Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

The Sobol decomposition

• Suppose! x = (x1, · · · , xp) ∈ [0, 1]p

! η : [0, 1]p → R a deterministic function

• ANOVA decomposition gives

η(x) = η0 +p∑

i=1

ηi(xi) + · · ·+ η1,··· ,p(x) (1)

• (1) exists and is unique iff∫

[0,1]pηu(xu)ηv(xv)dx = 0 ∀ u, v ⊆ 1, · · · , p, u %= v

with xu = (xu1, · · · , xut) if u = (u1, · · · , ut) ⊆ 1, · · · , p

4/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

The Sobol indexLet

Y = η(X1, · · · ,Xp) = η(X)

Assume that

• X independent, X ∼ U([0, 1]p)

• η ∈ L2R([0, 1]p) i.e.

〈h1, h2〉 =∫[0,1]p h1(x)h2(x)dx

= E(h1(X)h2(X)) ∀ h1, h2 ∈ L2R([0, 1]p)

By ANOVA decomposition

V (Y ) =∑

i

V (ηi) +∑

i<j

V (ηij) + · · · + V (η1,··· ,p)

5/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

The Sobol indexThe Sobol index of a group of variables Xu is

Su =V (ηu)

V (Y )

Moreover

η0 = E(Y )ηi = E(Y |Xi)− η0ηij = E(Y |Xi,Xj)− ηi − ηj − η0· · ·

Su =V [E(Y |Xu)]−

∑v⊂u(−1)|u|−|v|V [E(Y |Xu)]

V (Y )

and ∑

u

Su = 1

6/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

Context

• X = (X1, · · · ,Xp) can be non independent

• Y = η(X)

• Idea : Decompose η(X) into a sum of increasing dimensionfunctions to mimic the construction of Sobol indices

• Define! ν reference measure

! PX distribution of X = (X1, · · · , Xp)

! pX =dPX

dνthe density function of X w.r.t. ν

! η ∈ L2R(Rp,B(Rp), PX) c.a.d.

〈h1, h2〉 = E(h1(X)h2(X)) =

∫h1(x)h2(x)dPX

8/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

Conditions

PX << νwhere

ν(dx) = ν1(dx1)⊗ · · ·⊗ νp(dxp)(C.1)

Main assumption

∃ 0 < M ≤ 1, ∀ u ⊆ 1, .., p, pX ≥ M · pXupXuc(C.2)

with pXu et pXcu

marginal densities of Xu and Xcu = X \Xu

9/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

Generalized functional ANOVA

Define

H0u =

hu(Xu), ‖hu‖2 < ∞, 〈hu, hv〉 = 0,∀v ⊂ u, hv ∈ H0

v

H0∅ = h0 constant

Theorem

Let η ∈ L2R(Rp,B(Rp), PX). Under (C.1) and (C.2), there exists

η0, η1, · · · , η1,··· ,p ∈ H∅ ×H01 × · · ·H0

1,··· ,p such that :

η(X) = η0 +∑

i ηi(Xi) +∑

i<j ηij(Xi,Xj) + · · ·+ η1,··· ,p(X)

Moreover, this decomposition is unique.

10/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

Generalized sensitivity indicesWe have

• η(X) =∑

u ηu(Xu), ηu ∈ H0u

• H0u ⊥ H0

v , ∀ v ⊂ u if |u| > |v|

As a consequence,

V (Y ) =∑

u

[V (ηu) +∑

u∩v %=u,v

Cov(ηu, ηv)]

Sensitivity index for the group Xu is

Su =V (ηu(Xu)) +

∑u∩v "=u,v Cov(ηu(Xu), ηv(Xv))

V (Y )

and ∑

u

Su = 1

12/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

Illustration with p = 2 inputs

η(X) = η0 + η1(X1) + η2(X2) + η12(X1,X2)

Orthogonality relation

η12

η1 η2

η0

S1 =V (η1)+Cov(η1,η2)

V (η) ,

S2 =V (η2)+Cov(η1,η2)

V (η)

13/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

Application with p = 2 inputsAssume X ∼ αN(0, I2) + (1− α)N(0,Ω), with α = 0.2,

Ω =

(0.5 0.40.4 0.5

). L = 50 simulations, n = 1000 observations.

