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Gerhard-W ilhelm  W EBER   *  Ayşe ÖZMEN Zehra Çavuşoğlu Özlem Defterli Institute of Applied Mathematics, METU, Ankara, Tu...
<ul><li>Introduction </li></ul><ul><li>Regression </li></ul><ul><li>MARS </li></ul><ul><li>CMARS </li></ul><ul><li>Robust ...
<ul><li>learning  from data  has  become very important  </li></ul><ul><li>in every field of science and technology,  e.g....
Regression <ul><li>Regression is mainly based on the methods of </li></ul><ul><ul><li>Least squares estimation, </li></ul>...
<ul><li>To estimate general functions of  high-dimensional  arguments. </li></ul><ul><li>An  adaptive  procedure. </li></u...
<ul><li>Let us consider  </li></ul><ul><li>The goal is to construct reflected pairs for each input   </li></ul>MARS r egre...
<ul><li>Set of basis functions:   </li></ul><ul><li>Thus,  can be represented by </li></ul><ul><li>are basis functions fro...
<ul><li>Two subalgorithms: </li></ul><ul><li>(i)  Forward stepwise algorithm : </li></ul><ul><li>Search for the basis func...
PRSS for  MARS <ul><li>Tradeoff   between both  accuracy  and  complexity .  </li></ul><ul><li>Penalty parameters  .  </li...
L  is an  matrix.   CQP and Tikhonov Regularization for  MARS
<ul><li>For a short representation, we can rewrite the  approximate  relation as  </li></ul><ul><li>In case of the  same  ...
<ul><li>Conic quadratic  programming: </li></ul>CQP for  MARS
CQP for  MARS <ul><li>Moreover,  is a  primal dual   optimal  solution   if and only if  </li></ul>
<ul><li>CQPs belong to the  well -structured  convex problems .  </li></ul><ul><li>Interior Point Methods. </li></ul><ul><...
Robust Optimization Laurent El Ghaoui .   Robust Optimization and Applications , IMA Tutorial, March 11, 2003
d ual  of conic program ,   .  .   .   Robust Optimization
r obust  conic  programming .   .   .   .   Robust Optimization
p olytopic  uncertainty .  .  .  Robust Optimization
r obust   C Q P CQP   ,   .   ,  .   Robust Optimization
<ul><li>CMARS models depend on the parameters.  Small perturbations in data may give different model parameters.  </li></u...
General model on the relation between input and response : error term mean noisy  input data value is  random variable, an...
<ul><li>Robustification of  CMARS </li></ul><ul><li>CMARS  Model with Uncertainity </li></ul>To employ robust optimization...
<ul><li>Provided the observations represented by the  data with uncertainty  </li></ul><ul><li>the form of the  m th basis...
polyhedral  uncertainty  sets  Cartesian product . . . . vertex vertices . . . . . . . . . . . . Polyhedral Uncertainty
<ul><li>Based on polyhedral  uncertainty  sets  ,  t he robust counterpart  for CMARS </li></ul>Polyhedral Uncertainty and...
is  the  p olytope   with  vertices   w here   is the  convex hull . Polyhedral Uncertainty and  Robust Counterpart for  C...
Robust CQP with  the Polytopic Uncertainity Robust conic quadratic program ming   o f our  CMARS:  where  L   ice-cream (o...
<ul><li>T he class of  G eneralized  L inear  M odel s  ( GLM s )  has gained popularity as a statistical   modeling tool....
<ul><li>Some  other useful statistical models   such as with </li></ul><ul><li>Poisson, binomial ,  </li></ul><ul><li>G am...
<ul><li>A pa rticular  type of  s emiparametric  model s are  the   G eneralized  P artial  L inear  M odel s   ( GPLM s )...
<ul><li>The  general model can be written as follows: </li></ul><ul><li>with observation values  , </li></ul><ul><li>  and...
The Least-Squares Estimation with Tikhonov Regularization The procedure is as follows: The vector  is found by the applica...
<ul><li>After  obtaining  the regression coefficients, we subtract the linear least - square s  model  </li></ul><ul><li>(...
<ul><li>Tikhonov Regularization  is employed to find the approximate solution to the equation (1) by minimizing the quadra...
