Péter Elek and László MárkusPéter Elek and László Márkus

Dept. Probability Theory and StatisticsDept. Probability Theory and Statistics

Eötvös Loránd UniversityEötvös Loránd University

Budapest, HungaryBudapest, Hungary

River Tisza and its aquiferRiver Tisza and its aquifer

Water discharge at VásárosnaményWater discharge at Vásárosnamény(We have 5 more monitoring sites)(We have 5 more monitoring sites)

from1901-2000from1901-2000

Empirical and smoothed seasonal Empirical and smoothed seasonal componentscomponents

Autocorrelation functionAutocorrelation function is slowly is slowly decayingdecaying

Indicators of long memoryIndicators 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)

2 4 6 8 10

logido,

-20

24

6

logr

star

1[2:

len]

Hurst=0.6603, std.err=0.0009

0 20 40 60 80

Crossing upper critical bound at 44

02

46

8

Vq

N

0 1 2 3 4

Slope = -0.2913945, Hurst estimate = 0.7913945

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

lVq

N

5 6 7 8 9 10

R/S: 0.6807567

34

56

7

Rp

erS

[1:(

sza

mo

l - 2

), 1

]

3 4 5 6

-2.0

-1.5

-1.0

-0.5

log

(szo

ras[

1:4

00

])

Nameny Variance plot

Estimated Hurst coefficient=0.73782

-8 -6 -4 -2 0

logfreq

-15

-10

-50

logs

pec

-15 -10 -5 0

GPHfreq

-15

-10

-50

logs

pec

-8 -6 -4 -2 0

logfreq

-15

-10

-50

GP

Hfre

q

0 500 1000 1500 2000 2500

fi

0.5

0.6

0.7

0.8

0.9

hu

rstc

fs

Nameny log-periodogram, GPH-Hurst estimation

Estimated Hurst coeff= 0.84796, (mean of [22:55])

Linear long-memory model : Linear long-memory model : fractional fractional ARIMA-processARIMA-process

(Montanari et al., (Montanari et al., Lago Maggiore, Lago Maggiore, 1997)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)

ttd BXBB )()1()(

Results of Results of fractional fractional ARIMAARIMA fitfit

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)

tt BXBBB )21.01()1()12.080.01( 34.02

Reconstruct the innovation from the Reconstruct the innovation from the fitted modelfitted model

The density of the innovationsThe density of the innovations

Reconstructed innovations are uncorrelatedReconstructed innovations are uncorrelated......

But not independentBut not independent

Simulations using i.i.d. innovationsSimulations using i.i.d. innovations

If we assume that innovations are i.i.d, we can If we assume that innovations are i.i.d, we can generate synthetic series:generate synthetic series:– Use resampling to generate synthetic innovations Use resampling to generate synthetic innovations – Apply then the linear filter Apply then the linear filter – Add the sesonal components to get a synthetic Add the sesonal components to get a synthetic

streamflow series streamflow series

But: these series do not approximateBut: these series do not approximate well well the high the high quantiles of the original seriesquantiles of the original series

But: they fail to catch the densities and But: they fail to catch the densities and underestimate the high quantiles of the underestimate the high quantiles of the

original seriesoriginal series

Logarithmic linear modelLogarithmic 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

But the backtransformed quantiles are clearly unrealistic.

Let’s have a closer look at innovationsLet’s 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 modelThe classical GARCH is far too heavy

tailed to our purposes

SimulationSimulation from the GARCH-process 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

Need a model „between”– i.i.d.-driven FARIMA-series and– GARCH(1,1)-driven FARIMA-series

General form of GARCH-models:

General form of GARCH-modelsGeneral form of GARCH-models

),( 211

2

ttt

ttt

f

Z

A smooth transition GARCH-A smooth transition GARCH-modelmodel

. :large for

, :small For

))exp(1(

21110

221

211

2110

221

211

2110

2

ttt

tttt

ttt

ttt

baa

bkaa

bkaa

Z

ACF of GARCH-residualsACF of GARCH-residuals

Results of simulationsResults of simulationsat Vat Váássáárosnamrosnaményény

Back to the original GARCH philosophyBack to the original GARCH philosophy

The above described GARCH model is somewhat The above described GARCH model is somewhat artificial, and hard to find heuristic explanations for it:artificial, and hard to find heuristic explanations for it:– why does the conditional variance depend on the why does the conditional variance depend on the innovationsinnovations of the linear filter? of the linear filter?

– in the original GARCHin the original GARCH-context-context the variance is the variance is dependent on the lagged values of the process itself.dependent on the lagged values of the process itself.

A pA possible solution: cossible solution: condition the variance on the lagged ondition the variance on the lagged discharge process instead !discharge process instead !

The fractional integration does not seem to be necessaryThe fractional integration does not seem to be necessary– almost the same innovations as from an ARMA(3,1)almost the same innovations as from an ARMA(3,1)– In extreme value theory long memoryIn extreme value theory long memory in linear models in linear models does not does not

make a difference make a difference

Estimated variance of innovationEstimated variance of innovationss plotted against the lagged dischargeplotted against the lagged discharge

SpectacularSpectacularlyly linear linear relationshiprelationship

This approves the new This approves the new modelling attemptmodelling attempt

Distorted at sitesDistorted at sites with with damming damming along Tisza along Tisza RiverRiver

  The variance is not conditional on the lagged innovation but it is conditional on the lagged water discharge.

