Prediction of Stable Stochastic Processes

Transcription

1 Prediction of Stable Stochastic Processes Evgeny Spodarev Institute of Stochastics Workshop Probability, Analysis and Geometry,

2 Seite 2 Prediction of Stable Stochastic Processes Evgeny Spodarev Part 1: Introduction Stochastic prediction Applications Stationary random fields Kriging

3 Seite 3 Prediction of Stable Stochastic Processes Evgeny Spodarev Spatial data {X(t i )} n i=1 - spatial data in observation window W R d. They are interpreted as a realisation of a real valued random field X = {X(t) : t R d } which is a spatially indexed family of random variables defined on a joint probability space (Ω, F, P).

4 Seite 4 Prediction of Stable Stochastic Processes Evgeny Spodarev Stochastic prediction Let the observations X(t 1 ),..., X(t n ) of a random field X = {X(t), t R d } be given for t 1,... t n W, W R d being a compact set. Find a predictor X(t) for X(t), t {t 1,..., t n } that is optimal in some sense and has a number of nice properties such as exactness, continuity, etc.

5 Seite 5 Prediction of Stable Stochastic Processes Evgeny Spodarev Examples of stochastic prediction methods Kriging Geoadditive regression models Whittaker smoothing Randomly coloured mosaics...

6 Seite 6 Prediction of Stable Stochastic Processes Evgeny Spodarev Prediction of wide sense stationary random functions Mathematical foundations: extrapolation of stationary time series A.N. Kolmogorov (1941), N.Wiener (1949). Origins of geostatistics: D. Krige (1951), B. Mathérn (1960), L. Gandin (1963), G. Matheron ( ).

7 Seite 7 Prediction of Stable Stochastic Processes Evgeny Spodarev Stationary random fields Random field X = {X(t) : t R d } is (strictly) stationary if its probability law is translation invariant, i.e., all finite dimensional distributions are invariant with respect to any shifts in R d : for all h R d, n N, t 1,..., t n R d holds (X(t 1 + h),..., X(t n + h)) d = (X(t 1 ),..., X(t n )). Random field X = {X(t) : t R d } is stationary of 2nd order if E X 2 (t) < for all t R d and E(X(t)) = µ for all t. γ(h) = 1 2 E [(X(t + h) X(t)) 2] depends only on vector h, but not on t.

8 Seite 8 Prediction of Stable Stochastic Processes Evgeny Spodarev Stationary random fields Strict stationarity = stationarity of second order A second order stationary random field is called isotropic if C(h) = C( h ), h R d. Correlation structure: Let the random field X = {X(t)} be stationary of second order. Variogram: γ(h) = 1 2 E [ (X(t + h) X(t)) 2] Covariance function: C(h) = E [X(t) X(t + h)] µ 2 γ(h) = C(0) C(h)

9 Seite 9 Prediction of Stable Stochastic Processes Evgeny Spodarev Example: Gaussian random fields A random field {X(t)} is called Gaussian if the distribution of (X(t 1 ),..., X(t n )) is multivariate Gaussian for each 1 n < and t 1,..., t n R d. The distribution of X is completely defined by the mean value function µ(t) = E X(t) and covariance function C(s, t) = Cov ( X(s), X(t) ), s, t R d. Hence: strict stationarity stationarity of second order.

10 Seite 10 Prediction of Stable Stochastic Processes Evgeny Spodarev Ordinary Kriging (D. Krige (1951), G. Matheron ( )) Assumptions: X is stationary of second order. Notation t i : locations of the sample points X(t i ) : observed values of X n : number of sample points : weights λ i Estimator: X(t) = n i=1 λ ix(t i ), where n i=1 λ i = 1. The weights λ i are chosen such that the estimation variance σ 2 E = Var( X(t) X(t)) is minimized.

