The Sobol decomposition

Transcription

1 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution The Sobol decomposition Suppose x =(x1,,x p ) [0, 1] p η : [0, 1] p R adeterministicfunction ANOVA decomposition gives p η(x) =η 0 + η i (x i )+ + η 1,,p (x) (1) i=1 (1) exists and is unique iff [0,1] p η u (x u )η v (x v )dx =0 u, v {1,,p}, u v with x u =(x u1,,x ut ) if u =(u 1,,u t ) {1,,p} /44 Gaelle Chastaing Sensitivity analysis and dependent variables

2 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Let Assume that The Sobol index Y = η(x 1,,X p )=η(x) X independent, X U([0, 1] p ) η L 2 R ([0, 1]p ) i.e. h 1,h 2 = [0,1] p h 1 (x)h 2 (x)dx = E(h 1 (X)h 2 (X)) h 1,h 2 L 2 R ([0, 1]p ) By ANOVA decomposition V (Y )= i V (η i )+ i<j V (η ij )+ + V (η 1,,p ) /44 Gaelle Chastaing Sensitivity analysis and dependent variables

3 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution The Sobol index The Sobol index of a group of variables X u is Moreover S u = V (η u) V (Y ) η 0 = E(Y ) η i = E(Y X i ) η 0 η ij = E(Y X i,x j ) η i η j η 0 and S u = V [E(Y X u)] v u ( 1) u v V [E(Y X u )] V (Y ) S u =1 u 6/44 Gaelle Chastaing Sensitivity analysis and dependent variables

4 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Context X =(X 1,,X p ) can be non independent Y = η(x) Idea :Decompose η(x) into a sum of increasing dimension functions to mimic the construction of Sobol indices Define ν reference measure PX distribution of X =(X 1,,X p ) px = dp X dν the density function of X w.r.t. ν η L 2 R (R p, B(R p ),P X ) c.a.d. h 1,h 2 = E(h 1 (X)h 2 (X)) = h 1 (x)h 2 (x)dp X /44 Gaelle Chastaing Sensitivity analysis and dependent variables

5 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Conditions where P X << ν ν(dx) =ν 1 (dx 1 ) ν p (dx p ) (C.1) Main assumption 0 <M 1, u {1,..,p}, p X M p Xu p Xu c (C.2) with p Xu et p X c u marginal densities of X u and X c u = X \ X u /44 Gaelle Chastaing Sensitivity analysis and dependent variables

6 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Generalized functional ANOVA Define Hu 0 = { h u (X u ), h u 2 <, h u,h v =0, v u, h v Hv 0 } H 0 = {h 0 constant} Theorem Let η L 2 R (Rp, B(R p ),P X ).Under(C.1)and(C.2),thereexists η 0, η 1,, η 1,,p H H1 0 H0 1,,p such that : η(x) =η 0 + i η i(x i )+ i<j η ij(x i,x j )+ + η 1,,p (X) Moreover, this decomposition is unique. 0/44 Gaelle Chastaing Sensitivity analysis and dependent variables

7 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Generalized sensitivity indices We have η(x) = u η u(x u ), η u Hu 0 Hu 0 Hv 0, v u if u > v As a consequence, V (Y )= u [V (η u )+ Sensitivity index for the group X u is u v u,v Cov(η u, η v )] S u = V (η u(x u )) + u v u,v Cov(η u(x u ), η v (X v )) V (Y ) and S u =1 u 12/44 Gaelle Chastaing Sensitivity analysis and dependent variables

8 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Illustration with p =2inputs η(x) =η 0 + η 1 (X 1 )+η 2 (X 2 )+η 12 (X 1,X 2 ) Orthogonality relation η 12 η 1 η 2 S 1 = V (η 1)+Cov(η 1,η 2 ) V (η), S 2 = V (η 2)+Cov(η 1,η 2 ) V (η) η 0 3/44 Gaelle Chastaing Sensitivity analysis and dependent variables