General model on the relation between input and response :  error term mean is  random variable, and we assume that  it   ...
Variables of  Robust Conic GPLM Robustification of  Conic GPLM
Linear   Part of  Robust Conic GPLM A linear estimator is found as: As a Tikhonov Regularization form it can be written as...
Nonlinear Part of  Robust Conic GPLM By the help of the smooth function found by RCMARS the PRSS form is obtained: It can ...
Numerical Experience  To employ the robust optimization technique on the linear part of CGPLM model,  we include perturbat...
For nonlinear part, we constructed model functions for these data using  MARS Software ,  where we selected the maximum nu...
<ul><li>To apply the robust optimization technique on the nonlinear part of CGPLM model, we include perturbation (uncertai...
<ul><li>To solve  our problem : </li></ul><ul><li>T r an s f o r m it into the MOSEK format.  </li></ul><ul><li>W rite thi...
<ul><li>Data </li></ul><ul><li>1019 observations which belong to 45 emerging markets from 1980 to 2005 </li></ul><ul><li>1...
<ul><li>Variables’ scatterplot graphs are plotted in Minitab 14 to see linearity </li></ul><ul><li>4 of 13 variables show ...
<ul><li>Result of Linear Part </li></ul><ul><li>“ 1” or “0” </li></ul>The Remainder  (RCMARS Part) consists of  “ -1”, “0”...
<ul><li>Group I </li></ul><ul><li>consists of 378 observations,  </li></ul><ul><li>52 of which are “-1”,  </li></ul><ul><l...
Real-word Application for RCGPLM Results  and Comparision   Training Sample Validation Sample D-D ND-ND Correct Classifica...
Process Version of RCGPLM Bio-Systems   medicine food education health care development sustainability bio materials bio e...
DNA microarray   chip  experiments prediction  of gene patterns  based on with M.U. Akhmet,  H. Öktem  S.W. Pickl,  E. Que...
Financial Systems  Process Version of RCGPLM
Regulatory Networks:   Examples Further examples: Socio-econo-networks, stock markets, portfolio optimization, immune syst...
Modeling & Prediction  prediction,  anticipation least squares  –  max likelihood statistical learning expression data m a...
Process Version of RCGPLM Ex.: M We analyze the influence of  em   -parameters  on the dynamics  ( e xpression- m etabolic...
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
Process Version of RCGPLM Gene-Environment Networks
Process Version of RCGPLM The Model Class d- vector  of concentration levels of proteins and  of certain levels of environ...
Process Version of RCGPLM <ul><li>(i)  is an  constant ( n x n )-matrix </li></ul><ul><li>is an  ( n x 1)-vector of gene-e...
θ 1 θ 2 Regulatory Networks  under  Uncertainty Process Version of RCGPLM Errors uncorrelated Errors correlated Fuzzy valu...
Process Version of RCGPLM Robustification of GPLM Approach for Regulatory, Dynamical Systems   W e can represent their gen...
Process Version of RCGPLM Robustification of GPLM Approach for Regulatory, Dynamical Systems   W e represent the process v...
When the entiries of the matrix  are splines, to solve the problem given in ( ** ) ,   CGPLM  can be used for target-envir...
Aster, A., Borchers, B., and Thurber, C., Parameter Estimation and Inverse Problems, Academic Press, 2004. Ben-Tal, A.,   ...
Özmen, A., Weber, G.-W., Batmaz, I.  and  Kropat E .,  RCMARS:   Robustification of CMARS with  Different Scenarios under Po...
Weber, G.-W., Alparslan -Gök, S.Z., and Dikmen, N., Environmental and life sciences:  Gene-environment networks-optimizati...
