Application of linear stochastic models for drought forecasting in the Büyük Menderes river basin, western Turkey

In the present study, a seasonal and non-seasonal prediction of the Standardized Precipitation Index (SPI) time series is addressed by means of linear stochastic models. The methodology presented here is to develop adequate linear stochastic models known as autoregressive integrated moving average (ARIMA) and multiplicative seasonal autoregressive integrated moving average (SARIMA) to predict drought in the Büyük Menderes river basin using SPI as drought index. Temporal characteristics of droughts based on SPI as an indicator of drought severity indicate that the basin is affected by severe and more or less prolonged periods of drought from 1975 to 2006. Therefore, drought prediction plays an important role for water resources management. ARIMA modeling approach involves the following three steps: model identification, parameter estimation, diagnostic checking. In model identification step, considering the autocorrelation function (ACF) and partial autocorrelation function (PACF) results of the SPI series, different ARIMA models are identified. The model gives the minimum Akaike Information Criterion (AIC) and Schwarz Bayesian Criterion (SBC) is selected as the best fit model. Parameter estimation step indicates that the estimated model parameters are significantly different from zero. Diagnostic check step is applied to the residuals of the selected ARIMA models and the results indicated that the residuals are independent, normally distributed and homoscedastic. For the model validation purposes, the predicted results using the best ARIMA models are compared to the observed data. The predicted data show reasonably good agreement with the actual data. The ARIMA models developed to predict drought found to give acceptable results up to 2 months ahead. The stochastic models developed for the Büyük Menderes river basin can be employed to predict droughts up to 2 months of lead time with reasonably accuracy.     

Smoothing and high risk areas detection in space-time disease mapping: a comparison of P-splines,autoregressive, and moving average models

Recently, several models have been proposed for smoothing risks in disease mapping. These models consider different ways of introducing both spatial and temporal dependence as well as spatio-temporal interactions. In this work, a comparison among some autoregressive, moving average, and P-spline models is performed. Firstly, brain cancer mortality data are used to analyze the degree of smoothness introduced by these models. Secondly, two separate simulation studies (one model-based and the other model-free) are carried out to evaluate the model performance in terms of bias, variability, sensitivity, and specificity. We conclude that P-spline models seem to be a good alternative to autoregressive and moving average models when analyzing highly sparse disease mapping data.     

Author Index 1991

Abstract

Evapotranspiration (ET) is an important process in the hydrological cycle and needs to be accurately quantified for proper irrigation scheduling and optimal water resources systems operation. The time variant characteristics of ET necessitate the need for forecasting ET. In this paper, two techniques, namely a seasonal ARIMA model and Winter's exponential smoothing model, have been investigated for their applicability for forecasting weekly reference crop ET. A seasonal ARIMA model with one autoregressive and one moving average process and with a seasonality of 52 weeks was found to be an appropriate stochastic model. The ARIMA and Winter's models were compared with a simple ET model to assess their performance in forecasting. The forecast errors produced by these models were very small and the models would be promisingly of great use in real-time irrigation management.     

Real‐time recurrent learning neural network for stream‐flow forecasting

Various types of neural networks have been proposed in previous papers for applications in hydrological events. However, most of these applied neural networks are classified as static neural networks, which are based on batch processes that update action only after the whole training data set has been presented. The time variate characteristics in hydrological processes have not been modelled well. In this paper, we present an alternative approach using an artificial neural network, termed real‐time recurrent learning (RTRL) for stream‐flow forecasting. To define the properties of the RTRL algorithm, we first compare the predictive ability of RTRL with least‐square estimated autoregressive integrated moving average models on several synthetic time‐series. Our results demonstrate that the RTRL network has a learning capacity with high efficiency and is an adequate model for time‐series prediction. We also investigated the RTRL network by using the rainfall–runoff data of the Da‐Chia River in Taiwan. The results show that RTRL can be applied with high accuracy to the study of real‐time stream‐flow forecasting networks. Copyright © 2002 John Wiley & Sons, Ltd.     

