Conditional simulations of max-stable processes C. Dombry , . - PowerPoint PPT Presentation

Conditional simulations of max-stable processes C. Dombry † , . Éyi-Minko † , M. Ribatet ‡ F † Laboratoire de Mathématiques et Application, Université de Poitiers ‡ Institut de Mathématiques et de Modélisation, Université Montpellier 2 Conditional simulations of max-stable processes Mathieu Ribatet – 1 / 24

� Goal Max-stable processes are widely used for modelling spatial extremes since � they arise as the only possible (non degenerate) limit of pointwise maxima over independent replicates, i.e., X i ( x ) − b n ( x ) x ∈ X ⊂ R d , max → Z ( x ), n → ∞ , − a n ( x ) i = 1,..., n for some normalizing functions a n > 0 and b n and where X i are independent copies of a stochastic process X . Conditional simulations of max-stable processes Mathieu Ribatet – 2 / 24

Goal Max-stable processes are widely used for modelling spatial extremes since � they arise as the only possible (non degenerate) limit of pointwise maxima over independent replicates, i.e., X i ( x ) − b n ( x ) x ∈ X ⊂ R d , max → Z ( x ), n → ∞ , − a n ( x ) i = 1,..., n for some normalizing functions a n > 0 and b n and where X i are independent copies of a stochastic process X . � Can we get a procedure for conditional simulations of max-stable processes (with continuous spectral measure)? Conditional simulations of max-stable processes Mathieu Ribatet – 2 / 24

Setup Recall that any max-stable process has the spectral characterization � Z ( · ) = max i ≥ 1 ζ i Y i ( · ), where Y i ( · ) are independent copies of a non negative stochastic process such – that E [ Y ( x )] = 1 for all x ∈ X ; { ζ i } i ≥ 1 are the points of a Poisson process on (0, ∞ ) with intensity – d Λ ( ζ ) = ζ − 2 d ζ . � Given a study region X ⊂ R d , we want to sample from Z ( · ) | { Z ( x 1 ) = z 1 ,..., Z ( x k ) = z k }, for some z 1 ,..., z k > 0 and k conditioning locations x 1 ,..., x k ∈ X . Conditional simulations of max-stable processes Mathieu Ribatet – 3 / 24

1. Conditional ⊲ distributions Random partitions Sampling scheme Examples 2. MCMC sampler 3. Application 1. Conditional distributions of max-stable processes Conditional simulations of max-stable processes Mathieu Ribatet – 4 / 24

� Decomposition of Φ Let Φ a point process on (0, ∞ ) k whose atoms are � 1. Conditional distributions ϕ i ( x ) = ζ i Y i ( x ), x = ( x 1 ,..., x k ). Random partitions Sampling scheme Examples 2. MCMC sampler Consider the two following point processes � 3. Application Φ − = � � ϕ ∈ Φ : ϕ ( x i ) < z i , for all i ∈ {1,..., k } , (sub-extremal functions) Φ + = � � ϕ ∈ Φ : ϕ ( x i ) = z i , for some i ∈ {1,..., k } . (extremal functions) Clearly Φ = Φ − ∪ Φ + and � Φ + = { ϕ + 1 ,..., ϕ + k } = { ϕ + 1 ,..., ϕ + ℓ }, a.s. (1 ≤ ℓ ≤ k ). Conditional simulations of max-stable processes Mathieu Ribatet – 5 / 24

Decomposition of Φ Let Φ a point process on (0, ∞ ) k whose atoms are � 1. Conditional distributions ϕ i ( x ) = ζ i Y i ( x ), x = ( x 1 ,..., x k ). Random partitions Sampling scheme Examples 2. MCMC sampler Consider the two following point processes � 3. Application Φ − = � � ϕ ∈ Φ : ϕ ( x i ) < z i , for all i ∈ {1,..., k } , (sub-extremal functions) Φ + = � � ϕ ∈ Φ : ϕ ( x i ) = z i , for some i ∈ {1,..., k } . (extremal functions) Clearly Φ = Φ − ∪ Φ + and � Φ + = { ϕ + 1 ,..., ϕ + k } = { ϕ + 1 ,..., ϕ + ℓ }, a.s. (1 ≤ ℓ ≤ k ). � Key point #1: Conditionally on Z ( x ) = z , Φ − and Φ + are inde- pendent. Conditional simulations of max-stable processes Mathieu Ribatet – 5 / 24

