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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
Station m a0 a1
Komárom 789 1807.8 (2009.8) 26.06 (2.22)
Nagymaros 586 544.7 (154.1) 11.95 (0.57)
Budapest 580 907.1 (314.0) 10.29 (0.55)
Tivadar 23 24,49 (5,95) 18,80 (1,13)
Namény 30 82,45 (32,82) 20,71 (0,51)
Záhony 45 67,04 (17,31) 12,37 (1,19)
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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!