Expectile prediction through asymmetric kriging

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Expectile prediction through asymmetric kriging

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

To cite this version:

Expectile prediction through asymmetric kriging

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 expectiles for elliptical random fields. We also did a similar work for quantiles (see Maume-Deschamps et al. (2016b)).

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)

Slash β (a, 1)

Table 1: Some consistent distributions

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.

Figure 1: Slash random field with a Matérn 3₂ kernel Furthermore, if X = (X₁, X₂) is a (R, d₁ + d₂)−elliptical

random vector with parameters µ = (µ₁, µ₂), we know

that X₂|X₁ is (R_ ∗, d₂)−elliptical with parameters : µ_2|1 = µ₂ + Σ₂₁Σ−1₁₁ (x₁ − µ₁)

Σ_2|1 = Σ₂₂ − Σ₂₁Σ−1₁₁ Σ₁₂ (3)

Elliptical expectiles

If X is a random variable, the α−expectile of X is given by :

e_α(X) = arg min

x_∈R E[φα

(X − x)]

where the scoring function φ_α is

(1 − α)x2_1{x<0} + αx2_1{x>0} = αx2 + (1 − 2α)x2_1{x<0}. (4) Let X be a (R, 1)−elliptical random variable, with

param-eters µ and σ2. We define the function Ψ_R as follows :

Ψ_R(x) = x Z −∞ c₁g₁(y2)dy + 1 x +Z∞ x yc₁g₁(y2)dy

where c₁g₁(t) = 1₂f_R(√t), f_R(t) being the density of R. Then, the α−expectile of X is given by :

e_α(X) = µ + σΨ−1_R       α 2α − 1       (5)

We propose, in Maume-Deschamps et al. (2016a), several

ways to compute Ψ−1_R    α 2α−1   .

Figure 2: Computation of the 0.95 gaussian expectile

Now, we consider X = (X₁, X₂), X₁ ∈ RN, X₂ ∈ R a

(R, N + 1)−elliptical random vector. According to (5),

the α−expectile of X₂|X₁ is given by :

e_α(X) = µ_2|1 + v u u u tΣ2|1Ψ −1 R∗       α 2α − 1       (6)

In the general case, the term Ψ−1_R∗

   α 2α−1    is difficult to

compute. This is why we propose some other predictors.

Expectile Regression

From now one, we consider the following context:

(X(t))_t∈T is an R−elliptical random field. We

con-sider N observations at locations t₁, ..., t_N ∈ T , called

(X(t₁), ..., X(t_N)). In order to predict the value of X₂ =

X(t) given X₁ = X(t₁), ..., X(t_N), we approximate X(t)

by :

^

e_α(X₂|X₁ = x₁) = β∗Tx₁ + β∗₀, (7)

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)]. (8)

In the case α = 1₂, the solution of (8) is exactly the

krig-ing vector. Otherwise, this problem leads to an expectile regression, introduced by Newey and Powell (1987).

In our context of elliptical random fields, we are able to solve this minimization problem, and then define the Ex-pectile Regression Predictor :

^ e_α(X₂|X₁ = x₁) = µ_2|1 + vu u tΣ 2|1Ψ−1_R    α 2α−1    (9)

Furthermore, its distribution is ^ e_α(X₂|X₁) ∼ E₁      µ2 + v u u u tΣ2|1Ψ −1 R       α 2α − 1       , Σ21Σ −1 11 Σ12, R       (10)

Extremal expectiles

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

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

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

as :

lim

x→+∞

1 − Ψ_R∗(x)

1 − Ψ_R(xγ) = ` (11)

Under this assumption, we can define Extreme Conditional Expectiles Predictors :      ^ ^ e_α↑(X₂|X₁ = x₁) = µ_2|1 + σ_2|1    Ψ −1 R    1 − α−1 (2α−1)`         1 γ ^ ^ e<sub>α↓</sub>(X<sub>2</sub>|X<sub>1</sub> = x<sub>1</sub>) = µ<sub>2|1</sub> − σ<sub>2|1</sub>    Ψ −1 R    1 − α (2α−1)`         1 γ (12) Distribution γ ` Gaussian 1 1 Student, ν > 0 N_ν + 1 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 ν−1 ν+N−1 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     Slash, a > 0 N_a + 1 2 1−a 2 (a−1)Γ(N+1+a 2 )q N+a 2 1

(N+a)(N+a−1)Γ(N+a₂ )Γ(1+a₂ )χ2_N+a(q1)

Table 2: Some examples

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

re-spectively when α → 1 and α → 0.

 ^ ^ e_α↑(X₂|X₁ = x₁) ∼ α→1 eα(X2|X1 = x1) ^ ^ e_α↓(X₂|X₁ = x₁) ∼ α→0 eα(X2|X1 = x1) (13)

Numerical study

Figure 3: Levels of quantile α = 0.9995 and α = 0.0005

Figure 4: E-E plots for Slash 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.

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.

Maume-Deschamps, V., Rullière, D., and Usseglio-Carleve, A. (2016a). Spatial Expectile Predictions for Elliptical Random Fields. preprint.

Maume-Deschamps, V., Rullière, D., and Usseglio-Carleve, A. (2016b). Spatial Quantile Predictions for Elliptical Random Fields. preprint.

Newey, W. and Powell, J. (1987). Asymmetric Least

Squares Estimation and Testing. Econometrica, 55:819– 847.