A HYBRID MODELLING FRAMEWORK FOR FORECASTING MONTHLY RESERVOIR INFLOWS

Transcription

1 J. Hydrol. Hydromech., 56, 2008, 3, A HYBRID MODELLING FRAMEWORK FOR FORECASTING MONTHLY RESERVOIR INFLOWS MAGDA KOMORNÍKOVÁ 1), JÁN SZOLGAY 2), DANKA SVETLÍKOVÁ 2), DANUŠA SZÖKEOVÁ 1), STANISLAV JURČÁK 2) 1) Dept. of Mathematics, Faculty of Civil Engineering, Slovak University of Technology, Slovakia; mailto: magda@math.sk, szoke@math.sk 2) Dept. of Land and Water Resources Management, Faculty of Civil Engineering, Slovak University of Technology, Slovakia; mailto: jan.szolgay@stuba.sk, svetlikova@svt.stuba.sk, jurcak@stuba.sk The paper approaches problem of the flow forecasting for the Liptovská Mara reservoir with a hybrid modelling approach. The nonlinear hybrid modelling framework was investigated under the specific physiographic conditions of the High Core Mountains of Slovakia. The mean monthly flows of rivers used in this study are predominantly fed by snowmelt in the spring and convective precipitation in the summer. Therefore, their hydrological regime exhibits at least two clear seasonal patterns, which provide an intuitive justification for the application of nonlinear regime-switching time series models. Differences in the prevailing geology, orientation of slopes and their exposure to atmospheric circulation for the right and left-sided tributaries of the Váh River indicate that the hydrological regime of mean monthly discharge time series in this area with respect to seasonality and cyclicity may differ, too. Therefore, a simple deterministic water balance scheme was set up for estimating the reservoir inflow from the left and right-sided tributary flows separately. It consists of the linear combination of the measured tributary flows and estimated ungauged tributary flows. The contributions of the ungauged catchments were estimated from flows from gauged tributaries with similar physiographic conditions by weighting these by the ratio of the catchment areas. Separate nonlinear regime-switching time series models were identified for each gauged tributary. The forecasts of the tributaries were combined into a hybrid forecasting model by the water balance model. The performance of the combined forecast, which could better include the specific regime of the time series of tributaries, was compared with the single forecast of the overall reservoir inflow in several combinations. KEY WORDS: Time Series, Nonlinear Time Series Models, SETAR Model, Mean Monthly Flow, Hybrid Forecasting. Magda Komorníková, Ján Szolgay, Danka Svetlíková, Danuša Szökeová, Stanislav Jurčák: VYUŽITIE HYBRIDNÉHO PRÍSTUPU NA PREDPOVEDANIE A MODELOVANIE PRIEMERNÝCH MESAČNÝCH PRIETOKOV. J. Hydrol. Hydromech., 56, 2008, 3; 45 lit., 7 obr., 11 tab. V štúdii sme porovnávali kvalitu predpovede viacerých lineárnych a nelineárnych modelov časových radov pri predpovedaní prítokov do nádrže Liptovská Mara. Testovali sme výkonnosť modelov ARMA, SETAR na samotnej rieke Váh a v kombinácii jej prítokov do nádrže Liptovská Mara. Ďalej bol uplatnený jednoduchý deterministický model vodnej bilancie pre prítok do nádrže, ktorý pozostáva z lineárnej kombinácie meraných prietokov prítokov Váhu vážených plochou subpovodia. Výber analogónov sa vykonal vzhľadom na podobnosť fyzicko-geografických podmienok v meraných a nemeraných subpovodiach. Modely typu ARMA a SETAR boli zostavené pre každý prítok osobitne a predpovede prietokov na prítokoch boli skombinované modelom vodnej bilancie a do predpovede celkového prítoku do nádrže. Kombinovanú hybridnú predpoveď (stochasticko-deterministická), zachovávajúcu špecifický režim prítokov a vodnej bilancie v povodiach, sme porovnali s predpoveďou celkového prítoku do nádrže získanou pomocou modelov identifikovaných na hlavnom toku. KĽÚČOVÉ SLOVÁ: časové rady, nelineárne modely časových radov, model SETAR, priemerné mesačné prietoky, hybridné modely. 145

