Wavelet Regression Models Research Articles

Comparative Analysis of River Flow Modelling by Using Supervised Learning Technique

The goal of this research is to investigate the efficiency of three supervised learning algorithms for forecasting monthly river flow of the Indus River in Pakistan, spread over 550 square miles or 1800 square kilometres. The algorithms include the Least Square Support Vector Machine (LSSVM), Artificial Neural Network (ANN) and Wavelet Regression (WR). The forecasting models predict the monthly river flow obtained from the three models individually for river flow data and the accuracy of the all models were then compared against each other. The monthly river flow of the said river has been forecasted using these three models. The obtained results were compared and statistically analysed. Then, the results of this analytical comparison showed that LSSVM model is more precise in the monthly river flow forecasting. It was found that LSSVM has he higher r with the value of 0.934 compared to other models. This indicate that LSSVM is more accurate and efficient as compared to the ANN and WR model.

Optimal designs for spline wavelet regression models

In this article we investigate the optimal design problem for some wavelet regression models. Wavelets are very flexible in modeling complex relations, and optimal designs are appealing as a means of increasing the experimental precision. In contrast to the designs for the Haar wavelet regression model (Herzberg and Traves 1994; Oyet and Wiens 2000), the I-optimal designs we construct are different from the D-optimal designs. We also obtain c-optimal designs. Optimal (D- and I-) quadratic spline wavelet designs are constructed, both analytically and numerically. A case study shows that a significant saving of resources may be realized by employing an optimal design. We also construct model robust designs, to address response misspecification arising from fitting an incomplete set of wavelets.

Estimation of daily global solar radiation using wavelet regression, ANN, GEP and empirical models: A comparative study of selected temperature-based approaches

Although the sunshine-based models generally have a better performance than temperature-based models for estimating solar radiation, the limited availability of sunshine duration records makes the development of temperature-based methods inevitable. This paper presents a comparative study between Artificial Neural Networks (ANNs), Gene Expression Programming (GEP), Wavelet Regression (WR) and 5 selected temperature-based empirical models for estimating the daily global solar radiation. A new combination of inputs including four readily accessible parameters have been employed: daily mean clearness index (KT), temperature range (ΔT), theoretical sunshine duration (N) and extraterrestrial radiation (Ra). Ten statistical indicators in a form of GPI (Global Performance Indicator) is used to ascertain the suitability of the models. The performance of selected models across the range of solar radiation values, was depicted by the quantile-quantile (Q-Q) plots. Comparing these plots makes it evident that ANNs can cover a broader range of solar radiation values. The results shown indicate that the performance of ANN model was clearly superior to the other models. The findings also demonstrated that WR model performed well and presented high accuracy in estimations of daily global solar radiation.

A Comparison of Three Soft Computing Techniques, Bayesian Regression, Support Vector Regression, and Wavelet Regression, for Monthly Rainfall Forecast

Abstract Rainfall, being one of the most important components of the hydrological cycle, plays an extremely important role in agriculture-based economies like India. This paper presents a comparison between three soft computing techniques, namely Bayesian regression (BR), support vector regression (SVR), and wavelet regression (WR), for monthly rainfall forecast in Assam, India. A WR model is a combination of discrete wavelet transform and linear regression. Monthly rainfall data for 102 years from 1901 to 2002 at 21 stations were used for this study. The performances of different models were evaluated based on the mean absolute error, root mean square error, correlation coefficient, and Nash-Sutcliffe efficiency coefficient. Based on model statistics, WR was found to be the most accurate followed by SVR and BR. The efficiencies for the BR, SVR, and WR models were found to be 32.8%, 52.9%, and 64.03%, respectively. From the spatial analysis of model performances, it was found that the models performed best for the upper Assam region followed by lower, southern, and middle regions, respectively.

Wavelet regression and wavelet neural network models for forecasting monthly streamflow

This paper investigates the performance of wavelet-based regression models for monthly streamflow forecasting. The wavelet-based regression model combines wavelet transformation and multiple linear regression (LR). The wavelet-based regression forecasts are also compared to the wavelet-based neural network, which combines the wavelet transformation and feed forward neural network. The wavelet transformation has significantly positive effects on the modeling performance. In this study, the different approaches of the wavelet-based models were applied to forecast the monthly flow. The results show that the wavelet-based feed forward neural network and the wavelet-based linear regression (WLR) produce very good results for 1-month-ahead streamflow forecasting. Both techniques demonstrated an almost similar performance. Also, the result of the WLR5 model is better than the results of the other WLR models in terms of performance criteria.

