Extremal cluster characteristics of a regime switching model, with

hydrological applications

Péter Elek,Krisztina Vasas and András Zempléni

Eötvös Loránd University, Budapestelekpeti@cs.elte.hu

4th Conference on Extreme Value AnalysisGothenburg, 2005

Contents

• Outline of EVT for stationary series– extremal index

– limiting cluster size distribution (e.g. distribution of flood length)

– distribution of aggregate excesses (e.g. distribution of flood volume)

• Two models:– a light-tailed conditionally heteroscedastic model

– a regime switching autoregressive model

• Extremal behaviour of the regime switching model

• Application to the study of flood dynamics

Extremal index

• Conditions D(un) or (un) are always assumed.

• A stationary series has extremal index if there exists a real sequence un for which

• n(1-F(un))

• P(M1,nun) exp(-)

• where M1,n = max(X1,X2,...,Xn)

• Under D(un) the extremal index can be estimated as:

= lim P(M1,p(n) un | X0>un)

• where p(n) is an appropriately increasing sequence

• p(n) is regarded as the cluster size

Cluster size distribution and point process convergence

• Distribution of the number of exceedances in [1,pn]:

n(j) = P( 1{X1>un}+...+ 1{Xp(n)>un} = j | M1,p(n)>un )• The point process of exceedances:

Nn(.) = i/n(.)1{Xi>un} • Under appropriate conditions:

n converges to some limiting distribution – Nn(.) converges weakly to a compound Poisson process

whose underlying Poisson process has intensity and whose i.i.d clusters are distributed as

• High-level exceedances occur in clusters, with cluster size distribution . Moreover, E()=1/.

Distribution of aggregate excess

• Aggregate excess above u in time interval [k,l]:

Wk,l(u) = (Xk-u)++(Xk+1-u)++...+(Xl-u)+

• This value (called flood volume in hydrology) is a good

indicator of the severity of extreme events.

• Under appropriate conditions (Smith et al., 1997):

W1,n(un) d W1+W2+...+WK

where K~Poisson() and the variables Wi are i.i.d,

independent of K.

• The distribution of Wi can be regarded as the limiting

aggregate excess distribution during an extremal event.

Problems

• Estimation of limiting quantities (, , W) is difficult.• Often the subasymptotic behaviour is of interest,

too, since the convergence to the limit is very slow.• To overcome these problems, one can restrict

attention to certain families of models.• A large class of Markov-chains behaves like a

random walk at extreme levels• which can be used to simulate extremal clusters in a

Markov-chain, see e.g. Smith et al. (1997)

Water discharge series are non-Markovian – even above high thresholds

• If the series were Markovian,

(Xt-Xt-1 | Xt-1,Xt-1-Xt-2>0) ~ (Xt –Xt-1| Xt-1,Xt-1-Xt-2<0) would hold

• The following plots show Xt-Xt-1 as a function of Xt-1 (if Xt-1 is above the 98% quantile), conditionally on the sign of Xt-1-Xt-2

• The two plots are not similar!

A light-tailed conditionally heteroscedastic model

Xt-ct = ai(Xt-i-ct-i) + t + bjt-j

t = t Zt

t = [d0 + d1(Xt-1-m)+]1/2

• Zt is an i.i.d. sequence with zero mean and unit variance

• ct describes the deterministic seasonal behaviour in mean

• If all moments of Zt are finite, then all moments of Xt are finite

• However, the exact tail behaviour is unknown (a special case of a similar model has Weibull-like tails, see Robert, 2000)

• The model approximates the extremal properties of water discharge series well (see Elek and Márkus, 2005)

A regime switching (RS) autoregressive model

Xt = Xt-1 + 1t if It = 1 (rising regime)

Xt = aXt-1 + 0t if It = 0 (falling regime)

1t is an i.i.d noise, distributed as Gamma(,)

0t is an i.i.d noise, distributed as Normal(0,)

• 0<a<1

• Successive regime durations are independent and distributed as

– NegBinom(1,p1) in the rising regime

– NegBinom(0,p0) in the falling regime

Properties of the RS-model • Heuristic explanation:

– Xt gets independent positive shocks in the rising regime

– it develops as a mean-reverting autoregression in the falling regime

• If 1=0=1, then It is a Markov-chain and Xt is a Markov-switching autoregression

• The model is stationary by applying the result of Brandt (1986) for stochastic difference equations

• Regime switching models have deep roots in hydrology (see e.g. Bálint and Szilágyi, 2005)

The model gives back the asymmetric shape of the hydrograph

Tail behaviour of the stationary distribution

• Theorem: The process has Gamma-like upper tail:• P( Xt>u | It=1 ) ~ K1 u-1 exp{-u[1-(1-p1)1/]}

• P( Xt>u | It=0 ) ~ K0 u-1 exp{-u[1-(1-p1)1/]/a}

• thus: P( Xt>u ) ~ K1 u-1 exp{-u[1-(1-p1)1/]}.

• The proof is based on the observations that • the aggregate increment during a rising regime has

Gamma-like tail • which becomes “negligible” during the falling regime.

• Corollary: Exceedances above high thresholds are asymptotically exponentially distributed:• limu P(Xt>x+u | Xt>u) = exp{-x[1-(1-p1)1/]}

Limiting cluster quantities in the model I.

