Detecting seasonality changes in multivariate extremes of - - PowerPoint PPT Presentation

Detecting seasonality changes in multivariate extremes of climatological time series

Philippe Naveau Laboratoire des Sciences du Climat et l’Environnement, France joint work with Sebastian Engelke (Geneva University) and Chen Zhou (Erasmus University)

Motivation : heavy rainfall in Brittany

−4 −2 2 4 6 8 42 44 46 48 50

Two regions

Longitudes

Daily rainfall from 1976 to 2015

Seasons in 1976−1977 Rainfall in mmm 10 20 30 40 50

BREST

Seasons in 1976−1977 Rainfall in mmm 10 20 30 40

LANVEOC

Seasons in 1976−1977 Rainfall in mmm 10 20 30 40 50 60 70

QUIMPER

Our climatological objectives Are heavy rainfall dependence structures change from seasons to seasons? Are extreme precipitation dependence structures differ from regions to regions? Our statistical objective Detecting changes in the dependence structure in multivariate time series of extremes

Statistical desiderata Few assumptions and non-parametric models Fast, simple and general tools No complicated MEVT jargon for climatologists

Our tools Strengthen links between the MEVT (Multivariate Extreme Value Theory) and information theory communities by revisiting the Kullback-Leibler divergence Detecting changes in precipitation extremes structure

Our tools Strengthen links between the MEVT (Multivariate Extreme Value Theory) and information theory communities by revisiting the Kullback-Leibler divergence Detecting changes in precipitation extremes structure

Why the Kullback–Leibler divergence? Machine learning Supervised learning = minimizing the KL divergence objective Proper scoring rules in forecast The logarithm score Information theory Information gain for comparing two distributions Causality theory Bayesian causal conditionals Dynamical systems Statistical mechanics (Boltzmann thermodynamic) Climate science Detection & Attribution and compound events

Kullback–Leibler divergence Definition and notation Let X and Y two random variables with pdfs f and g DKL(X||Y) = Ef log {f(X)} − Ef log {g(X)}

Kullback–Leibler divergence Definition and notation Let X and Y two random variables with pdfs f and g DKL(X||Y) = Ef log {f(X)} − Ef log {g(X)} and D(X, Y) = DKL(X||Y) + DKL(X||Y)

Kullback–Leibler divergence Definition and notation Let X and Y two random variables with pdfs f and g DKL(X||Y) = Ef log {f(X)} − Ef log {g(X)} and D(X, Y) = DKL(X||Y) + DKL(X||Y) Properties DKL(X||Y) ≥ 0 DKL(X||Y) = DKL(X1||Y1) + DKL(X2||Y2) if X1 ⊥ X2 and Y1 ⊥ Y2 DKL is convex DKL is not a metric. Still, the total variation distance,δ(X, Y), satisfies δ(X, Y) ≤

• DKL(X||Y)

Kullback–Leibler divergence A simple example If X and Y two Bernoulli random variables with p = P(X = 1) and q = P(Y = 1), then D(X, Y) = (p − q) × log p q 1 − q 1 − p

• 0.2

0.4 0.6 0.8 0.2 0.4 0.6 0.8 p q 2 4 6 8 10 12 Bernoulli Kullback Liebler divergence 0.0 0.2 0.4 0.6 0.8 1.0 2 4 6 probability of sucess Bernoulli Kullback Liebler divergence

Kullback–Leibler divergence A more complicated example If X and Y are two multinomial random variables with p1, p2, . . . , pK and q1, q2, . . . , qK where q1 + · · · + qK = 1 = p1 + · · · + pK, then D := D(p1, . . . , pK; q1, . . . , qK) =

K

• j=1

(pj − qj)(log pj − log qj)

Kullback–Leibler divergence Inference Let X and Y two random variables with pdfs f and g DKL(X||Y) = Ef log f(X) g(X)

• Estimation hurdle

It seems that the densities f and g need to be known to estimate DKL(X||Y). This will be an issue for multivariate extremes.

Univariate case

Univariate regularly varying case P(X > x) = ¯ F(x) = x−αLX(x) and P(Y > x) = ¯ G(x) = x−βLY(x) where x > 0, LX, LY slowly varying and α, β > 0 are the tail indices. Examples : Cauchy, t-distribution, α-stable, Pareto,...

