Daily Runoff Forecasting Using a Hybrid Model Based on Variational Mode Decomposition and Deep Neural Networks

Accurate and reliable runoff forecasting plays an increasingly important role in the optimal management of water resources. To improve the prediction accuracy, a hybrid model based on variational mode decomposition (VMD) and deep neural networks (DNN), referred to as VMD-DNN, is proposed to perform daily runoff forecasting. First, VMD is applied to decompose the original runoff series into multiple intrinsic mode functions (IMFs), each with a relatively local frequency range. Second, predicted models of decomposed IMFs are established by learning the deep feature values of the DNN. Finally, the ensemble forecasting result is formulated by summing the prediction sub-results of the modelled IMFs. The proposed model is demonstrated using daily runoff series data from the Zhangjiashan Hydrological Station in Jing River, China. To fully illustrate the feasibility and superiority of this approach, the VMD-DNN hybrid model was compared with EMD-DNN, EEMD-DNN, and multi-scale feature extraction -based VMD-DNN, EMD-DNN and EEMD-DNN. The results reveal that the proposed hybrid VMD-DNN model produces the best performance based on the Nash-Sutcliffe efficiency (NSE?=?0.95), root mean square error (RMSE?=?9.92) and mean absolute error (MAE?=?3.82) values. Thus the proposed hybrid VMD-DNN model is a promising new method for daily runoff forecasting.

     

A Novel Hybrid Decompose-Ensemble Strategy with a VMD-BPNN Approach for Daily Streamflow Estimating

Streamflow estimation is highly significant for water resource management. In this work, we improve the accuracy and stability of streamflow estimation through a novel hybrid decompose-ensemble model that employs variational mode decomposition (VMD) and back-propagation neural networks (BPNN). First, the latest decomposition algorithm, namely, VMD, was used to extract multiscale features that were subsequently learned and ensembled by the BPNN model to obtain the final estimate streamflow results. The historical daily streamflow series of Laoyukou and Wushan hydrological stations in China were analysed by VMD-BPNN, by a single GBRT and BPNN model, ensemble empirical mode decomposition (EEMD) models. The results confirmed that the VMD outperformed a single-estimation model without any decomposition and EEMD-based models; moreover, ensemble estimations using the BPNN model development technique were consistently better than a general summation method. The VMD-BPNN model’s estimation performance was superior to that of five other models at the Wushan station (GBRT, BPNN, EEMD-BPNN-SUM, VMD-BPNN-SUM, and EEMD-BPNN) using evaluation criteria of the root-mean-square error (RMSE?=?2.62 m³/s), the Nash–Sutcliffe efficiency coefficient (NSE?=?0. 9792) and the mean absolute error (MAE?=?1.38 m³/s). The proposed model also had a better performance in estimating higher-magnitude flows with a low criterion for MAE. Therefore, the hybrid VMD-BPNN model could be applied as a promising approach for short-term streamflow estimating.

     

Monthly rainfall forecasting by a hybrid neural network of discrete wavelet transformation and deep learning

Rainfall forecast is critical to the management and allocation of water resources. Deep learning is used to predict rainfall time series with high temporal and spatial variability. Discrete wavelet transform (DWT), long-short term memory (LSTM) and dilated causal convolutional neural network (DCCNN) is integrated to build a hybrid model (DWT-CLSTM-DCCNN). Two methods of sample construction are used to train DWT-CLSTM-DCCNN and their effects on the model performance are analyzed. LSTM and DCCNN are built as benchmark models. The forecasting performance of DWT-CLSTM-DCCNN on monthly rainfall data from four major cities in China is evaluated. The results of DWT-CLSTM-DCCNN are compared with those of benchmark models in terms of the mean absolute error (MAE), root mean squared error (RMSE) and Nash-Sutcliffe efficiency (NSE) as well as the forecasting curves. The results show that DWT-CLSTM-DCCNN outperforms the benchmark models in model accuracy and peak and mutational rainfall capture.