2 M. Komorníková, J. Szolgay, D. Svetlíková, D. Szökeová, S. Jurčák 1. Introduction Approaches used for monthly river flow forecasting cover a wide range of different methods from deterministic (usually conceptual water balance models) to time series modelling approaches. Owing to the complexity of the meteorological and hydrological processes involved, reliable monthly and seasonal forecasts remain a difficult task for physically based monthly watershed models because quantitative information on future values of input processes is required (Todini, 2004). Therefore, in operational scheduling of monthly reservoir releases time series models are also used for flow forecasting for reservoir operation. Linear time series models (e.g., Box and Jenkins, 1970) became popular in hydrology in this respect, especially for the generation of monthly and annual flows the design and optimization of reservoirs (an extensive review of these can be found, e.g., Shrikanthan and McMahon, (2002)). Time series model were extensively used in the optimisation of water resources systems in the Czech Republic (e.g. Kos (1979), Nacházel (1998), Votruba, Heřman et al. (1993), Starý, Doležal, Králová, (2004), Nacházel, Zezulák, Starý, (2004)). Linear time series models have been also used for forecasting river flows with weak nonlinearities such as annual flows. Recently, e.g., Pekárová, Miklánek and Pekár (2003, 2007) and Pekárová and Pekár (2006), studied fluctuations of river runoff and reported on predictions of annual runoff, which can be regarded as nearly linear (Wang et al., 2005b) with autoregressive models. For strongly nonlinear and nonstationary processes in hydrology, models originating from the classical Box-Jenkins methodology have to be reconsidered and substituted by approaches that respect these properties. In the time series domain, several such models were proposed recently in financial mathematics, such as TAR (Threshold Autoregressive) and SETAR (Self-Exciting Threshold Autoregressive) or STAR (Smooth Transition Autoregressive) models (e.g., Clements and Smith, 1997; Hansen, 1997; Granger and Teräsvirta, 1993; Lin and Granger, 1994; Tjøstheim, 1994; Tong, 1990). There have also been attempts to conduct a nonlinear analysis in the time series domain in hydrology (e.g., Tsai, et al., 2000; Tamea, et al., 2005; Amendola, 2003; Wang et al., 2005a,b, Komorník et al., 2006). Several papers were devoted to testing nonlinearity in particular time series at different time scales; e.g., Wang et al. (2005b) focused on general nonlinearity and low-dimensional chaos; Tsai and Chan (2000) developed a test to detect SETAR bilinear and nonlinear continuous-time nonlinearity, Chen and Rao (2003) attempted to detect nonlinearity composed of linear stationary segments in streamflow, temperature, precipitation and Palmer drought index series; Amendola (2003) tested the forecasting performance of regime-switching models in hydrological time series; Wong et al. (2005) proposed a method for detecting change points in hydrological time series; and Szökeová, Komorník and Komorníková (2006) investigated the adequacy of regime-switching models based on aggregation operators. Applications have seemed to offer the potential of explaining the nonlinear behaviour of the modelled time series. However, as pointed out by Amendola (2003), nonlinear time series structures have often led to good fitting performances, but the good fitting results of nonlinear models do not guarantee an equally good forecasting performance (Chatfield, 2001), and this has to be tested on a case-by-case basis. This paper focuses on the development of a forecasting scheme for monthly flows based on regime-switching SETAR models and the evaluation of their predictive performance compared to linear models in the High Core Mountains region of Slovakia. Considerable literature has also accumulated over the years regarding the combination of forecasts from various sources. Some conclusions which can be drawn from these studies are that the accuracy of forecasts can be substantially improved through a combination of multiple individual forecasts and that simple combination methods often work reasonably well. See, e.g., Clemen (1989), who also provides a review and annotated bibliography of the literature on this subject. Zou and Yang (2004) give additional references. The possibility of combining the advantages of both deterministic and time series/datadriven models has also been proposed in various applications in meteorology and hydrology (see, e.g., Singh, 1995; Reyes-Aldasoro et al., 1999; Ganguly and Bras, 2001), where the term hybrid modelling approach was introduced to describe the approach. In the paper several hybrid flow forecasting models are suggested and compared with respect to their ability to forecast monthly flows into the Lip- 146

3 A hybrid modelling framework for forecasting monthly reservoir inflows tovská Mara reservoir. The hybrid modelling framework tested consists of a deterministic part and of a combination of linear and nonlinear time series models. The deterministic part of the hybrid modelling framework consists of a simple water balance scheme, which is set up for estimating the main reservoir inflow by the weighted combination of its measured and ungauged tributary flows. These are estimated by hydrological analogy with two alternatives. First, the ungauged tributary flows are estimated from the measured tributary flows weighted by the catchment area, taking into account the similarity of the physiographic conditions in the catchments. In the second alternative, the bias in the estimated overall reservoir inflow of this scheme is evaluated and added to the model. Since the hydrological regime of the tributaries of the main reservoir inflow exhibits clear seasonal patterns, and the left and right-side tributaries of the main river spring in different physiographic conditions, the nonlinear regime-switching SETAR time series models are proposed for the data-driven part of the hybrid scheme for each tributary, respectively. Data-driven forecasts of the tributary flows are then combined by the water balance model to the hybrid forecast of the overall reservoir inflow. The combined hybrid data driven deterministic forecast, which is expected to preserve the specific regime of the tributaries and the water balance in the catchments, is compared by the data-driven forecasts set up for the overall reservoir inflow on the main river. The paper is organized as follows: After a general introduction to the pilot sites in the High Core Mountains region of Slovakia, where the Liptovská Mara reservoir is situated, the description of the deterministic part of the hybrid framework follows. Next, the regime-switching time series methods suggested to be used for the forecasting of monthly flows are described. The paper continues with the application of the methods to a fairly realistic situation. First, the capabilities of the regime-switching models are tested for the main reservoir inflow from the Váh River. Next, six hybrid models are set up for the reservoir inflow, and the combined forecasts are compared through their predictive performance. Final comments and recommendations conclude the paper. 2. The pilot sites and the water balance model The High and Low Tatra mountains and the Váh River with its tributaries were selected as the pilot areas for testing the approaches. The Váh River is the main source of water for the Liptovská Mara reservoir. Forecasting of mean monthly reservoir inflows and mean monthly river flows is therefore of practical interest. The Váh catchment is a part of the High Core Mountains which comprise the inner arch of the West Carpathian Belt. The high core mountains in the catchment are composed of the Západné (Western), Vysoké (High) and the Nízke (Low) Tatry (Tatras). The Tatra Mountains belong among the highest mountains in Slovakia, with elevations rising from 800 to 2600 m above sea level. The longterm mean annual precipitation amounts vary from 900 mm in the lowest parts of the mountains to 2000 mm and more in the highest elevations. The average duration of the snow cover can reach approximately 200 days a year. The mean monthly temperature in January ranges from between 10 to 6 o C and in July between 11 and 15 o C. Due to the strong altitudinal zonality of runoff generation in Slovakia, the whole region is usually considered homogeneous with respect to the runoff regime in general (Hlavčová and Čunderlík, 1998). A cluster analysis of Slovak catchments based on physiographic and climatic variables indicated the presence of homogeneous pooling group located in this region (Kohnová, 1998). On the other hand, the left and right-hand tributaries of the main river, the Váh, come from different mountain ranges, which partly differ in their geology, orientation of slopes and exposure to atmospheric forcing with respect to runoff generation. It is therefore of interest to investigate the structural properties of mean the monthly discharge time series of the main river and its tributaries with respect to their nonlinear properties separately. A total of six catchments in this region were selected for the case study. These catchments are as follows: the Biely Váh River at Východná, the Belá River at Liptovský Hrádok, the Čierny Váh River at Čierny Váh, the Boca at Kráľova Lehota, the Štiavnica at Liptovský Ján and the Váh, the main river at Liptovský Mikuláš. Their physiographic and basic hydrologic characteristics are presented in Tab. 1, and their locations are shown in Fig. 1. The flow regime of these rivers has an alpine character with the highest rates of occurrence of annual maximum mean monthly flows in the spring 147

