Comparative Study of Stochastic Models for Seasonal Streamflow Generation

This paper presents a constructive neural network model for seasonal streamflow forecasting. This surface water hydrology is basic to the design and operation of the reservoir. A good example is the operation of a reservoir with an uncontrolled inflow but having a means of regulating the outflow. If information on the nature of the inflow is determinable in advance, then the reservoir can be operated by some decision rule to minimize downstream flood damage. For this reasons, several companies in the Brazilian Electrical Sector use the linear time-series models such as PARMA (Periodic Auto regressive Moving Average) models developed by Box-Jenkins. This paper provides for river flow prediction a numerical comparison between neural networks, called nonlinear sigmoidal regression blocks networks (NSRBN) and PARMA models. The model was implemented to forecast weekly average inflow on an step-ahead basis. It was tested on four hydroelectric plants located in different river basins in Brazil. The results obtained in the evaluation of the performance of NSRBN were better than the results obtained with PARMA models.

The seemingly complex nature of river flow and the significant variability it exhibits in both time and space, have largely led to the development and application of the stochastic process concept for its modelling, forecasting, and other ancillary purposes. Towards this end, in this study, attempt was made at stochastic modelling of the daily streamflow process of the Benue River. In this regard, Autoregressive Moving Average (ARMA) models and its derivative, the Periodic Autoregressive (PAR) model were developed and used for forecasting. Comparative forecast performances of the different models indicate that despite the shortcomings associated with univariate time series, reliable forecasts can be obtained for lead times, 1 to 5 day-ahead. The forecast results also showed that the traditional ARMA model could not robustly simulate high flow regimes unlike the periodic AR (PAR). Thus, for proper understanding of the dynamics of the river flow and its management, especially, flood defense, in the light of this study, the traditional ARMA models may not be suitable since they do not allow for real-time appraisal. To account for seasonal variations, PAR models should be used in forecasting the streamflow processes of the Benue River. However, since almost all mechanisms involved in the river flow processes present some degree of nonlinearity thus, how appropriate the stochastic process might be for every flow series may be called to question.

Periodic autoregressive forecasting of global solar irradiation without knowledge-based model implementation

This paper studies correlation and partial autocorrelation properties of periodic autoregressive moving‐average (PARMA) time series models. An efficient algorithm to compute PARMA autocovariances is first derived. An innovations based algorithm to compute partial autocorrelations for a general periodic series is then developed. Finally, periodic moving averages and autoregressions are characterized as periodically stationary series whose autocovariances and partial autocorrelations, respectively, are zero at all lags that exceed some periodically varying threshold.

Three-phase sampling can be a very effective design for the estimation of regional and national forest cover type frequencies. Simultaneous estimation of frequencies and sampling variances require estimation of a large number of parameters; often so many that consistency and robustness of results becomes an issue. A new stepwise estimation model, in which bias in phase one and two is corrected sequentially instead of simultaneously, requires fewer parameters. Simulated three-phase sampling tested the new model with 144 settings of sample sizes, the number of classes and classification accuracy. Relative mean absolute deviations and root mean square errors were, in most cases, about 8% lower with the stepwise method than with a simultaneous approach. Differences were a function of design parameters. Average expected relative root mean square errors, derived from the assumption of a Dirichlet distribution of cover-type frequencies, tracked the empirical root mean square errors obtained from repeated sampling with - 10%. Resampling results indicate that the relative bias of the most frequent cover types was slightly inflated by the stepwise method. For the least common cover type, the simultaneous method produced the largest relative bias.

In this paper we study the weighted least absolute deviations estimator (WLADE) for Periodic ARMA (PARMA) models with unspecified noises, asymptotic normality of the estimator is derived. The random weighting approach is proposed to estimate the asymptotic covariance matrices of WLADE. Simulation study is carried out to examine the performance of the proposed procedure.

ABSTRACTResults involving correlation properties and parameter estimation for autoregressive‐moving average models with periodic parameters are presented. A multivariate representation of the PARMA model is used to derive parameter space restrictions and difference equations for the periodic autocorrelations. Close approximation to the likelihood function for Gaussian PARMA processes results in efficient maximum‐likelihood estimation procedures. Terms in the Fourier expansion of the parameters are sequentially included, and a selection criterion is given for determining the optimal number of harmonics to be included. Application of the techniques is demonstrated through analysis of a monthly streamflow time series.