     

Development of Bio-Inspired- and Wavelet-Based Hybrid Models for Reconnaissance Drought Index Modeling

The present study aimed to model reconnaissance drought index (RDI) time series at three various time scales (i.e., RDI-6, RDI-9, RDI-12). Two weather stations located at Iran, namely Tehran and Dezful, were selected as the case study. First, support vector regression (SVR) was utilized as the standalone modeling technique. Then, hybrid models were implemented via coupling the standalone SVR with two bio-inspired-based techniques including firefly algorithm (FA) and whale optimization algorithm (WOA) as well as wavelet analysis (W). Accordingly, the hybrid SVR-FA, SVR-WOA, and W-SVR models were proposed. It is worth mentioning that six mother wavelets (i.e., Haar, Daubechies (db2, db4), Coifflet, Symlet, and Fejer-Korovkin) were employed in development of the hybrid W-SVR models. The performance of models was assessed through root mean square error (RMSE), mean absolute error (MAE), Willmott index (WI), and Nash-Sutcliffe efficiency (NSE). Generally, the implemented coupled models illustrated better results than the standalone SVR in modeling the RDI time series of studied locations. Besides, the Coifflet mother wavelet was found to be the best-performing wavelet. The most accurate results were achieved for RDI-12 modeling via the W-SVR utilizing db4(2) at Tehran station (RMSE = 0.253, MAE = 0.174, WI= 0.888, NSE = 0.934) and Coifflet(2) at Dezful station (RMSE = 0.301, MAE = 0.166, WI= 0.910, NSE = 0.936). As a result, the hybrid models developed in the current study, specifically W-SVR ones, can be proposed as suitable alternatives to the single SVR.

     

Reservoir Inflow Prediction by Ensembling Wavelet and Bootstrap Techniques to Multiple Linear Regression Model

In this study, a new hybrid model, bootstrap multiple linear regression (BMLR) is suggested to investigate the potential of bootstrap resampling technique for daily reservoir inflow prediction. The proposed model compares with three other models: Multiple linear regression (MLR), wavelet multiple linear regression (WMLR) and wavelet bootstrap multiple linear regression (WBMLR). River stage data of monsoon season (1st July 2010 to 30 September 2010) from three gauging stations of Chenab river basin are used. In wavelet transformation, input vectors are decomposed using discrete wavelet transformation (DWT) into discrete wavelet components (DWCs). Then suitable DWCs are used to provide input to MLR model to develop WMLR model. Bootstrap technique coupled with MLR model to build up BMLR model. While WBMLR model is the conjunction of suitable DWCs and bootstrap technique to MLR model. Performance indices namely root mean square error (RMSE), mean absolute error (MAE), Nash-Sutcliffe coefficient of efficiency (NSC), and persistence index (CP) are used in study to evaluate the performance of model. Results showed that hybrid model BMLR produce significantly better results on performance indices than other models MLR, WMLR and WBMLR.

     

Subset Modeling Basis ANFIS for Prediction of the Reference Evapotranspiration

The study investigates accuracy of a new modeling scheme, subset adaptive neuro fuzzy inference system (subset ANFIS), in estimating the daily reference evapotranspiration (ET₀). Daily weather data of relative humidity, solar radiation, air temperature, and wind speed from three stations in Central Anatolian Region of Turkey were utilized as input to the applied models. The input data set for modeling the ET₀ was divided to several subsets to calibrate the local data using a local modeling-based ANFIS. The estimates obtained from subset ANFIS models were compared with those of the M5 model tree (M5Tree), ANFIS models and ANN. Mean absolute error (MAE), root mean square error (RMSE), and model efficiency factor criteria were applied for analysis of models. The accuracy of M5Tree (from 15.3% to 32.5% in RMSE, from 14.4% to 24.2% in MAE), ANN (from 24.3% to 65.3% in RMSE, from 34.1% to 47% in MAE) and ANFIS (from 17.4% to 35.4% in RMSE, from 10.8% to 28.3% in MAE) models was significantly increased using subset ANFIS for estimating da ily ET_0.     

Monthly Rainfall Forecasting Using EEMD-SVR Based on Phase-Space Reconstruction

Rainfall links atmospheric and surficial processes and is one of the most important hydrologic variables. We apply support vector regression (SVR), which has a high generalization capability, to construct a rainfall forecasting model. Before construction of the model, a self-adaptive data analysis methodology called ensemble empirical mode decomposition (EEMD) is used to preprocess a rainfall data series. In addition, the phase-space reconstruction method is implemented to design input vectors for the forecasting model. The proposed hybrid model is applied to forecast the monthly rainfall at a weather station in Changchun, China as a case study. To demonstrate the capacity of the proposed hybrid model, a typical three-layer feed-forward artificial neural network model, an auto-regressive integrated moving average model, and a support vector regression model are constructed. Predictive performance of the models is evaluated based on normalized mean squared error (NMSE), mean absolute percent error (MAPE), Nash–Sutcliffe efficiency (NSE), and the coefficient of correlation (CC). Results indicate that the proposed hybrid model has the lowest NMSE and MAPE values of 0.10 and 14.90, respectively, and the highest NSE and CC values of 0.91 and 0.83, respectively, during the validation period. We conclude that the proposed hybrid model is feasible for monthly rainfall forecast and is better than the models currently in common use.     

