Some conditional extremes of a Markov chain

Transcription

1 Some conditional extremes of a Markov chain Seminar at Edinburgh University, November 2005 Adam Butler, Biomathematics & Statistics Scotland Jonathan Tawn, Lancaster University Acknowledgements: Janet Heffernan, Paul Fearnhead

2 Some motivating examples... The Environment Agency needs to design a plan for emergency response in the event of coastal flooding along the east coast of Britain A financial analyst wishes to calculate the risk that the value of an investment suffers one or more heavy falls over the course of a week An ecologist is interested in evaluating the probability that the diversity of an ecosystem will collapse as a result of regional climate change

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4 Common threads Consider the risk of catastrophic failure in a structured system which contains inter-dependent components Wish to base inferences on historical data, but are typically interested in events more extreme than any that have actually been recorded

5 A general class of problem We have a Markov chain X = X 1,..., X n We wish to estimate probabilities of the form where u is extreme P(X = x X i = u)

6 Structure of the talk 1. Multivariate extreme value theory; 2. Characterising the extremes of a Markov chain; 3. Statistical modelling & inference.

7 1. Multivariate extreme value theory EVT is a branch of probability theory Univariate EVT seeks to characterise the asymptotic distribution of X i (X i > u) as u Multivariate EVT seeks to characterise the asymptotic distribution of X i (X i = u) as u

8 A classical result in univariate EVT Under weak conditions on X i it can be shown that (X i u) (X i > u) converges to a Generalised Pareto Distribution as u A powerful result, somewhat analogous to the Central Limit Theorem... Motivates a statistical model for the exceedances of a high threshold

9 x time Frequency x

10 But complications arise... OK, can we generalise this result to the multivariate case? Answer: definitely no! A simple example explains why not... If X 1 and X 2 are independent then the distribution of X 2 (X 1 = u) as u is just that of X 2 and so could be absolutely anything!

11 x x1

12 A simple scenario We ignore the Markov assumption for the rest of this section. We look at the bivariate case, and consider the distribution of X 2 (X 1 = u) as u

13 Independence Normal X Logistic X1 Inverse logistic X

14 Dependence In order to make progress we need to seperate out the marginal and dependence characteristics of X = (X 1, X 2 )

15 Independence Normal X Logistic X1 Inverse logistic X

16 Copulas For any X it is known that we can uniquely define a corresponding copula function C, via P(X 1 x 1, X 2 x 2 ) = C(P(X 1 x 1 ), P(X 2 x 2 )) C is a DF for a bivariate random variable U defined on the set [0, 1] 2 C contains all information about the dependence structure of X.

17 Independence U2 Logistic U Normal U1 Inverse logistic

18 Standardised marginals Copulas standardise the marginals to be Uniform(0,1) But we could standardise the marginals to have any known distribution We transform to standard Gumbel margins, so that P(Y i y i ) = exp( exp( y i ))

19 Independence Y Normal Logistic Y1 Inverse logistic Y

20 A classical result A classic limit result in multivariate EVT states that P((Y 2 u) z Y 1 = u) V (1, exp(z)) as u V is a partial derivative of the exponent measure of a bivariate extreme value distribution

21 [The bivariate extreme value distribution] (Z 1, Z 2 ) has a bivariate EVD if it has a distribution function of the form exp[ V (z 1, z 2 )], where the exponent measure V is defined to be V (z 1, z 2 ) = max ( w, 1 w ) h(w) z 1 z 2 The spectral density h is the PDF of an arbitrary random variable W on [0, 1] which satisfies the constraint that E(W ) = 1 0 wh(w) = 1/2

22 Asymptotic (in)dependence Consider the limit of P(Y 2 > u Y 1 > u) as u If the limit is positive then X 1 and X 2 are said to be asymptotically dependent If the limit is zero then X 1 and X 2 are said to be asymptotically independent

23 Dependence structure Independence Inverted logistic Bivariate normal, cor. 0 < ρ < 1 Logistic Perfect dependence Limiting dependence structure Asymptotic independence Asymptotic independence Asymptotic independence Asymptotic dependence Asymptotic dependence

24 Independence Y2 Logistic Y Normal Y1 Inverse logistic

25 The trouble with asymptotic independence If X 1 and X 2 are asymptotically independent then the limit distribution V is degenerate So classical EVT cannot meaningfully quantify dependence at extreme levels for asymptotically independent random variables