4 M. Komorníková, J. Szolgay, D. Svetlíková, D. Szökeová, S. Jurčák T a b l e 1. Physiographic catchment characteristisc, descriptive statistics of test data and significant periods for their periodic components. T a b u ľ k a 1. Základné fyzicko-geografické a štatistické charakteristiky povodí. River Site Area Elevation Slope Mean flow St. deviation Significant periods St.deviation of residuals [km 2 ] [m a.s.l.] [%] [m 3 s -1 ] [m 3 s -1 ] [month] [m 3 s -1 ] Váh Liptovský Mikuláš , Belá Liptovský Hrádok , 4, Biely Váh Východná , 6, Čierny Váh Čierny Váh , 14.4, Boca Kráľova Lehota , Štiavnica Liptovský Ján , Fig. 1. The location of the catchments in test region. Obr. 1. Poloha povodí v pilotnom území. 148

5 A hybrid modelling framework for forecasting monthly reservoir inflows and the annual minimum monthly flows in the winter. The spring flows are predominantly fed by snowmelt. Relatively high mean monthly flows also occur in June, July and August, which are mainly caused by convective precipitation. The low-flow period starts in September and in general lasts till spring. The length of the observations available for this study in all the individual catchments was 36 years ( ). Thirty-two years were used for the identification and parameter estimation of the datadriven time series stochastic models; the rest of the data was used for the verification and comparison of the forecasting performance of the models. A simple, deterministic water balance scheme for estimating the monthly discharges of the Váh River at Liptovský Mikuláš from gauged and ungauged tributary flows is based on a hydrological analogy. In this method, which is widely used for estimating of flows at ungauged sites, an analogue basin with gauged flows is chosen which has similar physiographic characteristics to the ungauged site. Here, simply the nearest (adjacent) gauged catchments to the ungauged basins were used as the analogs. The measured tributary flows weighted by the ratio of the respective catchment areas were attributed to each ungauged tributary, and the sum of all the flows as taken as the estimate of the flow at Liptovský Mikuláš. The catchment areas of the gauged basins and their extended areas, including the analogs which were used in the weighting scheme, are as follows: Biely Váh / km 2, Belá / km 2, Čierny Váh / km 2, Boca / km 2, and Štiavnica 61.79/ km 2. The measured discharges and flows estimated by the water balance scheme were compared in the period between 1966 and 1997, as seen in Fig. 2. A comparison of the forecasted and measured values showed that the water balance model is biased by 1.98 m 3 s -1 and that the Mean Absolute Error (MAE, see Eq. (13)) and Root Mean Square Error (RMSE, see Eq. (14)) of the simulation were 2.55 and 3.23 m 3 s -1 respectively. The water balance model was used in two scenarios in the hybrid modelling framework (see Section 5): - In the first scenario (Scheme 1) it was assumed that the flows measured at Liptovský Mikuláš do not exist and that overall reservoir inflow has to be estimated by the water balance model established by a hydrological analogy on the basis of the known tributary flows. In this model the bias in the estimation of the flow at Liptovský Mikuláš was accepted. This scenario was used to indicate how the proposed hybrid forecasting framework would perform on a main river. - In the second scenario (Scheme 2), the bias was considered be known and was added to the estimate of the flows at Liptovský Mikuláš. This scenario represents a situation, when the combined forecast, which accounts for the specific regime of the tributaries and the water balance in the catchments is believed to be a better model. Fig. 2. Comparison of simulated and measured mean monthly flows in Liptovský Mikuláš in the validation period (right) and residuals of the simple water balance model of the Váh River in Liptovský Mikuláš from the period (simulations dashed line, measurements solid line). Obr. 2. Porovnanie simulovaných a meraných priemerných mesačných prietokov vo validačnom období a rozdiel medzi vypočítaným a odhadnutým priemerným mesačným prietokom Váhu pre bilančný model prietokov v Liptovskom Mikuláši v období (simulácia čiarkovaná a merané prietoky plná čiara). 149