Wavelet Neural Modeling for Hydrologic Time Series Forecasting with Uncertainty Evaluation

An approach, with the basic idea of resampling wavelet neural parameters, was proposed for probabilistic forecasting of hydrologic time series by the wavelet neural model. Parameters in wavelet neural model are assumed as following uniform distribution, and both proper convergence criterion and likelihood function are used to train the wavelet neural structure and judge the acceptance of parameter set. By training and learning wavelet neural structure as many times (i.e., resampling neural parameters) until becoming stable, all sets of wavelet neural parameters are composed as the resampling results, based on which probabilistic forecasting of hydrologic time series is attained. Optimal forecasting result can be gained by computing mathematical mean of the resampling results, and uncertainty can be described by proper confidence interval. Results of one runoff example indicated the identical performance of the proposed approach and wavelet regression model, but both perform better than conventional neural model. The proposed approach has similar efficiency as the Bayesian method for uncertainty evaluation, and both show higher efficiency than traditional Monte-Carlo method. Choice of proper convergence criterion is an important task when using the proposed approach, because it directly determines the convergence rate, accuracy and uncertainty level of probabilistic forecasting result. Overall, several key issues should be carefully considered for obtaining more reasonable probabilistic forecasting results by the proposed approach, including choice of proper likelihood function, accurate wavelet decomposition of series, and determination of proper wavelet neural structure.

A Wavelet-Based Second Order Nonlinear Model for Forecasting Monthly Rainfall

In this article, a rainfall forecasting model using monthly historical rainfall data and climate indices is developed by incorporating wavelet analysis (WA) and second order volterra nonlinear model. The monthly rainfall time series and large-scale climate index time series are decomposed using wavelets into a certain number of component subseries at different temporal scales. The lag relationship between the rainfall anomaly and each potential predictor is identified by cross correlation analysis with a lag time of at least 1 month at different temporal scales. The components of predictor variables with known lag times are then integrated using a second order Volterra model. Further, orthogonal least squares method is used to reduce the redundant variables and select the significant variables to be included into the final forecast model. The proposed multivariate wavelet nonlinear rainfall forecasting method is examined with over three places in India, and compared to the traditional ANN model based on the original time series and linear wavelet regression model. The models are trained with data from the 1916 to 1968 period and then tested in the 1968–1989 period. The results show that the proposed wavelet nonlinear model provides considerably more accurate monthly rainfall forecasts for the three selected places in India than the traditional regression model, neural networks model and the wavelet based linear model. It was seen that for the proposed models and other models also, both the past rainfall and the large-scale climate signals were useful in forecasting the future.

Modeling of Sediment Yield Prediction Using M5 Model Tree Algorithm and Wavelet Regression

The forecast of the sediment yield generated within a watershed is an important input in the water resources planning and management. The methods for the estimation of sediment yield based on the properties of flow and sediment have limitations attributed to the simplification of important parameters and boundary conditions. Under such circumstances, soft computing approaches have proven to be an efficient tool in modelling the sediment yield. The focus of present study is to deal with the development of decision tree based M5 Model Tree and wavelet regression models for modeling sediment yield in Nagwa watershed in India. A comparison is also performed with the artificial neural network (ANN) model for streamflow forecasting. The root mean square errors (RMSE), Nash-Sutcliff efficiency index (N-S Index), and correlation coefficient (R) statistics are used for the statistical criteria. A comparative evaluation of the performance of M5 Model Tree and wavelet regression versus ANN clearly shows that M5 Model Tree and wavelet regression can prove more useful than ANN models in estimation of sediment yield. Further, M5 model tree offers explicit expressions for use by design engineers.

Streamflow estimation at ungauged site using wavelet group method of data handling in peninsular Malaysia

This study investigates the ability of wavelet group method of data handling (WGMDH) conjunction model in the estimation of flood quantiles in ungauged sites in Peninsular Malaysia. The conjunction method was obtained by combining two methods, discrete wavelet transform and group method of data handling . Comparison between the WGMDH model, group method of data handling model (GMDH), wavelet regression (WR) model and linear regression (LR) model were done. To assess the effectiveness of this model, 70 catchments in the province of Peninsular Malaysia were used as case studies. The performance of WGMDH model was compared with the conventional LR, GMDH and WR models using various statistical measures such as the mean absolute error, root mean square error and Nash-Sutcliffe coefficient of efficiency. Jackknife procedure was required for the evaluation of the performance of the four approaches. The jackknife procedure was needed to simulate the ungauged sites. The results of the comparison indicate that the WGMDH model was more accurate and perform better than the traditional LR, GMDH and WR models. Thus, WGMDH model is a promising new method for estimation of flood quantiles in ungauged sites.