An Integration of Stationary Wavelet Transform and Nonlinear Autoregressive Neural Network with Exogenous Input for Baseline and Future Forecasting of Reservoir Inflow

For effective water resources management and planning, an accurate reservoir inflow forecast is essential not only in training and testing phases but also in particular future periods. The objective of this study is to develop a reservoir inflow integrated forecasting model, relying on nonlinear autoregressive neural network with exogenous input (NARX) and stationary wavelet transform (SWT), namely SWT-NARX. Due to the elimination of down-sampling operation, SWT provides influential reinforcement of efficiently extracting the hidden significant, temporal features contained in the nonstationary inflow time series without information loss. The decomposed SWT sub-time series are determined as input-output for NARX forecaster; where a multi-model ensemble global mean (MMEGM) of downscaled precipitation based on nine global climate models (GCMs) represents as a climate-change exogenous input. Two major reservoirs in Thailand, Bhumibol and Sirikit ones are focused. Pearson’s correlation coefficient (r) and root mean square error (RMSE) are employed for performance evaluation. The achieved results indicate that the SWT-NARX explicitly outperforms the comparable forecasting approaches regarding a historical baseline period (1980–1999). Therefore, such SWT-NARX is further employed for future projection of the reservoir inflow over near (2010–2039) -, mid (2040–2069) - and far (2070–2099) - future periods against the inflow of the baseline one.     

Artificial Neural Network and Support Vector Machine Models for Inflow Prediction of Dam Reservoir (Case Study: Zayandehroud Dam Reservoir)

Inflow prediction of reservoirs is of considerable importance due to its application in water resources management related to downstream water release planning and flood protection. Therefore, in this research, different new input patterns for predicting inflow to Zayandehroud dam reservoir is proposed employing artificial neural network (ANN) and support vector machine (SVM) models. Nine different models with different patterns of input data such as inflow to the dam reservoir considering time duration lags, time index, and monthly rainfall of Ghaleh-Shahrokh station have been proposed to predict the inflow to the dam reservoir. Comparison of the results indicates that the ninth proposed model has the least error for inflow prediction in which the results of SVM model outperform those of ANN model. That is, the least error has been obtained using the ninth SVM (ANN) model with correlation coefficient (R) values of 0.8962 (0.89296), 0.9303 (0.92983) and 0.9622 (0.95333) and root mean squared error (RMSE) values of 47.9346 (48.5441), 42.69093 (43.748) and 23.56193 (28.5125) for training, validation and test data, respectively.

     

Drought Forecasting Using Neural Networks,Wavelet Neural Networks,and Stochastic Models: Case of the Algerois Basin in North Algeria

Drought forecasting is a major component of a drought preparedness and mitigation plan. This paper focuses on an investigation of artificial neural networks (ANN) models for drought forecasting in the algerois basin in Algeria in comparison with traditional stochastic models (ARIMA and SARIMA models). A wavelet pre-processing of input data (wavelet neural networks WANN) was used to improve the accuracy of ANN models for drought forecasting. The standard precipitation index (SPI), at three time scales (SPI-3, SPI-6 and SPI-12), was used as drought quantifying parameter for its multiple advantages. A number of different ANN and WANN models for all SPI have been tested. Moreover, the performance of WANN models was investigated using several mother wavelets including Haar wavelet (db1) and 16 daubechies wavelets (dbn, n varying between 2 and 17). The forecast results of all models were compared using three performance measures (NSE, RMSE and MAE). A comparison has been done between observed data and predictions, the results of this study indicate that the coupled wavelet neural network (WANN) models were the best models for drought forecasting for all SPI time series and over lead times varying between 1 and 6 months. The structure of the model was simplified in the WANN models, which makes them very convenient and parsimonious. The final forecasting models can be utilized for drought early warning.

