Transcript How long does the river remember...?

László Márkus and Péter Elek
Dept. Probability Theory and Statistics
Eötvös Loránd University
Budapest, Hungary
River Tisza and its aquifer
Water discharge at Vásárosnamény
(We have 5 more monitoring sites)
from1901-2000
Empirical and smoothed seasonal
components
Autocorrelation function is slowly
decaying
Indicators of long memory

Nonparametric statistics
– Rescaled adjusted range or R/S
• Classical
• Lo’s (test)
• Taqqu’s graphical (robust)
– Variance plot
– Log-periodogram (Geweke-Porter Hudak)
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Linear long-memory model :
fractional ARIMA-process
(Montanari et al., Lago Maggiore, 1997)
 Fractional ARIMA-model:
(B)  (1  B)d X t  (B)  t
 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)
Results of fractional ARIMA fit
(1  0.80B  0.12B2 )  (1  B)0.34  X t  (1  0.21B)  t
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H=0.846 (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)
Reconstruct the innovation from the
fitted model
Reconstructed innovations are uncorrelated...
But not independent
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
But: they fail to catch the densities and
underestimate the high quantiles of the
original series
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
Simulation from the GARCH-process
 Simulations:
– Generate i.i.d. series from
the estimated GARCHresiduals
– Then simulate the
GARCH(1,1) process using
these residuals
– Apply the linear filter and
the seasonalities
 The simulated series are much
heavier-tailed than the original
series
A smooth transition GARCHmodel
 t   t Zt
 t2  a0  a1 (1  exp(k t21 ))  b1  t21
For  t21 small:  t2  a0  a1 k t21  b1  t21 ,
for  t21 large :  t2  a0  a1  b1  t21.
ACF of GARCH-residuals
Results of simulations
at Vásárosnamény
Back to the original GARCH philosophy
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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.
 Possible solution: condition the variance on the lagged
discharge process instead !
 Theoretical problems (e.g. on stationarity) arise but
heuristically clear explanation can be given more easily
Estimated variance of innovations
plotted against the lagged discharge
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Spectacularly linear
relationship
Distorted at sites with
damming
(lower row)
This motivates the next
modelling attempt
t   t Zt
  max(a0  a1  Qt 1 , var0 )
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Qt  syntheticwater discharge
Zt   t  FARIMA seasonal Qt
The variance is not conditional on the lagged innovation
but it is conditional on the lagged water discharge.
Estimation is carried out by
normal-based maximum likelihood.
(This is not uncommon in the GARCH-context,
even if the residuals are non-Gaussian. See McNeil and Frey, 2000)
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
The simulation process
resampling and GPD
Zt
GARCH-type model
t
FARIMA filter
Xt
Seasonal filter
Evaluating the model fit
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Independence of residual series
ACF, extremal clustering
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Fit of probability density and high quantiles
 Variance – lagged discharge relationship
 Extremal index
 Consistence of parameter estimates
ACF of original and squared
innovation series – residual series
Results of new simulations
at Vásárosnamény
Densities and quantiles at all 6 locations
Seasonalities of extremes
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The seasonal appearance
of the highest values
(upper 1%) of the
simulated processes
follows closely the same
for the observed one.
Estimated extremal indices displayed
Multivariate modelling
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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)
Issues of joint modelling
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Measures of linear interdependences (the crosscorrelations) 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
Basic statistics of
Tivadar (Tisza) and Csenger (Szamos)
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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
Displaying the nature of interdependence
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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, especially at
the upper right corner
(tail dependence)
– 2. at the upper left (and the lower
right) corner(s)
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Taking into account these properties is
crucial during joint simulation
The GARCH-residuals lack the second
type of irregularity
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A possible explanation of this type of interdependence
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The cond. variance process is
essentially common for the two
rivers (correlation = 0.84)
This gives a hint to explain the
interdependence of the
innovations:
– Generate two interdependent
residual series (correlation=0.48)
– Multiply by a common standard
deviation process
(distributed as Gamma)
– The obtained copula is very
similar to the observed copula of
the innovations
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This justifies the hypothesis that
the common variance causes the
interdependence of the given
type
Thank you for your attention!