Spatial quantile predictions for elliptical random fields

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Spatial quantile predictions for elliptical random fields

Véronique Maume-Deschamps, Didier Rullière, Antoine Usseglio-Carleve

To cite this version:

Spatial quantile predictions for elliptical random

fields

Véronique Maume-Deschamps

, Didier Rullière

, Antoine Usseglio-Carleve

1

_{Institut Camille Jordan, Lyon}

2

_{Laboratoire de Sciences Actuarielle et Financière, Lyon}

Introduction

Kriging (see Krige (1951)) aims at predicting the

condi-tional mean of a random field (Z_t)_t∈T given the values

Z_t1, ..., Z_tN of the field at some points t₁, ..., t_N ∈ T ,

where typically T ⊂ Rd. It seems natural to predict, in

the same spirit as Kriging, other functionals. In our study, we focus on quantiles for elliptical random fields.

Elliptical Distributions

Cambanis et al. (1981) give the representation : the

ran-dom vector X ∈ Rd is elliptical with parameters µ ∈ Rd

and Σ ∈ Rd×d, if and only if

X = µ + RΛU(d), (1)

where ΛΛT = Σ, U(d) is a d−dimensional random vector

uniformly distributed on Sd−1 (the unit disk of dimension

d), and R is a non-negative random variable independent

of U(d). Furthermore, X is said consistent if :

R =d χd  (2) Distribution  Gaussian 1 Student, ν > 0 _√χν ν

Unimodal Gaussian Mixture Pn

k=1

π_kδ_θk

Laplace, λ > 0 √1

E(_λ1)

Uniform Gaussian Mixture U (]0, 1[)

Table 1: Some consistent distributions

Now, we consider X = (X₁, X₂)T be a consistent

(R, d)−elliptical random vector with with X_{1 ∈ R}d1,

X₂ ∈ Rd2, d

1 + d2 = d and parameters µ and Σ. Let us

write: Σ =           Σ₁₁ Σ₁₂ Σ₂₁ Σ₂₂           , µ =           µ₁ µ₂           . (3)

The conditional distribution X₂|(X₁ = x₁) has

parame-ters: 

µ_2|1 = µ₂ + Σ₂₁Σ−1₁₁ (x₁ − µ₁)

Σ_2|1 = Σ₂₂ − Σ₂₁Σ−1₁₁ Σ₁₂ (4)

Furthermore, X₂|(X₁ = x₁) is elliptical, with radius R∗

given by : R∗ = Rd v u u t1 − β               R v u u tβU(d) = C−1 11 (x1 − µ1)    (5)

where C₁₁ is the Cholesky root of Σ₁₁, and β ∼

Beta(d1

2 , d2

2 ).

We can now define the notion of elliptical random fields.

Indeed, a random field (X(t))_t∈T is R−elliptical if ∀n ∈

N, ∀t1, ..., tn ∈ T , the vector (X(t1), ..., X(tn)) is

(R, n)−elliptical.

Conditional quantiles

From now, we consider the following context: (X(t))_t∈T

is an R−elliptical random field. We consider N observa-tions at locaobserva-tions t₁, ..., t_n ∈ T , called (X(t₁), ..., X(t_N)). Our aim is to predict, at a site t ∈ T , the

quan-tile of X(t) given X(t₁), ..., X(t_N). Notice that the

vector (X(t), X(t₁), ..., X(t_N)) is (R, N + 1)−elliptical.

Thus, we can denote X₂ = X(t) _{∈ R and X1} =

(X(t₁), ..., X(t_N)) ∈ RN and restrict ourselves to the

study of the q_α(X₂|X₁).

General case

Then the α−quantile of X₂|(X₁ = x₁) is given by :

q_α (X₂|X₁ = x₁) = µ_2|1 + vu u

tΣ

2|1Φ−1_R∗ (α) (7)

Gaussian case

Since a conditional Gaussian distribution is still Gaussian, we have :

X₂|(X₁ = x₁) ∼ N (µ_2|1, Σ_2|1) (8)

Then, the calculation of the conditional α−quantile of X₂|(X₁ = x₁) is immediate, and gives :

q_α(X₂|X₁ = x₁) = µ_2|1 + v u u u tΣ2|1Φ −1 (α) (9)

Student case

Even if it is more calculative, we can also get theoretical

formula. The conditional α−quantile of X₂|(X₁ = x₁)

has the following expression q_α(X₂|X₁ = x₁) = µ_2|1+ v u u u tΣ2|1 v u u u u u u u t ν ν + N v u u u u u u u u t1 + 1 νq1Φ −1 ν+N(α) . (10) We did not obtain such simple results for other elliptical distributions. It is why we propose, in what follows, two approaches.