6 M. Komorníková, J. Szolgay, D. Svetlíková, D. Szökeová, S. Jurčák for the reservoir inflow than the data-driven model of the Váh River itself (in this case, the forecasts of the tributary flows are combined by the water balance model into the overall forecast of the reservoir inflow). 3. Regime-switching time series models The modelled monthly time series are considered to be composed of following components: 1. trend, 2. seasonal component, 3. cyclical component, 4. irregular component. The first three ( systematic ) components can be eliminated by generally know approaches. The main interest here is devoted to the irregular component, which will be analyzed using nonlinear regime-switching models. We denote here the univariate time series of interest as y t, which is the residual left after the removal of the systematic components. The variable y t is observed for t = 1, 2,, n (where n is the total number of observations in a time series and is called the length of the time series), while we assume that the initial conditions or pre-sample values y 0, y -1,, y 1-p are available. We denote by Ω t-1 the history or information set at time t-1, which contains all the available information that can be exploited for forecasting the future values y t, y t+1,. In general, any time series y t can be thought of as being the sum of two parts: what can and what cannot be predicted using the knowledge from the past as gathered in Ω t-1. That is, y t can be interpreted as y t = E[y t Ω t-1 ] + ν t, (1) where ν t is called the unpredictable part with E[ν t Ω t-1 ] = 0, (see, e.g., Franses and van Dijk, 2000). A natural extension to the linear approach to modelling time series (e.g. Box and Jenkins, 1970) is to consider that certain properties of a time series, such as its mean, variance and autocorrelation, are different in the various regimes of the modelled processes. We restrict our attention here to models that assume that in each of the regimes, a linear autoregressive model (AR) can adequately describe the dynamic behaviour of the time series. Tong (1983) initially proposed this Threshold Autoregressive (TAR) model (see also Tong and Lim, 1980 and Tsay, 1989). It assumes that a regime that occurs at a time t can be determined by an observable threshold variable q t relative to a threshold value, which we denote here as c. Assume that the observed data are (y 1,, y n ). Let: ( ) x = (1, y,..., y ) and φ = φ, φ, L, φ ti, t 1 t pi i 0i 1i pi i for i = 1, 2. The TAR model is linear within a regime, but liable to move between regimes as the process crosses the threshold. The two-regime Threshold Autoregressive (TAR) model with the regimes AR(p 1 ) and AR(p 2 ) takes the form: y t = xt,1φ1 Ι [q c] + xt,2φ2 Ι [q t > c] + ε t, (3) where Ι [A] an indicator function with Ι [A] = 1 if event A occurs and Ι [A] = 0 otherwise, q t the transition variable, ε t i.i.d. (0, σ 2 ). Identification of the appropriate model orders p 1, p 2 and estimation of the threshold c and of the AR coefficients in the two regimes of the TAR model can be done with the help of information criteria, e. g.,: - the Akaike information criterion: 2 2( p1+ p2) AIC(p 1, p 2 ) = ln ˆ σ +, n - the Schwarz information criterion: 2 ( p1+ p2) ln n SIC(p 1, p 2 ) = ln ˆ σ +, n - the Hannan-Quinn information criterion: 2 2( p1+ p2) ln(ln n) - HQC(p 1, p 2 ) = ln ˆ σ + n (for details, see (Liew and Chong, 2003)), which are minimized. SETAR is a special case of the TAR models, where the threshold variable q t is taken to be a lagged value of the time series itself: q t = y t-d for a certain integer d > 0 (see, Petruccelli and Woolford, 1984; Chen and Tsay, 1991). For example, in a two-regime case with AR(p 1 ) and AR(p 2 ), the model is y t = (φ 0,1 + φ 1,1 y t φ p,1 yt p ) Ι [y t d c] p y 2 t p2 + (φ 0,2 + φ 1,2 y t φ,2 ) Ι [y t d > c] + ε t.. (4) 150

7 A hybrid modelling framework for forecasting monthly reservoir inflows Liptovský Mikuláš/Váh Liptovský Hrádok/Belá Východná/Biely Váh Čierny Váh/ Čierny Váh Kráľová Lehota/Boca Liptovský Ján/Štiavnica Fig. 3. Diagramm of the counts of switching between regimes of the SETAR models during model identification. The bar code represents the two regimes of the model. Obr. 3. Diagram znázorňujúci prepínanie režimov modelov SETAR v identifikačnom období. Čiarový kód znázorňuje jednotlivé režimy modelu. The least squares estimate of c can be obtained by minimizing: cˆ= arg min ˆ2 σ ( c) c C, (5) 2 1 n c = εt c n t= 1 where ˆ ( ) ˆ ( ) 2 σ is a residual variance, and C denotes the set of all the threshold values allowed. A popular choice for C is C = = {c y c y ( ) } where ( π0 n 1 ) ( ( 1 π 0)( n 1) ) 151

