Introduction Motivation Objectives Model description Estimation Moments method MCEM algorithm E-step M step Simulations Variational methods VEM algorithm VBEM algorithm Simulations Discussion Algorithmes MCEM VEM et VBEM pour l’estimation d’un processus de Cox log-Gaussien C´ eline Delmas - Julia Radoszycki - Nathalie Peyrard - R´ egis Sabbadin Unit´ e de Math´ ematiques et Informatique Appliqu´ ees, INRA, Toulouse
32
Embed
Algorithmes MCEM VEM et VBEM pour l'estimation d'un …genome.jouy.inra.fr/applibugs/applibugs.14_06_11.celine... · 2014-06-12 · Introduction Motivation Objectives Model description
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
Algorithmes MCEM VEM et VBEM pourl’estimation d’un processus de Cox log-Gaussien
Celine Delmas - Julia Radoszycki - Nathalie Peyrard - RegisSabbadin
Unite de Mathematiques et Informatique Appliquees, INRA, Toulouse
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
Motivation
Modeling of a spatial phenomenon when data are sampledcounts
• A regular grid of quadrats A1, · · ·AN on a D ⊂ R2
• We observe Yi the count in quadrati ∈ O ⊂ V = {1, · · · , N}
3 11
10 20 5 15
1 12 9
50 30
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
Motivation
Such type of data (counts associated with spatial pointprocesses) are encountered in various fields of applications :
• forestry (counts of trees of a given species)
• ecology (sightings of wild animals)
• epidemiology (disease mapping based on reported infectioncases)
• environmental sciences (radioactivity counts)
• agronomy (counts of weeds)
• etc ...
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
Objectives
The log-Gaussian Cox process is often used for modeling thistype of data.
1. We derive the parameters estimations by a momentsmethod
2. We propose a MCEM algorithm
3. We present a preliminary comparison
4. We propose a VEM and a VBEM algorithm
5. We present some simulation results
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
Poisson process
We consider a spatial Poisson process in R2 with intensityλ = {λ(x), x ∈ D}.• Yi ∼ P(Λi) with Λi =
∫Aiλ(x)dx
• Non stochastic Λi ⇒ Yi ⊥ Yj for i 6= j ⇒ No statisticalcorrelation between Yi and Yj for i 6= j
• In practice• there may exist a stochastic dependence between the
numbers of points observed in non-overlapping domains• the intensity of the point process is often uncertain in
areas without data
⇒ More convenient and more parsimonious to use a stochasticmodeling of this intensity
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
Cox process
We consider a spatial Poisson process in R2 with stochasticintensity λ = {λ(x), x ∈ D}.• λ(x) = exp(β) exp(S(x))
• S(·) is a Gaussian random field centered with variance σ2
and exponential covariance function
Cov(S(x1), S(x2)) = σ2 exp(−α||x1 − x2||)
• Yi|Λi ∼ P(Λi) with Λi =∫Aiλ(x)dx approximated by
Λi = |Ai| exp(β) exp(Si) where Si is the value of S at thecenter of Ai.
• Yi|Λi ⊥ Yj |Λj for i 6= j.
• 3 parameters in the model θ = (β, σ, α), β ∈ R,α, σ ∈ R?+.
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
Moments method
By straightforward calculations we obtain :
E(Yi) = |Ai| exp(σ2/2) exp(β) (1)
Var(Yi) = |Ai| exp(σ2/2) exp(β)
+ |Ai|2 exp(σ2) exp(2β)(exp(σ2)− 1) (2)
E(YiYj) = |A|2 exp(2β) exp(σ2(1 + e−αr)) (3)
where r = |i− j|
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
β and σ2 estimations
By equations (1) and (2) :
σ2 = ln
[Var(Y )− E(Y ) + E(Y )2
E(Y )2
](4)
β = ln
E(Y )
|A|√
Var(Y )−E(Y )+E(Y )2
E(Y )2
(5)
Yi, ∀i, are identically distributed. By the weak law of large
number, E(Yi) and Var(Yi) are estimated by Y =1
#O∑i∈O
Yi
and V (Y ) =1
#O∑i∈O
(Yi − Y )2
⇒ β and σ2 are obtained by replacing E(Yi) and Var(Yi) by Yand V (Y ) in (4) and (5).
Introduction
Motivation
Objectives
Modeldescription
Estimation
Momentsmethod
MCEMalgorithm
E-step
M step
Simulations
Variationalmethods
VEM algorithm
VBEMalgorithm
Simulations
Discussion
α estimation
Using equation (3) we obtain :
α = −1
rln
[1
σ2ln
[E(YiYj)
|Ai|2 exp(2β)
]− 1
]E(YiYj) is estimated by using the variogram estimation.