Pter Elek and Lszl Mrkus

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Nonlinear time series for modelling river flows
and their extremes

• Péter Elek and László Márkus
• Dept. Probability Theory and Statistics
• Eötvös Loránd University
• Budapest, Hungary

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River Tisza and its aquifer
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Tisza produces devastating floods
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Water discharge at Vásárosnamény(We have 5 more
monitoring sites)from1901-2000
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Basic properties of the series

• Series exhibit
• slight upward trend in mean
• strong seasonal component both in mean and
  variance
• Seasonal components can be removed by a local
  polinomial fitting (LOESS) procedure
• The distribution of the standardized series are
  still periodic.
• The distribution is skewed.

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Empirical and smoothed seasonal components
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Bi-monthly distributions
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Autocorrelation function is slowly decaying
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How long does the river remember...?

• Long memory - known ever since Hursts original
  work on the Aswan dam (1952) of the Nile River
• Short and long memory - autocorrelations die out
  at exponential versus hyperbolic rate
• Equivalent for long memory - spectral density has
  pole at zero, meaning it tends to infinity at
  zero at polynomial speed.

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Long memory processes

• Short and long memory - autocorrelations die out
  at exponential versus hyperbolic rate
• Equivalent for long memory - spectral density has
  pole at zero, meaning it tends to infinity at
  zero at polynomial speed.

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Indicators of long memory

• Nonparametric statistics
• Rescaled adjusted range or R/S
• Classical
• Los (test)
• Taqqus graphical (robust)
• Variance plot
• Log-periodogram (Geweke-Porter Hudak)

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• Classic R/S biased, lack of distribution theory,
  no test or confidence intervals, sensitive for
  short mem component if present.
• Los modification - convergence to Brownian
  Bridge. Test of long memory often accepts short
  memory when long is present, but reliably rejects
  long mem when only short is present.
• Taqqus graphical R/S get out the most of the
  data drop blocks from the beginnig, reestimate
  R/S from the shorter data, log-log plot all
  together and fit a straight line (regression)

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Variance plot

• The variance of the mean tends to zero as n2H-2
• Estimate the variance of means from the
  aggregated processes
• Fit a straight line on the log-log scale
• Determine the slope from a linear regression

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The log - spectrum

• The order of the pole of the spectral density is
  1-2H
• On the log-log scaleplot of the periodogram this
  means a straight line of steepness 1-2H
• The celebrated Geweke - Porter Hudak statistics
  uses log(sin2(?/2)) instead of the log of the
  frequencies.

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Linear long-memory model fractional
ARIMA-process (Montanari et al., Lago Maggiore,
1997)

• Fractional ARIMA-model
• Fitting is done by Whittle-estimator
• based on the empirical and theoretical
  periodogram
• quite robust consistent and asymptotically
  normal for linear processes driven by innovatons
  with finite forth moments (Giraitis and
  Surgailis, 1990)

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Results of fractional ARIMA fit

• H0.846 (standard error 0.014)
• p-value 0.558 (indicates goodness of fit)
• Innovations can be reconstructed using a linear
  filter (the inverse of the filter above)

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Reconstruct the innovation from the fitted model
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The density of the innovations
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Reconstructed innovations are uncorrelated...

• But not independent

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Simulations using i.i.d. innovations

• If we assume that innovations are i.i.d, we can
  generate synthetic series
• Use resampling to generate synthetic innovations
  
• Apply then the linear filter
• Add the sesonal components to get a synthetic
  streamflow series
• But these series do not approximate well the
  high quantiles of the original series

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But they fail to catch the densities and
underestimate the high quantiles of the original
series
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Logarithmic linear model

• It is quite common to take logarithm in order to
  get closer to the normal distribution
• It is indeed the case here as well
• Even the simulated quantiles from a fitted linear
  model seem to be almost acceptable

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• But the backtransformed quantiles are clearly
  unrealistic.

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Lets have a closer look at innovations

• Innovations can be regarded as shocks to the
  linear system
• Few properties
• Squared and absolute values are autocorrelated
• Skewed and peaked marginal distribution
• There are periods of high and low variance
• All these point to a GARCH-type model
• The classical GARCH is far too heavy tailed to
  our purposes

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Simulation from the GARCH-process

• Simulations
• Generate i.i.d. series from the estimated
  GARCH-residuals
• Then simulate the GARCH(1,1) process using these
  residuals
• Apply the linear filter and add the seasonalities
• The simulated series are much heavier-tailed than
  the original series

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General form of GARCH-models

• Need a model between
• i.i.d.-driven FARIMA-series and
• GARCH(1,1)-driven FARIMA-series
• General form of GARCH-models

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A smooth transition GARCH-model
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ACF of GARCH-residuals
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Results of simulationsat Vásárosnamény
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Back to the original GARCH philosophy