Abstract. The aim of this article is to give a complete description of the spectral measure of periodic autoregressive moving‐average (PARMA) system in terms of its coefficients. In the analysis we use the spectral theory of strongly harmonizable sequences presented in Hurd [Journal of Multivariate Analysis(1989) Vol. 29, pp. 53–67] and the form of the unique bounded solution of ARMA model with periodic coefficients. As an application of the theoretical results, we present some examples of the spectral measures for PARMA models.

수공구조물의 위험도에 관한 불확실성을 검토하기 위하여 본 연구에서는 빈도해석을 통하여 추정되는 설계홍수량의 분산량을 고려한 불확실성 해석을 실시하였다. Gumbel 분포형을 기본 분포형으로 가정하였으며, 모멘트법, 최우도법, 확률가중모멘트법을 이용하여 각 매개변수 추정방법별로 추정된 설계홍수량에 대한 이론적인 분산량을 산정하였다. 이론적으로 유도한 분산량의 특성을 규명하기 위하며 다양한 표본크기와 설계연한, 비초과확률 및 변동계수조건에 대하여 Monte-Carlo 모의를 실시하고 각 매개변수 추정방법별 비교를 실시하였다. 그 결과 확률 가중 모멘트법을 사용한 경우 위험도에 대하여 상대적으로 가장 작은 상대편의 및 상대제곱근오차를 발생시키는 것으로 나타났으며, 최우도법의 경우에는 상대적으로 큰 표본자료에 대해서는 설계연한 및 비초과확률에 관계없이 작은 상대편의 및 상대제곱근오차를 발생시키는 것으로 나타났다. 또한 다양한 변동계수 조건은 상대편의 및 상대제곱근오차의 측면에서 고려하여 볼 때 거의 영향을 주지 않는 것으로 나타났다. The uncertainty of the risk of failure of hydraulic structures can be determined by estimating the variance of the risk of failure based on the methods of moments, probability weighted moments, and maximum likelihood assuming that the underlying model is the Gumbel distribution. In this paper, the variance of the risk of failure was derived. Monte Carlo simulation was peformed to verify the characteristics of the derived formulas for various sample size, design life, nonexceedance probability, and variation coefficient. As the results, PWM showed the smallest relative bias and root mean square error than the others while ML showed the smallest ones for relatively large sample siBes regardless of design life and nonexceedance probability. Also, it was found that variation coefficient does not effect on the relative bias and relative root mean square error.

Determination of the extent of peak rainfall for different return periods is an essential ingredient for the accurate design of hydraulic structures such as drains, dams and culverts as well as detection of flood risk areas. The focus of this study is to analyze annual maximum daily rainfall series in some selected sites within the coastal region of Nigeria using three parameter probability distribution models, namely, Generalized Logistics (GLO), Generalized Extreme Value (GEV) and Generalized Pareto (GPA) with the view of identifying the best fit probability distribution model per station that can be employed to estimate the rainfall magnitude for selected return periods. Specific time series analysis test, namely, detection of outlier and homogeneity test were performed to certify that the data utilized are adequate and suitable. Descriptive statistics such as sample mean, variance, standard deviation, kurtosis, skewness, and coefficient of variation were computed using basic statistical equations. The probability weighted moment parameters (b0, b1, b2 and b3), L-moment values (λ1, λ2, λ3 and λ4) and ratios (τ2, τ 3 and τ4) including the distribution parameters, namely, shape (k), scale (α) and location (ξ) parameters were computed based on L-moments procedures. To select the best-fit probability distribution model per station, carefully chosen goodness-of-fit statistics, namely, root mean square error, relative root mean square error, maximum absolute deviation index, maximum absolute error and probability plot correlation coefficient were employed since they can adequately assess the fitted distribution at a site. Results obtained indicate that the GLO is the best fit distribution for analyzing annual maximum daily rainfall series from Warri and Calabar while GPA for Port Harcourt and Uyo.

Rainfall frequency analyses for four cities namely Benin City, Port Harcourt, Calabar and Uyo in Nigeria's Niger Delta area were carried out utilizing daily rainfall data from the yearly maxima series for 48 years (1965 – 2012) at each location. The study's goal was to identify the probability distribution model that fit the data the best and applicable to each location from among six candidate probability distribution models namely: Pearson Type III (PIII), log Normal (LN), log Pearson Type III, Generalized Extreme value (GEV), Extreme value type I (EVI), and Generalized Pareto (GPA). The method of moments (MOM) was used to estimate the distributions' parameters applying the outcomes of seven goodness of fit tests, the most optimal fit distribution model was chosen for each site namely Root Mean Squared Error (RMSE), Relative Root Mean Square Error (RRMSE), Maximum Absolute Derivation Index (MADI), Chi–Square etc. The best distribution model at each location was utilized to predict rainfall of desired return periods. Based on our findings, PIII for Benin City, GEV for Port Harcourt, GEV for Calabar and EVI and LN for Uyo are the distribution models that suit the data the best., The Best Fit Probability Distribution Model predicted rainfall return values (RT) at 200 years return period ranging from192.92mm at Uyo, 185mm at Port Harcourt; 218mm for Benin City; and 245 mm at Calabar.. The study's findings are helpful in the planning, designing, and maintaining of hydraulic structures for preventing flood damage and mitigating floods at the locations