8 M. Komorníková, J. Szolgay, D. Svetlíková, D. Szökeová, S. Jurčák y (0),, y (n-1) denotes the order statistics of the threshold variable y t-d, y (0) y (n-1), and [.] denotes the integer part. A safe choice for π 0 appears to be 0.15 (Franses and van Dijk, 2000). Testing the linearity against the alternative of a SETAR model is discussed in Chan (1990, 1991), Chan and Tong (1990), and Hansen (1997, 2000). In the test used in this paper the estimates of the SETAR model are used to define a likelihood ratio or F-statistics, which tests the restrictions as given by the null hypothesis, that is % 2 ˆ 2 () ˆ = 2 σ % Fc n σ σ, (6) 2 where σ% is an estimate of the residual variance under the null hypothesis of linearity. As ĉ minimizes the residual variance over the set C, F ĉ is ( ) equivalent to the supremum over this set C of the pointwise test-statistics F(c) with an asymptotic χ 2 distribution with p + 1 degrees of freedom. The distribution of F ( is then nonstandard. Because ĉ) the exact form of the dependence between the different F(c) s is difficult to analyze, the critical values are most easily determined by means of simulation (see Hansen, 1997, 2000, for more details). The applicability of a particular type of nonlinear time series model for a particular problem in hydrology has to be proved and tested case by case. In this paper regime-switching nonlinear models belonging to the TAR class, namely, the SETAR models, are examined with respect to their abilities to forecast monthly runoff. The theoretical aspects of forecasting with the TAR class of models have not been studied in the paper, since it is more practice oriented. Computing forecasts from nonlinear models is not such an easy task, and when the forecast horizon is longer than one time step, this becomes even more complicated. The good fitting results of nonlinear models do not guarantee an equally good forecasting performance (Chatfield, 2001), due to the current position in the state-space and to some factors which are beyond the variable specified in the prediction (Amendola, 2003). Computing point forecasts from nonlinear models is much more complicated than from a linear model (Box and Jenkins, 1970; Franses, 1998). Consider the case where y t is described by the general nonlinear autoregressive model y t = F(q t-1 ; θ) + ε t (7) for some nonlinear function F(q t-1 ; θ). The optimal h-step-ahead forecast of y t+h at time t is given by y ˆt+ h t = E[y t+h Ω t ], (8) where Ω t denotes the history of the time series up to and including the observation at time t. Using (4) and the fact that E[ ε t+1 Ω t ] = 0, the optimal 1- step-ahead forecast is y + = E[y t+1 Ω t ] = F(q t ; θ). (9) ˆt 1 t When the forecast horizon is longer than 1 period, the problem becomes more complicated, because in general, the linear conditional expectation operator E cannot be interchanged with the nonlinear operator F, that is E[F(.)] F (E[.]). (10) Thus the expected value of a nonlinear function is not equal to the function evaluated at the expected value of its arguments. Several methods have been developed to obtain more adequate multiple-step-ahead forecasts. An alternative approach to computing multiple-stepahead forecasts is to use Monte Carlo or bootstrap methods to approximate the conditional expectation (8). The h-step-ahead Monte Carlo forecast is given by ( ε θ ) 1 L yˆt+ h t F yˆ t+ h 1 t i; L i= 1 = +, (11) where L is some large number and the ε i are drawn from the presumed distribution of ε t+1. The bootstrap forecast is very similar, the only difference being that the residuals from the estimated model ˆε, t = 1,, n are used, t ( ˆ ε θ ) 1 L yˆt+ h t F yˆ t+ h 1 t i; L i= 1 = +. (12) In this paper the Monte Carlo method has been applied. For a comparison of the forecasting performance of alternative models, the Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) were applied: 1 P P t= 1 RMSE = ( ˆ ) 2 yt, (13) yt 152

9 A hybrid modelling framework for forecasting monthly reservoir inflows MAE = 1 P P t= 1 yˆ t yt, (14) where P is the number of forecast periods and the predicted value for yt. The time series models described above were applied to data describing the mean monthly flows on the Boca, Belá, Biely Váh, Čierny Váh, Štiavnica and Váh Rivers. The data from 32 years ( ) have been used for model building. Subsequent data from 4 years ( ) served for testing the quality of the prediction. In all the cases, we first have determined and removed the systematic components of the time series and modelled the remaining residua only. Harmonic regression was used instead of the more standard lag 12 differencing operator. We preferred this approach, since it preserves the complete information and allows for physical interpretation (which is not always the case when applying differentiation). Concerning the systematic components, we got these results: - In all cases, no trend was detected. - By each river, the significant period for the seasonal component is L = 12, as verified by means of the correlogram. - For the cyclical component, we determined by means of spectral analysis significant frequencies (significance was tested by the Fisher test). Descriptive statistics of the test data and significant periods are included in Tab. 1. The periods of 6, 4 and 3 months can be hydrologically explained by the grouping of high and low runoff periods in the series, caused by the mixed snowmelt and convective precipitation fed runoff. The major part of the modelling work was devoted to the residua modelling. Although all the rivers were analyzed exhaustively, the details of the all models tested are not given here. Here, the results of the best-fitting models will be reported. We started with the linear ARMA model using the standard model identification procedure (Box and Jenkins 1970) and diagnostic checking by the Portmanteau test. The results are given in Tab. 2. As can be seen from the values of the Portmanteau statistics in Tab. 2, the ARMA models do not adequately describe the modelled process in general, except of the flows of the Biely Váh and the Boca Rivers. The standard deviation of the residuals decreased in all the rivers, so the models represent an improvement versus the no model case. ŷ t In the next step the SETAR models were tested on all the rivers. By applying the AIC and BIC information criteria for the linear models, see, e. g., Franses and van Dijk, 2000, we determined an appropriate order for the AR models. After the estimation of parameters of the SETAR (p 1, p 2, d) c models for p 1, p 2, d 2 (and with step max cmin 100 when estimating the best fitting threshold parameter c), we tested linearity (besides p = 1 we also considered p = 2) against the alternative of a SETAR model with a heteroscedasticity-robust variant of the LM test, by applying the distribution derived by Hansen (1997). The null hypothesis of the linearity can be rejected at conventional significance levels (for all 6 rivers) for several combinations of p 1, p 2 and d. The best SETAR models were chosen based on the minimal p-values and on the minimal values of the AIC, SIC and HQC information criteria for the TAR models (we also took into account the threshold value c to be in the middle of the data, if possible). In Tab. 3 the p 1, p 2, d, c, p-value and the residual variance are given for the best SETAR models for all 6 rivers. Concerning the diagnostic checking, after modelling the residuals, the remainder was again tested for serial correlation and remaining non-linearity. In all the cases our models were accepted. The standard deviation of the residuals decreased again (see Tab. 3) versus the ARMA models, so the SETAR models represent a further improvement in modelling the residuals in all the cases. The last two columns in Tab. 3 contain the counts of switching between the regimes during the model identification; it can be seen that both regimes were used by the models, and Fig. 3 depicts the model state runs. It can be seen that the left and right-sided tributaries exhibit clearly similar patterns. This provides the intuitive justification for the application of nonlinear two-regime models for the modelling and forecasting of these time series. Further, it can be seen that the pattern attributed to the flows in Liptovský Mikuláš does not belong to these groups. This can be regarded as a motivation to test the performance of the combined forecasts of the tributaries against the at-site forecast, which may better reflect the peculiarities of the mixture of both distinct switching regimes. 153