Bayesian-combined wavelet regressive modeling for hydrologic time series forecasting

Wavelet regression (WR) models are used commonly for hydrologic time series forecasting, but they could not consider uncertainty evaluation. In this paper the AM-MCMC (adaptive Metropolis-Markov chain Monte Carlo) algorithm was employed to wavelet regressive modeling processes, and a model called AM-MCMC-WR was proposed for hydrologic time series forecasting. The AM-MCMC algorithm is used to estimate parameters' uncertainty in WR model, based on which probabilistic forecasting of hydrologic time series can be done. Results of two runoff data at the Huaihe River watershed indicate the identical performances of AM-MCMC-WR and WR models in gaining optimal forecasting result, but they perform better than linear regression models. Differing from the WR model, probabilistic forecasting results can be gained by the proposed model, and uncertainty can be described using proper credible interval. In summary, parameters in WR models generally follow normal probability distribution; series' correlation characters determine the optimal parameters values, and further determine the uncertain degrees and sensitivities of parameters; more uncertain parameters would lead to more uncertain forecasting results and hard predictability of hydrologic time series.

Wavelet regression models for predicting flood stages in rivers: a case study in Eastern India

AbstractCombining discrete wavelet transform (DWT) and autoregression (AR), two types of wavelet regression (WR) models were developed for forecasting 1‐day‐ahead river stages. In the first type of WR models, AR was applied on the DWT‐obtained subtime series while in the second type, AR was applied on the modified time series which was formed by recombining effective subtime series and ignoring the ‘noise’ subtime series. Depending upon different input combinations, five models in each type of WR models were developed. The efficiency of developed models was tested in forecasting monsoon stages of Kosi River in Bihar State of India. During monsoon (June to Oct), the Kosi carries large flow and makes the entire North Bihar unsafe for habitation or cultivation. When compared, WR models predicted river stages with greater accuracy than AR and artificial neural network (ANN) models, developed for the purpose. Between the two types of WR models, the first type gave slightly better results than the second type. The best performing WR model, with five previous days’ subtime series as inputs, predicted stages of the Kosi River, with the highest accuracy of 97.41%, the minimum root mean square error of 7.9 cm and the maximum coefficient of correlation of 0.952.

Reference evapotranspiration based on Class A pan evaporation via wavelet regression technique

Accurate estimation of reference evapotranspiration (ET0) is important for water resources engineering. Therefore, a large number of empirical or semi-empirical equations have been developed for assessing ET0 from numerous meteorological data. However, records of such weather variables are often incomplete or not always available for many locations, which is a shortcoming of these complex models. Therefore, practical and simpler methods are required for estimating the ET0. In this study, the efficiency of a wavelet regression (WR) model in estimating reference evapotranspiration based on only Class A pan evaporation is examined. The results of the WR model are compared with those of three pan-based equations, namely the FAO-24 pan, Snyder ET0 and Ghare ET0 equations and their calibrated versions. Daily Class A pan evaporation data from the Fresno and Bakersfield stations of the United States Environmental Protection Agency in California, USA, are used in the study. The WR model estimates are compared against those of the FAO-56 Penman–Monteith equation. Results showed that the WR model is capable of accurately predicting the ET0 values as a product of pan evaporation data.

Robust designs for Haar wavelet approximation models

In this paper, we discuss the construction of robust designs for heteroscedastic wavelet regression models when the assumed models are possibly contaminated over two different neighbourhoods: G1 and G2. Our main findings are: (1) A recursive formula for constructing D-optimal designs under G1; (2) Equivalency of Q-optimal and A-optimal designs under both G1 and G2; (3) D-optimal robust designs under G2; and (4) Analytic forms for A- and Q-optimal robust design densities under G2. Several examples are given for the comparison, and the results demonstrate that our designs are efficient. Copyright © 2010 John Wiley & Sons, Ltd.

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.

