Conventional ANN Research Articles

Enhancing short-term streamflow forecasting of extreme events: A wavelet-artificial neural network hybrid approach

Accurate short-term streamflow forecasting models are crucial for effective water resource management, enabling timely responses to extreme flood or drought events and mitigating potential socioeconomic damage. This study proposes robust hybrid Wavelet Artificial Neural Network (WANN) models for real-world hydrological applications. Two WANN variants, WANNone and WANNmulti, are proposed for short-term streamflow forecasting of extreme (high and low) flows at eight gauging stations within Brazil's Paraíba do Sul River basin. WANNone directly feeds both the original streamflow data and the decomposed components obtained through an À Trous wavelet transform into the ANN architecture. Conversely, WANNmulti utilizes separate ANNs for the original data, with the final streamflow estimate reconstructed via the inverse wavelet transform of the individual ANN outputs. The performance of these WANN models is then compared against conventional ANN models. In both approaches, Bayesian optimization is employed to fine-tune the hyperparameters within the ANN architecture. The WANN models achieved superior performance for 7-day streamflow forecasts compared to conventional ANN models. WANN models yielded high R2 values (>0.9) and low MAPE (4.8%–14.7%) within the expected RMSE range, demonstrating statistically significant improvements over ANN models (71% and 75% reduction in RMSE and MAPE, respectively, and 69% increase in R2). Further analysis revealed that WANNmulti models generally exhibited superior performance for low extreme flow predictions, while WANNone models achieved the highest accuracy for high extreme flows at most stations. WANN models' strong performance suggests their value for real-time flood warnings, enabling improved decision-making in areas like flood/drought mitigation and urban water planning.

A hybrid DMO-RERNN based UPFC controller for transient stability analysis in grid connected wind-diesel-PV hybrid system

AbstractIn this paper, a novel hybrid technique is proposed for transient stability analysis on grid connected Wind-Diesel-PV hybrid system. The proposed hybrid methodology is combination of the dwarf mongoose optimization algorithm (DMO) and the recalling enhanced recurrent neural network (RERNN) named DMO-RERNN. The main purpose of this work is to consider various elements on hybrid system for the analysis of transient stability according to different conditions. The voltage profile of hybrid system is enhanced using the proposed unified power flow controller (UPFC), which also has higher performance improving transient performance compared to the conventional ANN, PI and fuzzy-sliding mode controller. Considering the proposed technique, DMO is used to find the optimal global solution for the fault predicted by the RERNN approach. The proposed system is executed on MATLAB work platform; its performance with existing systems is analyzed. The result proves that the proposed hybrid technique based UPFC controller provides better results compared with other existing technique. The efficiency of the PI is 82.136, ANN is 77, Fuzzy Sliding Mode is 65.097% and proposed technique is 97.99038%.

Short-term load and price forecasting using artificial neural network with enhanced Markov chain for ISO New England

Nowadays, forecasting methods have gained significant attention, particularly with the design and development of energy systems. In fact, accurate load and price forecasting is crucial for effective planning, controlling, and operation of power systems, especially with renewable energy sources (RES). This paper implemented an improved Markov Chain Artificial Neural network (ANN-MC) for load forecasting. The proposed design involved a two-step implementation process, considering various statistical factors such as daily and weekly load, date/time of the year, environmental factors (e.g., dry bulb temperature and dew point), and user behaviour on weekdays and weekends. The test cases were conducted using historical data from ISO New England spanning the years 2004 to 2020. Moreover, the validation of the proposed model has been confirmed through comparing the results with those of Gaussian Process Regression (GPR), Regression Decision Tree (RDT), deep learning Bi-Long Short Memory (bi-LSTM), MLP, and conventional ANN. This article discusses the use of various performance indices such as MAPE, MPE, skewness, kurtosis, and risk indices for evaluating model performance. The performance of a developed model is compared with a conventional ANN model, and its performance is studied for both yearly and seasonal variations. In addition to existing indices, the article proposes two risk indices. The first is based on evaluating the standard deviation of load increment for each time, while the second is based on MC-ANN, the error between forecasted and actual loads. The risk assessment is compared between different cases such as actual load, load forecasting with ANN, and enhanced ANN-MC. Finally, the result confirms that the enhanced ANN-MC provides a higher yearly MPE value compared to other methods. In addition, it has a higher computational time than the conventional ANN-MC model, which is approximately 180.7s and 221.8s, respectively.