11 Seite 11 Prediction of Stable Stochastic Processes Evgeny Spodarev Ordinary Kriging n X(t) is unbiased: E X(t) = µ since λ i = 1 i=1 σe 2 min = σ2 OK : solve the Lagrange equations n λ j γ(t j t i ) + ν = γ(t t i ), i = 1,..., n, j=1 n λ j = 1. j=1 The minimal estimation variance: n σok 2 = ν + λ i γ(t i t) i=1

12 Seite 12 Prediction of Stable Stochastic Processes Evgeny Spodarev Ordinary Kriging Variogram fitting To find the weights λ i from the system of linear equations, the variogram γ(h) has to be known or estimated from the data X(t 1 ),..., X(t n ). Matheron s estimator: ˆγ(h) = 1 2N(h) i,j:t i t j h ( X(ti ) X(t j ) ) 2, N(h) is the number of pairs (t i, t j ) : t i t j h. Computations are made for h on a grid in R d. ˆγ(h) not conditionally negative definite a valid variogram model has to be fitted to ˆγ(h) e.g. by least squares

13 Seite 13 Prediction of Stable Stochastic Processes Evgeny Spodarev Variogram fitting Variogram point cloud and a fitted exponential variogram

14 Seite 14 Prediction of Stable Stochastic Processes Evgeny Spodarev Properties of ordinary kriging The kriging predictor exists and is unique. BLUE: best linear unbiased estimator by definition. Exactness: X(t i ) = X(t i ) a.s., i = 1,..., n If X is a stationary Gaussian random field, X(t) N(µ, σ 2 ), then X is Gaussian as well, and X(t) N(µ, σ0 2 (t)) with σ 2 0 (t) = σ2 + ν n λ i γ(t i t) i=1

15 Seite 15 Prediction of Stable Stochastic Processes Evgeny Spodarev Part 2: Stable laws and integration Motivation Stable distributions Covariation Random measures Stochastic integration

16 Seite 16 Prediction of Stable Stochastic Processes Evgeny Spodarev Random fields without a finite second moment Before: Extrapolation of 2nd order stationary random fields Now: need more flexible models and corresponding extrapolation methods for random fields with infinite variance. Why do we take care?

17 Seite 17 Prediction of Stable Stochastic Processes Evgeny Spodarev Motivation Natural disasters and their mapping (geosciences) Hundred year flood, 2002 Winter storm Kyrill, 2007

18 Seite 18 Prediction of Stable Stochastic Processes Evgeny Spodarev Motivation: storm insurance in Austria Centers of 2047 postal code regions in Austria

19 Seite 19 Prediction of Stable Stochastic Processes Evgeny Spodarev Motivation: storm insurance in Austria Histogram of the deviations Q-Q plot of the deviations Goal: Spatial modelling of the deviations Y (t) = X(t) µ(t) from the mean claim payments µ(t) = E X(t) with random fields. However, the distribution of the deviations is not Gaussian (rather skewed and heavy-tailed).

20 Seite 20 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable distributions Stable distributions (Kchinchine, Levy, 1930s): A random variable X is said to have a stable distribution if there is a sequence of i.i.d. random variables Y 1, Y 2,... and sequences of positive numbers {d n } and real numbers {a n }, such that Y Y n d n + a n d X where d denotes convergence in distribution.

21 Seite 21 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable distributions A random variable X is stable if and only if for A, B > 0 C > 0, D R: AX 1 + BX 2 d = CX + D where X 1 and X 2 are independent copies of X. There exists a number α (0, 2] (index of stability) such that C α = A α + B α Also referred to as (α )stable distribution For α = 2: normal distribution

22 Seite 22 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable distributions Characteristic function of an α-stable random variable X S α (σ, β, µ), 0 < α 2: E ( e iθx ) { e σ α θ α (1 iβ sgnθ tan πα 2 )+iµθ if α 1 = e σ θ (1+iβ 2 π sgnθ ln θ )+iµθ if α = 1 Parameters σ, β, µ are unique for α (0, 2): µ R: shift β [ 1, 1]: skewness (form), β = 0: symmetry σ 0: scale

23 Seite 23 Prediction of Stable Stochastic Processes Evgeny Spodarev Multivariate stable distributions A random vector X = (X 1,..., X d ) is called stable if for A, B > 0 C > 0, D R d : AX (1) + BX (2) d = CX + D Symmetric random vector A random vector X in R d is symmetric if P(X A) = P( X A) for any Borel set A R d.