Exceedances above a high threshold u Xu = (X/u | X > u), Yu = (Y/u | Y > u) with respective densities fu and gu on [1, ∞) Symmetric Kullback–Leibler divergence D(Xu, Yu) = E log fu(Xu) gu(Xu)

• + E log

gu(Yu) fu(Yu)

Exceedances above a high threshold u Xu = (X/u | X > u), Yu = (Y/u | Y > u) with respective densities fu and gu on [1, ∞) Symmetric Kullback–Leibler divergence D(Xu, Yu) = E log fu(Xu) gu(Xu)

• + E log

gu(Yu) fu(Yu)

• Extremes for univariate regularly varying functions

lim

u→∞ D(Xu, Yu) = (α − β)2

αβ

Survival functions versus densities?

PN, Guillou and Riestch (2014, JRSSB) The KL divergence has the representation D(Xu, Yu) = −

• 2 + E log
• G(uXu)

G(u)

• + E log
• F(uYu)

F(u)

• + ∆(u)

= L(Xu, Yu) + ∆(u), where ∆(u) → 0, u → ∞, under a second-order condition.

PN, Guillou and Riestch (2014, JRSSB) The KL divergence has the representation D(Xu, Yu) = −

• 2 + E log
• G(uXu)

G(u)

• + E log
• F(uYu)

F(u)

• + ∆(u)

= L(Xu, Yu) + ∆(u), where ∆(u) → 0, u → ∞, under a second-order condition. E ¯ xample for the GPD distribution

PN, Guillou and Riestch (2014, JRSSB) The KL divergence has the representation D(Xu, Yu) = −

• 2 + E log
• G(uXu)

G(u)

• + E log
• F(uYu)

F(u)

• + ∆(u)

= L(Xu, Yu) + ∆(u), where ∆(u) → 0, u → ∞, under a second-order condition.

PN, Guillou and Riestch (2014, JRSSB) The KL divergence has the representation D(Xu, Yu) = −

• 2 + E log
• G(uXu)

G(u)

• + E log
• F(uYu)

F(u)

• + ∆(u)

= L(Xu, Yu) + ∆(u), where ∆(u) → 0, u → ∞, under a second-order condition. Inference only based on cdf’s For two indep. samples, X (1), . . . , X (n) ∼ F and Y (1), . . . , Y (n) ∼ G, Ln(fu, gu) = −  2 + 1 Nn

• X(i)>u

log

• Gn(X (i))

Gn(u)

• + 1

Mn

• Y (i)>u

log F n(Y (i)) F n(u)   , where F n and Gn the empirical survival functions.

PN, Guillou and Riestch (2014, JRSSB) + Engelke, PN, Zhou (2020+)

Multivariate case

Classical EVT trick Let X in Rd, we don’t like to define extremes in a full multivariate space, so we condition unto one dimensional vector r(X) in R+

Definition of the tail region 1 Homogeneity condition of order one r(tx) = t × r(x), for any scalar t > 0 and x ∈ Rd

+

Examples : r(x) = max(x1, . . . , xd), r(x) = x1 + . . . xd, r(x) = min(x1, . . . , xd)

• 1. Dombry and Ribatet (2015, Statistics and its Interface)

Cutting the tail region into smaller regions Partition {x ∈ Rd : r(x) > 1} = K

j=1 Aj

Cutting the tail region into smaller regions Partition {x ∈ Rd : r(x) > 1} = K

j=1 Aj

Cutting the tail region into smaller regions Partition {x ∈ Rd : r(x) > 1} = K

j=1 Aj

Our main assumption under some type of partitioning lim

u→∞ pj(u) := lim u→∞

Pr(X ∈ uAj) Pr(r(X) > u) = pj ∈ (0, 1) lim

u→∞ qj(u) := lim u→∞

Pr(Y ∈ uAj) Pr(r(Y) > u) = qj ∈ (0, 1)

A special case with r(x) = max(x1, 0) with X1 = X2 in distribution and Xi > 0 Pr(r(X) > u) = Pr(X1 > u) and Pr(X ∈ uA1) = Pr(X1 > u, X2 > u) with A1 = {min(x1, x2) > 1}

A special case with r(x) = max(x1, 0) with X1 = X2 in distribution and Xi > 0 Pr(r(X) > u) = Pr(X1 > u) and Pr(X ∈ uA1) = Pr(X1 > u, X2 > u) with A1 = {min(x1, x2) > 1} χ the classical extremal dependence coefficient lim

u→∞ p1(u) := lim u→∞ Pr(X2 > u|X1 > u) = χ and

lim

u→∞ p2(u) = 1 − χ

Main objective Two sample hypothesis testing Given two independent samples of X in Rd and Y in Rd, we want to test