9 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Application with p =2inputs Assume ( X αn(0,i ) 2 )+(1 α)n(0, Ω), withα =0.2, Ω =. L =50simulations, n =1000observations Model Result Y = X 1 + X 2 + X 1 X 2 S 1 S 2 S 12 u S u New indices Analytical DVP of Sobol indices with LPE 31/44 Gaelle Chastaing Sensitivity analysis and dependent variables

10 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Under good conditions, Constrained minimization η(x) = u η u (X u ), η u, η v =0 v u Idea : Define the effects (η u ) u as solution of the minimization problem min (η(x) η u (x u )) 2 p X (x)dx {η u L 2 (R u )} u u under the orthogonality constraints η u, η v = η u (x u )η v (x v )p X (x)dx, v u, u {1,,p} 5/44 Gaelle Chastaing Sensitivity analysis and dependent variables

11 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Conclusions Conclusions and perspectives Existence and uniqueness of a generalized decomposition of the output Construction of generalized sensitivity indices Summed to 1 Include the case of independent inputs Copula representation allows for describing the required conditions Both projection method and minimization under constraints give satisfying results for simple models Perpectives Improve numerical methods Study convergence properties of estimators 43/44 Gaelle Chastaing Sensitivity analysis and dependent variables

12 Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution G. Chastaing, F. Gamboa, and C. Prieur. Generalized hoeffding-sobol decomposition for dependent variables -application to sensitivity analysis G. Hooker. Generalized functional anova diagnostics for high-dimensional functions of dependent variables. Journal of Computational and Graphical Statistics, 16(3),2007. R.B. Nelsen. An introduction to copulas. Springer, New York. I.M. Sobol. Sensitivity estimates for nonlinear mathematical models. Wiley Ed., 1(4),1993. C.J. Stone. The use of polynomial splines and their tensor products in multivariate function estimation. The Annals of Statistics, 22(1), /44 Gaelle Chastaing Sensitivity analysis and dependent variables

13 !! F( x, t) d x Ω R ( 0, T ]! Yij = F( x i, t j ), i = 1,.., N; j = 1,.. J! i F( x, t ), j = 1,! J i j, t IFP Energies nouvelles, Rueil-Malmaison, France!!!! f Yij = α j f ( ti θ j ) + vj + εij, j = 1,.., J, i = 1,.., N 12

14 IFP Energies nouvelles, Rueil-Malmaison, France 13!!! * # = (",!, v )!! Gamboa, F., Loubes, J.-M., Maza, E., Semi-parametric estimation of shifts. 2007, Electronic Journal of Statistics. Lawton, W., Sylvestre, E., Maggio, M., Self modelling nonlinear regression. Technometrics

15 IFP Energies nouvelles, Rueil-Malmaison, France!! " F( X, t) t X N " # =!! vˆ ( ) αˆ ( ) x 0 N ( 0, T] (",!, v) ( X, # N (X N ))! Fˆ ( x, t ) = ˆ( α x ) f ( t ˆ( θ x )) ˆ( x ) t i 0, T 0 i 0 i 0 + v! θˆ ( )! 0 ] ] 14

16 !!!! IFP Energies nouvelles, Rueil-Malmaison, France!!! 15

17 !! IFP Energies nouvelles, Rueil-Malmaison, France!! 16

18 Framework Modeling Road traffic (Mediamobile) : Activity: Real-time prediction of traveling time Aim: Understand the speed process on the road traffic network Observations : Fixed sensors: corrupted values Cars fleet: unobserved areas The graph is known Probem: Use the spatial dependency for: Spatial completion Spatio-temporal prediction F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 2 / 31

19 Framework Modeling Road traffic (Mediamobile) : Activity: Real-time prediction of traveling time Aim: Understand the speed process on the road traffic network Observations : Fixed sensors: corrupted values Cars fleet: unobserved areas The graph is known Probem: Use the spatial dependency for: Spatial completion Spatio-temporal prediction F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 2 / 31

20 Framework Modeling Road traffic (Mediamobile) : Activity: Real-time prediction of traveling time Aim: Understand the speed process on the road traffic network Observations : Fixed sensors: corrupted values Cars fleet: unobserved areas The graph is known Probem: Use the spatial dependency for: Spatial completion Spatio-temporal prediction F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 2 / 31