Selection of a kernel bandwidth for measuring dependence in hydrologic time series using the mutual information criterion

  Mutual information is a generalised measure of dependence between any two variables. It can be used to quantify non-linear as well as linear dependence between any two variables. This makes mutual information an attractive alternative to the use of the correlation coefficient, which can only quantify the linear dependence pattern. Mutual information is especially suited for application to hydrological problems, because the dependence between any two hydrologic variables is seldom linear in nature. Calculation of the mutual information score involves estimation of the marginal and joint probability density functions of the two variables. This paper uses nonparametric kernel density estimation methods to estimate the probability density functions. Accurate estimation of the mutual information score using kernel methods requires selection of appropriate smoothing parameters (bandwidths) for use with the kernels. The aim of this paper is to obtain a practical method for bandwidth selection for calculation of the mutual information score. In this paper, the lag-one dependence structures of several autocorrelated time series are analysed using mutual information (note that this produces the lag-one auto-MI score, the analog of the lag-one autocorrelation). Empirical trials are used to select appropriate bandwidths for a range of underlying autoregressive and autoregressive-moving average models with normal or near-normal parent distributions. Expressions for reasonable bandwidth choices under these conditions are proposed.     

Seismic wavelet estimation via a system identification method

On the assumption that the wavelet is causal and nonminimum phase, an autoregressive moving average (ARMA) model is introduced to fit the seismic trace. Seismic wavelet extraction is converted to parameters estimation of the ARMA model. Singular value decomposition (SVD) of an appropriate matrix formed by autocorrelation is exploited to determine the autoregressive (AR) order, and the cumulant-based SVD-TLS (total least squares) approach is proposed to obtain the AR parameters. The author proposes a new moving average (MA) model order determination method via combining the information theoretic criteria method and higher-order cumulant method. The cumulant approach is used to achieve the MA parameters. Theoretical analysis and numerical simulations demonstrate the feasibility of the wavelet extraction approach.     

Non‐stationary hydrological model parameters: a framework based on SOM‐B

The application of stationary parameters in conceptual hydrological models, even under changing boundary conditions, is a common yet unproven practice. This study investigates the impact of non‐stationary model parameters on model performance for different flow indices and time scales. Therefore, a Self‐Organizing Map based optimization approach, which links non‐stationary model parameters with climate indices, is presented and tested on seven meso‐scale catchments in northern Germany. The algorithm automatically groups sub‐periods with similar climate characteristics and allocates them to similar model parameter sets. The climate indices used for the classification of sub‐periods are based on (a) yearly means and (b) a moving average over the previous 61 days. Classification b supports the estimation of continuous non‐stationary parameters. The results show that (i) non‐stationary model parameters can improve the performance of hydrological models with an acceptable growth in parameter uncertainty; (ii) some model parameters are highly correlated to some climate indices; (iii) the model performance improves more for monthly means than yearly means; and (iv) in general low to medium flows improve more than high flows. It was further shown how the gained knowledge can be used to identify insufficiencies in the model structure. Copyright © 2015 John Wiley & Sons, Ltd.     

Simulation of earthquake ground motions using autoregressive moving average (ARMA) models

Parsimonious representations of recorded earthquake acceleration time series are obtained by fitting stationary autoregressive moving average models after a variance-stabilizing transformation. Simulated acceleration series are then constructed by generating realizations from the fitted stationary models and applying the reverse transformation. As demonstrated on three components of a typical series, the response spectra for the observed and simulated series show good agreement for periods of less than eight seconds. Further, the model parameters for the three components are very similar, suggesting a consistency which could be useful for identifying site-specific characteristics.     

Functional series TARMA modelling and simulation of earthquake ground motion

The non‐stationary Functional Series time‐dependent autoregressive moving average (TARMA) modelling and simulation of earthquake ground motion is considered. Full Functional Series TARMA models, capable of modelling both resonances and antiresonances, are examined for the first time via a novel mixed parametric/non‐parametric estimation scheme, and critical comparisons with pure TAR and recursive ARMA (RARMA)‐recursive maximum likelihood (RML) adaptive filtering type modelling are made. The study is based upon two California ground motion signals: a 1979 El Centro accelerogram and a 1994 Pacoima Dam accelerogram. A systematic analysis, employing various functional subspaces and model orders, leads to two Haar function based models: a TARMA(2,4)₈ model for the El Centro case and a TARMA(6,2)₁₀ model for the Pacoima Dam case. Both models are formally validated and their simulation (synthesis) capabilities are demonstrated via Monte Carlo experiments focusing on important time domain signal characteristics. The Functional Series TAR/TARMA models are shown to achieve parsimony, as well as superior accuracy and simulation capabilities, over their RARMA counterparts. Copyright © 2001 John Wiley & Sons Ltd.     