H0 : pj = qj, ∀j

with lim

Pr(X∈uAj ) Pr(r(X)>u) = pj ∈ (0, 1) and lim Pr(Y∈uAj ) Pr(r(Y)>u) = qj ∈ (0, 1)

Back to the multinomial Kullback–Leibler divergence Reminder If X and Y are two multinomial random variables with p1, p2, . . . , pK and q1, q2, . . . , qK where q1 + · · · + qK = 1 = p1 + · · · + pK, then D := D(p1, . . . , pK; q1, . . . , qK) =

K

• j=1

(pj − qj)(log pj − log qj)

Multinomial Kullback-Liebler divergence Estimation ˆ D(u, v) :=

K

• j=1

(ˆ pj(u) − ˆ qj(v))(log ˆ pj(u) − log ˆ qj(v)). with ˆ pj(u) = n

i=1 1Xi ∈uAj

n

i=1 1r(Xi )>u

and ˆ qj(v) = n

i=1 1Yi ∈vAj

n

i=1 1r(Yi )>v

.

Our main result

Multinomial Kullback-Liebler divergence Choose two sequences un and vn such that mn = n Pr (r(X) > un) = n Pr (r(Y) > vn) → ∞, and mn/n → 0, and assume some second order conditions of pj(u) towards pj (same for qj(u))

Multinomial Kullback-Liebler divergence Choose two sequences un and vn such that mn = n Pr (r(X) > un) = n Pr (r(Y) > vn) → ∞, and mn/n → 0, and assume some second order conditions of pj(u) towards pj (same for qj(u)) Under H0 : pj = qj mn 2 ˆ D(un, vn) → χ2(K − 1), as n ↑ ∞.

Multinomial Kullback-Liebler divergence Choose two sequences un and vn such that mn = n Pr (r(X) > un) = n Pr (r(Y) > vn) → ∞, and mn/n → 0, and assume some second order conditions of pj(u) towards pj (same for qj(u)) Under H0 : pj = qj mn 2 ˆ D(un, vn) → χ2(K − 1), as n ↑ ∞. If pj = qj for some j D is positive and √mn(ˆ D(un, vn) − D) → N(0, σ2), as n ↑ ∞, where σ2 an explicit function of pi and qj

Dealing with marginals and random thresholds In theory We have mn = n Pr (r(X) > un) = n Pr (r(Y) > vn) and ˆ pj(un) = n

i=1 1Xi ∈uAj

n

i=1 1r(Xi )>un

and ˆ qj(vn) = n

i=1 1Yi ∈vAj

n

i=1 1r(Yi )>vn

Dealing with marginals and random thresholds In theory We have mn = n Pr (r(X) > un) = n Pr (r(Y) > vn) and ˆ pj(un) = n

i=1 1Xi ∈uAj

n

i=1 1r(Xi )>un

and ˆ qj(vn) = n

i=1 1Yi ∈vAj

n

i=1 1r(Yi )>vn

In practice : the distribution of r(X) and r(Y) unknown ˆ pj = 1 mn

n

• i=1

1Xi ∈R(X)

n−mn,nAj and ˆ

qj = 1 mn

n

• i=1

1Yi ∈R(Y)

n−mn,nAj ,

where R(X) and R(Y) denote the ranks

Effect of the marginals and number of sets KNOWN MARGINS UNKNOWN MARGINS

• 2

3 4 5 6 7 8 9 0.0 0.2 0.4 0.6 0.8 nb.sets prob

• Type 1 error, weak dependence

Type 1 error, strong dependence power of test significance level of 5%

Effect of the marginals and number of sets KNOWN MARGINS UNKNOWN MARGINS

• 2

3 4 5 6 7 8 9 0.0 0.2 0.4 0.6 0.8 nb.sets prob

• Type 1 error, weak dependence

Type 1 error, strong dependence power of test significance level of 5%

• 2

3 4 5 6 7 8 9 0.0 0.2 0.4 0.6 0.8 nb.sets prob

• Type 1 error, weak dependence

Type 1 error, strong dependence power of test significance level of 5%

Dealing with marginals and random thresholds Case A (rainfall) : unknown marginals with same RV tail index Under some second RV order conditions on r(X) and r(Y), we still have mn 2 ˆ D(un, vn) → χ2(K − 1), under H0

Seasonality in Brittany rainfall extremes dependence?