21 Framework Modeling Road traffic (Mediamobile) : Activity: Real-time prediction of traveling time Aim: Understand the speed process on the road traffic network Observations : Fixed sensors: corrupted values Cars fleet: unobserved areas The graph is known Probem: Use the spatial dependency for: Spatial completion Spatio-temporal prediction F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 2 / 31

22 Steps Modeling Modeling : Random process (X (n) i ) n Z,i G F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 3 / 31

23 Steps Modeling Modeling : Random process (X (n) i ) n Z,i G Indexed by (discrete) time Z and the graph G of the road traffic network Gaussian Centered Stationary Extension of classical tools from time series to graphs F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 3 / 31

24 Steps Modeling Modeling : Random process (X (n) i ) n Z,i G Indexed by (discrete) time Z and the graph G of the road traffic network Gaussian Centered Stationary Extension of classical tools from time series to graphs F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 3 / 31

25 Steps Modeling Modeling : Random process (X (n) i ) n Z,i G Indexed by (discrete) time Z and the graph G of the road traffic network Gaussian Centered Stationary Extension of classical tools from time series to graphs F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 3 / 31

26 Steps Modeling Modeling : Random process (X (n) i ) n Z,i G Indexed by (discrete) time Z and the graph G of the road traffic network Gaussian Centered Stationary Extension of classical tools from time series to graphs F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 3 / 31

27 Steps Modeling Modeling : Random process (X (n) i ) n Z,i G Indexed by (discrete) time Z and the graph G of the road traffic network Gaussian Centered Stationary Extension of classical tools from time series to graphs F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 3 / 31

28 Steps Modeling Modeling : Random process (X (n) i ) n Z,i G Indexed by (discrete) time Z and the graph G of the road traffic network Gaussian Centered Stationary Extension of classical tools from time series to graphs Objective: Yield a parametric model (K θ ) θ Θ for covariance operators of X F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 3 / 31

29 Problem Modeling Speed of vehicles on the road network at a fixed time: zero-mean Gaussian field (X i ) i G indexed by the vertices of a graph. Aim: Chose a model for covariance operators Modeling constraints Adaptability to physical modeling Compatibility with classical cases (time series, Z d, homogeneous tree...) Extension of classical tools from time series (spectral representation, Whittle s estimation...) Define covariance operators from a spectral construction F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 6 / 31

30 Problem Modeling Speed of vehicles on the road network at a fixed time: zero-mean Gaussian field (X i ) i G indexed by the vertices of a graph. Aim: Chose a model for covariance operators Modeling constraints Adaptability to physical modeling Compatibility with classical cases (time series, Z d, homogeneous tree...) Extension of classical tools from time series (spectral representation, Whittle s estimation...) Define covariance operators from a spectral construction F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 6 / 31

31 Problem Modeling Speed of vehicles on the road network at a fixed time: zero-mean Gaussian field (X i ) i G indexed by the vertices of a graph. Aim: Chose a model for covariance operators Modeling constraints Adaptability to physical modeling Compatibility with classical cases (time series, Z d, homogeneous tree...) Extension of classical tools from time series (spectral representation, Whittle s estimation...) Define covariance operators from a spectral construction F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 6 / 31

32 A few bibliography Modeling Spectral representation of stationary process Z d : X. Guyon Homogeneous tree: J-P. Arnaud Distance transitive graphs: H. Heyer Maximum likelihood Z: R. Azencott and D. Dacunha-Castelle Z d : X. Guyon, R. Dahlhaus F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 7 / 31

33 A few bibliography Modeling Spectral representation of stationary process Z d : X. Guyon Homogeneous tree: J-P. Arnaud Distance transitive graphs: H. Heyer Maximum likelihood Z: R. Azencott and D. Dacunha-Castelle Z d : X. Guyon, R. Dahlhaus F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 7 / 31

34 Graph Modeling Model: Zero-mean Gaussian field (X i ) i G indexed by the vertices G of a graph G. Definition (Unoriented weigthed graph) G =(G, W ) : G set of vertices (countable) W [ 1, 1] G G Weighted adjacency operator (symmetric) Neighbors: i j if W ij = 0 Degree of a vertex: D i = {j, i j}. Assumption (H 0 ) D := sup i G D i < +, G has bouded degree i G, j G W ij 1 even renormalizing F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 8 / 31