The role of anthropogenic activities in karst spring discharge volatility

The traditional hydrological time series methods tend to focus on the mean of whichever variable is analysed but neglect its time‐varying variance (i.e. assuming the variance remains constant). The variances of hydrological time series vary with time under anthropogenic influence. There is evidence that extensive well drilling and groundwater pumping can intercept groundwater run‐off and consequently induce spring discharge volatility or variance varying with time (i.e. heteroskedasticity). To investigate the time‐varying variance or heteroskedasticity of spring discharge, this paper presents a seasonal autoregressive integrated moving average with general autoregressive conditional heteroskedasticity (SARIMA‐GARCH) model, whose the SARIMA model is used to estimate the mean of hydrological time series, and the GARCH model estimates its time‐varying variance. The SARIMA‐GARCH model was then applied to the Xin'an Springs Basin, China, where extensive groundwater development has occurred since 1978 (e.g. the average annual groundwater pumping rates were less than 0.20 m³/s in the 1970s, reached 1.20 m³/s at the end of the 1980s, surpassed 2.0 m³/s in the 1990s and exceeded 3.0 m³/s by 2007). To identify whether human activities or natural stressors caused the heteroskedasticity of Xin'an Springs discharge, we segmented the spring discharge sequence into two periods: a predevelopment stage (i.e. 1956–1977) and a developed stage (i.e. 1978–2012), and set up the SARIMA‐GARCH model for the two stages, respectively. By comparing the models, we detected the role of human activities in spring discharge volatility. The results showed that human activities caused the heteroskedasticity of the Xin'an Spring discharge. The predicted Xin'an Springs discharge by the SARIMA‐GARCH model showed that the mean monthly spring discharge is predicted to continue to decline to 0.93 m³/s in 2013, 0.67 m³/s in 2014 and 0.73 m³/s in 2015. Copyright © 2014 John Wiley & Sons, Ltd.     

Annual streamflow modelling with asymmetric distribution function

Classical autoregressive models (AR) have been used for forecasting streamflow data in spite of restrictive assumptions, such as the normality assumption for innovations. The main reason for making this assumption is the difficulties faced in finding model parameters for non‐normal distribution functions. However, the modified maximum likelihood (MML) procedure used for estimating autoregressive model parameters assumes a non‐normally distributed residual series. The aim in this study is to compare the performance of the AR(1) model with asymmetric innovations with that of the classical autoregressive model for hydrological annual data. The models considered are applied to annual streamflow data obtained from two streamflow gauging stations in K?z?l?rmak Basin, Turkey. Copyright © 2008 John Wiley & Sons, Ltd.     

Performance of detrending models of crop yield risk assessment: evaluation on real and hypothetical yield data

Detrending is a widely used technique for obtaining stationary time series data in residual analysis and risk assessment. The technique is frequently applied in crop yield risk assessment and insurance ratings. Although several trend models have been proposed in the literature, whether these models achieve consistent detrending results and successfully extract the true yield trends is rarely discussed. In the present article, crop insurance pricing is evaluated by different trend models using real and historical yield data, and hypothetical yield data generated by Monte Carlo simulations. Applied to real historical data, the linear, loglinear, autoregressive integrated moving average trend models produce different risk assessment results. The differences among the model outputs are statistically significant. The largest deviation in the county crop assessment reaches 6–8 %, substantially larger than the present countrywide gross premium rate of 5–7 %. In performance tests on simulated yield trends, popular detrending methods based on smoothing techniques proved overall superior to linear, loglinear, and integrated autoregression models. The best performances were yielded by the moving average and robust locally weighted regression models.     