Evapotranspiration modeling using a wavelet regression model

The present study proposes a simple wavelet regression (WR) approach for modeling reference evapotranspiration (ET0). The WR model was improved combining two methods: discrete wavelet transform (DWT) and a linear regression model. The accuracy of the WR models was compared with that of the single linear regression (LR) models. The daily climatic data from three stations in central California are used as inputs to the WR models to estimate ET0 obtained using the FAO-56 Penman–Monteith equation. The comparison of these results revealed that the WR models could increase the forecast accuracy of the LR models. A comparison is also made between the estimates provided by the WR models and those of the following empirical models: CIMIS Penman, Hargreaves, Ritchie and Turc. Based on a comparison of these results, the WR models were found to perform better than the empirical models in daily ET0 modeling.

Wavelet regression model for short-term streamflow forecasting

Summary Wavelet regression (WR) technique is proposed for short-term streamflow forecasting in this study. The WR model is improved combining two methods, discrete wavelet transform and linear regression model. The proposed model is applied to the daily streamflow data of two stations, Karabuk and Derecikviran, on the Filyos River in the Western Black Sea region of Turkey. Root mean square error (RMSE), mean absolute error (MAE) and correlation coefficient (R) statistics are used for evaluating the accuracy of the WR models. In the first part of the study, the accuracy of the WR models is compared with the artificial neural network (ANN) and autoregressive moving average (ARMA) models in 1-day ahead streamflow forecasting. Comparison results reveal that the WR model performs better than the ANN and ARMA models. The ARMA model is also found to be slightly better than the ANN. For the Karabuk and Derecikviran stations, it was found that WR models with RMSE = 8.48 m3/s, MAE = 2.46 m3/s, R = 0.978 and RMSE = 33.3 m3/s, MAE = 10.2 m3/s, R = 0.976 in the validation stage are superior in forecasting 1-day ahead streamflows than the most accurate ARMA models with RMSE = 13.5 m3/s, MAE = 3.44 m3/s, R = 0.942 and RMSE = 46.5 m3/s, MAE = 13.2 m3/s, R = 0.953, respectively. In the second part of study, WR and ANN models are compared in 2- and 3-day ahead streamflow forecasting. Based on the comparison results, WR models are found to be more accurate than the ANN models.

Combination of Models Obtained by Regression in the Wavelet Domain

The wavelet transform is a useful tool to preprocess and compress datasets for linear regression modelling. However, the prediction performance of the resulting model depends on the choice of wavelet filter and number of decomposition levels, which may not be a straightforward task. This paper proposes an alternative approach, which consists of combining models obtained from different wavelet decompositions of the dataset. For this purpose, a method is developed to convert wavelet regression models back to the original domain. The proposed approachis illustrated in a case study involving the determination of density in gasoline samples by using infrared spectroscopy. The results are favourably compared to those obtained by using individual wavelet decompositions.

Universal optimality of digital nets and lattice designs

This article considers universal optimality of digital nets and lattice designs in a regression model. Based on the equivalence theorem for matrix means and majorization theory, the necessary and sufficient conditions for lattice designs being φp- and universally optimal in trigonometric function and Chebyshev polynomial regression models are obtained. It is shown that digital nets are universally optimal for both complete and incomplete Walsh function regression models under some specified conditions, and are also universally optimal for complete Haar wavelet regression models but may not for incomplete Haar wavelet regression models.

Wavelet regression model as an alternative to neural networks for monthly streamflow forecasting

AbstractThe accuracy of the wavelet regression (WR) model in monthly streamflow forecasting is investigated in the study. The WR model is improved combining the two methods—the discrete wavelet transform (DWT) model and the linear regression (LR) model—for 1‐month‐ahead streamflow forecasting. In the first part of the study, the results of the WR model are compared with those of the single LR model. Monthly flow data from two stations, Gerdelli Station on Canakdere River and Isakoy Station on Goksudere River, in Eastern Black Sea region of Turkey are used in the study. The comparison results reveal that the WR model could increase the forecast accuracy of the LR model. In the second part of the study, the accuracy of the WR model is compared with those of the artificial neural networks (ANN) and auto‐regressive (AR) models. On the basis of the results, the WR is found to be better than the ANN and AR models in monthly streamflow forecasting. Copyright © 2009 John Wiley & Sons, Ltd.

Formula omitted]-optimal designs for a hierarchically ordered system of regression models

I L r -optimal designs are described for a kind of hierarchically ordered system of regression models with an r -dimensional response variable y . The components of y may be correlated with a known variance–covariance matrix Σ . The present results show that I L r -optimal designs for this system of regression models do not depend on Σ . The I L r -optimal designs are given for the systems of trigonometric and Haar wavelet regression models, respectively.