ModelY = X1 +X2 +X1X2

Result

S1 S2 S12∑

u Su

Newindices

Analytical 0.39 0.39 0.22 1

Estimation 0.42 0.41 0.17 1

DVP 1 Estimation 0.64 0.65 0.41 1.7

1. Estimation of Sobol indices with LPE

31/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

Constrained minimization

Under good conditions,

η(X) =∑

u

ηu(Xu), 〈ηu, ηv〉 = 0 ∀ v ⊂ u

Idea : Define the effects (ηu)u as solution of the minimizationproblem

minηu∈L2(Ru)u

∫(η(x) −

∑

u

ηu(xu))2pX(x)dx

under the orthogonality constraints

〈ηu, ηv〉 =∫

ηu(xu)ηv(xv)pX(x)dx, ∀ v ⊂ u, ∀ u ⊆ 1, · · · , p

35/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

Conclusions and perspectives

Conclusions

• Existence and uniqueness of a generalized decomposition ofthe output

• Construction of generalized sensitivity indices! Summed to 1! Include the case of independent inputs

• Copula representation allows for describing the requiredconditions

• Both projection method and minimization underconstraints give satisfying results for simple models

Perpectives

• Improve numerical methods

• Study convergence properties of estimators

43/44 Gaelle Chastaing Sensitivity analysis and dependent variables

Introduction Generalized decomposition Generalized sensitivity indices Examples of distribution Estimation

G. Chastaing, F. Gamboa, and C. Prieur.Generalized hoeffding-sobol decomposition for dependent variables-application to sensitivity analysis.2012.

G. Hooker.Generalized functional anova diagnostics for high-dimensional functions ofdependent 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 multivariatefunction estimation.The Annals of Statistics, 22(1), 1994.

44/44 Gaelle Chastaing Sensitivity analysis and dependent variables

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Lawton, W., Sylvestre, E., Maggio, M., 1972. Self modelling nonlinear regression. Technometrics

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ModelingEstimation

Applications

Framework

Road traffic (Mediamobile) :Activity: Real-time prediction of traveling timeAim: Understand the speed process on the road trafficnetworkObservations :

Fixed sensors: corrupted valuesCars fleet: unobserved areasThe graph is known

Probem: Use the spatial dependency for:Spatial completionSpatio-temporal prediction

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 2 / 31

ModelingEstimation

Applications

Framework

Road traffic (Mediamobile) :Activity: Real-time prediction of traveling timeAim: Understand the speed process on the road trafficnetworkObservations :

Fixed sensors: corrupted valuesCars fleet: unobserved areasThe graph is known

Probem: Use the spatial dependency for:Spatial completionSpatio-temporal prediction

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 2 / 31

ModelingEstimation

Applications

Framework

Road traffic (Mediamobile) :Activity: Real-time prediction of traveling timeAim: Understand the speed process on the road trafficnetworkObservations :

Fixed sensors: corrupted valuesCars fleet: unobserved areasThe graph is known

Probem: Use the spatial dependency for:Spatial completionSpatio-temporal prediction

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 2 / 31

ModelingEstimation

Applications

Framework

Road traffic (Mediamobile) :Activity: Real-time prediction of traveling timeAim: Understand the speed process on the road trafficnetworkObservations :

Fixed sensors: corrupted valuesCars fleet: unobserved areasThe graph is known

Probem: Use the spatial dependency for:Spatial completionSpatio-temporal prediction

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 2 / 31

ModelingEstimation

Applications

Steps

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

ModelingEstimation

Applications

Steps

Modeling : Random process (X (n)i

)n∈Z,i∈G

Indexed by (discrete) time Z and the graph G of the roadtraffic networkGaussianCentered“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