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Forecasting Default Probabilities in Emerging Markets and Dynamical Regulatory Networks through New Robust Conic GPLMs and Optimization

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

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Forecasting Default Probabilities in Emerging Markets and Dynamical Regulatory Networks through New Robust Conic GPLMs and Optimization

  1. 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
  2. 2. <ul><li>Introduction </li></ul><ul><li>Regression </li></ul><ul><li>MARS </li></ul><ul><li>CMARS </li></ul><ul><li>Robust Optimization </li></ul><ul><li>Robustification of CMARS </li></ul><ul><li>CMARS Model with Polyhedral Uncertainty </li></ul><ul><li>Generalized Linear Models (GLMs) </li></ul><ul><li>Generalized Partial Linear Models (GPLMs) </li></ul><ul><li>Conic GPLMs </li></ul><ul><li>Robust Conic GPLMs (RCGPLMs) </li></ul><ul><li>Numerical Experience with RCGPLM </li></ul><ul><li>Real-word Application for RCGPLM </li></ul><ul><li>Process Version of RCGPLM </li></ul><ul><li>Conclusion </li></ul>Content
  3. 3. <ul><li>learning from data has become very important </li></ul><ul><li>in every field of science and technology, e.g., in </li></ul><ul><li>financial sector , </li></ul><ul><li>quality improvent in manufacturing , </li></ul><ul><li>computational biology , </li></ul><ul><li>medicine and </li></ul><ul><li>engineering . </li></ul><ul><li>Learning enables for doing estimation and prediction . </li></ul>Introduction
  4. 4. Regression <ul><li>Regression is mainly based on the methods of </li></ul><ul><ul><li>Least squares estimation, </li></ul></ul><ul><ul><li>Maximum likelihood estimation. </li></ul></ul><ul><li>There are many regression models </li></ul><ul><ul><li>Linear regression models, </li></ul></ul><ul><ul><li>Nonlinear regression models, </li></ul></ul><ul><ul><li>Generalized linear models, </li></ul></ul><ul><ul><li>Nonparametric regression models, </li></ul></ul><ul><ul><li>Additive models, </li></ul></ul><ul><ul><li>Generalized additive models. </li></ul></ul>
  5. 5. <ul><li>To estimate general functions of high-dimensional arguments. </li></ul><ul><li>An adaptive procedure. </li></ul><ul><li>A nonparametric regression procedure. </li></ul><ul><li>No specific assumption about the underlying functional relationship between the dependent and independent variables. </li></ul><ul><li>Ability to estimate the contributions of the basis functions so that both the additive and the interactive effects of the predictors are allowed to determine the response variable. </li></ul><ul><li>Uses expansions in piecewise linear basis functions of the form </li></ul>MARS: Multivariate Adaptive Regression Spline
  6. 6. <ul><li>Let us consider </li></ul><ul><li>The goal is to construct reflected pairs for each input </li></ul>MARS r egression w ith
  7. 7. <ul><li>Set of basis functions: </li></ul><ul><li>Thus, can be represented by </li></ul><ul><li>are basis functions from or products of two or more such functions; interaction basis functions are created by multiplying an existing basis function with a truncated linear function involving a new variable. </li></ul><ul><li>Provided the observations represented by the data </li></ul><ul><li>where subvectors of . </li></ul>MARS
  8. 8. <ul><li>Two subalgorithms: </li></ul><ul><li>(i) Forward stepwise algorithm : </li></ul><ul><li>Search for the basis functions. </li></ul><ul><li>Minimization of some “lack of fit” criterion . </li></ul><ul><li>The process stops when a user-specified value is reached. </li></ul><ul><li>Overfitting. </li></ul><ul><li>This model typically overfits the data; so a backward deletion procedure is applied. </li></ul><ul><li>(ii) Backward stepwise algorithm : </li></ul><ul><li>Prevents from over-fitting by decreasing the complexity of the model without degrading the fit to the data . </li></ul><ul><li>Alternative : </li></ul>MARS
  9. 9. PRSS for MARS <ul><li>Tradeoff between both accuracy and complexity . </li></ul><ul><li>Penalty parameters . </li></ul>
  10. 10. L is an matrix. CQP and Tikhonov Regularization for MARS
  11. 11. <ul><li>For a short representation, we can rewrite the approximate relation as </li></ul><ul><li>In case of the same penalty parameter , then: </li></ul><ul><li>Tikhonov regularization </li></ul>CQP and Tikhonov Regularization for MARS