24 Seite 24 Prediction of Stable Stochastic Processes Evgeny Spodarev Multivariate stable distributions Characteristic function of an α-stable random vector X = (X 1,..., X d ), 0 < α 2: θ ( ) T s α (1 i sgnθ T s tan πα 2 )Γ(ds)+iθ T µ S E e i θt X e d if α 1 = θ T s (1+i 2 π sgnθt s ln θ T s )Γ(ds)+iθ T µ S e d if α = 1 where Γ is a finite (spectral) measure on the unit sphere S d of R d and µ R d. Parameters Γ and µ are unique for α (0, 2): µ R: shift Γ: skewness (form) and scale together.

25 Seite 25 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable distributions Symmetric stable random vector A symmetric α-stable random vector X (SαS) in R d has a characteristic function ϕ X (θ) = e S d θ,s α Γ(ds), θ R d where the spectral measure Γ is symmetric on S d. If Γ is not concentrated on a great sub-sphere of S d, then X is called full-dimensional.

26 Seite 26 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable distributions Properties and characteristics Moments: if p < α then E X p <. For p α, it holds E X p =. Covariation: for an α-stable random vector (X 1, X 2 ), 1 < α 2 with spectral measure Γ define [X 1, X 2 ] α = s 1 s <α 1> 2 Γ(ds 1, ds 2 ) S 1 where a <p> := a p sgn(a) for a R and p 0. Gaussian case α = 2: if (X 1, X 2 ) is a centered Gaussian random vector then [X 1, X 2 ] 2 = 1 2 Cov(X 1, X 2 ).

27 Seite 27 Prediction of Stable Stochastic Processes Evgeny Spodarev Covariation and moments Lemma (Karcher, Shmileva, S. (2013)) Let 1 < α < 2 and suppose that (X, Y ) T is an α-stable random vector with spectral measure Γ such that X S α (σ X, β X, 0) and Y S α (σ Y, β Y, 0). For 1 p < α, it holds E ( XY <p 1>) E Y p = [X, Y ] α(1 c β Y ) + c (X, Y ) α σy α, where (X, Y ) α := S 1 s 1 s 2 α 1 Γ(ds) and c := c α,p (β Y ) is a constant. If Y is symmetric, i. e. β Y = 0, then c = 0.

28 Seite 28 Prediction of Stable Stochastic Processes Evgeny Spodarev Random measures Let (E, E, m) be a measurable space with a σ finite measure m, E 0 = {A E : m(a) < }, L 0 (Ω) = {random variables on (Ω, F, P)}. An independently scattered stable random measure M with control measure m and skewness intensity β : E R is a random measure with independent α stable increments, i.e., an a.s. σ additive function M : E 0 L 0 (Ω) with ) M(A) S α ((m(a)) 1/α, β(x) m(dx)/m(a), 0 A for any A E 0.

29 Seite 29 Prediction of Stable Stochastic Processes Evgeny Spodarev Stochastic integration For f L α (E), construct I(f ) = E f (x) M(dx), where M is an independently scattered α-stable random measure on (E, E) with control measure m and skewness intensity β. Simple functions: for f (x) = n j=1 c j1i(x A j ), x E, with A j E 0 : A i A j, i j, we set I(f ) = n c j M(A j ). j=1 General functions: for any f L α (E), there exists a sequence of simple functions f n f a.e. on E. Set I(f ) = p lim n I(f n ).