     

Long Term Streamflow Forecasting Using a Hybrid Entropy Model

In this paper, the development and evaluation of an entropy based hybrid data driven model coupled with input selection approach and wavelet transformation is investigated for long-term streamflow forecasting with 10 years lead time. To develop and test the models, data including 45 years of monthly streamflow time series from Taleghan basin, located in northwest of Tehran, are employed. For this purpose, first the performance of a maximum entropy forecasting model is evaluated. To boost the accuracy, an auto-correlation method with %95 confidence levels was carried out to determine the optimum order of the entropy model. Nevertheless, the basic entropy model, as expected, was only able to reach Nash-Sutcliffe efficiency (NSE) index of 0.35 during the test period. On the other hand, data driven models such as artificial neural networks (ANN) have shown to yield good accuracy in modeling complicated and nonlinear systems. Thus, to improve the performance of the maximum entropy model, an entropy-based hybrid model using evolutionary ANN (ENN) was proposed for further investigation. The proposed model with seasonality index substantially improved the test NSE to 0.51 and provided more accurate results than the basic entropy model. Moreover, when wavelet transform was applied to preprocess the input data, the model shows a slight improvement (NSE?=?0.54).     

A Wavelet-ANFIS Hybrid Model for Groundwater Level Forecasting for Different Prediction Periods

Artificial neural network (ANN) and Adaptive Neuro-Fuzzy Inference System (ANFIS) have an extensive range of applications in water resources management. Wavelet transformation as a preprocessing approach can improve the ability of a forecasting model by capturing useful information on various resolution levels. The objective of this research is to compare several data-driven models for forecasting groundwater level for different prediction periods. In this study, a number of model structures for Artificial Neural Network (ANN), Adaptive Neuro-Fuzzy Inference System (ANFIS), Wavelet-ANN and Wavelet-ANFIS models have been compared to evaluate their performances to forecast groundwater level with 1, 2, 3 and 4 months ahead under two case studies in two sub-basins. It was demonstrated that wavelet transform can improve accuracy of groundwater level forecasting. It has been also shown that the forecasts made by Wavelet-ANFIS models are more accurate than those by ANN, ANFIS and Wavelet-ANN models. This study confirms that the optimum number of neurons in the hidden layer cannot be always determined by using a specific formula but trial-and-error method. The decomposition level in wavelet transform should be determined according to the periodicity and seasonality of data series. The prediction of these models is more accurate for 1 and 2 months ahead (for example RMSE?=?0.12, E?=?0.93 and R ²?=?0.99 for wavelet-ANFIS model for 1 month ahead) than for 3 and 4 months ahead (for example RMSE?=?2.07, E?=?0.63 and R ²?=?0.91 for wavelet-ANFIS model for 4 months ahead).     

三峡水库实时调度水文气象预报应用风险及控制

为量化水文气象预报误差对水库实时预报调度的影响,以三峡水库为例,基于其历年水文气象预报信息的误差分析成果,假定入库过程,并叠加对应不同概率的预报误差,以此作为水库调洪演算模型的输入。分析了不同预见期、不同保证率水文气象误差条件下,不超过风险控制点的库水位指标,提出了三峡水库实时预报调度风险控制策略,可为水库科学调度提供参考。     

Daily Forecasting of Dam Water Levels: Comparing a Support Vector Machine (SVM) Model With Adaptive Neuro Fuzzy Inference System (ANFIS)

Reservoir planning and management are critical to the development of the hydrological field and necessary to Integrated Water Resources Management. The growth of forecasting models has resulted in an excellent model known as the Support Vector Machine (SVM). This model uses linearly separable patterns based on an optimal hyperplane, which are extended to non-linearly separable patterns by transforming the raw data to map into a new space. SVM can find a global optimal solution equipped with Kernel functions. These Kernel functions have high flexibility in the forecasting computation, enabling data to be mapped at a higher and infinite-dimensional space in an implicit manner. This paper presents a new solution to the expert system, using SVM to forecast the daily dam water level of the Klang gate. Four categories are identified to determine the best model: the input scenario, the type of SVM regression, the number of V-fold cross-validation and the time lag. The best input scenario employs both the rainfall R(t-i) and the dam water level L(t-i). Type 2 SVM regression is selected as the best regression type, and 5-fold cross-validation produces the most accurate results. The results are compared with those obtained using ANFIS: all the RMSE, MAE and MAPE values prove that SVM is a superior model to ANFIS. Finally, all the results are combined to determine the best time lag, resulting in R(t-2) L(t-2) for the best model with only 1.64 % error.     