� Conditional intensity function Z ( x ) = max i ≥ 1 ζ i Y i ( x ) = max i ≥ 1 ϕ i ( x ) 1. Conditional distributions Random partitions Sampling scheme The Poisson point process { ϕ i ( x )} i ≥ 1 has intensity measure � Examples � ∞ 2. MCMC sampler Pr{ ζ Y ( x ) ∈ A } ζ − 2 d ζ , Borel set A ⊂ R k . Λ x ( A ) = 3. Application 0 We assume that Φ is regular, i.e., Λ x (d z ) = λ x ( z )d z , for all � x ∈ X k . Conditional simulations of max-stable processes Mathieu Ribatet – 6 / 24

Conditional intensity function Z ( x ) = max i ≥ 1 ζ i Y i ( x ) = max i ≥ 1 ϕ i ( x ) 1. Conditional distributions Random partitions Sampling scheme The Poisson point process { ϕ i ( x )} i ≥ 1 has intensity measure � Examples � ∞ 2. MCMC sampler Pr{ ζ Y ( x ) ∈ A } ζ − 2 d ζ , Borel set A ⊂ R k . Λ x ( A ) = 3. Application 0 We assume that Φ is regular, i.e., Λ x (d z ) = λ x ( z )d z , for all � x ∈ X k . � Key point #2: The conditional intensity function λ s | x , z ( u ) = λ ( s , x ) ( u , z ) λ x ( z ) is the (regular) conditional distribution of Z ( x )—if we integrate w.r.t. all possible partitions of x . But not that of Z ( · )!!! Conditional simulations of max-stable processes Mathieu Ribatet – 6 / 24

Random partitions? 1. Conditional distributions 2.5 Random ⊲ partitions 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 7 / 24

Random partitions? 1. Conditional ϕ 1 + distributions 2.5 Random ⊲ partitions 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 7 / 24

Random partitions? 1. Conditional ϕ 1 + distributions 2.5 Random ϕ 2 + ⊲ partitions 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 7 / 24

Random partitions? 1. Conditional ϕ 1 + distributions 2.5 Random ϕ 2 + ⊲ partitions ϕ 3 + 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 7 / 24

Random partitions? 1. Conditional ϕ 1 + distributions 2.5 Random ϕ 2 + ⊲ partitions ϕ 3 + 2.0 Sampling scheme ϕ 4 + Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 7 / 24

Random partitions? 1. Conditional ϕ 1 + distributions 2.5 Random ϕ 2 + ⊲ partitions ϕ 3 + 2.0 Sampling scheme ϕ 4 + Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Here the set { x 1 ,..., x 5 } is partitioned into ({ x 1 , x 3 },{ x 2 },{ x 4 },{ x 5 }) Conditional simulations of max-stable processes Mathieu Ribatet – 7 / 24

Random partitions? 1. Conditional ϕ 1 + distributions 2.5 Random ϕ 2 + ⊲ partitions ϕ 3 + 2.0 Sampling scheme ϕ 4 + Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Here the set { x 1 ,..., x 5 } is partitioned into ({ x 1 , x 3 },{ x 2 },{ x 4 },{ x 5 }) The hitting bounds { z i } i = 1,..., k might be reached by several � extremal functions, i.e., Φ + = { ϕ + 1 ,..., ϕ + k } = { ϕ + 1 ,..., ϕ + ℓ } a.s., 1 ≤ ℓ ≤ k . So we need to take into account all possible ways these hitting � bounds are reached: the hitting scenarios Conditional simulations of max-stable processes Mathieu Ribatet – 7 / 24

Why should we bother about Φ − ? 1. Conditional distributions 2.5 Random ⊲ partitions 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 8 / 24

Why should we bother about Φ − ? 1. Conditional max Φ + distributions 2.5 Random ⊲ partitions 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 8 / 24

Why should we bother about Φ − ? 1. Conditional max Φ + distributions 2.5 Random max Φ − ⊲ partitions 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 8 / 24

Why should we bother about Φ − ? 1. Conditional max Φ + distributions 2.5 Random max Φ − ⊲ partitions 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x Conditional simulations of max-stable processes Mathieu Ribatet – 8 / 24

Why should we bother about Φ − ? 1. Conditional max Φ + distributions 2.5 Random max Φ − ⊲ partitions 2.0 Sampling scheme Examples 1.5 Z ( x ) 2. MCMC sampler 1.0 3. Application 0.5 0.0 x 5 x 3 x 2 x 1 x 4 x The atoms of Φ + are only of interest if we restrict our attention � to the conditioning points x ; But most often one would like to get realizations at s �= x . � � The atoms of Φ − are needed since it is likely that max Φ − ( s ) > max Φ + ( s )! Conditional simulations of max-stable processes Mathieu Ribatet – 8 / 24