ModelingEstimation

Applications

Steps

Modeling : Random process (X (n)i

)n∈Z,i∈G

Indexed by (discrete) time Z and the graph G of the roadtraffic networkGaussianCentered“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

ModelingEstimation

Applications

Steps

Modeling : Random process (X (n)i

)n∈Z,i∈G

Indexed by (discrete) time Z and the graph G of the roadtraffic networkGaussianCentered“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

ModelingEstimation

Applications

Steps

Modeling : Random process (X (n)i

)n∈Z,i∈G

Indexed by (discrete) time Z and the graph G of the roadtraffic networkGaussianCentered“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

ModelingEstimation

Applications

Steps

Modeling : Random process (X (n)i

)n∈Z,i∈G

Indexed by (discrete) time Z and the graph G of the roadtraffic networkGaussianCentered“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

ModelingEstimation

Applications

Steps

Modeling : Random process (X (n)i

)n∈Z,i∈G

Indexed by (discrete) time Z and the graph G of the roadtraffic networkGaussianCentered“Stationary“Extension of classical tools from time series to graphs

Objective: Yield a parametric model (Kθ)θ∈Θ for covarianceoperators of X

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 3 / 31

ModelingEstimation

Applications

Problem

Speed of vehicles on the road network at a fixed time:zero-mean Gaussian field (Xi)i∈G indexed by the vertices of agraph.

Aim: Chose a model for covariance operators

Modeling constraintsAdaptability to physical modelingCompatibility with classical cases (time series, Zd ,homogeneous tree...)Extension of classical tools from time series (spectralrepresentation, 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

ModelingEstimation

Applications

Problem

Speed of vehicles on the road network at a fixed time:zero-mean Gaussian field (Xi)i∈G indexed by the vertices of agraph.

Aim: Chose a model for covariance operators

Modeling constraintsAdaptability to physical modelingCompatibility with classical cases (time series, Zd ,homogeneous tree...)Extension of classical tools from time series (spectralrepresentation, 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

ModelingEstimation

Applications

Problem

Speed of vehicles on the road network at a fixed time:zero-mean Gaussian field (Xi)i∈G indexed by the vertices of agraph.

Aim: Chose a model for covariance operators

Modeling constraintsAdaptability to physical modelingCompatibility with classical cases (time series, Zd ,homogeneous tree...)Extension of classical tools from time series (spectralrepresentation, 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

ModelingEstimation

Applications

A few bibliography

Spectral representation of stationary processZd : X. GuyonHomogeneous tree: J-P. ArnaudDistance transitive graphs: H. Heyer

Maximum likelihoodZ: R. Azencott and D. Dacunha-CastelleZd : X. Guyon, R. Dahlhaus

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 7 / 31

ModelingEstimation

Applications

A few bibliography

Spectral representation of stationary processZd : X. GuyonHomogeneous tree: J-P. ArnaudDistance transitive graphs: H. Heyer

Maximum likelihoodZ: R. Azencott and D. Dacunha-CastelleZd : X. Guyon, R. Dahlhaus

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 7 / 31

ModelingEstimation

Applications

Graph

Model: Zero-mean Gaussian field (Xi)i∈G indexed by thevertices 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 Wij = 0Degree of a vertex: Di = j , i ∼ j.

Assumption (H0)D := supi∈G Di < +∞, G has bouded degree

∀i ∈ G,

j∈G

Wij

≤ 1 even renormalizing

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 8 / 31

ModelingEstimation

Applications

Graph

Model: Zero-mean Gaussian field (Xi)i∈G indexed by thevertices 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 Wij = 0Degree of a vertex: Di = j , i ∼ j.

Assumption (H0)D := supi∈G Di < +∞, G has bouded degree

∀i ∈ G,

j∈G

Wij

≤ 1 even renormalizing

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 8 / 31

ModelingEstimation

Applications

Graph

Model: Zero-mean Gaussian field (Xi)i∈G indexed by thevertices 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 Wij = 0Degree of a vertex: Di = j , i ∼ j.