• The above described GARCH model is somewhat
  artificial, and hard to find heuristic
  explanations for it
• why does the conditional variance depend on the
  innovations of the linear filter?
• in the original GARCH-context the variance is
  dependent on the lagged values of the process
  itself.
• A possible solution condition the variance on
  the lagged discharge process instead !
• The fractional integration does not seem to be
  necessary
• almost the same innovations as from an ARMA(3,1)
• In extreme value theory long memory in linear
  models does not make a difference

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Estimated variance of innovations plotted against
the lagged discharge

• Spectacularly linear relationship
• This approves the new modelling attempt
• Distorted at sites with damming along Tisza River

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  The variance is not conditional on the lagged
innovation but it is conditional on the lagged
water discharge.
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• Theoretical problems arise in the new model
• Existence of stationary solution
• Finiteness of all moments
• Consistence and asymptotic normality of quasi
  max-likelihood estimators
• Heuristically clearer explanation can be given
• The discharge is indicative of the saturation of
  the watershed
• A saturated watershed results in more
  straightforward reach for precipitation to the
  river, hence an increase in the water supply.
• A saturated watershed gives away water quicker.
• The possible changes are greater and so is the
  uncertainty for the next discharge value.

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An example ZtN(0,1) c20, a10.95, ?01, ?12,
m1
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Existence and moments of the stationary solution

• We assume that ct constant
• The model has a unique stationary solution if the
  corresponding ARMA-model is stationary
• i.e. all roots of the characteristic equation lie
  within the unit circle
• Moreover, if the m-th moment of Zt is finite then
  the same holds for the stationary distribution of
  Xt, too.
• These are in contrast to the usual, quadratic
  ARCH-type innovations. There the condition for
  stationarity is more complicated and not all
  moments of the stationary distribution are finite.

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Sketch of the proof I.

• The process can be embedded into a
  (pq)-dimensional Markov-chain YtAYt-1Et
• where Yt(Xt-c, Xt-1-c,...Xt-p1-c, et, et-1,...,
  et-q1) and Et(et, 0,...).
• Yt is aperiodic and irreducible (under some
  technical conditions).
• General condition for geometric ergodicity and
  hence for existence of a unique stationary
  distribution (Meyn and Tweedie, 1993) there
  exists a V?1 function with 0lt?lt1, blt? and C
  compact set
• E( V(Y1) Y0y ) ? (1-?) V(y) b IC(y)
• In other words V is bounded on a compact set
  and is a contraction outside it.
• Moreover E?(V(Yt)) is finite (? is the
  stationary distribution).

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Sketch of the proof II.

• In the given case
• if E(Ztm) is finite,
• V(y) 1 QPymm will suffice
• where
• BPAP-1 is the real valued block
  Jordan-decomposition of A
• and Q is an appropriately chosen diagonal matrix
  with positive elements.
• This also implies the finiteness of the mth
  moment of Xt.

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Estimation

• Estimation of the ARMA-filter can be carried out
  by least squares.
• Essentially only the uncorrelatedness of
  innovations is needed for consistency.
• Additional moment condition is needed for
  asymptotic normality (e.g. Francq and Zakoian,
  1998).
• The ARCH-equation is estimated by quasi maximum
  likelihood (assuming that Zt is Gaussian), using
  the ?t innovations calculated from the
  ARMA-filter.
• The QML estimator of the ARCH-parameters is
  consistent and asymptotically normal under mild
  conditions (Zt does not need to be Gaussian).

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Estimation of the ARCH-equation in the case of
known ?t innovations(along the lines of
Kristensen and Rahbek, 2005)

• Maximising the Gaussian log-likelihood
• we obtain the QML-estimator of ?.
• For simplicity we assume that ?0? ?mingt0 in the
  whole parameter space.

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Consistency of the estimator

• By the ergodic theorem
• It is easy to check that L(a)ltL(a0) for all a?a0
  where a0 denotes the true parameter value.
• All other conditions of the usual consistency
  result for QML (e.g. Pfanzagl, 1969) are
  satisfied hence the estimator is consistent.

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Asymptotic normality I.

• Using the notations
• for the derivatives of the log-likelihood
• for the information matrix
• and for the expected Hessian

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Asymptotic normality II.

• A standard Taylor-expansion implies
• Finiteness of the fourth moment with a martingale
  central limit theorem yields
• Moreover, the asymptotic covariance matrix can be
  consistently estimated by the empirical
  counterparts of H and V.

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Estimation of the ARCH-equation when ?t is not
known

• In this case the innovations of the ARMA-model
  are calculated using the estimated
  ARMA-parameters
• If the ARMA-parameter vector is estimated
  consistently, the mean difference of squared
  innovations tends to zero
• If the ARMA parameter estimate is asymptotically
  normal, a stronger statement holds

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Consistency in the case of estimated innovations

• Now the following expression is maximised
• But the difference tends to zero (uniformly on
  the parameter space)
• which then yields consistency of the estimate of
  ?.