Modeling time series of ground water head fluctuations subjected to multiple stresses

The methods behind the predefined impulse response function in continuous time (PIRFICT) time series model are extended to cover more complex situations where multiple stresses influence ground water head fluctuations simultaneously. In comparison to autoregressive moving average (ARMA) time series models, the PIRFICT model is optimized for use on hydrologic problems. The objective of the paper is twofold. First, an approach is presented for handling multiple stresses in the model. Each stress has a specific parametric impulse response function. Appropriate impulse response functions for other stresses than precipitation are derived from analytical solutions of elementary hydrogeological problems. Furthermore, different stresses do not need to be connected in parallel in the model, as is the standard procedure in ARMA models. Second, general procedures are presented for modeling and interpretation of the results. The multiple-input PIRFICT model is applied to two real cases. In the first one, it is shown that this model can effectively decompose series of ground water head fluctuations into partial series, each representing the influence of an individual stress. The second application handles multiple observation wells. It is shown that elementary physical knowledge and the spatial coherence in the results of multiple wells in an area may be used to interpret and check the plausibility of the results. The methods presented can be used regardless of the hydrogeological setting. They are implemented in a computer package named Menyanthes (www.menyanthes.nl).     

Assessment of regional estimation error by canonical residual kriging

Various regional flood frequency analysis procedures are used in hydrology to estimate hydrological variables at ungauged or partially gauged sites. Relatively few studies have been conducted to evaluate the accuracy of these procedures and estimate the error induced in regional flood frequency estimation models. The objective of this paper is to assess the overall error induced in the residual kriging (RK) regional flood frequency estimation model. The two main error sources in specific flood quantile estimation using RK are the error induced in the quantiles local estimation procedure and the error resulting from the regional quantile estimation process. Therefore, for an overall error assessment, the corresponding errors associated with these two steps must be quantified. Results show that the main source of error in RK is the error induced into the regional quantile estimation method. Results also indicate that the accuracy of the regional estimates increases with decreasing return periods. Copyright © 2010 John Wiley & Sons, Ltd.     

Prediction of hydrological drought durations based on Markov chains: case of the Canadian prairies

Abstract

Hydrological drought durations (lengths) in the Canadian prairies were modelled using the standardized hydrological index (SHI) sequences derived from the streamflow series at annual, monthly and weekly time scales. The rivers chosen for the study present high levels of persistence (as indicated by values exceeding 0.95 for lag-1 autocorrelation in weekly SHI sequences), because they encompass large catchment areas (2210–119 000 km²) and traverse, or originate in, lakes. For such rivers, Markov chain models were found to be simple and efficient tools for predicting the drought duration (year, month, or week) based on annual, monthly and weekly SHI sequences. The prediction of drought durations was accomplished at threshold levels corresponding to median flow (Q50) (drought probability, q?=?0.5) to Q95 (drought probability, q?=?0.05) exceedence levels in the SHI sequences. The first-order Markov chain or the random model was found to be acceptable for the prediction of annual drought lengths, based on the Hazen plotting position formula for exceedence probability, because of the small sample size of annual streamflows. On monthly and weekly time scales, the second-order Markov chain model was found to be satisfactory using the Weibull plotting position formula for exceedence probability. The crucial element in modelling drought lengths is the reliable estimation of parameters (conditional probabilities) of the first- and second-order persistence, which were estimated using the notions implicit in the discrete autoregressive moving average class of models. The variance of drought durations is of particular significance, because it plays a crucial role in the accurate estimation of persistence parameters. Although, the counting method of the estimation of persistence parameters was found to be unsatisfactory, it proved useful in setting the initial values and also in subsequent adjustment of the variance-based estimates of persistence parameters. At low threshold levels corresponding to q < 0.20, even the first-order Markov chain can be construed as a satisfactory model for predicting drought durations based on monthly and weekly SHI sequences.

Editor D. Koutsoyiannis; Associate editor C. Onof

Citation Sharma, T.C. and Panu, U.S., 2012. Prediction of hydrological drought durations based on Markov chains in the Canadian prairies. Hydrological Sciences Journal, 57 (4), 705–722.     