26 The Heffernan-Tawn model (Heffernan & Tawn, 2004) Assume that there exist functions α and β such that ( ) Y2 α(u) P = z β(u) Y 1 = u G(z) as u where G is nondegenerate. Then α, β and G together characterise (Y 2 Y 1 = u) as u

27 About the H+T model Suitable functions α and β do not always exist......but H+T show that α and β do exist for a range of important cases H+T are not able to characterise G......but are able to suggest plausible parametric families for α and β

28 Independence Y Normal Logistic Y1 Inverse logistic Y

29 Dependence structure α(u) β(u) Independence 0 1 Inverted logistic 0 u λ Bivariate normal, cor. 0 < ρ < 1 ρ 2 x u Logistic u 1 Perfect dependence u 1 General form for positive extremal dependence: α(u) = au, β(u) = u b where 0 a 1 and 0 b < 1.

30 2. Characterising the extremes of a Markov chain Assume that Y = (Y 1,..., Y n ) is a first-order Markov chain with Gumbel margins We aim to characterise dependence at extreme levels between the components of Y We aim to use the H+T model to find the distribution of Y 1 (Y 1 = u) as u

31 A previous result (Smith, 1992; Perfekt, 1994) Assume that Y is a stationary Markov chain Then under weak conditions P((Y 1 u) z Y 1 = u) G(z) as u, where the limit distribution G is a random walk The increments of this random walk have distributions of the form H (z) = V (1, exp(z))

32 Our assumptions A1 Y is a Markov chain A2 For each j {2,..., n} there exist functions α j and β j such that ( ) Yj α j (y) P = z β j (y) Y j 1 = u g j (z) as u, where the limit density g j is nondegenerate A3 Extremal dependence is positive

33 Main result If we can find functions αj, β j, m j which fulfil S1-S4 for j = 2,..., n, then: ( Y2 α P 2(u) β2 (u) = z 2,..., Y n αn(u) ) = z βn(u) n Y 1 > u g(z) as u, where g is the density of a Markov chain, g(z 2,..., z n ) = d m j (z j 1, z j ) z j g ( j m j (z j 1, z j ) ) j=2

34 [Sufficient conditions S1-S4] S1 [ ] αj (y) + βj (y)z j α j α j 1 (y) + βj 1(y)z j 1 [ ] β j α j 1 (y) + βj 1(y)z j 1 m j (z j 1, z j ) = o [ βj (y) ] S2 β j (y) m j (z j 1, z j ) z j β j (α j 1(y) + β j 1(y)z j 1 ) = o [ β j (y) ] S3 The inverse of M j (z) = ( z 2, m 3 (z 2, z 3 ),..., m j (z j 1, z j ) ) is continuously differentiable & bijective, with non-zero Jacobian. S4 [ m ] j (z j 1, z j ) [ g j m z j (z j 1, z j ) ], j is uniformly continuous as a function of z j

35 [Extension to asymptotic independence: an example] If and P((Y 2 au) = z Y 1 = u) g 2 (z) as u P((Y 3 au) = z Y 2 = u) g 3 (z) as u then P((Y 3 a 2 u) = z 3, (Y 2 au) = z 2 Y 1 = u) g 2 (z)g 3 (z 3 az 2 ) as u

36 3. Statistical modelling & inference (e.g. Coles, 1999) Statistical applications of EVT assume that asymptotic results remain approximately valid at finite but extreme levels Univariate example: assume that the Generalised Pareto Distribution applies to exceedances of a high threshold u estimate the parameters of the GPD model using only observed exceedances of u

37 The Heffernan-Tawn model revisited Transform multivariate data to have Gumbel margins via: a fitted GDP model at extreme levels the empirical distribution function at non-extreme levels Assume a parametric form for α and β, but do not make strong assumptions about G Use a pseudo-likelihood scheme to obtain point estimates for α and β Quantify uncertainty using the semiparametric bootstrap

38 Application to a Markov chain Under a Markov assumption the results in this paper can be used to simplify the structure of α, β and G within the H+T model Yields parsimonious models for dependence at extreme levels...

39 Conclusions We have presented sufficient conditions that the H+T model is able describe dependence within a Markov chain at extreme levels When the conditions are satisfied our result allows us to evaluate α, β and G Motivates parsimonious statistical models for extreme values in temporal/spatial contexts