30 Seite 30 Prediction of Stable Stochastic Processes Evgeny Spodarev Stochastic integral This limit exists and does not depend on the choice of the sequence {f n } tending to f. Distribution: I(f ) S α (σ f, β f, µ f ), where ( σ f = E f (x) α m(dx)) 1/α = f L α, E β f = f (x)<α> β(x) m(dx) E f (x) α β(x) m(dx), { 0, α 1, µ f = f (x)β(x) log f (x) m(dx), α = 1. 2 π E

31 Seite 31 Prediction of Stable Stochastic Processes Evgeny Spodarev Example: Spatial modelling of storm data (Austria) Parameter estimation for the one-dimensional case: The field X of deviations from the mean claim sizes has the univariate distribution X(t) S α (σ, β, µ) with α β σ µ

32 Seite 32 Prediction of Stable Stochastic Processes Evgeny Spodarev Part 3: Stable random fields Definition Spectral representation Examples Subgaussian random fields Stable motion Their properties

33 Seite 33 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable random fields A random field {X(t), t R d } is called α-stable if the distribution of (X(t 1 ),..., X(t n )) is multivariate α stable for any 1 n < and t 1,..., t n R d. A random field {X(t), t R d } is called separable if a countable subset T 0 R d s.t. for all t R d X(t) = p lim k X(t k ) with {t k, k N} T 0. Consider a stable random field X(t) = f t (x)m(dx), t R d, where f t L α (E), t R d, and M is an α-stable independently scattered random measure with control measure m and skewness β. E

34 Seite 34 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable random fields Spectral representation: for separable in probability α-stable fields with 0 < α 2, α 1 it holds { } 1 {X(t), t R d } = d f t (x)m(dx) + µ(t), t R d where f t L α (0, 1) for all t R d, 0 M is an α-stable independently scattered random measure on (0, 1) with Lebesgue control measure and skewness intensity β(x) = 1, x (0, 1), µ : R d R is some function. Case α = 1: open.

35 Seite 35 Prediction of Stable Stochastic Processes Evgeny Spodarev Examples: α stable random fields Sub-Gaussian random fields: Let A S α/2 ((cos(πα/4)) 2/α, 1, 0) and let G = {G(t), t R d } be a stationary zero mean Gaussian random field with covariance function C. Assume that A is independent of G. The SαS random field X = {X(t), t R d } with X(t) = A 1/2 G(t), t R d is called sub-gaussian. Characteristic function of X t1,...,t n = (X(t 1 ),..., X(t n )) : for any n N, t 1,..., t n R d it holds ϕ (s Xt1,...,tn 1,..., s n ) = exp 1 n α/2 2 C(t i t j )s i s j. i,j=1

36 Seite 36 Prediction of Stable Stochastic Processes Evgeny Spodarev Examples: Simulation Subgaussian random field X = {A 1/2 G(t), t [0, 1] 2 } with α = 1.5, A S α/2 ((cos(πα/4)) 2/α, 1, 0) and G being a stationary isotropic Gaussian random field with covariance function C(h) = 7 exp{ (h/0.1) 2 }, h 0.

37 Seite 37 Prediction of Stable Stochastic Processes Evgeny Spodarev Examples: Simulation Realization of the sub-gaussian random field

38 Seite 38 Prediction of Stable Stochastic Processes Evgeny Spodarev Examples: α stable random fields SαS Lévy motion X(t) = 1I{x 1 t 1,..., x d t d }M(dx), [0,1] d where t = (t 1,..., t d ) [0, 1] d, and M is a SαS random measure with Lebesgue control measure. Simulation: Two-dimensional SαS Lévy motion X(t) = 1I{x 1 t 1, x 2 t 2 }M(dx), t [0, 1] 2, [0,1] 2 where α = 1.5.