Improved Thomas–Fiering and wavelet neural network models for cumulative errors reduction in reservoir inflow forecast

Inflow forecasting for reservoirs is a key component for reservoir operation, water resources management, and inter-regional water transfer. In this paper, to obtain enhanced accuracy in forecasting reservoir inflow, the Thomas–Fiering (TF) model as a typical stochastic model was improved through pre-defining an array in the model that was previously supposed to be a random one to find the suitable parameters for the data series for forecast errors' reduction. Also, a wavelet neural network (WNN) model was utilized to represent a machine learning approach which was improved through using an input–output matrix for the model parameter training to timely adjust input parameters according to the errors produced in the model application process, so as to reduce overall forecast errors as much as possible. Through comparing the performance of the two models within a test year, the TF model was performing better in the wet season and the WNN model had better performance in the dry seasons at the Danjiangkou Reservoir, an important water-providing area of the South-North Water Transfer Project in China. Through comparing the abilities of the improved TF and the WNN models to forecast reservoir inflow, consistencies, differences and reasonable suggestions could be obtained for helping select appropriate models over different hydrologic seasons. Thus, these two models were combined to support the forecast of reservoir inflow within an entire hydrological year. Using these models improved in this paper, the forecasted inflow to the Danjiangkou Reservoir was 4.52 × 10¹⁰ m³ in the coming year (here is 2006), which is approximately 70% of incoming frequency based on daily observed inflow data from 1980 to 2005.     

Wavelet Regression Model as an Alternative to Neural Networks for River Stage Forecasting

River stage forecasting is an important issue in water resources management and real-time prediction of extreme floods. The present study investigates the performance of the wavelet regression (WR) technique in daily river stage forecasting. The WR model was improved combining two methods, discrete wavelet transform and a linear regression model. Two different WR models were developed using the stage sub-time series, and these were compared with each other. The data from two stations on the Schuylkill River in Philadelphia were used. The root mean square errors (RMSE), mean absolute errors (MAE) and correlation coefficient (R) statistics were used for evaluating the accuracy of the WR models. The accuracy of the WR models was then compared with those of the artificial neural networks (ANN) models. Based on a comparison of these results, the WR models were found to perform better than the ANN models. For the upstream and downstream stations, it was found that the WR models with upstream readings of with RMSE = 0.070, MAE = 0.027, R = 0.937 and with downstream readings of RMSE = 0.048, MAE = 0.024, R = 0.969 in the validation stage performed better in forecasting daily river stages than the best accurate ANN models with upstream readings of RMSE = 0.168, MAE = 0.052, R = 0.802 and with downstream readings of RMSE = 0.115, MAE = 0.051, R = 0.807, respectively.     

ANN、ANFIS和AR模型在日径流时间序列预测中的应用比较

水文预测是水文学为经济和社会服务的重要方面。其预报结果不仅能为水库优化调度提供决策支持,而且对水电系统的经济运行、航运以及防洪等方面具有重大意义。自回归模型(AR模型)、人工神经网络(ANN)和自适应神经模糊推理系统(ANFIS)在日径流时间序列中应用广泛。将这三种模型应用于桐子林的日径流时间序列预测中,不仅采用纳什系数(NS系数)、均方根误差(RMSE)和平均相对误差(MARE)为评价指标,对三种模型的综合性能进行了比较。而且,在对三种模型预测结果的平均相对误差的阈值统计基础上,分析了三种模型的预测误差分布。同时,通过研究模型性能指标随预见期的变化过程评价了三种模型不同预见期下的预测能力。结果表明ANFIS相对于ANN和AR模型不仅具有更好的模拟能力、泛化能力,而且在相同的预见期下具有更优的模型性能,可以作为日径流时间序列预测的推荐模型。     

Improving Hydrological Process Modeling Using Optimized Threshold-Based Wavelet De-Noising Technique