  12. 12. <ul><li>Conic quadratic programming: </li></ul>CQP for MARS
  13. 13. CQP for MARS <ul><li>Moreover, is a primal dual optimal solution if and only if </li></ul>
  14. 14. <ul><li>CQPs belong to the well -structured convex problems . </li></ul><ul><li>Interior Point Methods. </li></ul><ul><li>Better complexity bounds. </li></ul><ul><li>Better practical performance. </li></ul>CQP for MARS C-MARS
  15. 15. Robust Optimization Laurent El Ghaoui . Robust Optimization and Applications , IMA Tutorial, March 11, 2003
  16. 16. d ual of conic program , . . . Robust Optimization
  17. 17. r obust conic programming . . . . Robust Optimization
  18. 18. p olytopic uncertainty . . . Robust Optimization
  19. 19. r obust C Q P CQP , . , . Robust Optimization
  20. 20. <ul><li>CMARS models depend on the parameters. Small perturbations in data may give different model parameters. </li></ul><ul><li>This may cause unstable solutions. </li></ul><ul><li>In CMARS, the aim is to reduce the estimation error, while keeping efficiency as high as possible. </li></ul><ul><li>In order to achieve this aim, we use some approaches: </li></ul><ul><ul><li>scenario optimization, </li></ul></ul><ul><ul><li>robust counterpart, </li></ul></ul><ul><ul><li>usage of more robust estimators. </li></ul></ul><ul><li>By using robustification in CMARS, the estimation variance will decrease . </li></ul>Robustification of CMARS The Idea of Robust CMARS (RCMARS)
  21. 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 ,
  22. 22. <ul><li>Robustification of CMARS </li></ul><ul><li>CMARS Model with Uncertainity </li></ul>To employ robust optimization on CMARS , a “perturbation” ( uncertainty ) is incorporate d into the input data (for each dimension ) and into the output data : will be represented as after perturbation Here , is the mean of the vector in each dimension is restricted by , which is the semi-length of confidence interval . ; the amount of perturbation
  23. 23. <ul><li>Provided the observations represented by the data with uncertainty </li></ul><ul><li>the form of the m th basis function with uncertainty set is defined by </li></ul><ul><li>The PRSS has the following representation: </li></ul>CMARS Model with Uncertainity Tikhonov regularization
  24. 24. polyhedral uncertainty sets Cartesian product . . . . vertex vertices . . . . . . . . . . . . Polyhedral Uncertainty
  25. 25. <ul><li>Based on polyhedral uncertainty sets , t he robust counterpart for CMARS </li></ul>Polyhedral Uncertainty and Robust Counterpart for CMARS Model is a poly tope with vertices where is the convex hull . : ,
  26. 26. is the p olytope with vertices w here is the convex hull . Polyhedral Uncertainty and Robust Counterpart for CMARS Model :
  27. 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
  28. 28. <ul><li>T he class of G eneralized L inear M odel s ( GLM s ) has gained popularity as a statistical modeling tool. </li></ul><ul><li>It has some advantages like: </li></ul><ul><li>GLM has t he flexibility in addressing a variety of statistical p roblems , </li></ul><ul><li>GLM has an advantage in the case of t he availability of software ( Stata, SAS, S-PLUS, R ). </li></ul><ul><li>The class of GLM is an extension of traditional linear models allows: </li></ul><ul><li>T he mean of a dependent variable to depend on a linear predictor by a n onlinear link function . </li></ul><ul><li>T he probability distribution of the response, to be any member of an exponential family of distributions. </li></ul><ul><li>GLM contains m any widely used statistical models : </li></ul><ul><li>linear models with normal errors , </li></ul><ul><li>logistic and probit models for binary data, </li></ul><ul><li>log-linear models for multinomial data. </li></ul>Generalized Linear Models
  29. 29. <ul><li>Some other useful statistical models such as with </li></ul><ul><li>Poisson, binomial , </li></ul><ul><li>G amma or normal distribution s , </li></ul><ul><li>can be formulated as GLM by the selection of an appropriate link function </li></ul><ul><li>and response probability distribution. </li></ul><ul><li>A GLM has the following form: </li></ul><ul><li>where </li></ul><ul><li>: expected value of the response variable , </li></ul><ul><li>: smooth monotonic link function , </li></ul><ul><li>: observed value of explanatory variable for the i th case , </li></ul><ul><li>: vector of unknown parameter s . </li></ul>Generalized Linear Models