39 Seite 39 Prediction of Stable Stochastic Processes Evgeny Spodarev Examples: Simulation Realization of the Lévy stable motion

40 Seite 40 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable random fields: Properties and characteristics Let X(t) = f t (x)m(dx), t R d as above. E Symmetry: if β(x) = 0 x then the field X is symmetric. Scale parameter of X(t): σ X(t) = f t L α where ( E X(t) p ) 1/p = cα,β (p) σ X(t) for 0 < p < α, 0 < α < 2 and some constant c α,β (p). Covariation function: for t 1, t 2 R d and 1 < α 2 κ(t 1, t 2 ) = [X(t 1 ), X(t 2 )] α = f t1 (x)f t2 (x) <α 1> m(dx). E

41 Seite 41 Prediction of Stable Stochastic Processes Evgeny Spodarev Stable random fields: Properties and characteristics Stationarity: if E = R d, f t (x) = f (t x), x, t R d, β(x) = const and m(dx) = dx then X is stationary (moving average) and κ(s, t) = κ(s t, o) = κ(h), h = s t, s, t R d. Linear dependence: For a d-dimensional α-stable random vector X = (X 1,..., X d ) T with integral representation ( E T f 1 (x)m(dx),..., f d (x)m(dx)), E let Γ be its spectral measure. X is not full-dimensional (i.e., Γ is concentrated on a great sub sphere of S d ) iff d i=1 c ix i = 0 a.s. for some (c 1,..., c d ) T R d \ {0}. This is equivalent to d i=1 c if i (x) = 0 m-a. e.

42 Seite 42 Prediction of Stable Stochastic Processes Evgeny Spodarev Part 4: Prediction of stable random functions (Non)linear predictors and their properties Least scale predictor Covariation orthogonal predictor Maximization of covariation Numerical results Open problems Literature

43 Seite 43 Prediction of Stable Stochastic Processes Evgeny Spodarev Prediction of stable random functions Random functions without finite second moments: discrete stable processes: minimization of dispersion (Cambanis, Soltani (1984); Brockwell, Cline (1985); Kokoszka (1996); Brockwell, Mitchell (1998); Gallardo et al. (2000); Hill (2000)) fractional stable motion: conditional simulation (Painter(1998)) subgaussian random functions: maximum likelihood (ML) (Painter(1998)), linear regression (Miller (1978)), conditional simulation stable moving average processes: minimization of L 1 -distance (Mohammadia, Mohammadpour (2009)) α-stable random fields with integral spectral repr.: three methods (Karcher, Shmileva, S. (2013))

44 Seite 44 Prediction of Stable Stochastic Processes Evgeny Spodarev Prediction Let X be a centered (E X(t) = 0, t R d ) α-stable random field, 1 < α 2, with skewness intensity β satisfying the spectral representation X(t) = f t (x)m(dx), t R d. E Let X(t 1 ),..., X(t n ) be the observations of X for t 1,... t n W, W R d being a compact set. Non-linear predictors for X(t), t {t 1,..., t n }: for some particular random functions (e.g. subgaussian ones) one can use Maximum likelihood (ML) predictors Conditional simulators

45 Seite 45 Prediction of Stable Stochastic Processes Evgeny Spodarev Linear predictors Linear predictor for X(t), t {t 1,..., t n }: X(t) = n λ i X(t i ), i=1 where λ i = λ i (t, t 1,..., t n ) for i = 1,..., n. Properties X is unbiased since E X(t) = 0, t R d X is exact if X(ti ) = X(t i ) a.s., i = 1,..., n. X is continuous if λi = λ i (, t 1,..., t n ) are continuous as functions of t, i = 1,..., n

46 Seite 46 Prediction of Stable Stochastic Processes Evgeny Spodarev Linear predictors X(t) should be optimal in a sense that it minimizes the scale parameter σ X(t) X(t) = Least Scale Linear (LSL) Predictor mimics the covariation structure between X(t) and X(t j ), j = 1,..., n = Covariation Orthogonal Linear (COL) Predictor maximizes the covariation between X(t) and X(t) = Maximization of Covariation Linear (MCL) Predictor

47 Seite 47 Prediction of Stable Stochastic Processes Evgeny Spodarev Least Scale Linear Predictor Generalization of Kriging techniques: σ α X(t) X(t) = E f t(x) with respect to λ 1,..., λ n. n λ i f ti (x) i=1 α m(dx) min Non-linear optimization problem = numerical methods for its solution.