ttt XseasonalARMAεZ

trendperiodic , discharge water synthetic)(

tt cX

1)(

0)(

),max()(

)(

2

1101102

11

t

t

ttt

ttt

q

iiti

p

iitititt

ZD

ZE

mmXmX

Z

bcXacX

Theoretical problems arise in the new modelTheoretical problems arise in the new model – Existence of stationary solutionExistence of stationary solution– Finiteness of all momentsFiniteness of all moments– Consistence and asymptotic normality of quasi max-likelihood Consistence and asymptotic normality of quasi max-likelihood

estimatorsestimators

Heuristically clearer explanation canHeuristically clearer explanation can be givenbe given– The discharge is indicative of the saturation of the watershedThe discharge is indicative of the saturation of the watershed– A saturated watershed results in more straightforward reach for A saturated watershed results in more straightforward reach for

precipitation to the river, hence an increase in the water supply.precipitation to the river, hence an increase in the water supply.– A saturated watershed gives away water quicker.A saturated watershed gives away water quicker.– The possible changes are greater and so is the uncertainty for the The possible changes are greater and so is the uncertainty for the

next discharge value.next discharge value.

An example: ZAn example: Ztt~N(0,1)~N(0,1)

c=20, a c=20, a11=0.95, =0.95, 00=1, =1, 11=2, m=1=2, m=1

Existence and moments Existence and moments of the stationary solutionof 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.

Sketch of the proof I.Sketch of the proof I. The process can be embedded into a (p+q)-dimensional Markov-

chain: Yt=AYt-1+Et

where Yt=(Xt-c, Xt-1-c,...Xt-p+1-c, εt, εt-1,..., εt-q+1) and Et=(εt, 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 V1 function with 0<<1, b< and C compact set

E( V(Y1) | Y0=y ) (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).

Sketch of the proof II.Sketch of the proof II.

In the given case:

if E(|Zt|m) is finite,

V(y) = 1 + ||QPy||mm will suffice

where:– B=PAP-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.

EstimationEstimation 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).

Estimation of the ARCH-equation in the case of Estimation of the ARCH-equation in the case of knownknown tt innovations innovations

(along the lines of Kristensen and Rahbek, 2005)(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 min>0 in the

whole parameter space.

)( 1102 mX

Z

tt

ttt

n

t t

ttnn n

LL1

2

22

10 2)log(

2

1)2log(

2

11),()(

α

Consistency of the estimatorConsistency of the estimator

By the ergodic theorem

It is easy to check that L(α)<L(α0) for all αα0 where α0 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.

),,(2

)log(2

1)2log(

2

1)()( 12

22.. ααα

tt

t

tt

san XlEELL

Asymptotic normality I.Asymptotic normality I.

Using the notations for the derivatives of the log-likelihood:

for the information matrix

and for the expected Hessian:

α

αα

)(

)( nn

LS

α

αα

2

2 )()(

nn

LH

α

αα

21

2 ),,()( tt Xl

EH

T

tttt XlXlEV

α

α

α

αα

),,(),,()( 11

Asymptotic normality II.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.

)ˆ()()()ˆ(0 00 ααααα nnnnnn HSS

))()()(,0(

)()()ˆ(10010

010

ααα

αααα

HVHN

SnHn nnnn

Estimation of the ARCH-equation Estimation of the ARCH-equation when when tt is not known 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:

)ˆ,...,ˆ,ˆ,..,ˆ(ˆˆ 11 qptt bbaa

0ˆ1 ..

1

22

san

tttn

0ˆ1 ..

1

22

san

ttt

n

Consistency in the case of estimated Consistency in the case of estimated innovationsinnovations

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 .

02

ˆ1)()(ˆ ..

12

22

san

t t

ttnn n

LL

αα

n

t t

ttn n

L1

2

22

2

ˆ)log(

2

1)2log(

2

11)(ˆ

α

Asymptotic normality in the case of Asymptotic normality in the case of estimated innovationsestimated 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.

0)()(ˆ .. sann SSn αα

0ˆ1 ..

1

22

san

ttt

n

0)()(ˆ .. sann HH αα

Estimation resultsEstimation results

Station m α0 α1

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)

How to simulate the residuals of How to simulate the residuals of the new GARCH-the new GARCH-typetype modelmodel

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 processThe simulation process

t

Zt

Xt

resampling and GPD

FARIMA filter

smoothed GARCH

Seasonal filter

Evaluating the modelEvaluating the model fit fit

Independence of residual series ACF, extremal clustering

Fit of probability density and high quantilesVariance – lagged discharge relationshipExtremal indexLevel exceedance timesFlood volume distribution

ACF of original and squared ACF of original and squared innovation series – residual seriesinnovation series – residual series

Results of Results of new new simulationssimulationsat Vat Váássáárosnamrosnaményény

Densities and quantiles at all 6 locations Densities and quantiles at all 6 locations

Reestimated (from the fitted model) Reestimated (from the fitted model) discharge-variance relationshipdischarge-variance relationship

Seasonalities of extremesSeasonalities of extremes

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

Extremal indexExtremal index to measure the clustering of high values 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%.

Estimated extremal indices displayedEstimated extremal indices displayed

Extremal indices in theExtremal indices in the discharge discharge dependentdependent GARCH model GARCH model

Level exceedanceLevel exceedance times 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.

Flood volume distributionFlood volume distributionThe match of the empirical and simulated flood

volume distributions also approve the good fit.

Multivariate modellingMultivariate 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)

Issues of joint modellingIssues 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

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

Displaying the nature of interdependenceDisplaying 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

12

A possible explanation of this type of interdependenceA 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 (correlation=0.48)

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

Thank you for your attentionThank you for your attention!!