A Real-Time C-V2X Beamforming Selector Based on Effective Sequence to Sequence Prediction Model Using Transitional Matrix Hard Attention

For C-V2X systems, the selection of the best beam in a real-time mode becomes an increasingly critical and yet open topic. Most of the existing approaches adopt either conventional ARIMA or ANN. Recently, there has been research on adopting sequence-to-sequence (Seq2Seq) predictors with attentions to extract time series features and emphasis on critical information to achieve data prediction. In this paper, a Seq2Seq predictor integrating with a Transitional Matrix based Hard attention is presented and validated through an artificial test dataset with predefined transitional states. At first, the transition probability matrix is generated from previous time series data and fed into the “hard” attention module of Seq2Seq predictor to determine the weights during the training phase. Secondly, the presented Seq2Seq predictor was implemented and adopted to predict the best beams of a C-V2X beamforming selector built up by the authors. Experiments were conducted and captured data were used to validate the performance of the predictor. When compared with baseline models, the presented predictor can achieve an enhanced prediction accuracy in a gain of 10-12%.

Integrating LA and EDM for Improving Students Success in Higher Education Using FCN Algorithm

EDM and LA are two fields that study how to use facts to get more academic learning and enhance the students’ entire performance. Both areas are concerned with a broad range of issues such as curriculum strategies, coaching, mental well-being of students, learning motivation, and academic achievement. The COVID-19 pandemic highly disrupted the higher education sector and shifted the old, chalk-talk teaching-learning model to an online learning format. This meant that the structure and nature of teaching, learning, assessment, and feedback methodologies also changes. With the empowerment in technology, timely and effective feedback is provided by the teachers to achieve greater learning. Through these studies, it is noted that negative feedback discourages the effort and achievement of learners, so it should be carefully crafted and delivered. In this work, a new methodology is planned based on an improved FCN (fully connected network). The key impartial of the proposed method is to regulate the assessment of the quality of students in Higher Education HE. The proposed methodology is composed of different phases: The first phase is data acquisition, in which the data are gathered from various sources for training and testing of the proposed method. The second phase is data orientation, in which the information is oriented in a specific file format. After that, data are cleaned, and preprocessing methods are applied. In the fourth phase, a machine learning-based model is developed to predict student’s academic performance. The fully connected neural network is enhanced with LA to better assess student quality in higher education. The proposed work is evaluated with the OULAD database, which was gathered from the students of Open University. The proposed methodology has attained an accuracy of 84%, more significant than the conventional ANN model accuracy rate. The proposed methodology’s Recall, F1-score, and precision rates are 0.88, 0.91, and 0.93, respectively.

Functional-Hybrid modeling through automated adaptive symbolic regression for interpretable mathematical expressions

Mathematical models used for the representation of (bio)-chemical processes can be grouped into two broad paradigms: white-box or mechanistic models, completely based on knowledgeor black-box data-driven models based on patterns observed in data. However, in the past two-decade, hybrid modeling that explores the synergy between the two paradigms has emerged as a pragmatic compromise. The data-driven part of these has been largely based on conventional machine learning algorithms (e.g., artificial neural network, support vector regression), which prevents interpretability of the finally learnt model by the domain experts. In this work, we present a novel hybrid modeling framework, the Functional-Hybrid model, that uses the ranked domain-specific functional beliefs together with symbolic regression to develop dynamic models. We demonstrate the successful implementation of the Functional-Hybrid model and its interpretability, focusing on applying chemical reaction kinetic principles to classical chemical reactions, biochemistry, ecology, physiology, and a bioreactor. Furthermore, we demonstrate that during interpolation, the Functional-Hybrid model performs similarly to a Hybrid-ANN hybrid model implementing a conventional ANN. However, it provides the advantage of being –to some extent– interpretable, unlike the conventional Hybrid-ANN model. Additionally, it is shown that the Functional-Hybrid model outperforms the Hybrid-ANN model for a very low number of experiments, making it more suitable when data is scarce. Finally, the Functional-Hybrid models show superior extrapolation capabilities compared to the Hybrid-ANN model. This improved performance can be attributed to the structure imposed by the functional transformations introduced in the Functional-Hybrid model.