Quantile Regression

Quantile regression, introduced by Koenker and Bassett (1978), approximates the conditional quantile as follows :

^

q_α(X₂|X₁ = x₁) = β∗Tx₁ + β∗₀, (11)

where β∗ and β∗₀ are solutions of the following

minimiza-tion problem

(β∗, β∗₀) = arg min

β_∈RN,β0∈R

E[φα(X2 − βTX1 − β0)]. (12)

and where the scoring function φ_α is

φ_α(x) = (α − 1)x_1{x<0} + αx_1{x>0} = αx − x_1{x<0}. (13) In our context of elliptical random fields, we are able to solve this minimization problem, and then define the Quantile Regression Predictor :

^

q_α(X₂|X₁ = x₁) = µ_2|1 + vu u

tΣ

2|1Φ−1_R (α) (14)

Furthermore, its distribution is ^ q_α(X₂|X₁) ∼ E₁   µ2 + v u u u tΣ2|1Φ −1 R (α), Σ21Σ −1 11 Σ12, R    (15)

Extremal quantiles

In this section, the aim is to establish a relation between

Φ−1_R and Φ−1_R∗ for extremal values of α. For that, we do

an assumption : Their exist 0 < ` < +∞ and γ ∈ R

such as :

lim

x→+∞

1 − Φ_R∗(x)

1 − Φ_R(xγ) = ` (16)

Under this assumption, we can define Extreme Conditional Quantiles Predictors :      ^ ^ q_α↑(X₂|X₁ = x₁) = µ_2|1 + σ_2|1     Φ −1 R     1 − 1 ` 1−α+2(1−`)           1 γ ^ ^ q_α↓(X₂|X₁ = x₁) = µ_2|1 − σ_2|1     Φ −1 R     1 − 1 ` α+2(1−`)           1 γ (17) Distribution γ ` Gaussian 1 1 Student, ν > 0 N+ν_ν Γ( ν+N+1 2 )Γ( ν 2) Γ(ν+N₂ )Γ(ν+1₂ )  1 + q1 ν   N+ν 2 ν N 2+1 ν+N Unimodal GM 1 min(θ1,...,θn) N _exp   − min(θ1,...,θn)2 2 q1    n P k=1 π_kθN_k exp    − θ2 k 2 q1     Uniform GM N + 1 Γ( N+2 2 )q N+1 2 1 √ 2 Γ(N+1₂ )(N+1)χ2_N+1(q1)

Table 2: Some examples

Thanks to the paper of Djurčić and Torgašev (2001), we are able to prove that these predictors ^q^_α↑ and ^q^_α↓ are asymptoticaly equivalent to the theoretical quantiles

re-spectively when α → 1 and α → 0.

 ^ ^ q_α↑(X₂|X₁ = x₁) ∼ α→1 qα(X2|X1 = x1) ^ ^ q_α↓(X₂|X₁ = x₁) ∼ α→0 qα(X2|X1 = x1) (18)

Numerical study

Figure 1: Levels of quantile α = 0.995 and α = 0.005

Figure 2: Q-Q plots for Student example

References

Cambanis, S., Huang, S., and Simons, G. (1981). On the theory of elliptically contoured distributions. Journal of

Multivariate Analysis, (11):368–385.

Djurčić, D. and Torgašev, A. (2001). Strong

asymp-totic equivalence and inversion of functions in the class kc. Journal of Mathematical Analysis and Applications, 255:383–390.

Koenker, R. and Bassett, G. J. (1978). Regression quan-tiles. Econometrica, 46(1):33–50.

Krige, D. (1951). A statistical approach to some basic mine valuation problems on the witwatersrand.

Jour-nal of the Chemical, Metallurgical and Mining Society,

52:119–139.