48 Seite 48 Prediction of Stable Stochastic Processes Evgeny Spodarev Least Scale Linear Predictor Lemma Let α (1, 2). A solution of the above minimization problem resolves the system of equations [ ] n X(t j ), X(t) λ i X(t i ) = 0, j = 1,... n, i=1 which can be written as ( n f tj (x) f t (x) λ i f ti (x) E i=1 α ) <α 1> m(dx) = 0, j = 1,... n. This is a system of non linear equations in λ 1,..., λ n.

49 Seite 49 Prediction of Stable Stochastic Processes Evgeny Spodarev Least Scale Linear Predictor Theorem Existence: The LSL estimator exists. Uniqueness: Assume that the random vector (X(t 1 ),..., X(t n )) T is full-dimensional. Then the LSL estimator is unique. Exactness: If there is a unique LSL estimator, then it is obviously exact. Continuity: If the random field X is stochastically continuous and (X(t 1 ),..., X(t n )) T is full-dimensional then the LSL estimator is continuous.

50 Seite 50 Prediction of Stable Stochastic Processes Evgeny Spodarev Least Scale Linear Predictor Example: SαS Lévy motion X(t) = 0 1I(x t)m(dx), where M is a SαS random measure with Lebesgue control measure. Let t = 3/4 and t 1 = 1. Then the optimization problem for the LSL predictor is σ α X(t) X(t) = 3/4 0 1 λ 1 α dx + 1 3/4 λ 1 α dx We obtain the LSL predictor = λ 1 α λ 1 α min λ 1. X(t) = (1/3) 1/(α 1) X(t 1).

51 Seite 51 Prediction of Stable Stochastic Processes Evgeny Spodarev Covariation Orthogonal Linear Predictor Let X be a random field as above. The linear predictor with weights λ 1,..., λ n being a solution of the following system of equations [X(t), X(t j )] α = [ X(t), X(t j )] α, j = 1,..., n is the COL predictor. It is a linear system of equations [ X(t), X(tj ) ] n α [ λ i X(ti ), X(t j ) ] = 0, j = 1,..., n. α i=1 The COL predictor is obviously exact.

52 Seite 52 Prediction of Stable Stochastic Processes Evgeny Spodarev Covariation Orthogonal Linear Predictor The regression of X(t) on (X(t 1 ),..., X(t n )) T is called linear if there exists some (λ 1,..., λ n ) R n such that it holds a.s. E(X(t) X(t 1 ),..., X(t n )) = n λ i X(t i ). i=1 The regression of X(t) on (X(t 1 ),..., X(t n )) T is linear if X is e.g. a (sub)gaussian random function. Lemma If the regression of X(t) on the random vector (X(t 1 ),..., X(t n )) T is linear then the vector (λ 1,..., λ n ) T is a solution of the COL system of equations.

53 Seite 53 Prediction of Stable Stochastic Processes Evgeny Spodarev Covariation Orthogonal Linear Predictor Theorem Let X be an α-stable moving average. If the kernel function f : R d R + is positive semi-definite, then the covariation function κ is positive semi-definite. If f : R d R + is positive definite and positive on a set with positive Lebesgue measure, then κ is positive definite. If the covariation function is positive definite then the COL predictor exists and is unique. If the covariation function is positive definite and continuous, then the COL predictor is continuous.