35 Graph Modeling Model: Zero-mean Gaussian field (X i ) i G indexed by the vertices G of a graph G. Definition (Unoriented weigthed graph) G =(G, W ) : G set of vertices (countable) W [ 1, 1] G G Weighted adjacency operator (symmetric) Neighbors: i j if W ij = 0 Degree of a vertex: D i = {j, i j}. Assumption (H 0 ) D := sup i G D i < +, G has bouded degree i G, j G W ij 1 even renormalizing F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 8 / 31

36 Graph Modeling Model: Zero-mean Gaussian field (X i ) i G indexed by the vertices G of a graph G. Definition (Unoriented weigthed graph) G =(G, W ) : G set of vertices (countable) W [ 1, 1] G G Weighted adjacency operator (symmetric) Neighbors: i j if W ij = 0 Degree of a vertex: D i = {j, i j}. Assumption (H 0 ) D := sup i G D i < +, G has bouded degree i G, j G W ij 1 even renormalizing F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 8 / 31

37 Graph Modeling Model: Zero-mean Gaussian field (X i ) i G indexed by the vertices G of a graph G. Definition (Unoriented weigthed graph) G =(G, W ) : G set of vertices (countable) W [ 1, 1] G G Weighted adjacency operator (symmetric) Neighbors: i j if W ij = 0 Degree of a vertex: D i = {j, i j}. Assumption (H 0 ) D := sup i G D i < +, G has bouded degree i G, j G W ij 1 even renormalizing F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 8 / 31

38 Modeling Modeling for covariance operators Models for covariance operators (of the speed field) K(f )=f (W ) F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 10 / 31

39 Modeling Modeling for covariance operators Models for covariance operators (of the speed field) K(f )=f(w) W acts on l 2 (G) : u l 2 (G), i G, (Wu) i := W ij u j. j G F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 10 / 31

40 Modeling Modeling for covariance operators Models for covariance operators (of the speed field) K(f )=f (W ) Under H 0 W is a bounded Hilbertian self-adjoint operator in B G := l 2 (G) l 2 (G): W 2,op 1. F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 10 / 31

41 Modeling Modeling for covariance operators Models for covariance operators (of the speed field) K(f )=f (W ) Definition (Identity resolution) M σ-algebra E : M B G such that ω, ω M, 1 E(ω)self-adjoints projectors. 2 E(ø) = 0, E(Ω) =I 3 E(ω ω )=E(ω)E(ω ) 4 Si ω ω =ø, alors E(ω ω )=E(ω)+E(ω ) F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 10 / 31

42 Modeling Modeling for covariance operators Models for covariance operators (of the speed field) K(f )=f(w) Spectral decomposition E, M, W = M λde(λ) F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 10 / 31

43 Modeling Models for covariance operators, spectral density Definition (Construction of the covariance operators) Let g be an positive function, analytic over Sp(W ), K(g) = g(λ)de(λ), g polynomial: MA (W ) q 1 g polynomial: AR(W ) p Sp(W ) F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

44 Modeling Models for covariance operators, spectral density Definition (Construction of the covariance operators) Let g be an positive function, analytic over Sp(W ), K(g) = g(λ)de(λ), g polynomial: MA (W ) q 1 g polynomial: AR(W ) p Sp(W ) F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

45 Modeling Models for covariance operators, spectral density Definition (Construction of the covariance operators) Let g be an positive function, analytic over Sp(W ), K(g) = g(λ)de(λ), g polynomial: MA (W ) q 1 g polynomial: AR(W ) p Sp(W ) F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

46 Modeling Models for covariance operators, spectral density Definition (Construction of the covariance operators) Let g be an positive function, analytic over Sp(W ), K(g) = g(λ)de(λ), g polynomial: MA (W ) q 1 g polynomial: AR(W ) p K(g) ij := Sp(W ) Sp(W ) g(λ)dµ ij (λ). F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

