The paper approaches problem of the flow forecasting for the Liptovska 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 Vah 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. and 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.
This study examines two long-term time series of nitrate-nitrogen concentrations from the River Ouse and Stour situated in the Eastern England. The time series of monthly averages were decomposed into trend, seasonal and cyclical components and residuals to create a simple additive model. Residuals were then modelled by linear time series models represented by models of the ARMA (autoregressive moving average) class and nonlinear time series models with multiple regimes represented by SETAR (self-exciting threshold autoregressive) and MSW (Markov switching) models. The analysis showed that, based on the minimal value of residual sum of squares (RSS) of one-step ahead forecast in both datasets, SETAR and MSW models described the time series better than models ARMA. However, the relative improvement of SETAR models against ARMA models was low ranging between 1% and 4% with the exception of the three-regime model for the River Stour where the improvement was 48.9%. In comparison, the relative improvement of MSW models was between 44.6% and 52.5 for two-regime and from 60.4% to 75% for three-regime models. However, the visual assessment of models plotted against original datasets showed that despite a high value of RSS, some ARMA models could describe the analyzed time series better than AR (autoregressive), MA (moving average) and SETAR models with lower values of RSS. In both datasets MSW models provided a very good visual fit describing most of the extreme values. The results of this work could be used as a base for construction of other time series models used to describe or predict nitratenitrogen concentrations. and Štúdia sa zaoberá analýzou dlhých časových radov koncentrácií dusičnanového dusíka v rieke Ouse a Stour vo Východnom Anglicku. Časové rady priemerných mesačných koncentrácií dusičnanov boli rozložené na trendovú, sezónnu a cyklickú zložku a reziduá pripočítané k sebe a tvoriace jednoduchý aditívny model. Reziduá boli ďalej modelované zložitejšími lineárnymi modelmi reprezentovanými modelmi triedy ARMA a nelineárnymi viacrežimovými modelmi SETAR a MSW. Výsledky analýzy ukázali, že na základe minimálnej hodnoty sumy štvorcov reziduí (SSR) jednokrokovej predpovede, v oboch prípadoch SETAR aj MSW modely opísali časové rady lepšie ako modely triedy ARMA. Vo väčšine prípadov relatívne zlepšenie modelov SETAR oproti jednoduchým AR(1) modelom bolo malé v rozmedzí od 1 do 4 % s výnimkou trojrežimového modelu pre rieku Stour, kde to bolo až 48,9 %. Naopak, relatívne zlepšenie modelov MSW oproti AR(1) modelom bolo v rozmedzí 44,6 až 52,5 % pre dvojrežimové a 60,4 až 75 % pre trojrežimové modely. Vizuálne posúdenie jednotlivých modelov však ukázalo, že napriek vysokým hodnotám SSR, niektoré ARMA modely dokázali lepšie opísať dané časové rady ako modely AR, MA a SETAR s nižšími hodnotami SSR. V oboch prípadoch MSW modely dokázali dostatočne dobre opísať aj extrémne hodnoty oboch časových radov. Výsledky práce môžu byť použité pri tvorbe iných opisných alebo predpovedných modelov koncentrácie dusičnanového dusíka vo vodách.
A synthesis of recent development of regime-switching models based on aggregation operators is presented. It comprises procedures for model specification dans identification, parameter estimation and model adequacy testing. Constructions of models for real life data from hydrology and finance are presented.
The autocorrelation function describing the linear dependence is not suitable for description of residual dependence of the regime-switching models. In this contribution, inspired by Rakonczai (\cite{Rak09}), we will model the residual dependence of the regime-switching models (SETAR, LSTAR and ESTAR) with the autocopulas (Archimedean, EV and their convex combinations) and construct improved quality models for the original real time series.
We have intensified studies of reflections of copulas (that we introduced recently in \cite{Kom}) and found that their convex combinations exhibit potentially useful fitting properties for original copulas of the Normal, Frank, Clayton and Gumbel types. We show that these properties enable us to construct interesting models for the relations between investment in stocks and gold.
The univariate conditioning of copulas is studied, yielding a construction method for copulas based on an a priori given copula. Based on the gluing method, g-ordinal sum of copulas is introduced and a representation of copulas by means of g-ordinal sums is given. Though different right conditionings commute, this is not the case of right and left conditioning, with a special exception of Archimedean copulas. Several interesting examples are given. Especially, any Ali-Mikhail-Haq copula with a given parameter λ > 0 allows to construct via conditioning any Ali-Mikhail-Haq copula with parameter μ \in [0,λ].