4th International Summer School
Achievements and Applications of Contemporary
Informatics, Mathematics and Physics
Nationa...
Bio-Systems
Bio-Systems
Bio-Systems




              sustainability
Bio-Systems




              sustainability
Outline



• Computational Biology, Medicine, Health Care         and Environment

• Gene-Environment Networks and Eco-Fin...
Comp. Bio. & Med.

prediction of gene patterns based on

                            DNA microarray chip experiments




 ...
Comp. Bio. & Med.

 Ex.:     yeast data



   GENE        time        0      9.5    11.5       13.5     15.5     18.5     ...
Comp. Bio. & Med.


                    Comp. Bio. & Med.
Gene Patterns Modeling & Prediction

          least squares – ML
           statistical learning


  time-contin.
       ...
Gene Patterns Modeling & Prediction



                                           
                                    ...
Gene Patterns Modeling & Prediction



                                     
                                     
   ...
Gene Patterns Model. & Pred.


  For which parameters, i.e., for which set   M   (or: dynamics),
  is stability guaranteed...
Stability Analysis




                     Bk




                                  Yılmaz 2004      Yılmaz, Öktem, W. 20...
Analysis with Polytopes


Theorem   (Brayton, Tong 1979) :


Given a set     M : = {M 0 , ... , M m −1 }   of m distinct c...
Extremal Points




The ‘‘discrete” power of the algorithm is based on using
polyhedra     Bk    and focussing on the extr...
Construction Principle
Stopping Criterion



Theorem 2:
 Let         zi = M kj ′ ui      ( as above, i ∈ {1,2,..., r}).
 Then,

  H {z1,...,zr } ...
Construction Principle



                            " stopping " (at step k0 ),
                                    ∞
...
Stability Analysis


Ex.:


                               a 1           0 1
        M = {M 0 , M1}   M0 = 
         ...
Genetic Network
                                    •
Ex. :                          E = M E,                             ...
Genetic Network



                       0.4 x1
      gene1                            gene2


              0.2 x2      ...
Gene-Environment Networks - Hybrid Systems




E (k + 1) = M s ( k ) E (k ) + Cs ( k )
                                   ...
Gene-Environment Networks




 E (k + 1) = M s ( k ) E (k ) + Cs ( k )


                                 IE (k + 1) = IM ...
Gene-Environment Networks



                                        )
E (k + 1) = M s ( k ) { (k ) + Cs ( k ) { (k ) + Ds...
Gene-Environment Networks




    
    
                  
                           IE (k + 1) = IM k IE (k )
     ...
Gene-Environment Networks


 •                                       )
 E ( t ) = M s ( t ) E ( t ) + C s ( t ) E (t ) + D...
Gene-Environment Networks


 •                                       )
 E ( t ) = M s ( t ) E ( t ) + C s ( t ) E (t ) + D...
Gene-Environment Networks




                                                                 2
                        l...
Gene-Environment Networks
Gene-Environment Networks




                                    1 if gene j regulates gene i
                          ...
Gene-Environment Networks


                mixed integer programming
                                                    ...
Gene-Environment Networks


                     TPS3                          GSY2
      Trehalose                 UDP-Gl...
Gene-Environment Networks

GSIP relaxation
                                                                               ...
General. Semi-Infinite Programming




           C2




                                     I, K, L finite
GSIP – Structural Stability

∃           ∀         ∈               ∃ ψ (⋅), ϕ (⋅,⋅)∈ C 0 :



                            ...
GSIP – Structural Stability

Thm.   (W. 1999/2003, 2006):




                               ⇔
                           ...
GSIP – Structural Stability
Spline Approximation

                            l −1                                         .                          ...
Spline Approximation
                                                                                                     ...
Spline Approximation
                                                                                                     ...
TEM Model
TEM Model




            Example of stability
            Article 2, Kyoto Protocol
Dynamics and Control in CO2-Emission Reduction

      ( +1)                ()     ()
    0
       (k
                   ...
Dynamics and Control in CO2-Emission Reduction

      ( +1)                ()     ()
    0
       (k
                   ...
Gene-Environmental and Financial Dynamics



   d E i( t ) = ai (E ( t ) , t ) dt + bi (E ( t ) , t ) dWi (t )


. (k )
  ...
Gene-Environmental and Financial Dynamics



   d E i( t ) = ai (E ( t ) , t ) dt + bi (E ( t ) , t ) dWi (t )