Seasonal evolution of meltwater generation,storage and discharge at a non‐temperate glacier in Svalbard

In glacierized catchments, meteorological inputs driving surface melting are translated into runoff outputs mediated by the glacier hydrological system: analysis of the relationship between meteorology and diurnal and seasonal patterns of runoff should reflect the functioning of that system, with the role of meltwater storage likely to be of particular importance. Daily meltwater storage is determined for a glacier at 78 °N in the Svalbard archipelago, by comparing inputs calculated from a surface energy balance model with measured outputs (proglacial discharge). Solar radiation, air temperature, wind speed and proglacial discharge are then analysed by regression and time‐series methods, in order to assess the meteorology–discharge relationship and its variation at diurnal and seasonal time‐scales. The recorded discharge time‐series can be divided into two contrasting intervals: up to early August, proglacial discharge was high and variable, mean hydrographs showed little indication of diurnal cycling, ARIMA models of discharge indicated a non‐seasonal, moving‐average generating process, and there was a net loss of meltwater from storage; from early August, proglacial discharge was low and relatively invariable, but with clearer diurnal cycles, regression models of discharge showed substantially improved correlations with air temperature and solar radiation, ARIMA models indicated a non‐seasonal, autoregressive generating process, and eventually a seasonal component, and there was a net gain in meltwater storage. The transition between the two periods is brief compared with the duration of the melt season. The runoff response to meteorology therefore lacks the strongly progressive element previously identified in mid‐latitude glacierized catchments. In particular, the glacier hydrological system only appears responsive to diurnal forcing following the depletion of the seasonal snowpack meltwater store. Copyright © 2001 John Wiley & Sons, Ltd.     

Multi-step SETARMA predictors in the analysis of hydrological time series

The performance of the self-exciting threshold autoregressive moving average model in forecasting river flow data is investigated. Multi-step forecasts of two daily time series are generated through three different nonlinear predictors. The model adequacy to capture the main features of the data under study and its forecasting performance are analysed and discussed.     

Verification of ARMA identification for modelling temporal correlations of GNSS observations using the ARMASA toolbox

The classical least-squares (LS) algorithm is widely applied in practice of processing observations from Global Satellite Navigation Systems (GNSS). However, this approach provides reliable estimates of unknown parameters and realistic accuracy measures only if both the functional and stochastic models are appropriately specified. One essential deficiency of the stochastic model implemented in many available GNSS software products consists in neglecting temporal correlations of GNSS observations. Analysing time series of observation residuals resulting from the LS evaluation, the temporal correlation behaviour of GNSS measurements can be efficiently described by means of socalled autoregressive moving average (ARMA) processes. For a given noise realisation, a well-fitting ARMA model can be automatically estimated and identified using the ARMASA toolbox available free of charge in MATLAB® Central.In the preliminary stage of applying the ARMASA toolbox to residual-based modelling of temporal correlations of GNSS observations, this paper presents an empirical performance analysis of the automatic ARMA estimation tool using a large amount of simulated noise time series with representative temporal correlation properties comparable to the GNSS residuals. The results show that the rate of unbiased model estimates increases with data length and decreases with model complexity. For large samples, more than 80% of the identified ARMA models are unbiased. Additionally, the model error representing the deviation between the true data-generating process and the model estimate converges rapidly to the associated asymptotical value for a sufficiently large sample size with respect to the correlation length.     

Exploratory spectral analysis of hydrological times series

Current methods of estimation of the univariate spectral density are reviewed and some improvements are made. It is suggested that spectral analysis may perhaps be best thought of as another exploratory data analysis (EDA) tool which complements, rather than competes with, the popular ARMA model building approach. A new diagnostic check for ARMA model adequacy based on the nonparametric spectral density is introduced. Additionally, two new algorithms for fast computation of the autoregressive spectral density function are presented. For improving interpretation of results, a new style of plotting the spectral density function is suggested. Exploratory spectral analyses of a number of hydrological time series are performed and some interesting periodicities are suggested for further investigation. The application of spectral analysis to determine the possible existence of long memory in natural time series is discussed with respect to long riverflow, treering and mud varve series. Moreover, a comparison of the estimated spectral densities suggests the ARMA models fitted previously to these datasets adequately describe the low frequency component. Finally, the software and data used in this paper are available by anonymous ftp from fisher.stats.uwo.ca.