Soutenance d Habilitation à Diriger des Recherches. Théorie spatiale des extrêmes et propriétés des processus max-stables

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1 Soutenance d Habilitation à Diriger des Recherches Théorie spatiale des extrêmes et propriétés des processus max-stables CLÉMENT DOMBRY Laboratoire de Mathématiques et Applications, Université de Poitiers. Poitiers, le 8 Novembre 2012 C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 1 / 39

2 Overview of the research interests Structure of the talk 1 Overview of the research interests 2 Conditional distribution of max-i.d. random fields 3 Strong mixing properties of max-i.d. processes 4 Intermediate regime for aggregated ON/OFF sources 5 A stochastic gradient algorithm for the p-means C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 2 / 39

3 Overview of the research interests Theme 1 : Stochastic models in biology, geometry... PhD dissertation : "Applications of large deviation theory". Supervision by C.Mazza and N.Guillotin-Plantard. Asymptotics of stochastic models from biology and informatics. Interest in discrete stochastic models arising from various domains : Biology and population dynamics : A stochastic model for DNA denaturation. Asymptotic study of a mutation/selection genetic algorithm. Phenotypic diversity and population growth in a fluctuating environment (with V.Bansaye and C.Mazza). Statistical Physics : The Curie-Weiss model with quasiperiodic external random field (with N.Guillotin). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 3 / 39

4 Overview of the research interests Theme 1 : Stochastic models in biology, geometry... Informatic and stochastic algorithms : Data structures with dynamical random transitions (with R.Schott and N.Guillotin). The stochastic k-server problem on the circle (with E.Upfal and N.Guillotin). Geometry : Betti numbers of random polygon surfaces (with C.Mazza). Stochastic gradient algorithm for the p-mean on a manifold (with M.Arnaudon, A.Phan and L.Yang). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 4 / 39

5 Overview of the research interests Theme 2 : Limit theorems, heavy tails and LRD Interest in (functional) limit theorems for stochastic models in presence of heavy tails and/or long range dependence (LRD). Asymptotic theory is often very rich with different regimes and nice limit processes such as : - fractional Brownian motion, - self-similar non-gaussian stable processes, - fractional Poisson process, - telecom process... Some motivations from telecommunication models (Internet traffic, transmission network...) C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 5 / 39

6 Overview of the research interests Theme 2 : Limit theorems, heavy tails and LRD Particular models investigated : Random walks in random sceneries and random reward schemas (with N.Guillotin and S.Cohen). Weighted random ball models (with J.-C. Breton). Aggregation of sources based on ON/OFF or renewal processes (with I.Kaj). Le Page series in Skohorod space (with Y.Davydov). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 6 / 39

7 Overview of the research interests Theme 3 : Spatial EVT and max-stable processes Interest in spatial extreme value theory (EVT) and max-stable random fields. Deep connections with the previous theme via : - the theory of regular variation, - a parallel between sum-stable and max-stable processes, - importance of limit theorems. Some motivations from models in environmental science (heat waves, flood, storm...) Supervision of the PhD thesis of Frédéric Eyi-Minko. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 7 / 39

8 Overview of the research interests Theme 3 : Spatial EVT and max-stable processes Results obtained : Properties and asymptotics of extremal shot noises. A point process approach for the maxima of i.i.d. random fields (with F.Eyi-Minko). Conditional distribution of max-i.d. random fields (with F.Eyi-Minko). Conditional simulation of max-stable processes (with M.Ribatet and F.Eyi-Minko). Strong mixing properties of continuous max-i.d. random fields (with F.Eyi-Minko). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 8 / 39

9 Conditional distribution of max-i.d. random fields Structure of the talk 1 Overview of the research interests 2 Conditional distribution of max-i.d. random fields 3 Strong mixing properties of max-i.d. processes 4 Intermediate regime for aggregated ON/OFF sources 5 A stochastic gradient algorithm for the p-means C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 9 / 39

10 Conditional distribution of max-i.d. random fields Motivations Needs for modeling extremes in environmental sciences : - maximal temperatures in a heat wave, - intensity of winds during a storm, - water heights in a flood... Spatial extreme value theory - geostatistics of extremes : Schlather ( 02), Models for stationary max-stable random fields. de Haan & Pereira ( 06), Spatial extremes : Models for the stationary case. Davison, Ribatet & Padoan ( 11), Statistical modelling of spatial extremes. Max-stable random fields play a crucial role as possible limits of normalized pointwise maxima of i.i.d. random fields. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 10 / 39