. (k )
  ...
Regulatory Networks: Errors and Uncertainty



  Errors uncorrelated      Errors correlated       Fuzzy values


   Interv...
Regulatory Networks
        and
Ellipsoidal Calculus
Regulatory Networks ― Ellipsoidal Calculus


Assumption:
• Clustered variables (errors) are correlated
  Interacting group...
Regulatory Networks ― Ellipsoidal Calculus



1. Clustering
   (Groups of genes / groups of environmental items)
2. Assign...
Regulatory Networks ― Ellipsoidal Calculus


1) Clustering
Identify groups (clusters) of jointly acting genetic and enviro...
Regulatory Networks ― Ellipsoidal Calculus


2) Interaction of Genetic Clusters
Regulatory Networks ― Ellipsoidal Calculus


3) Interaction of Environmental Clusters
Regulatory Networks ― Ellipsoidal Calculus


3) Interaction of Genetic & Environmental Clusters




   ⇒   Determine the d...
Regulatory Networks ― Ellipsoidal Calculus


Task:

• Identify and analyze highly data based on ellipsoidalofmeasurement d...
Regulatory Networks ― Ellipsoidal Calculus


Clusters and Ellipsoids:
Genetic clusters:           C1,C2,…,CR
Environmental...
Regulatory Networks ― Ellipsoidal Calculus
Regulatory Networks ― Ellipsoidal Calculus




      r=1
Regulatory Networks ― Ellipsoidal Calculus


The Regression Problem:




                                                 ...
Regulatory Networks ― Ellipsoidal Calculus


Measures for the size of intersection:

• Volume   (→ ellipsoid matrix determ...
Thank you very much for your attention!


    References:

     http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf

...
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Modelling, Dynamics and Development of Gene-Environment and Eco-Finance Networks

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AACIMP 2009 Summer School lecture by Gerhard Wilhelm Weber. "Modern Operational Research and Its Mathematical Methods" course.

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Modelling, Dynamics and Development of Gene-Environment and Eco-Finance Networks