10 M. Komorníková, J. Szolgay, D. Svetlíková, D. Szökeová, S. Jurčák T a b l e 2. Results of the Portmanteau test for the applicability of linear models. T a b u ľ k a 2. Výsledky testovania vhodnosti lineárnych modelov. Portmanteau statistics Critical value St. deviation of residuals [ ] [ ] [m 3 s -1 ] Váh Belá Biely Váh Čierny Váh Boca Štiavnica T a b l e 3. Parameters of the best SETAR models. T a b u ľ k a 3. Parametre najlepších SETAR modelov. p 1 p 2 d c p-value St.deviation of residual Regime counts Regime counts [ ] [ ] [ ] [ ] [ ] [m 3 s -1 ] 1 st regime 2 nd regime Váh Belá Biely Váh Čierny Váh Boca Štiavnica The hybrid modelling framework and the model comparison Two scenarios described as Scheme 1 and Scheme 2 in Section 2 were used to set up the following hybrid modelling framework: - Hybrid Model 1 consisting of Scheme 1 and the SETAR models for the tributaries, - Hybrid Model 2 consisting of Scheme 2 and the SETAR models for the tributaries, - Hybrid Model 3 consisting of Scheme 1 and the combination of the ARMA and SETAR models for the tributaries according to the results of the Portmanteau test, - Hybrid Model 4 consisting of Scheme 2 and the combination of the ARMA and SETAR models for the tributaries according to the results of the Portmanteau test, - Hybrid Model 5 consisting of Scheme 1 and the combination of the best performing linear and nonlinear models, - Hybrid Model 6 consisting of Scheme 2 and the combination of the best performing linear and nonlinear models. All six hybrid schemes were also compared to the two reference cases, namely the ARMA and SETAR forecasts of the flows of the Váh River at Liptovský Mikuláš. Next we present an overview of the predictive performance of all six Hybrid Models. Data from 4 years ( ) served for testing the quality of the prediction. It has to be noted that after a period of two average years with fairly normal within the year distribution of mean monthly flows, two consecutive years with high monthly flows were observed (2000, 2001). These represented a special challenge in testing the performance of the models, especially when the forecast lead time grows and clearly influences the numerical values of the results. One-, 3-, 6- and 12-step ahead forecasts were computed, and in each case the prediction performance measures were described by the RMSE and MAE for the original discharge data (i. e., after summing the residual forecasts with the removed systematic components). The statistical characteristics of the predictions are described in Tabs. 4, 5, 6, 7, 8, 9, 10 and 11, and the predicted and measured flow for selected cases are shown in Figs. 4, 5, 6 and 7. As a basis for comparison of the predictive performance of the models, the one-, 3-, 6- and 12-step ahead forecasts of both the ARMA and SETAR models at Liptovský Mikuláš were used (see Tabs. 4 and 5). The forecasts of the ARMA model were computed in each case, despite the fact, that the linear model was not regarded as satisfactory in general by the Portmanteau test. In this case these 154

11 A hybrid modelling framework for forecasting monthly reservoir inflows were used as a criterion for the performance of the nonlinear predictions, since it is known from the literature that nonlinear time series structures have often led to good fitting performances (as in our case, compare the standard deviation of the residuals in Tab. 2 and 3); however these good fitting results do not guarantee an equally good forecasting performance (Chatfield, 2001; Amendola, 2003). It can be seen from Tabs. 4 and 5 that in general, the linear forecasts are slightly better, that is, in line with the findings in the literature. However, one has to consider that the SETAR models may not necessarily be the best nonlinear models for the given time series; other models, such as GARCH or STAR, could perform better. These models were not considered here; they will be explored in a follow-up study. T a b l e 4. The prediction errors for 1-, 3-, 6- and 12-step-ahead forecasts of the linear reference model for Liptovský Mikuláš. T a b u ľ k a 4. Chyby predpovede pre lineárny referenčný model v Liptovskom Mikuláši. 1-step-ahead forecast 3-step-ahead forecast 6-step-ahead forecast 12-step-ahead forecast [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] MAE RMSE MAE RMSE MAE RMSE MAE RMSE ARMA T a b l e 5. The prediction errors for 1-, 3-, 6- and 12-step-ahead forecasts of the SETAR reference model for Liptovský Mikuláš. T a b u ľ k a 5. Chyby predpovede pre referenčný model SETAR v Liptovskom Mikuláši. 1-step-ahead forecast 3-step-ahead forecast 6-step-ahead forecast 12-step-ahead forecast [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] MAE RMSE MAE RMSE MAE RMSE MAE RMSE SETAR T a b l e 6. The prediction errors for 1-, 3-, 6- and 12-step-ahead forecasts of Hybrid Model 1 for Liptovský Mikuláš. T a b u ľ k a 6. Chyby predpovede pre model Hybrid 1 v Liptovskom Mikuláši. Hybrid Model 1 1-step-ahead ahead forecast 3-step-ahead ahead forecast 6-step-ahead ahead forecast 12-step-ahead ahead forecast MAE RMSE MAE RMSE MAE RMSE MAE RMSE [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] T a b l e 7. The prediction errors for 1-, 3-, 6- and 12-step-ahead forecasts of Hybrid Model 2 for Liptovský Mikuláš. T a b u ľ k a 7. Chyby predpovede pre model Hybrid 2 v Liptovskom Mikuláši. Hybrid Model 2 1-step-ahead forecast 3-step-ahead forecast 6-step-ahead forecast 12-step-ahead forecast MAE RMSE MAE RMSE MAE RMSE MAE RMSE [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] T a b l e 8. The prediction errors for 1-, 3-, 6- and 12-step-ahead forecasts of the Hybrid 3 model for Liptovský Mikuláš (mixed ARMA (Boca and the Biely Váh) and SETAR models (the remaining catchments) as suggested by the Portmanteau test)). T a b u ľ k a 8. Chyby predpovede pre model Hybrid 3 v Liptovskom Mikuláši (kombinácia modelov ARMA (Boca a Biely Váh) a SETAR (ostatné povodia) podľa testu Portmanteau)). Hybrid Model 3 1-step-ahead forecast 3-step-ahead forecast 6-step-ahead forecast 12-step-ahead forecast MAE RMSE MAE RMSE MAE RMSE MAE RMSE [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ]