54 Seite 54 Prediction of Stable Stochastic Processes Evgeny Spodarev Covariation Orthogonal Linear Predictor Proof. The weights of the COL predictor satisfy the system of equations κ(0) κ(t n t 1 )..... κ(t n t 1 ) κ(0) λ 1. λ n = κ(t t 1 ). κ(t t n )

55 Seite 55 Prediction of Stable Stochastic Processes Evgeny Spodarev Covariation Orthogonal Linear Predictor Example: SαS Ornstein-Uhlenbeck process. X(t) = e λ(t x) 1I(t x 0)M(dx), t R, R for some λ > 0, where M is a SαS random measure with Lebesgue control measure. If t 1 < t 2 <... < t n < t, then the regression of X(t) on (X(t 1 ),..., X(t n )) T is linear, and X(t) = e λ(t tn) X(t n ).

56 Seite 56 Prediction of Stable Stochastic Processes Evgeny Spodarev Covariation Orthogonal Linear Predictor Let X be a centered (sub)gaussian α-stable random field with covariance function C of the Gaussian part. Then [X(t i ), X(t j )] α = 2 α/2 C(t i t j )C(0) (α 2)/2. The COL predictor is the solution of the system C(0) C(t n t 1 ) λ 1 C(t t 1 ) =. C(t n t 1 ) C(0) λ n C(t t n ) and thus coincides with simple kriging.

57 Seite 57 Prediction of Stable Stochastic Processes Evgeny Spodarev Covariation Orthogonal Linear Predictor Theorem Let X be a centered (sub)gaussian α-stable random field with positive definite covariance function C of the Gaussian part. The COL predictor exists and is unique. If the covariance function is continuous, then the COL predictor is continuous. Theorem For (sub)gaussian random fields, the COL and LSL predictors for X(t) coincide (with the maximum likelihood (ML) estimator of X(t)).

58 Seite 58 Prediction of Stable Stochastic Processes Evgeny Spodarev Maximization of Covariation Linear Predictor Let X be an α-stable random field with spectral integral representation and α > 1. To construct the MCL predictor, solve [ ] X(t), X(t) = n i=1 λ i [X(t i ), X(t)] α max, α λ 1,...,λ n σ X(t) = σ X(t), where the condition σ X(t) = σ X(t) means X(t) d = X(t) for SαS random fields.

59 Seite 59 Prediction of Stable Stochastic Processes Evgeny Spodarev Maximization of Covariation Linear Predictor Theorem Assume that the random vector (X(t 1 ),..., X(t n )) T is full-dimensional. Existence: The MCL predictor exists. Uniqueness: If [X(t i ), X(t)] α 0 for some i {1,..., n} then the MCL predictor is unique. Exactness: If the MCL predictor is unique then it is exact. Continuity: If X is a moving average, the covariation function κ is continuous and κ(t i t) 0 for some i {1,..., n} then the MCL predictor is continuous.

60 Seite 60 Prediction of Stable Stochastic Processes Evgeny Spodarev Numerical results Two-dimensional SαS Lévy motion X(t) = 1I{x 1 t 1, x 2 t 2 }M(dx), t [0, 1] 2, [0,1] 2 where M is a SαS random measure with m = Lebesgue control measure and α = 1.5. Method 5%-Quantile 1st Quartile Median 3rd Quartile 95%-Quantile LSL COL MCL Summary statistics for the deviations X(t) X(t).

61 Seite 61 Prediction of Stable Stochastic Processes Evgeny Spodarev Numerical results Realization of the Lévy stable motion (top left) and the extrapolations (out of 9 observation points) based on the LSL method (top right), the COL method (bottom left) and the MCL method (bottom right).

62 Seite 62 Prediction of Stable Stochastic Processes Evgeny Spodarev Numerical results Subgaussian random field X = {A 1/2 G(t), t [0, 1] 2 } with α = 1.5, A S α/2 ((cos(πα/4)) 2/α, 1, 0) and G being a stationary isotropic Gaussian random field with covariance function C(h) = 7 exp{ (h/0.1) 2 }, h 0. Method 5%-Quantile 1st Quartile Median 3rd Quartile 95%-Quantile LSL (COL, ML) MCL CS Summary statistics for the deviations X(t) X(t).