  1. 1. 4th International Summer School Achievements and Applications of Contemporary Informatics, Mathematics and Physics National University of Technology of the Ukraine Kiev, Ukraine, August 5-16, 2009 Modelling, Dynamics and Development of Gene-Environment and Eco-Finance Networks Gerhard-Wilhelm Weber *, Ba şak Akteke-Öztürk Zeynep Alparslan-Gök, Ömür Uğur, Hakan Öktem Institute of Applied Mathematics, METU, Ankara, Turkey Pakize Taylan Dicle University, Diyarbakır, Turkey Erik Kropat University of Erlangen-Nuremberg, Germany Özlem Deferli Department of Mathematics, Cankaya University, Ankara, Turkey * Faculty of Economics, Management Science and Law, University of Siegen, Germany Center for Research on Optimization and Control, University of Aveiro, Portugal
  2. 2. Bio-Systems
  3. 3. Bio-Systems
  4. 4. Bio-Systems sustainability
  5. 5. Bio-Systems sustainability
  6. 6. Outline • Computational Biology, Medicine, Health Care and Environment • Gene-Environment Networks and Eco-Finance Networks • Dynamical Systems • Hybrid and Anticipatory Systems • Stability • Optimization and Control Theory • Regression and Clustering • Financial Mathematics and Risk Management • Regulatory Networks under Uncertainty and Ellipsoidal Calculus • Conclusion
  7. 7. Comp. Bio. & Med. prediction of gene patterns based on DNA microarray chip experiments with M.U. Akhmet, H. Öktem S.W. Pickl, E. Quek Ming Poh T. Ergenç, B. Karasözen J. Gebert, N. Radde Ö. Uğur, R. Wünschiers M. Taştan, A. Tezel, P. Taylan F.B. Yılmaz, B. Akteke-Öztürk S. Özöğür, Z. Alparslan-Gök A. Soyler, B. Soyler, M. Çetin
  8. 8. Comp. Bio. & Med. Ex.: yeast data GENE time 0 9.5 11.5 13.5 15.5 18.5 20.5 'YHR007C' 0.224 0.367 0.312 0.014 -0.003 -1.357 -0.811 'YAL051W' 0.002 0.634 0.31 0.441 0.458 -0.136 0.275 'YAL054C' -1.07 -0.51 -0.22 -0.012 -0.215 1.741 4.239 'YAL056W' 0.09 0.884 0.165 0.199 0.034 0.148 0.935 'PRS316' -0.046 0.635 0.194 0.291 0.271 0.488 0.533 'KAN-MX' 0.162 0.159 0.609 0.481 0.447 1.541 1.449 'E. COLI #10' -0.013 0.88 -0.009 0.144 -0.001 0.14 0.192 'E. COLI #33' -0.405 0.853 -0.259 -0.124 -1.181 0.095 0.027 http://genome-www5.stanford.edu/
  9. 9. Comp. Bio. & Med. Comp. Bio. & Med.
  10. 10. Gene Patterns Modeling & Prediction least squares – ML statistical learning time-contin. Expression data • E = M (E) E + C(E) E ( 0 ) = E0 { environmental effects time-discr. Ex.: Euler, Runge-Kutta E k +1 = M k E k Μk = (emi j ) ∈ M
  11. 11. Gene Patterns Modeling & Prediction     M ( E ), Μ k =             E =      Uğur, W. 2006, W., Taylan, Alparsan-Gök, Özöğür, Akteke-Öztürk 2006
  12. 12. Gene Patterns Modeling & Prediction     M ( E ), Μ k =             E =      Kropat, W. , Tezel, Özöğür-Akyüz 2008
  13. 13. Gene Patterns Model. & Pred. For which parameters, i.e., for which set M (or: dynamics), is stability guaranteed ? Def.: M is stable : ⇔ ∃ B: (complex) bounded neighbourhood of Οn , ∀ k ∈ ΙΝ , M 0, M1 ,..., M k −1 ∈ M : (M k −1 M k − 2 ... M 0 ) Β ⊆ Β .
  14. 14. Stability Analysis Bk Yılmaz 2004 Yılmaz, Öktem, W. 2005 Gebert, Radde, W. 2005 Akhmet, Gebert, Pickl, Öktem, W. 2005 Öktem 2005, Akçay 2005 Uğur, Pickl, Taştan, W. 2005 Bk +1 Weber, Tezel 2006 Uğur, W. 2006 W, Tezel, Taylan, Soyler, Çetin 2007 W, Ugur, Taylan, Tezel 2007 Stability Theorems Uğur, Pickl, W, Wünschiers 2007