An evaluation of the thermal behaviour of a lithium-ion battery pack with a combination of pattern-based artificial neural networks (PBANN) and numerical simulation

Thermal management is an important factor in extending the battery's life time and ensuring the quality of the output current. A numerical evaluation of the effect of varying the configuration of the battery cells and liquid-cooled channels was conducted in this case study. For the first time, a physics-informed neural network is coupled with visual tracking and commercial software for prediction of the thermal behaviour of a battery package. Combination of physics-informed neural network and visual tracking present as pattern-based neural networks (PBANNs). This method was used to predict the surface temperature at several cooling rates (Vinlet=0.1, 0.3, and 0.5 m/s) in response to variations in the surface temperature of the battery. Compared with conventional ANN methods, PBANN can significantly reduce the computational cost of transient case studies. Furthermore, PBANN can be directly coupled with commercial software in real-time. The complexity of coding for numerical simulation could be reduced by this coupling algorithm. Based on the results of this coupling, battery configurations may affect temperature profiles. By distributing cooling tubes evenly, the average temperature of the battery and phase change material (PCM) could be reduced by 25.3%. According to the results, the combination of liquid-cooled and PCM could guarantee that the battery temperature would not exceed the limits.

Data mining and design of electromagnetic properties of Co/FeSi filled coatings based on genetic algorithms optimized artificial neural networks (GA-ANN)

It is difficult to establish analytical relationships between coating parameters (type and filling ratios of absorbent) and electromagnetic (EM) properties based on traditional trial-and-error methods. Data mining is expected to predict the EM properties of coatings filled with arbitrary set type/content of absorbent based on the measured parameters within a few samples, to solve this problem. In this study, a method for predicting EM properties of coatings was established based on GA-ANN algorithm. The proposed model was found be capable of predicting EM properties of coatings, not only in wide range of filling ratio but also in case of mixed absorbents. The GA-ANN model presented lower error and higher accuracy to compare with the conventional ANN methods. Additionally, GA was used to optimize the design of coatings via using of mixed absorbent aiming to obtain broadband electromagnetic absorption (EMA) characteristics. The optimized mixture coating with the thickness of 2.06 mm contained 13.5 wt% of Co and 55.6 wt% of FeSi particles, whose effective absorption bandwidth with reflection loss less than −10 dB ( ERL 10 ) could reach 9.3 GHz. • A predictive model for electromagnetic properties of coating was established. • The model was built based on genetic algorithm optimized artificial neural network. • Electromagnetic property for arbitrarily C02/F25 content were accurately predicted. • A broadband absorbing coating was designed based on the predictive model.

Implementation of coupled pattern recognition and regression artificial neural networks for mass estimation of headless‐shell‐on shrimp with random postures

AbstractIn an actual sorting process, shrimps are fed to a machine‐vision‐based sorter at random postures. This study proposed an Enhanced Artificial Neural Network (E‐ANN) coupled with a Pattern Recognition ANN (P‐ANN) model to overcome the posture‐specificity of the regression ANN model commonly used for mass estimation of the headless‐shell‐on (HSO) shrimps. Images of 103 shrimps with seven different postures were used. The similarity of any shrimp image to the reference shrimp postures (i.e., extended‐legs, collapsed‐legs, curl body, and dorsal body) was determined by the P‐ANN model and used as additional input, besides the area and perimeter. The coupled‐ANN model could accurately estimate the mass of shrimps with random postures (R2 = 0.70 to 0.88 and MRE = −2.62 to 2.97%) within ~10 ms per shrimp, which is practical to use in an automatic shrimp sorting system based on machine vision technique. Further enhancement of the model performance could be achieved by adding color and texture features to distinguish different shrimp parts (e.g., body, legs, and tail).Practical ApplicationsThis research shows the potential of using a pattern recognition artificial neural network (P‐ANN) to cut the limitation of the conventional regression ANN model based on area and perimeter for shrimp mass estimation, which is specific to shrimp posture (or shape). Since the coupled P‐ANN and enhanced artificial neural network (E‐ANN) used a relatively short processing time (~10 ms per shrimp) and yielded less than 1 g error in the mass estimation of the headless‐shell‐on (HSO) shrimps with random postures, it is possible to incorporate this coupled ANN model in an automatic shrimp sorting system based on machine vision technique.