12 M. Komorníková, J. Szolgay, D. Svetlíková, D. Szökeová, S. Jurčák T a b l e 9. The prediction errors for 1-, 3-, 6- and 12-step-ahead forecasts of the Hybrid 4 model for Liptovský Mikuláš (mixed ARMA (Boca and the Biely Váh) and SETAR models (the remaining catchments as suggested by the Portmanteau test)). T a b u ľ k a 9. Chyby predpovede pre model Hybrid 4 v Liptovskom Mikuláši (kombinácia modelov ARMA (Boca a Biely Váh) a SETAR (ostatné povodia) podľa testu Portmanteau)). Hybrid Model 4 1-step-ahead forecast 3-step-ahead forecast 6-step-ahead forecast 12-step-ahead forecast MAE RMSE MAE RMSE MAE RMSE MAE RMSE [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] T a b l e 10. The prediction errors for 1-, 3-, 6- and 12-step-ahead forecasts of Hybrid model 5 at Liptovský Mikuláš (the mixed best performing ARMA (Čierny Váh, Boca and Štiavnica) and SETAR models (the remaining catchments)). T a b u ľ k a 10. Chyby predpovede pre model Hybrid 5 v Liptovskom Mikuláši (kombinácia najlepších modelov ARMA (Čierny Váh, Boca a Štiavnica) a SETAR (ostatné povodia)). Hybrid Model 5 1-step-ahead forecast 3-step-ahead forecast 6-step-ahead forecast 12-step-ahead forecast MAE RMSE MAE RMSE MAE RMSE MAE RMSE [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] T a b l e 11. The prediction errors for 1-, 3-, 6- and 12-step-ahead forecasts of Hybrid model 6 at Liptovský Mikuláš (the mixed best performing ARMA (Čierny Váh, Boca and Štiavnica) and SETAR models (the remaining catchments)). T a b u ľ k a 11. Chyby predpovede pre model Hybrid 6 v Liptovskom Mikuláši (kombinácia najlepších modelov ARMA (Čierny Váh, Boca a Štiavnica) a SETAR (ostatné povodia). Hybrid Model 6 1-step-ahead forecast 3-step-ahead forecast 6-step-ahead forecast 12-step-ahead forecast MAE RMSE MAE RMSE MAE RMSE MAE RMSE [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] [m 3 s -1 ] Fig. 4. Measured data and 1-, 3-, 6- and 12-step-ahead forecasts in the validation period with the reference ARMA model at Liptovský Mikuláš (forecast dashed line, measurement solid line). Obr. 4. Merané prietoky a 1-, 3-, 6- a 12-krokové predpovede priemerných mesačných prietokov vo validačnom období pomocou referenčného modelu ARMA v Liptovskom Mikuláši (predpoveď čiarkovaná a merané prietoky plná čiara). 156

13 A hybrid modelling framework for forecasting monthly reservoir inflows Fig. 5. Measured data and 1-, 3-, 6- and 12-step-ahead forecasts in the validation period with the reference SETAR model at Liptovský Mikuláš (forecast dashed line, measurement solid line). Obr. 5. Merané prietoky a 1-, 3-, 6- a 12-krokové predpovede priemerných mesačných prietokov v referenčnom období pomocou referenčného modelu SETAR v Liptovskom Mikuláši (predpoveď čiarkovaná a merané prietoky plná čiara). Fig. 6. Measured data and 1-, 3-, 6- and 12-step-ahead forecasts in the validation period with the mixed ARMA (Boca and the Biely Váh) and SETAR (the remaining catchments) models in the Hybrid Model 4 at Liptovský Mikuláš (forecast dashed line, measurement solid line). Obr. 6. Merané prietoky a 1-, 3-, 6- a 12-krokové predpovede priemerných mesačných prietokov vo validačnom období pomocou kombinácie modelov ARMA (Boca a Biely Váh) a SETAR (ostatné povodia) v modeli Hybrid 4 v Liptovskom Mikuláši (predpoveď čiarkovaná a merané prietoky plná čiara). 157