The extent of the noise on hydrological data is inevitable, which reduces the efficiency of Data-Driven Models (DDMs). Despite of this fact that the DDMs such as Artificial Neural Network (ANN) are capable of nonlinear functional mapping between a set of input and output variables, but refining of the time series through data pre-processing methods can provide with the possibility to increase the performance of these set of models. The main objective of this study is to propose a new method called Optimized Threshold-Based Wavelet De-noising technique (OTWD) to de-noise hydrological time series and improve the prediction accuracy while the DDM is being used. For this purpose, in the first step, Wavelet-ANN (WNN) model was developed for identifying suitable wavelet function and maximum decomposition level. Afterward, sub-signals of original precipitation time series which were determined in the first step were de-noised by using of OTWD technique. Therefore, these clean sub-signals of precipitation time series were imposed as input data to the ANN to predict the precipitation one time step ahead. The results showed that OTWD technique could improve the efficiency of WNN model dramatically; this outcome was reported by the different efficiency criterions such as Nash-Sutcliffe Efficiency (NSE?=?0.92), Root Mean Squared Error (RMSE?=?0.0103), coefficient correlation of linear regression (R?=?0.93), Peak Value Criterion (PVC?=?0.021) and Low Value Criterion (LVC?=?0.026). The best fitted WNN model in comparison by proposed model showed weaker performance by the NSE, RMSE, R, PVC and LVC values of 0.86, 0.043, 0.87, 0.034 and 0.045, respectively.     

Two-step daily reservoir inflow prediction using ARIMA-machine learning and ensemble models

The reservoirs play a crucial role in the development of civilisation as they facilitate the storage of water for multiple purposes like hydroelectric power generation, flood control, irrigation, and drinking water etc. In order to effectively meet these multiple purposes, the knowledge of the inflow in the reservoir is essential. Apart from the historical data, future prediction of the inflows is also necessary specially in context of climate change. A two-step algorithm for the prediction of reservoir inflow to enable meticulous planning and execution of daily reservoir operation keeping the historical variation of inflow in account has been proposed. The developed algorithm takes into account the patterns in the historic inflow data using the time series analysis along with the variability in the climatic patterns using the different predictors in the machine learning model. The first step uses time series model, ARIMA method to forecast the monthly inflows, which are then used as the targets in the second step for the month-wise daily forecasting of the inflows using the two types of ensemble models, namely, averaging and boosting models in machine learning. The test results show that for both the monthly models and daily models the NRMSE and NMAE values were low for the monsoon periods compared to the non-monsoon periods. The averaging ensemble models were found to perform better than the boosting ensemble models for maximum number of months. The yearly results show an error of less than 5% between actual and predicted values for all the test cases, showing the precision in the developed algorithm. Further, the uncertainty analysis shows that the prediction done using the weighted average of the different inflow scenarios performs better than the prediction against the single inflow scenario.     

Short-term River streamflow modeling using Ensemble-based additive learner approach

Accurate streamflow (Q_t) prediction can provide critical information for urban hydrological management strategies such as flood mitigation, long-term water resources management, land use planning and agricultural and irrigation operations. Since the mid-20th century, Artificial Intelligence (AI) models have been used in a wide range of engineering and scientific fields, and their application has increased in the last few years. In this study, the predictive capabilities of the reduced error pruning tree (REPT) model, used both as a standalone model and within five ensemble-approaches, were evaluated to predict streamflow in the Kurkursar basin in Iran. The ensemble-approaches combined the REPT model with the bootstrap aggregation (BA), random committee (RC), random subspace (RS), additive regression (AR) and disjoint aggregating (DA) (i.e. BA-REPT, RC-REPT, RS-REPT, AR-REPT and DA-REPT). The models were developed using 15 years of daily rainfall and streamflow data for the period 23 September 1997 to 22 September 2012. A set of eight different input scenarios was constructed using different combinations of the input variables to find the most effective scenario based on the linear correlation coefficient. A comprehensive suite of graphical (time-variation graph, scatter-plot, violin plot and Taylor diagram) and quantitative metrics (root mean square error (RMSE), mean absolute error (MAE), Nash-Sutcliff efficiency (NSE), Percent of BIAS (PBIAS) and the ratio of RMSE to the standard deviation of observation (RSR)) was applied to evaluate the prediction accuracy of the six models developed. The outcomes indicated that all models performed well but the AR-REPT outperformed all the other models by rendering lower errors and higher precision across a number of statistical measures. The use of the BA, RC, RS, AR and DA models enhanced the performance of the standalone REPT model by about 26.82%, 18.91%, 7.69%, 28.99% and 28.05% respectively.