Assumption (H0)D := supi∈G Di < +∞, G has bouded degree

∀i ∈ G,

j∈G

Wij

≤ 1 even renormalizing

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 8 / 31

ModelingEstimation

Applications

Graph

Model: Zero-mean Gaussian field (Xi)i∈G indexed by thevertices 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 Wij = 0Degree of a vertex: Di = j , i ∼ j.

Assumption (H0)D := supi∈G Di < +∞, G has bouded degree

∀i ∈ G,

j∈G

Wij

≤ 1 even renormalizing

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 8 / 31

ModelingEstimation

Applications

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

ModelingEstimation

Applications

Modeling for covariance operators

Models for covariance operators (of the speed field)

K(f ) = f (W )

W acts on l2(G) :

∀u ∈ l2(G), ∀i ∈ G, (Wu)i :=

j∈G

Wijuj .

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 10 / 31

ModelingEstimation

Applications

Modeling for covariance operators

Models for covariance operators (of the speed field)

K(f ) = f (W )

Under H0

W is a bounded Hilbertian self-adjoint operator inBG := l2(G) → l2(G):

W2,op≤ 1.

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 10 / 31

ModelingEstimation

Applications

Modeling for covariance operators

Models for covariance operators (of the speed field)

K(f ) = f (W )

Definition (Identity resolution)M σ-algebra E : M → BG 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

ModelingEstimation

Applications

Modeling for covariance operators

Models for covariance operators (of the speed field)

K(f ) = f (W )

Spectral decomposition

∃E ,M,W =

MλdE(λ)

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 10 / 31

ModelingEstimation

Applications

Models for covariance operators, spectral density

Definition (Construction of the covariance operators)Let g be an positive function, analytic over Sp(W ),

K(g) =

Sp(W )g(λ)dE(λ),

g polynomial: MA(W )q

1g

polynomial: AR(W )p · · ·

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

ModelingEstimation

Applications

Models for covariance operators, spectral density

Definition (Construction of the covariance operators)Let g be an positive function, analytic over Sp(W ),

K(g) =

Sp(W )g(λ)dE(λ),

g polynomial: MA(W )q

1g

polynomial: AR(W )p · · ·

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

ModelingEstimation

Applications

Models for covariance operators, spectral density

Definition (Construction of the covariance operators)Let g be an positive function, analytic over Sp(W ),

K(g) =

Sp(W )g(λ)dE(λ),

g polynomial: MA(W )q

1g

polynomial: AR(W )p · · ·

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

ModelingEstimation

Applications

Models for covariance operators, spectral density

Definition (Construction of the covariance operators)Let g be an positive function, analytic over Sp(W ),

K(g) =

Sp(W )g(λ)dE(λ),

g polynomial: MA(W )q

1g

polynomial: AR(W )p · · ·

K(g)ij :=

Sp(W )g(λ)dµij(λ).

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

ModelingEstimation

Applications

Models for covariance operators, spectral density

Definition (Construction of the covariance operators)Let g be an positive function, analytic over Sp(W ),

K(g) =

Sp(W )g(λ)dE(λ),

g polynomial: MA(W )q

1g

polynomial: AR(W )p · · ·

Remarks:

K(g) = g(W )

Dependency in W

Analogy with Z

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 11 / 31

ModelingEstimation

Applications

G = Z: compatibility with time seriesAdjacency operator

Wij =12

11|i−j|=1.

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 12 / 31

ModelingEstimation

Applications

G = Z: compatibility with time seriesAdjacency operator

Wij =12

11|i−j|=1.

Local measure

∀i , j ∈ G, ∀k ∈ Z,

Wk

ij

=1π

[−1,1]λk

T|j−i|(λ)√1 − λ2

dλ.

Tk : k ieme Chebychev polynomials

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 12 / 31

ModelingEstimation

Applications

G = Z: compatibility with time seriesAdjacency operator

Wij =12

11|i−j|=1.