For analysis and design of water resource systems, it is sometimes useful to generate high-resolution (e.g., weekly) synthetic river flows. Periodic autoregressive moving average (PARMA) time series models provide a powerful tool for generating synthetic flows. Periodically stationary models are indicated when the basic statistics (mean, variance, and autocorrelation) of the time series exhibit significant seasonal variations. Parameter estimation for high-resolution PARMA models involves numerous parameters, which can lead to overfitting. Thus, this paper develops a parsimonious method of parameter fitting for high-resolution PARMA models, using discrete Fourier transforms to represent the set of periodic autoregressive and moving average model coefficients. Model parameters are computed via the innovations algorithm, and the asymptotic distributions of the discrete Fourier transform coefficients are obtained. Those asymptotic results are useful to determine the statistically significant Fourier coeffici...

Amplification analysis at Nador city and surrounding area (NE of Morocco)

In this paper, we address the problem of modeling data with periodic autoregressive (PAR) time series and additive noise. In most cases, the data are processed assuming a noise-free model (i.e., without additive noise), which is not a realistic assumption in real life. The first two steps in PAR model identification are order selection and period estimation, so the main focus is on these issues. Finally, the model should be validated, so a procedure for analyzing the residuals, which are considered here as multidimensional vectors, is proposed. Both order and period selection, as well as model validation, are addressed by using the characteristic function (CF) of the residual series. The CF is used to obtain the probability density function, which is utilized in the information criterion and for residuals distribution testing. To complete the PAR model analysis, the procedure for estimating the coefficients is necessary. However, this issue is only mentioned here as it is a separate task (under consideration in parallel). The presented methodology can be considered as the general framework for analyzing data with periodically non-stationary characteristics disturbed by finite-variance external noise. The original contribution is in the selection of the optimal model order and period identification, as well as the analysis of residuals. All these findings have been inspired by our previous work on machine condition monitoring that used PAR modeling.

Investigation on drought characteristics such as severity, duration, and frequency is crucial for water resources planning and management in a river basin. While the methodology for multivariate drought frequency analysis is well established by applying the copulas, the estimation on the associated parameters by various parameter estimation methods and the effects on the obtained results have not yet been investigated. This research aims at conducting a comparative analysis between the maximum likelihood parametric and non-parametric method of the Kendall \(\tau \) estimation method for copulas parameter estimation. The methods were employed to study joint severity–duration probability and recurrence intervals in Karkheh River basin (southwest Iran) which is facing severe water-deficit problems. Daily streamflow data at three hydrological gauging stations (Tang Sazbon, Huleilan and Polchehr) near the Karkheh dam were used to draw flow duration curves (FDC) of these three stations. The \(Q_{75}\) index extracted from the FDC were set as threshold level to abstract drought characteristics such as drought duration and severity on the basis of the run theory. Drought duration and severity were separately modeled using the univariate probabilistic distributions and gamma–GEV, LN2–exponential, and LN2–gamma were selected as the best paired drought severity–duration inputs for copulas according to the Akaike Information Criteria (AIC), Kolmogorov–Smirnov and chi-square tests. Archimedean Clayton, Frank, and extreme value Gumbel copulas were employed to construct joint cumulative distribution functions (JCDF) of droughts for each station. Frank copula at Tang Sazbon and Gumbel at Huleilan and Polchehr stations were identified as the best copulas based on the performance evaluation criteria including AIC, BIC, log-likelihood and root mean square error (RMSE) values. Based on the RMSE values, nonparametric Kendall-\(\tau \) is preferred to the parametric maximum likelihood estimation method. The results showed greater drought return periods by the parametric ML method in comparison to the nonparametric Kendall \(\tau \) estimation method. The results also showed that stations located in tributaries (Huleilan and Polchehr) have close return periods, while the station along the main river (Tang Sazbon) has the smaller return periods for the drought events with identical drought duration and severity.