14 M. Komorníková, J. Szolgay, D. Svetlíková, D. Szökeová, S. Jurčák Fig. 7. Measured data and 1-, 3-, 6- and 12-step-ahead forecasts in the validation period with the mixed best perfoming ARMA (Čierny Váh, Boca and Štiavnica) and SETAR (the remaining catchments) models in the Hybrid Model 6 at Liptovský Mikuláš (forecast dashed line, measurement solid line). Obr. 7. Merané prietoky a 1-, 3-, 6- a 12-krokové predpovede priemerných mesačných prietokov vo validačnom období pomocou kombinácie najlepších modelov ARMA (Čierny Váh, Boca a Štiavnica) a SETAR (ostatné povodia) a modelu Hybrid 6 v Liptovskom Mikuláši (predpoveď čiarkovaná a merané prietoky plná čiara). In Tab. 6 the Hybrid Model 1 performance measures are given for Liptovský Mikluláš for both nonlinear cases. These values can be regarded as indicators for the prediction errors in the situation, where no measured flow information is available in the pilot cross section, and the forecast of the reservoir inflow has to be estimated solely based on tributary inflows and an analogy. The overall performance of the hybrid scheme is worse than that of the reference models, but the differences are small, and the hybrid schemes could be used for the same purposes as the reference models in practice. Tab. 7 contains the prediction performance measures of Hybrid Model 2, which is corrected for the bias of Scheme 1. The overall performance of the model comes close to that of the nonlinear reference model, and in one case slightly out performs it. In the following, based on the results of the diagnostic checking in Tab. 2, a combination of the ARMA models in the Boca and Biely Váh catchments with the best SETAR models in the other catchments was used with both Scheme 1 and 2 in Hybrid Models 3 and 4 (see Tabs. 8 and 9). Next, the best performing models were combined with both Scheme 1 and 2 in the Hybrid Models 5 and 6, that is, the SETAR models for the Belá and Biely Váh catchments and the ARMA models for the Čierny Váh, Boca and Štiavnica catchments respectively (see Tabs. 10 and 11). It can be seen that both new model combinations outperformed the previous ones and are mutually comparable. In the case of Hybrid Models 3 and 5 the latter combination performed better; in the case of Hybrid Models 4 and 6, both seem to be almost identical in their performance. When compared with the reference nonlinear model (see Tab. 5), Hybrid Models 4 and 6 outperformed the SETAR model in terms of MAE in general (except the 12 step ahead forecasts), and for the one step ahead forecast, also outperformed the reference ARMA model. The practical relevance of this result is that it indicates that the proposed concept of combination of partial forecasts from different models can be regarded as equivalent to the reference model and seen as promising in the sense that these partial forecasts could reflect the runoff generation in the 158

15 A hybrid modelling framework for forecasting monthly reservoir inflows Váh catchment in a more differentiating way. However, for all the models, only a smaller part of the variability of the residuals was explained (see Tabs. 1, 2 and 3 for comparison); therefore, other nonlinear models (such as LSTAR, GARCH) and more differentiating hybrid schemes (such as multiple nonlinear regression or artificial neural networks) should be explored in the future. 5. Conclusions The paper aimed to approach the flow forecasting problem for reservoir operation with a hybrid modelling approach by combining nonlinear component processes in a linear simple water balance model. This nonlinear hybrid modelling framework was investigated under the specific physiographic conditions of the High Core Mountains of Slovakia. The testing of some technical aspects of the proposed nonlinear time series models and the hybrid framework through their fitting and a comparison of the predictive performance was attempted in a way which would allow both for deciding on their practical applicability and for further development. The mean monthly flows of the pilot region used in this study are predominantly fed by snowmelt in the spring and convective precipitation in the summer. Therefore, their hydrological regime exhibits at least two clear seasonal patterns, which provided an intuitive justification for the application of nonlinear two-regime time series models. This expectation was justified by the fact that in almost all the rivers examined in the region, the possible linearity of the time series models of the monthly flows was rejected. The differences in the prevailing geology, orientation of slopes and their exposure to atmospheric circulation for the right and left-sided tributaries of the Váh River indicated that the regime-switching properties of the mean monthly discharge time series in this area with respect to seasonality and cyclicity may differ. Examining the counts of switching between the two regimes of the SETAR models considered showed, that both regimes were accordingly present and that the left and right-sided tributaries exhibited clearly different switching patterns. Further, it was shown that the pattern attributed to the flows of the main river in Liptovský Mikuláš did not belong among these clearly identifiable groups. Therefore, in addition to testing the capability of the regime-switching models to model and forecast the main reservoir inflow from the Váh River to the Liptovská Mara reservoir, a simple deterministic water balance scheme was set up for estimating the reservoir inflow from the tributary flows. It consists of a linear combination of the measured tributary flows and the estimated ungauged tributary flows. The contributions of the ungauged catchments were estimated from the flows from the gauged tributaries with similar physiographic conditions by weighting these by the ratio of the catchment areas. Separate regime-switching models were identified for each gauged tributary. The forecasts of the tributaries were combined into six hybrid forecasting models by the water balance model. The combined forecasts, which may better include the specific regime of the time series of the tributaries, were compared with the forecasts for the overall reservoir inflow in several combinations. It was shown that the proposed concept of a combination of partial forecasts from different models can be regarded as nearly equivalent to the reference nonlinear model. This can be seen as promising in the sense that these partial forecasts obviously reflect the runoff generation in the catchment in a more differentiating way and that the method can be applied to ungauged basins. The choice of the partial nonlinear models and the hybrid scheme should, however, be further refined and tested. With regard to the optimization of reservoir operation based on the results the forecasting models, one can conclude that probably only the one step ahead forecasts could be of practical relevance at this stage of the model development. For all the tested models only a rather small part of the variability of the residuals was explained. The potential of the tested hybrid framework to provide the user with uncertainty estimates of a forecast was not utilised in this study; these estimates could also influence the model choice. It was not shown that nonlinear models could perform better in this sense; further attempts also with other model types seem to be worth considering and to be tested. Acknowledgements. This work was supported by the Slovak Research and Development Agency under the contract No. APVV The research was also supported by the VEGA Agency Grants 1/0496/08 and 1/4209/07. The authors gratefully acknowledge these supports. 159

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