  30. 30. <ul><li>A pa rticular type of s emiparametric model s are the G eneralized P artial L inear M odel s ( GPLM s ) : </li></ul><ul><li>GPLMs are extend ed version of the GLMs by adding a </li></ul><ul><li>single nonparametric component to the usual parametric terms : </li></ul><ul><li>is a vector of parameter s, and </li></ul><ul><li>is a smooth function , which we try to estimate by CMARS. </li></ul><ul><li>Assumption: m -dimensional random vector which represents ( typically discrete ) covariates , </li></ul><ul><li>q -dimensional random vector of continuous covariates , </li></ul><ul><li>which come s from a decomposition of explanatory variables . </li></ul>Generalized Partial Linear Models
  31. 31. <ul><li>The general model can be written as follows: </li></ul><ul><li>with observation values , </li></ul><ul><li> and and is a smooth function. </li></ul><ul><li>There are different kinds of estimation methods for GPLM . </li></ul><ul><li>Generally, the estimation methods for model are based on kernel methods and test procedures on the correct specification of this model . </li></ul><ul><li>Now, we will try to concentrate on special types of GPLM ( Conic GPLM ) estimation based on CMARS . </li></ul>Estimation for GPLM
  32. 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)
  33. 33. <ul><li>After obtaining the regression coefficients, we subtract the linear least - square s model </li></ul><ul><li>(without intercept) from corresponding responses : </li></ul><ul><li>T he resulting values become new response variables of the input data. </li></ul><ul><li>Then, the knots for nonparametric part of MARS can be found based on th e s e new data . </li></ul><ul><li>In this the model : </li></ul><ul><li> and is a smooth function which will be estimated by CMARS </li></ul><ul><li>that is an alternative to multivariate adaptive regression splines (MARS) . </li></ul>Conic GPLM The Least-Squares Estimation with Tikhonov Regularization
  34. 34. <ul><li>Tikhonov Regularization is employed to find the approximate solution to the equation (1) by minimizing the quadratic functional </li></ul><ul><li>(2) </li></ul><ul><li>where is a regularization parameter between the first and the second part. </li></ul><ul><li>The term is the response vector and shows unknown coefficients. </li></ul><ul><li>Tikhonov regularization problem (2) helps for finding the parameters. </li></ul>Conic GPLM The Least-Squares Estimation with Tikhonov Regularization
  35. 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. 36. Variables of Robust Conic GPLM Robustification of Conic GPLM
  37. 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. 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. 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. 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
  41. 41. <ul><li>To apply the robust optimization technique on the nonlinear part of CGPLM model, we include perturbation (uncertainty) into the real input data in each dimension, and into the output data ( i =1,2,…,24). </li></ul><ul><li>S imilar to linear part, the uncertainty matrices and vectors based on polyhedral uncertainty sets are obtained . </li></ul><ul><li>U n ce r t a i n ty is calculated for all input and output values which are represented by CIs and we transform the variables into the standard normal distribution . </li></ul><ul><li>T he CI is obtained to be [ -3, 3 ] . </li></ul><ul><li>T he uncertainty matrix for input data has a huge size </li></ul><ul><li>W e do not have enough computer capacity . </li></ul><ul><li>Tradeoff between tractibility and robustification . </li></ul>Robustification of Conic GPLM Numerical Experience
  42. 42. <ul><li>To solve our problem : </li></ul><ul><li>T r an s f o r m it into the MOSEK format. </li></ul><ul><li>W rite this formulation each value of our sample ( N= 2 4 ). </li></ul><ul><li>We formulate our models as a CQP problem for each sample value (observation) </li></ul><ul><li>using the combinatorial approach , which we call weak robustification . </li></ul><ul><li>As a result, we obtain 2 4 different weak RC GPLM ( WR CGPLM ) sub models </li></ul><ul><li>for each linear and nonlinear part. </li></ul><ul><li>S o lve them separately by using MOSEK program. </li></ul><ul><li>After the MOSEK result for each of our observation is found , we select a MOSEK model which has the maximum t value s for linear and nonlinear part , and we continue with these MOSEK model s to calculate our parameter values </li></ul>Robustification of Conic GPLM Numerical Experience