11 Conditional distribution of max-i.d. random fields Motivations Observations of a max-stable process η at some stations only : η(s i ) = y i, i = 1,..., k. (O) How to predict what happens at other locations? We are naturally lead to consider the conditional distribution of η given the observations (O). Different goals : - theoretical formulas for the conditional distribution, - sample from the conditional distribution, - compute (numerically) the conditional median or quantiles... Results for spectrally discrete max-stable processes : Wang & Stoev ( 11), Conditional sampling of spectrally discrete max-stable processes. New results for max-stable and even max-i.d. processes. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 11 / 39

12 Conditional distribution of max-i.d. random fields Structure of max-i.d. processes Theorem (de Haan 84, Giné Hahn & Vatan 90) For any continuous, max-i.d. random process η = (η(t)) t T satisfying ess inf η(t) 0, there exists a unique Borel measure µ on C 0 = C(T, [0, + )) \ {0} such that ( ) η(t) t T L = ( φ Φ ) φ(t), t T with Φ PPP(µ). FIGURE: A realization of Φ and η = max(φ) for Smith 1D storm process. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 12 / 39

13 Conditional distribution of max-i.d. random fields Hitting scenario and extremal functions Observations {η(s i ) = y i, 1 i k} with η(s) = φ Φ φ(s). Assume that the law of η(s i ) has no atom, 1 i k. Then, with probability 1,! φ i Φ, φ i (s i ) = η(s i ). Definition of the following random objects : the hitting scenario Θ, a partition of S = {s 1,..., s k } with l blocks, the extremal functions ϕ + 1,, ϕ+ l Φ, the subextremal functions Φ S Φ. Example with k = 4 : Θ = ({s 1 }, {s 2 }, {s 3, s 4 }) Θ = ({s 1, s 2 }, {s 3, s 4 }) C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 13 / 39

14 Conditional distribution of max-i.d. random fields Joint distribution Let P k be the set of partitions of {s 1,, s k }. We note s = (s 1,..., s k ). Theorem For τ = (τ 1,..., τ l ) P k, A C0 l and B M p(c 0 ) measurable = P [ Θ = τ, (ϕ + 1,..., ϕ+ l ) A, Φ S B] 1 { j [[1,l]], fj > τj j j f j }1 {(f1,,f l ) A} C l 0 P[{Φ B} { φ Φ, φ < S l j=1 f j}] µ(df 1 ) µ(df l ) Furthermore, the law ν s of η(s) is equal to ν s = τ P k ν τ s with ν τ s (dy) = exp ( µ(f (s) y) ) l j=1 µ ( f (s τ c j ) < y τ c j, f (s τj ) dy τj ). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 14 / 39

15 Conditional distribution of max-i.d. random fields Conditional distribution A three step procedure for the conditional law of η given η(s) = y : Step 1 : sample Θ from the conditional law w.r.t. η(s) = y. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 15 / 39

16 Conditional distribution of max-i.d. random fields Conditional distribution Step 2 : sample (ϕ + j ) from the conditional law w.r.t. η(s) = y, Θ = τ. Step 3 : sample Φ s from the conditional law w.r.t. η(s) = y, Θ = τ, (ϕ + j ) = (f j ). Finally, set η(t) = Θ j=1 ϕ+ j (t) φ Φ s φ(t). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 16 / 39

17 Conditional distribution of max-i.d. random fields Main theorem Theorem Let s T k and y (0, + ) k. 1 For all τ P K, P[Θ = τ η(s) = y] = dντ s dν s (y). 2 Conditionally on η(s) = y and Θ = τ, the extremal functions (ϕ + j ) 1 j τ are independent and ϕ + j has distribution µ ( df f (s τj ) = y τj, f (s τ c j ) < y τ c j ). 3 The conditional distribution of Φ s given η(s) = y, Θ = τ and (ϕ + j ) = (f j ) is a PPP with intensity 1 {f (s)<y} µ(df ). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 17 / 39