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Asymptotic normality in the case of estimated
innovations

• Under some moment conditions the least squares
  estimate of the ARMA-parameters is asymptotically
  normal, hence
• This way the differences between the first and,
  respectively, the second derivatives both
  converge to zero
• As a result, all the arguments for asymptotic
  normality given above remain valid.

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Estimation results
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How to simulate the residuals of the new
GARCH-type model

• Residuals are highly skewed and peaked.
• Simulation
• Use resampling to simulate from the central
  quantiles of the distribution
• Use Generalized Pareto distribution to simulate
  from upper and lower quantiles
• Use periodic monthly densities

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The simulation process

resampling and GPD
Zt
smoothed GARCH
?t
FARIMA filter
Xt
Seasonal filter
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Evaluating the model fit

• Independence of residual series
• ACF, extremal clustering
• Fit of probability density and high quantiles
• Variance lagged discharge relationship
• Extremal index
• Level exceedance times
• Flood volume distribution

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ACF of original and squared innovation series
residual series
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Results of new simulationsat Vásárosnamény
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Densities and quantiles at all 6 locations
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Reestimated (from the fitted model)
discharge-variance relationship
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Seasonalities of extremes

• The seasonal appearance of the highest values
  (upper 1) of the simulated processes follows
  closely the same for the observed one.

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Extremal index to measure the clustering of high
values

• Estimated for the observed and simulated
  processes containing all seasonal components
• Estimation by block method
• Value of block length changes from 0.1 to 1.
• Value of threshold ranges from 95 to 99.9.

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Estimated extremal indices displayed
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Extremal indices in the threshold GARCH model
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Extremal indices in the discharge dependent GARCH
model
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Level exceedance times

• The distribution of the level exceedance times
  may serve as a further goodness of fit measure.
• It has an enormous importance as it represents
  the time until the dams should stand high water
  pressure.

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Flood volume distribution

• The match of the empirical and simulated flood
  volume distributions also approve the good fit.

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Multivariate modelling

• Final aim to model the runoff processes
  simultaneously
• Nonlinear interdependence and non-Gaussianity
  should be addressed here, too
• First, the joint behaviour of the discharges
  inflowing into Hungary should be modelled
• Differential equation-oriented models of
  conventional hydrology may be used to describe
  downstream evolution of runoffs
• Now we concentrate on joint modelling of two
  rivers Tisza (at Tivadar) and Szamos (at
  Csenger)

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Issues of joint modelling

• Measures of linear interdependences (the
  cross-correlations) are likely to be
  insufficient.
• High runoffs appear to be more synchronized on
  the two rivers than small ones
• The reason may be the common generating weather
  patterns for high flows
• This requires a non-conventional analysis of the
  dependence structure of the observed series

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Basic statistics of Tivadar (Tisza) and Csenger
(Szamos)

• The model described previously was applied to
  both rivers
• Correlations between the series of raw values,
  innovations and residuals are highest when either
  series at Tivadar are lagged by one day
• Correlations
• Raw discharges 0.79
• Deseasonalized data 0.77
• Innovations 0.40
• Residuals 0.48
• Conditional variances 0.84

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Displaying the nature of interdependence

• The joint plot may not be informative because of
  the highly non-Gaussian distributions
• Transform the marginals into uniform
  distributions (produce the so-called copula),
• then the scatterplot is more informative on the
  joint behaviour
• The strange behaviour of the copula of the
  innovations is characterized by the concentration
  of points
• 1. at the main diagonal, and especially at the
  upper right corner (tail dependence)
• 2. at the upper left (and the lower right)
  corner(s)
• Linearly dependent variables do not display this
  type of copula
• The GARCH-residuals lack the second type of
  irregularity

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1
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A possible explanation of this type of
interdependence

• The variance process is essentially common for
  the two rivers
• This is justified by the high correlation (0.84)
• Generate two linearly interdependent residual
  series (correlation0.48)
• Multiply by the common standard deviation
  distributed as Gamma
• Observe the obtained copula
• This justifies the hypothesis that the common
  variance causes the nonlinear interdependence of
  the given type

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Tail dependence
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Conclusion

• Long range dependency does not have much impact
  on extremes of river discharges
• Nonlinearities are influental and have to be
  accounted for
• The proposed model has a stationary solution
• Its estimation is consistent and asymptotically
  normal
• The suggested model catches densities and
  quantiles well
• The model fitting procedure does not optimise on
  quantiles and maxima clustering, therefore their
  fit really shows model adequacy
• Other fit-independent measures include level
  exceedence and flood volume distributions
• Something different has to be done with dammed
  water

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Possible refinements of the model

• The simulated process is still slightly heavier
  tailed than the original one.
• The conditional distribution of residuals depends
  on the lagged water discharge
• very high residual values occur at low discharge
• Possible solutions
• innovation as a mixture of a GARCH-part and of a
  discharge-independent part
• Markov-switching model with discharge-dependent
  transition probabilities

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Thank you for your attention!