• Even when the regime lengths are negative binomial,

• the extremal index is p1,

• and the limiting cluster size distribution is geometric with parameter p1.

Limiting cluster quantities in the model II.

• If =1, the limiting aggregate excess distribution is W = E1 + 2E2 + ... + NEN

– where N is geometric with parameter p1

– the variables Ei are exponential with parameter , independent from each other and from N

• The exponential moments are infinite, but all polynomial moments are finite.

• Anderson and Dancy (1992) suggested to model the aggregate excesses of a hydrological data set with Weibull-distribution.

Slow convergence to the limiting quantities

• The plot gives (u,p)

• if =p1=0.5, p0=0.1, a=0.5 and =0=1=1– for p=100 and 200 and– for u ranging from the

99% to the 99.99% quantile

= limp limu P( M1,pu | X0>u ) = (u,p)

Parameter estimation• Estimation of the whole model with hidden regimes:

– (reversible jump) MCMC

– maximum likelihood if 1=0=1 (i.e. in the Markov-switching case) – but it is computationally infeasible

• However, if we focus only on extremal dynamics

and assume that the regime durations (at least above a high level) are geometrically distributed

we can write down the likelihood based solely on data during floods (i.e. above a high threshold)

=1 is also assumed (in accordance with the empirical data)

Exponential QQ-plot for the positive increments above the threshold 900 m3/s

Likelihood computations

• Likelihood can be determined recursively:– qt=P( It=1 | Xt, Xt-1, …)

– q1cond = P( It=1 | Xt-1,…) = (1-p1)qt-1 + p0(1-qt-1)

– q0cond = P( It=0 | Xt-1,…) = p1qt-1 + (1-p0)(1-qt-1)

– f1 = f(Xt , It=1 | Xt-1,…) = q1cond fExp() (Xt-Xt-1)

– f0 = f(Xt , It=0 | Xt-1,…) = q0cond fN(0,) (Xt-aXt-1)

– f(Xt | Xt-1,…) = f0 + f1

– qt = f1/(f0 + f1)

• Some care is needed:– at the beginning of the floods qt

is determined from the tail behaviour of the model

– at the end of the floods the observation is censored

Advantages of using only the data over a threshold

• Model dynamics may be different at lower levels– For physical reasons, the rate of decay in the falling

regime (characterised by a) is varying over the decay

• Fast maximum likelihood estimation– Smaller sample size– Regimes separate very well at high levels

Application to flood analysis

• Data: 50 years of daily water discharge series at Tivadar (river Tisza) – about 18000 observations

• We assume =0=1=1

• Threshold: 900m3/s (about 98% quantile)• Parameter estimates and asymptotic standard errors:

– p1=0.598 (0.037)

• on average 1.7 days of further increase – in accordance with emp. value

– p0=0.027 (0.011)

• has a negligible effect on the dynamics over the threshold

– a=0.823 (0.007) • high persistence even in the falling regime

=0.0044 (0.0003) =137.1 (8.0)

Empirical and simulated flood dynamics

• Shape of the empirical and simulated floods are very similar.• Subasymptotic behaviour is important:

– Simulated water discharge remains over the threshold for 1.4 days in

average after the peak

Exceedances over a threshold

• Maximal exceedance over a threshold is approximately exponential with parameter p1=1/392 in the model,

• in good accordance with the empirical distribution.

• The plot shows the exceedance over the threshold 1250m3/s.

Aggregate excess (flood volume)

• Threshold = 1250 m3/s• Operational definition: two

floods are separated when the water discharge goes below a lower threshold (900 m3/s) between them

• There are only 48 such floods in 50 years

• Emp. mean: 72.1 mill. m3

• Sim. mean: 76.9 mill m3

• The QQ-plot shows the fit of the distribution, too.

Dependence of p1 on the threshold

Conclusions

• The limiting cluster quantities can be determined in our physically motivated regime switching model

• Simulations are still needed since the subasymptotic behaviour is important at the relevant thresholds

• To determine return levels of, e.g., flood volume, the occurence of extreme events should also be modelled, by a Poisson-process.

• Further work: what parametric multivariate extreme value distribution does a reasonable multivariate regime switching model suggest?

References

• Anderson, C.W. and Dancy, G.P. (1992): The severity of extreme events, Research Report 409/92 University of Sheffield.

• Bálint, G. and Szilágyi, J. (2005): A hybrid, Markov-chain based model for daily streamflow generation, Journal of Hydrol. Engineering, in press.

• Brandt, A. (1986): The stochastic equation Yn+1=AnYn+Bn with stationary coefficients, Adv. in Appl. Prob., 18, 211-220.

• Elek, P. and Márkus, L. (2004): A long range dependent model with nonlinear innovations for simulating daily river flows, Natural Hazards and Earth Systems Sciences, 4, 277-283.

• Elek, P. and Márkus, L. (2005): A light-tailed conditionally heteroscedastic model with applications to river flows, in preparation.

• Robert, C. (2000): Extremes of alpha-ARCH models, in: Measuring Risk in Complex Stochastic Systems (ed. by Franke et al.), XploRe e-books.

• Segers, J. (2003): Functionals of clusters of extremes, Adv. in Appl. Prob., 35, 1028-1045.

• Smith, R.L., Tawn, J.A. and Coles, S.G. (1997): Markov chain models for threshold exceedances, Biometrika, 84, 249-268.

Thank you for your attention!