Extremal cluster characteristics of a regime switching model, with hydrological applications

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Extremal cluster characteristics of a regime
switching model, with hydrological applications

• Péter Elek,
• Krisztina Vasas and András Zempléni
• Eötvös Loránd University, Budapest
• elekpeti_at_cs.elte.hu
• 4th Conference on Extreme Value Analysis
• Gothenburg, 2005

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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

3
Quantities of interest in the analysis of time
series extremes

• Some are determined by the marginal distribution
• probability of exceeding a high threshold
• distribution of exceedances of a high threshold
• Others are determined by the clustering dynamics
  of extreme values
• average length of an extremal event (e.g. length
  of a flood)
• distribution of the length of an extremal event
• distribution of aggregate excesses (e.g.
  distribution of the flood volume)

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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,n?un) ? exp(-??)
• where M1,n max(X1,X2,...,Xn)
• Under D(un) the extremal index can be estimated
  as
• ? lim P(M1,p(n) ? un X0gtun)
• where p(n) is an appropriately increasing
  sequence
• p(n) is regarded as the cluster size

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Cluster size distribution and point process
convergence

• Distribution of the number of exceedances in
  1,pn
• ?n(j) P( 1X1gtun... 1Xp(n)gtun j
  M1,p(n)gtun )
• The point process of exceedances
• Nn(.) ? ?i/n(.)1Xigtun
• 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/?.

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Distribution of aggregate excess

• Aggregate excess above u in time interval k,l
• Wk,l(u) (Xk-u)(Xk1-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 W1W2...WK
• where KPoisson(??) 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.

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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)

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Water discharge series are non-Markovian even
above high thresholds

• If the series were Markovian,
• (Xt-Xt-1 Xt-1,Xt-1-Xt-2gt0) (Xt Xt-1
  Xt-1,Xt-1-Xt-2lt0) 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!

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A light-tailed conditionally heteroscedastic model

• Xt-ct ? ai(Xt-i-ct-i) ?t ? bj?t-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)

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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,?)
  
• 0ltalt1
• Successive regime durations are independent and
  distributed as
• NegBinom(?1,p1) in the rising regime
• NegBinom(?0,p0) in the falling regime

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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?01, 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)

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The model gives back the asymmetric shape of the
hydrograph
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Tail behaviour of the stationary distribution

• Theorem The process has Gamma-like upper tail
• P( Xtgtu It1 ) K1 u?-1 exp-?u1-(1-p1)1/?
• P( Xtgtu It0 ) K0 u?-1 exp-?u1-(1-p1)1/?/a
  
• thus P( Xtgtu ) K1 u?-1 exp-?u1-(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(Xtgtxu Xtgtu) exp-?x1-(1-p1)1/?

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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.

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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.

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Slow convergence to the limiting quantities

• ? limp?? limu?? P( M1,p?u X0gtu ) ?(u,p)

• The plot gives ?(u,p)
• if ?p10.5, p00.1, a0.5 and ??0?11
• for p100 and 200 and
• for u ranging from the 99 to the 99.99 quantile

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Parameter estimation

• Estimation of the whole model with hidden
  regimes
• (reversible jump) MCMC
• maximum likelihood if ?1?01 (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)

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

• Likelihood can be determined recursively
• qtP( It1 Xt, Xt-1, )
• q1cond P( It1 Xt-1,) (1-p1)qt-1
  p0(1-qt-1)
• q0cond P( It0 Xt-1,) p1qt-1
  (1-p0)(1-qt-1)
• f1 f(Xt , It1 Xt-1,) q1cond fExp(?)
  (Xt-Xt-1)
• f0 f(Xt , It0 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

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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

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Application to flood analysis

• Data 50 years of daily water discharge series at
  Tivadar (river Tisza) about 18000 observations
• We assume ??0?11
• Threshold 900m3/s (about 98 quantile)
• Parameter estimates and asymptotic standard
  errors
• p10.598 (0.037)
• on average 1.7 days of further increase in
  accordance with emp. value
• p00.027 (0.011)
• has a negligible effect on the dynamics over the
  threshold
• a0.823 (0.007)
• high persistence even in the falling regime
• ?0.0044 (0.0003)
• ?137.1 (8.0)

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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

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Exceedances over a threshold

• Maximal exceedance over a threshold is
  approximately exponential with parameter
  ?p11/392 in the model,
• in good accordance with the empirical
  distribution.
• The plot shows the exceedance over the threshold
  1250m3/s.

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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.

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Dependence of p1 on the threshold
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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?

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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
  Yn1AnYnBn 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.

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• Thank you for your attention!