18 Conditional distribution of max-i.d. random fields The regular case The model is called regular at s if µ(f (s) dz) = λ s (z) dz. The conditional hitting scenario distribution is then P[Θ = τ η(s) = y] = 1 C(s, y) If the model is regular at (s, t) τ j=1 {u j <y τ c j } λ ((sτj,s τ c j )(y τj, u j )du j. = P[ϕ + j (t) dz η(s) = y, Θ = τ] 1 λ (sτj,s τ c,t)(y τj, u, z) j C λ s (y) u<y τ c j du dz. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 18 / 39

19 Conditional distribution of max-i.d. random fields Conditional sampling for Brown-Resnick processes A Brown-Resnick process is a 1-Fréchet max-stable process η(t) = i 1 U i e W i (t) γ(t)/2 with {U i, i 1} PPP(u 2 du) and (W i ) i 1 i.i.d. copies of a Gaussian process with stationary increment and variogram γ. If the covariance of W (t) is nonsingular, η is regular at t : ( λ t (z) = C t exp 1 ) k 2 log zt Q t log z + L T t log z z 1 i. Step 1 : a Gibbs sampler for the conditional hitting scenario. Step 2 : extremal functions are (conditionned) log-normal process. i=1 C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 19 / 39

20 Conditional distribution of max-i.d. random fields Conditional sampling for Brown-Resnick processes Conditional sampling for 1D Brown-Resnick process driven by fractional Brownian motion : Numerical study for a 2D Brown-Resnick process calibrated on real data : extreme temperature and rainfall in Switzerland. Mored details in Mathieu s talk tomorrow! C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 20 / 39

21 Conditional distribution of max-i.d. random fields Perspectives Perfect simulation of the conditional hitting scenario? Further classes of models with effective conditional simulations? See Wang & Stoev (spectrally discrete processes) and Oesting (M3 processes). Characterization of the max-stable processes with the Markov property? Maximum likelihood estimation for multivariate max-stable distributions : consistency, asymptotic normality, efficiency of the MLE? a criterion for local asymptotic normality? design of efficient algorithm? an EM procedure with the hitting scenario as a hidden variable? C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 21 / 39

22 Strong mixing properties of max-i.d. processes Structure of the talk 1 Overview of the research interests 2 Conditional distribution of max-i.d. random fields 3 Strong mixing properties of max-i.d. processes 4 Intermediate regime for aggregated ON/OFF sources 5 A stochastic gradient algorithm for the p-means C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 22 / 39

23 Strong mixing properties of max-i.d. processes Motivations Statistics of max-stable process based on non i.i.d. observations but rather on stationary weakly dependent observations. Recent results for ergodic and mixing properties of stationary max-stable and max-i.d. processes : Weintraub ( 91) Sample and ergodic properties of some min-stable processes. Stoev ( 10) Max-stable processes : representations, ergodic properties and statistical applications. Kabluchko & Schlather ( 10) Ergodic properties of max-infinitely divisible processes. Ergodicity and mixing are important to derive strong law of large numbers but not enough to get central limit theorems. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 23 / 39

24 Motivations Strong mixing properties of max-i.d. processes CLTs for stationary weakly dependent processes are available under strong mixing assumptions. We consider here β-mixing (Volkonskii et Rozanov 59) : for random variables X 1, X 2, β(x 1, X 2 ) = P (X1,X 2 ) P X1 P X2 var. Consider a continuous max-i.d. process η on a locally compact set T such that ( ) ( ) L η(t) = φ(t), with Φ PPP(µ). t T t T φ Φ with µ the exponent measure on C 0 (T ) = C(T, [0, + )) \ {0}. For disjoint subsets S 1, S 2 T, can we get an estimate for β(s 1, S 2 ) = β(η S1, η S2 )? C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 24 / 39

25 Strong mixing properties of max-i.d. processes A natural decomposition of the point process For any S T, Φ = Φ + S Φ S with Φ + S = {φ Φ; s S, φ(s) = η(s)} Φ S = {φ Φ; s S, φ(s) < η(s)}. FIGURE: Realizations of the decomposition Φ = Φ + S Φ S, with S = [0, 5] (left) or S = {3} (right). Clearly for s S, η(s) = φ Φ + φ(s) whence S β(s 1, S 2 ) β(φ + S 1, Φ + S 2 ). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 25 / 39