63 Seite 63 Prediction of Stable Stochastic Processes Evgeny Spodarev Numerical results Realization of the sub-gaussian random field (top left) and the extrapolations (out of 9 observation points) based on the LSL (COL, ML) method (top right), the MCL method (bottom left) and the CS method (bottom right).

64 Seite 64 Prediction of Stable Stochastic Processes Evgeny Spodarev Open problems Extrapolation methods and their properties for stable random fields with α (0, 1] Control of skewness of known predictors for non symmetric stable random fields (β 0) Characterization of the covariation function

65 Seite 65 Prediction of Stable Stochastic Processes Evgeny Spodarev Literature Adler, R. J. and Taylor, J. (2007) Random fields and geometry. Springer. Chiles, J.-P., Delfiner. P. (1999) Geostatistics. Modelling spatial uncertainty. J. Wiley & Sons. Cressie, N. A. C. (1993) Statistics for spatial data. J. Wiley & Sons. Ivanov, A. V., Leonenko N. N. (1989) Statistical analysis of random fields. Kluwer. Schlather, M. (1999) Introduction to positive definite functions and to unconditional simulation of random fields. Technical report ST 99-10, Lancaster University. Spodarev, E. (ed.) Stochastic geometry, spatial statistics and random fields. Asymptotic methods, Volume 2068 of the Lecture Notes in Mathematics, Springer, Wackernagel, H. (1998) Multivariate geostatistics. Springer. Yaglom, A. M. (1987) Correlation theory of stationary and related random functions. Springer.

66 Seite 66 Prediction of Stable Stochastic Processes Evgeny Spodarev Literature Feller, W. An introduction to the probability theory and its applications. 2nd ed., Vol. 2, Wiley, Gnedenko, B.V. and Kolmogorov, A. N. Limit distributions for sums of independent random variables. Addison-Wesley, Reading, Karcher, W., Shmileva, E. and Spodarev, E. Extrapolation of stable random fields. J. Multivar. Anal, Vol. 115 (2013), p Rachev, S.T., Mittnik, S. Stable Paretian Models in Finance. Wiley, New York, Samorodnitsky, G., Taqqu, M. Stable Non-Gaussian Processes, Chapman & Hall, Spodarev, E., Shmileva, E. and Roth, S. Extrapolation of stationary random fields. Preprint, Uchaikin, V. V., Zolotarev, V. M. Chance and Stability. Stable Distributions and their Applications. VSP, Utrecht, Zolotarev, V. M. One dimensional stable distributions. AMS, 1986.

67 Seite 67 Prediction of Stable Stochastic Processes Evgeny Spodarev Literature Brockwell, P. J., Cline, D. B. H., Linear prediction of ARMA processes with infinite variance, Stoch. Process. Appl. 19: (1985) Brockwell, P. J., Mitchell, H., Linear prediction for a class of multivariate stable processes, Stoch. Models 14(1): (1998) Cambanis, S., Soltani, A. R., Prediciton of stable Processes: Spectral and moving average representations, Z. Wahrsch. verw. Gebiete 66: (1984) Gallardo, J. R., Matrakis, D., Orozco-Barbosa, L., Prediction of alpha-stable long-range dependent stochastic processes, Adv. Perform. Anal. 3(1): (2000) Hill, J. B., Minimum Dispersion and Unbiasedness: Best Linear Predictors for Stationary ARMA α-stable Processes, Preprint (2000) Kokoszka, P. S., Prediction of infinite variance fractional ARIMA: Probab. Math. Statist. 16(1): (1996) Mohammadia M., Mohammadpour A., Best linear prediction for α-stable random processes, Statist. Probab. Lett. 79(21): (2009) Painter, S., Numerical method for conditional simulation of Lévy random fields, Math. Geol. 30(2): (1998)