Uncertainty , preprint at Institute of Applied Mathematics, METU ,submitted to JOGO, 2010 . Özmen, A., Weber, G.-W. , Çavuşoglu Z., and Defterli Ö., The New Robust Conic GPLM Method with an Application to Finance and Regulatory Systems: Prediction of Credit Default and a Process Version , preprint at Institute of Applied Mathematics, METU ,submitted to JOGO, 2010 . Özmen, A., and Weber, G.-W.: Robust Conic Generalized Partial Linear Models Using RCMARS Method – A Robustification of CGPLM. preprint at Institute of Applied Mathematics, METU, in Proceedings of Fifth Global Conference on Power Control and Optimization PCO , June 1 – 3, 2011, Dubai, ISBN: 983-44483-49 . 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. 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. 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. Tastan, M., Pickl, S.W., and Weber, G.-W., Mathematical modeling and stability analysis of gene-expression patterns in an extended space and with Runge-Kutta discretization, Proceedings of Operations Research, Bremen, 2006, 443-450. Weber, G.-W., Batmaz, I., Köksal G., Taylan P., and Yerlikaya F., 2009. CMARS: A New Contribution to Nonparametric Regression with Multivariate Adaptive Regression Splines Supported by Continuous Optimisation, preprint at IAM, METU, submitted for publication. Weber, G.-W., Çavuşoğlu Z., and Özmen A. , Predicting Default Probabilities in Emerging Markets by New Conic Generalized Partial Linear Models and Their Optimization. To appear in Advances in Continuous Optimization with Applications in Finance, Special Issue Optimization ,2010

64. Weber, G.-W., Alparslan -Gök, S.Z., and Dikmen, N., Environmental and life sciences: Gene-environment networks-optimization, games and control - a survey on recent achievements, deTombe, D. (guest ed.), special issue of Journal of Organizational Transformation and Social Change 5, 3 (2008) 197-233. Weber, G.-W., Taylan, P., Alparslan-Gök, S.Z., Özögur, S., and Akteke-Öztürk, B., Optimization of gene-environment networks in the presence of errors and uncertainty with Chebychev approximation, TOP 16, 2 (2008) 284-318. Weber, G.-W., Alparslan-Gök, S.Z., and Söyler, B., A new mathematical approach in environmental and life sciences: gene-environment networks and their dynamics,Environmental Modeling & Assessment 14, 2 (2009) 267-288. Weber, G.-W., and Ugur, O., Optimizing gene-environment networks: generalized semi-infinite programming approach with intervals, Proceedings of International Symposium on Health Informatics and Bioinformatics Turkey '07, HIBIT, Antalya, Turkey, April 30 - May 2 (2007). Yılmaz, F.B., A Mathematical Modeling and Approximation of Gene Expression Patterns by Linear and Quadratic Regulatory Relations and Analysis of Gene Networks, MSc Thesis, Institute of Applied Mathematics, METU, Turkey, 2004. Weber, G.-W., Kropat, E., Tezel, A., and Belen, S., Optimization applied on on regulatory and eco-finance networks – survey and new development. Pacific J. Optim. 6(2), 319-340 (2010) .

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