  15. 15. Analysis with Polytopes Theorem (Brayton, Tong 1979) : Given a set M : = {M 0 , ... , M m −1 } of m distinct complex matrices. Then, ∞ M is stable ⇔ B* = U k =0 Bk is bounded . Here, B0 is a bounded neighbourhood of 0n , and for k > 0 ∞   Bk := H  U M i B k −1  ,  i=0 k'  where   k ′ := ( k − 1) mod m , H: convex hull .
  16. 16. Extremal Points The ‘‘discrete” power of the algorithm is based on using polyhedra Bk and focussing on the extremal points of the sets Bk . Theorem 1: If z is an extremal point of B k , then there exists a j ∈ ΙΝ 0 and an extremal point u of B k − 1 , in short : u ∈ E ( B k −1 ) , such that z = M k′ u . j
  17. 17. Construction Principle
  18. 18. Stopping Criterion Theorem 2: Let zi = M kj ′ ui ( as above, i ∈ {1,2,..., r}). Then, H {z1,...,zr } = Bk ⇔ M k ′ zi ∈ H {z1,...,zr } ∀ i = 1,2,..., r.
  19. 19. Construction Principle  " stopping " (at step k0 ),  ∞  • Bk = Bk0 (k ≥ k0 ) ⇒  B∗ = U Bi bounded, i =0   stability of matrices / dynamics   B ∗ unbounded • ∂B0 ∩ ∂Bk = 0 / ⇒   instability Gebert, Laetsch, Pickl, We. Wünschiers 2004 Ergenc, We. 2004
  20. 20. Stability Analysis Ex.:  a 1  0 1 M = {M 0 , M1} M0 =   b 0 ,  M1 =   b 0      region of stability algorithm instability
  21. 21. Genetic Network • Ex. : E = M E, h = 1, κ scalar - valued case  E1 (t 0 ) E 2 (t 0 ) E 3 (t 0 ) E 4 (t 0 )   255 250 0 255       E1 (t1 ) E 2 (t1 ) E 3 (t1 ) E 4 (t1 )   255 200 50 0   E (t ) = E 2 (t 2 ) E 3 (t 2 ) E 4 (t 2 )   255 180 70 255   1 2     E (t ) E 2 (t 3 ) E 3 (t 3 ) E 4 (t 3 )   255 0   1 3   170 80  9  0 0 0 0  8 ö5   Expression level, ö 7 ö3  0 .4 − 0 . 61 0 0  M = 6 ö1 0  5 4 ö2 ö4 0 0 .2 − 0 . 39 3   2 ö0  1 − 2 1 0  0 0  0 2 4 6 8 Time, t
  22. 22. Genetic Network 0.4 x1 gene1 gene2 0.2 x2 1 x1 gene3 gene4
  23. 23. Gene-Environment Networks - Hybrid Systems E (k + 1) = M s ( k ) E (k ) + Cs ( k ) s (k ) := FB Q( E (k − 1)) 1 if Ei (k ) > Ωi Qi ( E (k )) :=  0 else Akhmet, Gebert, Pickl, Öktem, W. 2005 Öktem 2005, Akçay 2005 Gebert, Radde, W. 2005 Yılmaz 2004 Yılmaz, Öktem, W. 2005 Uğur, Pickl, Taştan, W. 2005 Weber, Tezel 2006
  24. 24. Gene-Environment Networks E (k + 1) = M s ( k ) E (k ) + Cs ( k ) IE (k + 1) = IM k IE (k ) • IE (t) = IM IE (t) locally • ) E(t ) = Ms(t ) E(t ) + Cs(t ) E(t ) + Ds(t )
  25. 25. Gene-Environment Networks ) E (k + 1) = M s ( k ) { (k ) + Cs ( k ) { (k ) + Ds ( k ) E E IE (k + 1) = IM k IE (k ) • IE (t) = IM IE (t) locally • ) E(t ) = Ms(t ) E(t ) + Cs(t ) E(t ) + Ds(t )
  26. 26. Gene-Environment Networks     IE (k + 1) = IM k IE (k )         • IE (t) = IM IE (t) modules
  27. 27. Gene-Environment Networks • ) E ( t ) = M s ( t ) E ( t ) + C s ( t ) E (t ) + Ds ( t ) where s (t ) := F (Q ( E (t ))) Q( E (t )) = (Q1 ( E (t )),..., Qn ( E (t ))) 0 for Ei (t ) < θ i ,1 1 for θ i ,1 < Ei (t ) < θ i , 2  Qi ( E (t )) :=  ... di for  θ i ,d < Ei (t ) i
  28. 28. Gene-Environment Networks • ) E ( t ) = M s ( t ) E ( t ) + C s ( t ) E (t ) + Ds ( t ) where s (t ) := F (Q ( E (t ))) Q( E (t )) = (Q1 ( E (t )),..., Qn ( E (t ))) parameter estimation: 0 for Ei (t ) < θ i ,1 1 for θ i ,1 < Ei (t ) < θ i , 2  (i) estimation of thresholds Qi ( E (t )) :=  ... di for  θ i ,d < Ei (t ) i (ii) calculation of matrices and vectors describing the system in between thresholds
  29. 29. Gene-Environment Networks 2 l ∗ −1 ) ∗ min ∗ ∗ ∑ α =0 & M Eκα + C E κα + D∗ − Eκα ∗ ∗ (mij ), (cil ), (di ) ∞ Chebychev (maximum) norm
  30. 30. Gene-Environment Networks