26 Strong mixing properties of max-i.d. processes A simple upper bound for β(s 1, S 2 ) Theorem The following upper bound holds true β(s 1, S 2 ) 2 P[Φ + S 1 Φ + S 2 ] 2 P[f S1 η, f S2 η] µ(df ). C 0 In the particular case when η is a 1-Fréchet process, β(s 1, S 2 ) 2 [C(S 1 ) + C(S 2 )] [θ(s 1 ) + θ(s 2 ) θ(s 1 S 2 )] with C(S) = E [ sup S η 1] and θ(s) = log P [ sup S η 1 ] the areal coefficient (Coles & Tawn 96). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 26 / 39

27 Strong mixing properties of max-i.d. processes A CLT for max-i.d. random fields Theorem Let η be stationary max-i.d. on Z d and define X(h) = g(η(t 1 + h),..., η(t p + h)), h Z d. Assume that there is δ > 0 such that E[X(0) 2+δ ] < and β(η(0), η(h)) = o( h b ) ( for some b > d max 2, 2 + δ δ Then S n = h n X(h) satisfies the central limit theorem : ( ) c 1/2 n S n E[S n ] = N (0, σ 2 ) with c n = card{ h n} and σ 2 = h Zd cov(x(0), X(h)). ). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 27 / 39

28 Strong mixing properties of max-i.d. processes Perspectives Systematic study of other notions of dependence. In the spirit of the result by Samorodnitsky, Stoev, Kabluchko : we conjecture η ergodic iff η generated by a null flow, η beta-mixing iff η generated by a dissipative flow? Relationships with the tesselation It holds C(x) = {y R d ; Φ + x Φ + y }, x R d. C(0) bounded iff η generated by a dissipative flow, C(0) "with 0 density" iff η generated by a null flow. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 28 / 39

29 Intermediate regime for aggregated ON/OFF sources Structure of the talk 1 Overview of the research interests 2 Conditional distribution of max-i.d. random fields 3 Strong mixing properties of max-i.d. processes 4 Intermediate regime for aggregated ON/OFF sources 5 A stochastic gradient algorithm for the p-means C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 29 / 39

30 Intermediate regime for aggregated ON/OFF sources Motivations Modelisation of packet traffic on high-speed links : traffic measures show LRD and self-similarity. Many models developed to explain these features, considering traffic as the aggregation of a large number of streams generated by independent sources : Mikosh & al. ( 02), Is network traffic approximated by stable Lévy motion or fractional Brownian motion? Mikosch & Samorodnitsky ( 07), Scaling limits for cumulative input processes. Suitable heavy-tail assumptions and scalings account for long range dependence and self-similarity. Asymptotic theory (when both the number of sources and the time scale go to infinity) is very rich : Many results in the so-called "heavy traffic" or "light traffic regime". We focus here on the intermediate scaling. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 30 / 39

31 Intermediate regime for aggregated ON/OFF sources The ON/OFF model A single ON/OFF source Input process I(t) Workload process W(t) Aggregation of sources Cumulative workload of a network fed by m sources : 1 0 X0 Y0 X1 Y1 X2 Y2 X3 Y3 Time t W m (t) = m W j (t). j=1 Bivariate renewal (X i,y i ) i 0. Input and workload process : I(t)=1 [0,X0 )(t)+ n 0 1 [Tn,Tn+X n+1 )(t), W (t)= t 0 I(s)ds. Asymptotic of the form W m (at) E[W m (at)] b(a, m) when both a, m +? =??? C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 31 / 39

32 Intermediate regime for aggregated ON/OFF sources The ON/OFF model in intermediate scaling Theorem Assume the following heavy-tail assumptions F on RV( α on ), Foff RV( α off ) with 1 < α on < α off < 2. In the intermediate scaling, a +, m +, ma F on (a) µ c αon 1 (c > 0), the following convergence of processes holds in C(R +, R) : W m (at) E[W m (at)] a = σ 1 (α on ) µ off µ c P H(t/c), with P H fractional Poisson motion of Hurst index H = (3 α on )/2. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 32 / 39