# excesses KL 0.00 0.05 0.10 0.15 0.20 200 150 100 50 Spring vs Winter

Seasonality in Brittany rainfall extremes dependence?

# excesses KL 0.00 0.05 0.10 0.15 0.20 200 150 100 50 Spring vs Winter # excesses KL 0.00 0.05 0.10 0.15 0.20 200 150 100 50 Summer vs Winter # excesses KL 0.00 0.05 0.10 0.15 0.20 200 150 100 50 Fall vs Winter # excesses KL 0.00 0.05 0.10 0.15 0.20 200 150 100 50 Fall vs Spring # excesses KL 0.00 0.05 0.10 0.15 0.20 200 150 100 50 Fall vs Summer # excesses KL 0.00 0.05 0.10 0.15 0.20 200 150 100 50 Summer vs Spring

Seasonality in Eastern rainfall extremes dependence?

# excesses KL 0.00 0.02 0.04 0.06 0.08 0.10 0.12 200 150 100 Spring vs Winter in East # excesses KL 0.00 0.02 0.04 0.06 0.08 0.10 0.12 200 150 100 Summer vs Winterin East # excesses KL 0.00 0.02 0.04 0.06 0.08 0.10 0.12 200 150 100 Fall vs Winter in East # excesses KL 0.00 0.02 0.04 0.06 0.08 0.10 0.12 200 150 100 Fall vs Spring in East # excesses KL 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 200 150 100 Fall vs Summer in East # excesses KL 0.00 0.02 0.04 0.06 0.08 0.10 0.12 200 150 100 Summer vs Spring in East

Dealing with marginals and random thresholds Case B (Compound events) : unknown marginals with different tails

• 2. Ledford and Tawn (1996) and Draisma et al. (2004), with transformed standard Pareto

marginals and corresponding partition Aj

Dealing with marginals and random thresholds Case B (Compound events) : unknown marginals with different tails For d = 2, r(x) = min(x1, x2) and under some second order AI 2 condition (η < 1), we still have mn 2 ˆ D(un, vn) → χ2(K − 1), under H0

• 2. Ledford and Tawn (1996) and Draisma et al. (2004), with transformed standard Pareto

marginals and corresponding partition Aj

Dealing with marginals and random thresholds Case B (Compound events) : unknown marginals with different tails For d = 2, r(x) = min(x1, x2) and under some second order AI 2 condition (η < 1), we still have mn 2 ˆ D(un, vn) → χ2(K − 1), under H0 For η = 1 (AD), more complex limiting result.

• 2. Ledford and Tawn (1996) and Draisma et al. (2004), with transformed standard Pareto

marginals and corresponding partition Aj

Rainfall and max wind speed : unknown marginals with different tails

Summary The Kullback Liebler divergence is an efficient way of comparing multivariate tails Under various setups, the asymptotic limit under the null hypothesis is a simple χ2 distribution Fast inference of this divergence (no need to assume a parametric pdf) The dependence structure in heavy rainfall do not seem to depend on seasonality

Summary The Kullback Liebler divergence is an efficient way of comparing multivariate tails Under various setups, the asymptotic limit under the null hypothesis is a simple χ2 distribution Fast inference of this divergence (no need to assume a parametric pdf) The dependence structure in heavy rainfall do not seem to depend on seasonality Open questions and future work How to choose r(x) and the number of sets? How to deal with AD case with different tails indices?

Very short biblio Engelke, S., P . Naveau and C. Zhou. Multivariate Kullback Liebler Divergence for extremes (in preparation). Zscheischler, J., Naveau, P ., Martius, O., Engelke, S., and Raible, C. C. : Evaluating the dependence structure of compound precipitation and wind speed extremes, Earth Syst. Dynam. Discuss. 2020. Naveau P ., A. Guillou and T. Riestch (2014), Non-parametric entropy-based approach to detect changes in climate extremes JRSSB)

Simulations tables Coming figures Number of false positive (wrongly rejecting that f = g) and negative out of 1000 replicas of two samples of sizes n = m for a 95% level where f and g GP densities with shape parameter ξf and ξg, respectively.

An example Paris, station Montsouris Daily maxima of temperatures (detrented) Years from 1900 to 2010 40 515 maxima Urban island effect (?) Temperature maxima are often of Weibull type (precip light Fr´ echet) Clustering versus declustering 30 years = the climatology yard stick