47 Modeling Models for covariance operators, spectral density Definition (Construction of the covariance operators) Let g be an positive function, analytic over Sp(W ), K(g) = g(λ)de(λ), g polynomial: MA (W ) q 1 g Remarks: polynomial: AR(W ) p K(g) =g(w ) Dependency in W Analogy with Z Sp(W ) F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

48 Modeling G = Z: compatibility with time series Adjacency operator W ij = i j =1. F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 12 / 31

49 Modeling G = Z: compatibility with time series Adjacency operator Local measure W ij = i j =1. i, j G, k Z, W k = 1 ij π T k : k ième Chebychev polynomials [ 1,1] λ k T j i (λ) 1 λ 2 dλ. F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 12 / 31

50 Modeling G = Z: compatibility with time series Adjacency operator Local measure W ij = i j =1. i, j G, k Z, W k = 1 ij π [ 1,1] λ k T j i (λ) 1 λ 2 dλ. Model (K(g)) ij = 1 π [ 1,1] g(λ) T j i (λ) 1 λ 2 dλ. F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 12 / 31

51 Modeling G = Z: compatibility with time series Adjacency operator Local measure W ij = i j =1. i, j G, k Z, W k = 1 ij π [ 1,1] λ k T j i (λ) 1 λ 2 dλ. Spectral density f (t) =g(cos(t)) K(g) ij = 1 f (t) cos ((j i)t) dt := (T (f )) ij 2π [ π,π] F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 12 / 31

52 Modeling The concrete problem F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 14 / 31

53 Modeling The concrete problem F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 15 / 31

54 Ideas Modeling Framework: Parametric model of covariances operators K(f θ )=f θ (W ). Aim: Parametric estimation Remark: Spectral density Asymptotic eigendistribution of the covariance operators Computational issues log det Term of the log-likelihood Γ 1 term of the log-likelihood Other important ideas Trace measure Tappered periodogram F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

55 Ideas Modeling Framework: Parametric model of covariances operators K(f θ )=f θ (W ). Aim: Parametric estimation Remark: Spectral density Asymptotic eigendistribution of the covariance operators Computational issues log det Term of the log-likelihood Γ 1 term of the log-likelihood Other important ideas Trace measure Tappered periodogram F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

56 Ideas Modeling Framework: Parametric model of covariances operators K(f θ )=f θ (W ). Aim: Parametric estimation Remark: Spectral density Asymptotic eigendistribution of the covariance operators Computational issues log det Term of the log-likelihood Γ 1 term of the log-likelihood Other important ideas Trace measure Tappered periodogram F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

57 Ideas Modeling Framework: Parametric model of covariances operators K(f θ )=f θ (W ). Aim: Parametric estimation Remark: Spectral density Asymptotic eigendistribution of the covariance operators Computational issues log det Term of the log-likelihood Γ 1 term of the log-likelihood Other important ideas Trace measure Tappered periodogram F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

58 Ideas Modeling Framework: Parametric model of covariances operators K(f θ )=f θ (W ). Aim: Parametric estimation Remark: Spectral density Asymptotic eigendistribution of the covariance operators Computational issues log det Term of the log-likelihood Γ 1 term of the log-likelihood Other important ideas Trace measure Tappered periodogram F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

59 Ideas Modeling Framework: Parametric model of covariances operators K(f θ )=f θ (W ). Aim: Parametric estimation Remark: Spectral density Asymptotic eigendistribution of the covariance operators Computational issues log det Term of the log-likelihood Γ 1 term of the log-likelihood Other important ideas Trace measure Tappered periodogram F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

60 Ideas Modeling Framework: Parametric model of covariances operators K(f θ )=f θ (W ). Aim: Parametric estimation Remark: Spectral density Asymptotic eigendistribution of the covariance operators Computational issues log det Term of the log-likelihood Γ 1 term of the log-likelihood Other important ideas Trace measure Tappered periodogram F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

61 Modeling Spectrum of the road network F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 26 / 31

62 Real datas Modeling Aim: Predict missing values on FRC 0 in Toulouse F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 27 / 31

63 Modeling Merci! F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 31 / 31