33 Intermediate regime for aggregated ON/OFF sources The fractional Poisson motion P H Definition of fractional Poisson motion : P H (t)= 1 t σ 1 (αon) R R [x,x+u](y) dy (N(dx,du) dx α onu αon 1 du) where N(dx, du) PPP(dx α on u αon 1 du) on R R +. Properties of P H : infinitely divisible process with the same covariance structure as FBM with Hurst index H, but positive skewness ; long range dependence ; aggregate self-similarity ; paths are "almost" H-hölder continuous. P H realizes a Bridge between H-FBM and α on -stable Lévy motion (interpolation between "heavy traffic" and "light traffic" limits). C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 33 / 39

34 Intermediate regime for aggregated ON/OFF sources A moment measure approach Theorem Let (T n ) n 1 be a stationary renewal sequence with arrival law F, F RV( α), 1 < α < 2. Then, the k-th centered moment measure M k of the point process {T n, n 1} satisfies (α 1)µ k+1 lim Mk [f ( /a)] = ( 1) a F(a)a k Q k+1 k [f ], where f is a regular test function on R k + and Q k is the limit measure Q k (dx) = dx 1 dx k max(x i ) min(x i ) α 1. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 34 / 39

35 Intermediate regime for aggregated ON/OFF sources Perspectives Generalize the results to regenerative random measure, i.e. random mesaures satisfying the renewal property ξ( ) L = η 0 ( ) + ξ( T 1 ) where ξ and (η 0, T 1 ) are independent. Framework covering most usual models : ON/OFF, cluster models, renewal/reward, semi-markovian models... Computations of all moment measures via renewal theory and recursive formulas. Asymptotics under heavy tail assumptions? C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 35 / 39

36 A stochastic gradient algorithm for the p-means Structure of the talk 1 Overview of the research interests 2 Conditional distribution of max-i.d. random fields 3 Strong mixing properties of max-i.d. processes 4 Intermediate regime for aggregated ON/OFF sources 5 A stochastic gradient algorithm for the p-means C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 36 / 39

37 Motivations A stochastic gradient algorithm for the p-means Computation of the p-mean (p 1) of a probability measure µ on a Riemanian manifold (M, ρ) : e p = argmin x M ρ(x, y) p µ(dy). Condition for existence and unicity of the p-mean by Afsari 10 : (A 1 ) µ has support in a geodesic ball of radius r < r 0 (p, M), (A 2 ) if p = 1, the support of µ is not contained in a geodesic. Computation of the p-mean? Case p = 2 : Riemanian barycenter or Karcher mean e 2. A gradient descent algorithm by Le 04. Case p = 1 : Fermat-Werber problem for the median e 1. Algorithms by Weiszfeld 37, Fletcher 09, Yang 10. M C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 37 / 39

38 A stochastic gradient algorithm for the p-means A stochastic gradient algorithm (convergence) Theorem Suppose Afsari conditions (A 1 )-(A 2 ) are satisfied. Let (P k ) k 0 be an i.i.d. sequence on M with distribution µ. Let (t k ) k 1 be positive reals such that t k δ 1 (p, M, µ), t k = k 1 Then, the sequence (X k ) k 0 defined by and tk 2 <. k 1 { X0 = P 0, X k+1 = exp Xk ( tk+1 grad Xk ρ p (P k+1, ) ), k 0. converges almost surely and in L 2 to e p. Remark : with M = R d, p = 2, t k = 1 2k, X k = k i=1 P i. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 38 / 39

39 A stochastic gradient algorithm for the p-means A stochastic gradient algorithm (fluctuations) Theorem Suppose H p : x ρ p (x, y)µ(dy) is C 2 on a neighbourhood of e p. Set t k = δ k δ 1 with δ > δ 2 (p, M, µ). Then the normalized process Y n (t) = [nt] n exp 1 e p (X [nt] ), t 0, converges weakly in D((0, + ), T ep M) to the solution of the SDE { y0 + = 0, dy t = (y t δ dh p (y t, ) )t 1 dt + db t, t > 0, where dh p (y t, ) is the dual vector of dh p (y t, ) and (B t ) t 0 is a Brownian motion with covariance Γ = E[grad ep ρ p (P 1, ) grad ep ρ p (P 1, )]. C.Dombry (Université de Poitiers) Théorie spatiale des extrêmes Poitiers 08/11/12 39 / 39