  31. 31. Gene-Environment Networks 1 if gene j regulates gene i χi j :=  0 otherwise ξi l , ζ i
  32. 32. Gene-Environment Networks mixed integer programming 2 l ∗ −1 ) min ∑ α =0 ∗ & M Eκα + C E κα + D∗ − Eκα ∗ (mij ∗ ), (cil∗ ), (di ∗ ), ( χ ij ), (ξil ), (ζ i ) ∞ subject to ( j = 1, 2,..., n) n ∑χ i =1 ij ≤αj n ∑ξ il ≤ βl (l = 1, 2,..., m) i =1 n ∑ζ i =1 i ≤γ mii ≥ δ i ,min (i = 1, 2,..., n) & overall box constraints
  33. 33. Gene-Environment Networks TPS3 GSY2 Trehalose UDP-Glucose GLC3 Glycogen UGP1 NTH2 Glucose-1-Phosphate GPH1 PGM1 HXK1 Glucose Glucose-6-Phosphate Glycolysis pathway knockout glycogen metabolism pathway in yeast Saccharomyces cerevisiae
  34. 34. Gene-Environment Networks GSIP relaxation 2 l ∗ −1 ) min ∑ α =0 ∗ & M Eκα + C E κα + D∗ − Eκα ∗ ∞ (mij ∗ ), (cil∗ ), (di ∗ ) subject to n ∑i =1 p ij ( m ij ∗ , y ) ≤ α j ( y ) ( j = 1, ..., n ) n ∑i =1 q il ( c il ∗ , y ) ≤ β l ( y ) ( l = 1, ..., m ) ( y ∈ Y (C ∗ , D∗ )) n ∑ ζ i ( d i∗ , y ) ≤ γ ( y ) set of combined environmental effects : i =1 Y (C ∗ , D∗ ) := m ii ≥ δ i , m in ( i = 1, . . . , n ) ( ∏ i =1,..., n 0, ci∗l  ) × (   ∏ i =1,..., n 0, d i∗  )   & o v e r a ll b o x c o n s t r a in t s l =1,..., m
  35. 35. General. Semi-Infinite Programming C2 I, K, L finite
  36. 36. GSIP – Structural Stability ∃ ∀ ∈ ∃ ψ (⋅), ϕ (⋅,⋅)∈ C 0 : ψ (τ ) τ Jongen, W. ψ ϕ (⋅,τ ) homeom. ⇔ asymptotic : structurally stable effect ε (⋅) IR n global local global
  37. 37. GSIP – Structural Stability Thm. (W. 1999/2003, 2006): ⇔ ξ
  38. 38. GSIP – Structural Stability
  39. 39. Spline Approximation l −1 . 2 ) min PRSS ( M , C, D) := ∑ κ =0 M ( Eκ ) Eκ + C( Eκ ) Eκ + D( Eκ ) − Eκ ∞ + penalty term  n 1,i , j ′′ dE  2 ∫ (f ) n n U 2 penalty term := ∑∑  ∑ λα Eα  1,i , j L α ( Eα ) E j α i=1 i=1  α=1 ∞ Eα Eα    n 2,i ,l Eα 2,i ,l ) ′′2 2  ( ) n m U + ∑∑  ∑ µα ∫ L f α ( Eα ) El dEα  i=1 l=1  α=1 ∞ Eα Eα   n  n ′′ dE  2 ( ) U 2 + ∑  ∑ ςα ∫ L f α ( Eα ) Eα  3,i 3,i α i=1  α=1 ∞ Eα Eα   Tikhonov regularization (ridge regression)
  40. 40. Spline Approximation . 2 l −1 ) min PRSS ( M , C, D) := ∑ κ =0 M ( Eκ ) Eκ + C( Eκ ) Eκ + D( Eκ ) − Eκ ∞ + penalty term  n 1,i , j ′′ dE  2 ∫ (f ) n n U 2 penalty term := ∑∑  ∑ λα Eα  1,i , j L α ( Eα ) E j α i=1 i=1  α=1 ∞ Eα Eα    n 2,i ,l Eα 2,i ,l ) ′′2 2  ( ) m in t n m U 2 + ∑∑  ∑ µα ∫ L f α ( Eα ) El dEα  U (θ ,θ ,θ ) ≤ t i=1 l=1  α=1 ∞ 1 2 3 2 Eα Eα s .t. ∞   n  n ′′ dE  2 ( ) 2 2 V i α j (θ 1 ) U ≤ M 2 + ∑  ∑ ςα ∫ L f α ( Eα ) Eα  3,i 3,i ∞ iα j α i=1  α=1 ∞ Eα Eα 2   W i α l (θ 2 ) ≤ N 2 ∞ iα l 2 Z i α (θ 3 ) ≤ R 2 conic quadratic programming ∞ iα 0 ≤ t interior points methods
  41. 41. Spline Approximation . 2 l −1 ) min PRSS ( M , C, D) := ∑ κ =0 M ( Eκ ) Eκ + C( Eκ ) Eκ + D( Eκ ) − Eκ ∞ + penalty term m in t 2 s .t. U (θ ,θ ,θ ) 1 2 3 ≤ t2 ∞ 2 V i α j (θ 1 ) ≤ M 2 ∞ iα j 2 W i α l (θ 2 ) ≤ N 2 ∞ iα l 2 Z i α (θ 3 ) ≤ R 2 conic quadratic programming ∞ iα 0 ≤ t interior points methods
  42. 42. TEM Model
  43. 43. TEM Model Example of stability Article 2, Kyoto Protocol
  44. 44. Dynamics and Control in CO2-Emission Reduction ( +1) () ()  0 (k      ( k (k   = M ( k )   0   0         ( k +1)     ( k )   ( k )  0    = M (k )       +  (k )        u    IE ( k +1) = IM ( k ) IE ( k )
  45. 45. Dynamics and Control in CO2-Emission Reduction ( +1) () ()  0 (k      ( k (k   = M ( k )   0   0         ( k +1)     ( k )   ( k )  0    = M (k )       +  (k )        u    IE ( k +1) = IM ( k ) IE ( k )