  43. 43. <ul><li>Data </li></ul><ul><li>1019 observations which belong to 45 emerging markets from 1980 to 2005 </li></ul><ul><li>13 explanatory variables : </li></ul><ul><ul><li>Bank liquid reserves to bank assets ratio, </li></ul></ul><ul><ul><li>Changes in net reserves / GDP ( Gross Domestic Product), </li></ul></ul><ul><ul><li>Current account balance (% of GDP), </li></ul></ul><ul><ul><li>Exports of goods and services (% of GDP), </li></ul></ul><ul><ul><li>External debt total / Total Reserves, </li></ul></ul><ul><ul><li>Long-term debt / GDP, </li></ul></ul><ul><ul><li>GDP growth (annual %), </li></ul></ul><ul><ul><li>Liquid liabilities as % of GDP, </li></ul></ul><ul><ul><li>Total debt service (% of exports of goods services and income), </li></ul></ul><ul><ul><li>Short-term debt (% of exports of goods services and income), </li></ul></ul><ul><ul><li>Trade (% of GDP), </li></ul></ul><ul><ul><li>Use of IMF credit / GDP, </li></ul></ul><ul><ul><li>Inflation consumer prices (annual %). </li></ul></ul>Real-word Application for RCGPLM
  44. 44. <ul><li>Variables’ scatterplot graphs are plotted in Minitab 14 to see linearity </li></ul><ul><li>4 of 13 variables show a linear relationship with default values “ y ”. </li></ul><ul><ul><li>Liquid liabilities as % of GDP, </li></ul></ul><ul><ul><li>Total debt service (% of exports of goods services and income), </li></ul></ul><ul><ul><li>Use of IMF credit / GDP, </li></ul></ul><ul><ul><li>Inflation consumer prices (annual %). </li></ul></ul>Real-word Application for RCGPLM
  45. 45. <ul><li>Result of Linear Part </li></ul><ul><li>“ 1” or “0” </li></ul>The Remainder (RCMARS Part) consists of “ -1”, “0” and “1” values Response Variable shows defaults and nondefaults consists of “0” and “1” values Real-word Application for RCGPLM
  46. 46. <ul><li>Group I </li></ul><ul><li>consists of 378 observations, </li></ul><ul><li>52 of which are “-1”, </li></ul><ul><li>326 of which are “0”. </li></ul><ul><li>Group II </li></ul><ul><li>consists of 379 observations, </li></ul><ul><li>104 of which are “1”, </li></ul><ul><li>275 of which are “0”. </li></ul>Subgroups of Training Sample which consists of 757 observations Real-word Application for RCGPLM
  47. 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. 48. Process Version of RCGPLM Bio-Systems medicine food education health care development sustainability bio materials bio energy environment
  49. 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. 50. Financial Systems Process Version of RCGPLM
  51. 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. 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. 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. 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. 55. Process Version of RCGPLM Gene-Environment Networks
  56. 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)
  57. 57. Process Version of RCGPLM <ul><li>(i) is an constant ( n x n )-matrix </li></ul><ul><li>is an ( n x 1)-vector of gene-expression levels </li></ul><ul><li>represents the dynamical system of the n genes </li></ul><ul><li>and their interaction alone. </li></ul><ul><li>: (n x n)-matrix with entries as functions of polynomials, exponential, trigonometric, </li></ul><ul><li>(iii) </li></ul>environmental effects n genes , m environmental effects are ( n+m )-vector and ( n+m ) x ( n+m )-matrix, respectively. Weber et al. (2008c), Tastan (2005), Tastan et al. (2006), Ugur et al. (2009), Tastan et al. (2005), Yilmaz (2004), Yilmaz et al. (2005), Weber et al. (2008b), Weber et al. (2009b) The Model Class splines or wavelets containing some parameters to be optimized .
  58. 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. 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. 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. 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. 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. 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. 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) .
  65. 65. Thank you

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