Local measure

∀i , j ∈ G, ∀k ∈ Z,

Wk

ij

=1π

[−1,1]λk

T|j−i|(λ)√1 − λ2

dλ.

Model

(K(g))ij=

1π

[−1,1]g(λ)

T|j−i|(λ)√1 − λ2

dλ.

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 12 / 31

ModelingEstimation

Applications

G = Z: compatibility with time seriesAdjacency operator

Wij =12

11|i−j|=1.

Local measure

∀i , j ∈ G, ∀k ∈ Z,

Wk

ij

=1π

[−1,1]λk

T|j−i|(λ)√1 − λ2

dλ.

Spectral density

f (t) = g(cos(t))

K(g)ij =1

2π

[−π,π]f (t) cos ((j − i)t) dt := (T (f ))ij

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 12 / 31

ModelingEstimation

Applications

The concrete problem

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

ModelingEstimation

Applications

The concrete problem

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

ModelingEstimation

Applications

Ideas

Framework: Parametric model of covariances operators

K(fθ) = fθ(W ).

Aim: Parametric estimationRemark: Spectral density ∼ Asymptotic eigendistribution of thecovariance operatorsComputational issues

log det Term of the log-likelihoodΓ−1 term of the log-likelihood

Other important ideas

Trace measureTappered periodogram

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

ModelingEstimation

Applications

Ideas

Framework: Parametric model of covariances operators

K(fθ) = fθ(W ).

Aim: Parametric estimationRemark: Spectral density ∼ Asymptotic eigendistribution of thecovariance operatorsComputational issues

log det Term of the log-likelihoodΓ−1 term of the log-likelihood

Other important ideas

Trace measureTappered periodogram

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

ModelingEstimation

Applications

Ideas

Framework: Parametric model of covariances operators

K(fθ) = fθ(W ).

Aim: Parametric estimationRemark: Spectral density ∼ Asymptotic eigendistribution of thecovariance operatorsComputational issues

log det Term of the log-likelihoodΓ−1 term of the log-likelihood

Other important ideas

Trace measureTappered periodogram

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

ModelingEstimation

Applications

Ideas

Framework: Parametric model of covariances operators

K(fθ) = fθ(W ).

Aim: Parametric estimationRemark: Spectral density ∼ Asymptotic eigendistribution of thecovariance operatorsComputational issues

log det Term of the log-likelihoodΓ−1 term of the log-likelihood

Other important ideas

Trace measureTappered periodogram

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

ModelingEstimation

Applications

Ideas

Framework: Parametric model of covariances operators

K(fθ) = fθ(W ).

Aim: Parametric estimationRemark: Spectral density ∼ Asymptotic eigendistribution of thecovariance operatorsComputational issues

log det Term of the log-likelihoodΓ−1 term of the log-likelihood

Other important ideas

Trace measureTappered periodogram

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

ModelingEstimation

Applications

Ideas

Framework: Parametric model of covariances operators

K(fθ) = fθ(W ).

Aim: Parametric estimationRemark: Spectral density ∼ Asymptotic eigendistribution of thecovariance operatorsComputational issues

log det Term of the log-likelihoodΓ−1 term of the log-likelihood

Other important ideas

Trace measureTappered periodogram

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

ModelingEstimation

Applications

Ideas

Framework: Parametric model of covariances operators

K(fθ) = fθ(W ).

Aim: Parametric estimationRemark: Spectral density ∼ Asymptotic eigendistribution of thecovariance operatorsComputational issues

log det Term of the log-likelihoodΓ−1 term of the log-likelihood

Other important ideas

Trace measureTappered periodogram

F. Gamboa et al Modeling and estimation for Gaussian fields indexed by graphs, application to road traffic prediction 16 / 31

ModelingEstimation

Applications

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

ModelingEstimation

Applications

Real datasAim: 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

ModelingEstimation

Applications

Merci !

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