  46. 46. Gene-Environmental and Financial Dynamics d E i( t ) = ai (E ( t ) , t ) dt + bi (E ( t ) , t ) dWi (t ) . (k ) ∆Wi ( k ) 1 ( k )  ( ∆Wi (k ) 2 )  E i ≈ ai (E , t (k ) (k ) ) + bi (E , t (k ) (k ) ) ( k ) + (b'bi )(E , t )  i (k ) (k ) − 1 h 2  h  • Modeling • Testing IE ( k +1) = IM ( k ) IE ( k ) • Prediction • Stability
  47. 47. Gene-Environmental and Financial Dynamics d E i( t ) = ai (E ( t ) , t ) dt + bi (E ( t ) , t ) dWi (t ) . (k ) ∆Wi ( k ) 1 ( k )  ( ∆Wi (k ) 2 )  E i ≈ ai (E , t (k ) (k ) ) + bi (E , t (k ) (k ) ) ( k ) + (b'bi )(E , t )  i (k ) (k ) − 1 h 2  h  • Modeling • Testing IE ( k +1) = IM ( k ) IE ( k ) • Prediction • Stability
  48. 48. Regulatory Networks: Errors and Uncertainty Errors uncorrelated Errors correlated Fuzzy values Interval arithmetics Ellipsoidal calculus Fuzzy arithmetics θ2 θ1
  49. 49. Regulatory Networks and Ellipsoidal Calculus
  50. 50. Regulatory Networks ― Ellipsoidal Calculus Assumption: • Clustered variables (errors) are correlated Interacting groups (clusters) of genetic and environmental variables • How can we model the time-discrete dynamics of the ellipsoidal states of clusters?
  51. 51. Regulatory Networks ― Ellipsoidal Calculus 1. Clustering (Groups of genes / groups of environmental items) 2. Assign ellipsoids (Center = measurement value, configuration matrix = covariances) 3. Regulatory system (Interaction of clusters defined by affine-linear coupling rules) 4. Parameter identification
  52. 52. Regulatory Networks ― Ellipsoidal Calculus 1) Clustering Identify groups (clusters) of jointly acting genetic and environmental variables disjoint overlapping
  53. 53. Regulatory Networks ― Ellipsoidal Calculus 2) Interaction of Genetic Clusters
  54. 54. Regulatory Networks ― Ellipsoidal Calculus 3) Interaction of Environmental Clusters
  55. 55. Regulatory Networks ― Ellipsoidal Calculus 3) Interaction of Genetic & Environmental Clusters ⇒ Determine the degree of connectivity
  56. 56. Regulatory Networks ― Ellipsoidal Calculus Task: • Identify and analyze highly data based on ellipsoidalofmeasurement data. genetic and environmental interconnected systems clusters of • Calculate predictions of the ellipsoidal states. • Assume: Affine-linear coupling rules. ⇒ Ellipsoidal Calculus
  57. 57. Regulatory Networks ― Ellipsoidal Calculus Clusters and Ellipsoids: Genetic clusters: C1,C2,…,CR Environmental clusters: D1,D2,…,DS Genetic ellipsoids: X1,X2,…,XR Xi = E (µi,Σi) Environmental ellipsoids: E1,E2,…,ES, Ej = E (ρj,Πj)
  58. 58. Regulatory Networks ― Ellipsoidal Calculus
  59. 59. Regulatory Networks ― Ellipsoidal Calculus r=1
  60. 60. Regulatory Networks ― Ellipsoidal Calculus The Regression Problem: measurement Maximize (overlap of ellipsoids) T  R ˆ (κ ) R ˆ (κ ) ∩ E ( κ )  ∑ κ =1  ∑ X r ∩ X r + ∑ Er  r =1 (κ ) r =1 r   prediction
  61. 61. Regulatory Networks ― Ellipsoidal Calculus Measures for the size of intersection: • Volume (→ ellipsoid matrix determinant) • Sum of squares of semiaxes (→ trace of configuration matrix) • Length of largest semiaxes (→ eigenvalues of configuration matrix) E (µ r , Π r ) µr
  62. 62. Thank you very much for your attention! References: http://www3.iam.metu.edu.tr/iam/images/7/73/Willi-CV.pdf gweber@metu.edu.tr

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