A Convex Combination Approach for Artificial Neural Network of Interval Data

As the conventional models for time series forecasting often use single-valued data (e.g., closing daily price data or the end of the day data), a large amount of information during the day is neglected. Traditionally, the fixed reference points from intervals, such as midpoints, ranges, and lower and upper bounds, are generally considered to build the models. However, as different datasets provide different information in intervals and may exhibit nonlinear behavior, conventional models cannot be effectively implemented and may not be guaranteed to provide accurate results. To address these problems, we propose the artificial neural network with convex combination (ANN-CC) model for interval-valued data. The convex combination method provides a flexible way to explore the best reference points from both input and output variables. These reference points were then used to build the nonlinear ANN model. Both simulation and real application studies are conducted to evaluate the accuracy of the proposed forecasting ANN-CC model. Our model was also compared with traditional linear regression forecasting (information-theoretic method, parametrized approach center and range) and conventional ANN models for interval-valued data prediction (regularized ANN-LU and ANN-Center). The simulation results show that the proposed ANN-CC model is a suitable alternative to interval-valued data forecasting because it provides the lowest forecasting error in both linear and nonlinear relationships between the input and output data. Furthermore, empirical results on two datasets also confirmed that the proposed ANN-CC model outperformed the conventional models.

Focus shaping of high numerical aperture lens using physics-assisted artificial neural networks.

We present a physics-assisted artificial neural network (PhyANN) scheme to efficiently achieve focus shaping of high numerical aperture lens using a diffractive optical element (DOE) divided into a series of annular regions with fixed widths. Unlike the conventional ANN, the PhyANN does not require the training using labeled data, and instead output the transmission coefficients of each annular region of the DOE by fitting weights of networks to minimize the delicately designed loss function in term of focus profiles. Several focus shapes including sub-diffraction spot, flattop spot, optical needle, and multi-focus region are successfully obtained. For instance, we achieve an optical needle with 10λ depth of focus, 0.41λ lateral resolution beyond diffraction limit and high flatness of almost the same intensity distribution. Compared to typical particle swarm optimization algorithm, the PhyANN has an advantage in DOE design that generates three-dimensional focus profile. Further, the hyperparameters of the proposed PhyANN scheme are also discussed. It is expected that the obtained results benefit various applications including super-resolution imaging, optical trapping, optical lithography and so on.

ANN based LFC with Coordination strategies of DERs in Hybrid Isolated Micro-Grid Environment

This paper provides a load frequency control (LFC) of a micro grid with renewable energy resources (RES). The operation of micro grid with a low inertia system leds to disturbances in power system. The disturbances in frequency is more in micro grid than conventional power system. So there should be a fast recovery of changes in frequency with existing system and interconnected system (RES). Active power injection is the main scheme to control frequency of a system. The matlab simulink tells us that different active power injection system contribute for the fast control of grid frequency with PID controller. The use of ANN technology to this system the load frequency control can be illustrated in faster rate of its recovery. An ANN controller is investigated which handles the inputs collectively in each sector of the power system. Back-transmission time is normally used in the study for neural network education. The performance of the power system is simulated independently with a typically integrated conventional controller and ANN controller. A complete spectrum of small signals is introduced for RESs in the isolated microgrid and a correct role in frequency control studies is taken into account.

An integrated Bayesian least-squares-support-vector-machine factorial-analysis (B-LSVM-FA) method for inferring inflow from the Amu Darya to the Aral Sea under ensemble prediction

An integrated Bayesian least-squares-support-vector-machine factorial-analysis (B-LSVM-FA) method is developed through integrating techniques of Bayesian inference, least squares support vector machine (LSVM), and factorial analysis (FA) into a general framework. B-LSVM-FA has advantages in: (i) capturing the complicated nonlinear relationship between input factors and streamflow, (ii) optimizing the key parameters of LSVM through a maximum posterior density estimation, and (iii) quantifying the contributions of individual and interactive effects of multiple factors to streamflow variation. B-LSVM-FA is then applied to inferring the changes in inflow from the Amu Darya to the Aral Sea (named as IATA). Results obtained cannot only identify the key impact factors reducing IATA during the period of 1960–2015, but also predict future trends in IATA for 2020–2050. Comparing to the conventional ANN, SVM and LSVM, the proposed method performs better in describing the IATA changes with anthropogenic, hydrometeorological, and ecological factors in terms of NSE, RMSE, and PBS. Results disclose that the major factors affecting IATA at annual/seasonal scale are upstream streamflow, agricultural water use in Uzbekistan, reservoir water storage, and evapotranspiration. The significant differences in the contributions of main factors to IATA at seasonal scale are observed because each season has unique characteristics of human activities, meteorological conditions, and vegetation coverages. In order to seek the feasible strategies of recovering the IATA level in the future, 162 scenarios based on ensemble prediction are analyzed. Results indicate that the IATA would restore to its 1970s condition if the drip irrigation rate reaches 50% at the end of 2050, and the reservoir water storage level reduces to the average value of 1960–1970. The findings can provide valuable suggestions for decision makers to increase IATA, and thus ameliorating the ecological crisis within the Aral Sea Basin.

