the sobol decomposition - math.univ-toulouse.fr
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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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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
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