A novel clustering-enhanced adaptive artificial neural network model for predicting day-ahead building cooling demand

To accurately predict hourly day-ahead building cooling demand, year-round historical weather profile needs to be evaluated. The daily weather profiles among different time periods result in various features of historical datasets. The different appropriate structure and parameters of artificial neural network models may be identified for training datasets with different features. In this study, a novel clustering-enhanced adaptive artificial neural network (C-ANN) model is proposed to forecast 24h-ahead building cooling demand in subtropical areas. The uniqueness of the proposed adaptive model is that k-means clustering is implemented to recognise representative patterns of daily weather profile and thus categorize the annual datasets into featuring clusters. Each cluster of the weather profile, along with the corresponding time variables and cooling demand, is adopted to train one ANN sub-model. The optimal structure and parameters of each ANN sub-model are selected according to its featuring training datasets; thus the ANN sub-models are adaptive. The proposed C-ANN model is tested on a representative office building in Hong Kong. It is found that the mean absolute percentage error of the training and testing cases of the proposed predictive model is 3.59% and 4.71%, which has 4.2% and 3.1% improvement compared to conventional ANN model with a fixed structure. The proposed adaptive predictive model can be applied in building energy management system to accurately predict day-ahead building cooling demand using the latest forecast weather profile.

A new W-SVM kernel combining PSO-neural network transformed vector and Bayesian optimized SVM in GDP forecasting

Considering that in the literature there is a very limited number of studies proposing new SVM kernels especially in regression problems, the scope of this research is to investigate the development of a novel Support Vector Machine Kernel. The proposed new W-SVM (Weighted-SVM) kernel was developed by applying a suitably transformed weight vector derived from particle swarm optimized neural networks in order to satisfy the kernel conditions of Mercer’s theorem and then incorporated to a Bayesian Optimized (BO) kernel for building the new proposed W-SVM kernel. The proposed SVM kernel was applied in Gross Domestic Product growth forecasting. The new kernel has led to significantly improved forecasting results compared to all the other conventional ANN, SVM, and optimized BO-SVM, PSO-ANN machine learning models.

A hybrid wavelet neural network (HWNN) for forecasting rainfall using temperature and climate indices

Rainfall forecasting plays an important role in water resources management and also for controlling the unusual events related to the rainfall. This study aims to forecast monthly rainfall from antecedent monthly rainfall, temperature and climate indices using a hybrid wavelet neural network (HWNN) model. The discrete wavelet transform is used incorporation with a conventional ANN model. The skilfulness of the proposed model is compared with the observed rainfall and the ANN model. The results show that the HWNN model provides a good fit with the observed rainfall data particularly in facing the extreme rainfall. The decomposed sub-series obtained by wavelet transform can extract invaluable information which is enormously useful for future rainfall prediction. The results confirm that the hybrid model considerably improves the neural network ability to predict future rainfall.

Real-time prognosis of flowing bottom-hole pressure in a vertical well for a multiphase flow using computational intelligence techniques

An accurate prediction of well flowing bottom-hole pressure (FBHP) is highly needed in petroleum engineering applications such as for the field production optimization, cost per barrel of oil reduction, and quantification of workover remedial operations. A good number of empirical correlations and mechanistic models exist in the literature and are frequently used in oil industry to estimate FBHP. But majority of the empirical models were developed under a laboratory scale and are therefore inaccurate when scaled up for the field applications. The objective of this study is to present a new computational intelligence-based model to predict FBHP for a naturally flowing vertical well with multiphase flow. The present study shows that the accuracy of FBHP estimation using PSO-ANN is better than the conventional ANN model. A small average absolute percentage error of less than 2.1% is observed with the proposed model, while comparing the previous empirical correlations and mechanistic models on the same data gives more than 15% error. The new model is trained on a surface production data, which makes the prediction of FBHP in a real time. A group trend analysis tests were also carried out to assure that the proposed model is accurately capturing the underline physics behind the problem.

Application of PSO–ANN modelling for predicting the exergetic performance of a building integrated photovoltaic/thermal system

The main objective of this study is to examine the feasibility of hybrid PSO–ANN technique to estimate the exergetic performance of a building integrated photovoltaic/thermal (BIPV/T) system. A performance evaluation criterion (PEC) is defined in this study to assess the overall performance of a BIPV/T system from exergy point of view. Then, the mentioned method is utilized to identify a relationship between the input and output parameters of the BIPV/T system. The parameter PEC was taken as the essential output of the BIPV/T system, while the input parameters were channel length, channel depth, channel width, and air mass flow rate. Prior to PSO, variables of ANN algorithm were optimized. In addition, PSO influential parameters such as swarm size, personal learning coefficient, global learning coefficient, and inertia weight were optimized using a series of trial-and-error process. The predicted results for data sets from ANN and PSO–ANN models were evaluated according to several known statistical indices and novel ranking systems of color intensity rating and total ranking method. The obtained RMSE and R2 in the training (RMSE of 0.010274 and 0.006112, and R2 of 0.9968 and 0.9989, respectively, for the PSO and ANN methods) and testing (RMSE of 0.011146 and 0.005927, and R2 of 0.9967 and 0.9990, respectively, for the PSO and ANN methods) phases. The results revealed that the PSO–ANN network model could slightly accomplish a better performance when it is compared to the conventional ANN.

Short Term Price Forecasting Using Adaptive Generalized Neuron Model

This article described how in the competitive deregulated electricity market forecasting has become one of the essential planning tool that assists the planners in preparing the power systems for future demands. The commercial success of the market players depends on their competitive bidding strategy which is suffuicient enough to meet the regulatory requirements and minimize the cost. Artificial neural networks due to their capability of non-linear mapping finds extensive application in the field of price forecasting. Although, they are extensively used as forecasting model, they have certain limitations which are detrimental to system performance. The training time of the artificial neural network is affected by the complexity of the system, and moreover, they require a large amount of data for complex problems. The worl presented in this article deals with the application of the generalized neuron model for forecasting the electricity price. The generalized neuron model overcomes the limitation of the conventional ANN. The electricity price of the New South Wales electricity market is forecast to test the performance of the proposed model. The free parameters of the proposed model are trained using fuzzy tuned genetic algorithms to increase efficacy of the model.

Kernel machines and firefly algorithm based dynamic modulus prediction model for asphalt mixes considering aggregate morphology

Artificial Intelligence algorithm support vector regression (SVR) has proved successful in outperforming conventional Witczak and ANN models for estimation of dynamic modulus (E∗) of asphalt mixes. However, there were two issues related to the development of E∗ prediction models that the present study addresses. Firstly, since aggregates occupy almost 95% by weight of HMA, it is quite possible that the morphology of these aggregates play an important role in influencing the E∗ values. To address this issue, aggregate shape parameters, namely, angularity, sphericity, texture and form were used with aggregate gradation for stiffness estimation. Secondly, to fine tune the hyper-parameters firefly algorithm (FA) was coupled with SVR. E∗ tests of 20 HMA mixes having different sources, sizes of aggregates, and volumetric properties were conducted at 4 temperatures and 6 frequencies. Aggregate shape parameters were measured using the automated aggregate image measurement system (AIMS). SVR-FA models were developed that predicted the E∗ with an R2 of 0.98. SVR-FA models were compared with SVR and ANN models for E∗ prediction. Further, a sensitivity analysis was conducted to identify the important input parameters. Lastly, an approach for formulation of SVR-FA model equations for direct prediction of HMA stiffness is also discussed. FA proved instrumental in improving the efficiency of SVR by optimizing the hyper-parameters with lesser manual effort. Finally, it was concluded that SVR-FA algorithm is capable of successfully predicting the E∗ values using the aggregate shape parameters.