Scaled Conjugate Gradient Algorithm Research Articles

Artificial Neural Networks Based Red Palm Weevil (<i>Rynchophorus Ferrugineous, Olivier</i>) Recognition System

Problem statement: The most dangerous insect for the existence of palm trees in entire world is Red Palm Weevil (scientifically named as Rynchophorus Ferrugineous, Oliveir). The proposed research is conducted to develop an identification system for Automated Wireless Red Palm Weevil Detection and exterminated. The core idea of the proposed research is to develop software that can utilize image processing and Artificial neural network techniques to identify Red Palm Weevil and distinguishes it from other insects found in palm trees habitat. Approach: Images are taken and processed with image processing techniques. Afterwards, Artificial neural network is used to recognize the presence of Red Palm Weevil in an image. Two different feed-forward supervised learning algorithms of Artificial neural network are used i.e., scaled conjugate gradient and Conjugate Gradient with Powell/Beale Restarts Algorithms. Different Artificial neural network sizes are tested using both algorithms and are compared to find an optimal algorithm and network. The training, verification and testing of the Artificial neural network is accomplished by using a database of 319 images of Red Palm Weevil and 93 images of other insects which are usually found around palm trees. Images are randomly selected from database for training, verification and testing with a fixed percentage of 80, 10 and 10 respectively. Training for every selected set of configuration is repeated 10 times. Results: The best results for scaled conjugate gradient Algorithm is obtained by three layers ANN consuming 221 sec and 167 Epochs while its average success in identification of Red Palm Weevil and other insect is 99 and 93% respectively. On the other hand, best performance of Conjugate Gradient with Powell/Beale Restarts Algorithm is observed by using three layers ANN which consumed 183 sec and 109 Epochs for training while its average success in identification of Red Palm Weevil and other insect is 99.5 and 93.5% respectively. Conclusion: It is gleaned out that 3-layers Artificial neural network using Conjugate Gradient with Powell/Beale Restarts Algorithm for feed-forward supervised learning is optimal for identification of Red Palm Weevil.

The effects of intercooling on performance of a turbocharged diesel engines specific fuel consumption with neural network

Currently, fuel economy and thermal efficiency are more important to all engines. Efficiency is increased with cooled air by intercooler. Most of researches regarding engineering problems generally deal with experimental studies. But, the experimental researches are quite expensive and time consuming. In the last decades, Neural Networks (NN) had been used increasingly in a variety of engineering applications. The objective of the study is to investigate the adequacy of neural networks (NN) as a quicker, more secure and more robust method to determine the effects of intercooling on performance of a turbocharged diesel engine’s specific fuel consumption. The data are obtained from experimental research that is performed by the author. NN based model is developed, trained and tested through a based MATLAB program by using of these data. In the study, break specific fuel consumption (BSFC, g/kWh) was analysed with intercooling and without intercooling. The statistical analysis is performed to explain the performance of the NN based model. NN based model outputs are also compared with the experimental results. The statistical results and the comparison demonstrated that the NN based model is highly successful to determine the effects of intercooling on performance of a turbocharged diesel engine’s specific fuel consumption. Key words: Neural networks, intercooling, specific fuel consumption, scaled conjugate gradient algorithm, diesel engine.

An application of neural networks for harmonic coefficients and relative phase shifts detection

The varying the phase shifts will completely change the shape of the distorted wave, and may thus greatly affect the ability of the neural network to recognize harmonics. In this study, feed forward neural networks were used for the detection of the harmonic coefficients and relative phase shifts. The distorted wave including uniform distributed 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th harmonics with up to 20° relative phase shifts were simulated and used. Two neural networks were used for this purpose. One of the neural networks was used for the detection of the 5th, 7th, 11th, 13th harmonic coefficients and the other was used for the detection of the relative phase shifts of these harmonics. Scaled conjugate gradient algorithm was used as training algorithm for the weights update of the neural networks. The results show that these neural networks are applicable to detect each harmonic coefficient and relative phase shift effectively.

Neural network diagnostic system for dengue patients risk classification

With the dramatic increase of the worldwide threat of dengue disease, it has been very crucial to correctly diagnose the dengue patients in order to decrease the disease severity. However, it has been a great challenge for the physicians to identify the level of risk in dengue patients due to overlapping of the medical classification criteria. Therefore, this study aims to construct a noninvasive diagnostic system to assist the physicians for classifying the risk in dengue patients. Systematic producers have been followed to develop the system. Firstly, the assessment of the significant predictors associated with the level of risk in dengue patients was carried out utilizing the statistical analyses technique. Secondly, Multilayer perceptron neural network models trained via Levenberg-Marquardt and Scaled Conjugate Gradient algorithms was employed for constructing the diagnostic system. Finally, precise tuning for the models' parameters was conducted in order to achieve the optimal performance. As a result, 9 noninvasive predictors were found to be significantly associated with the level of risk in dengue patients. By employing those predictors, 75% prediction accuracy has been achieved for classifying the risk in dengue patients using Scaled Conjugate Gradient algorithm while 70.7% prediction accuracy were achieved by using Levenberg-Marquardt algorithm.

An Expert Model for Estimation of Distillation Sieve Tray Efficiency Based on Artificial Neural Network Approach

Knowledge of the efficiency of sieve tray columns as most common distillation equipments is necessary for the interpretation of separation and purification processes performance. In this study a new method based on Artificial Neural Network (ANN) for estimation of sieve tray efficiency has been proposed. In this case to develop data base several experimental data were collected from literatures. The network inputs are liquid and vapor density, liquid and vapor viscosity, liquid and vapor diffusivity, surface tension, slope of the equilibrium curve, hole diameter, weir height, weir length, liquid and gas flux, ratio of hole area to active area of the tray while the output is point efficiency. In order to find the best efficiency estimator of sieve tray, different training schemes for the back-propagation learning algorithm, such as; Scaled Conjugate Gradient (SCG), Levenberg-Marquardt (LM), Gradient Descent with Momentum (GDM), variable learning rate BP (GDA) and Resilient BP (RP) methods were examined. Finally among those trained networks, the SCG algorithm with ten neurons in the hidden layer shows the best suitable algorithm with the minimum average absolute relative error 0.029817. Finally, the capability of ANN and two recently published empirical models were compared. This ANN model reduced the prediction error by 64.03 and 92.64% relative to Garcia and Fair and Chan and Fair models, respectively. This is further proof that the proposed procedure can build a useful and robust model.

Application of Artificial Neural Network for Prediction of Liquid Viscosity

Viscosity is one of the important quality properties of fluids. It is a measure of the resistance of a fluid to being deformed by either shear or extensional stresses. Change in viscosity with temperature is very important in engineering calculations. In the present work a new method based on artificial neural network (ANN) for prediction of the viscosity of a compound has been proposed. The required data were taken from Perry's Chemical Engineers' Handbook. The accuracy and trend stability of the trained networks were tested against unseen data. Different training schemes for the back-propagation learning algorithm, such as scaled conjugate gradient (SCG), Levenberg-Marquardt (LM), and resilient back propagation (RP) methods were used. The SCG algorithm with 19 neurons in the hidden layer was found to be the most suitable algorithm with minimum mean square error (MSE) of 0.000381. Results of the ANN model were evaluated against the unseen data and then compared with the empirical models. The ANN's capability to predict viscosity is one of the best estimation methods with high performance.

Multifacility Location Problem using Scaled Conjugate Gradient Algorithm under Triangular Area Constraints

The problem of finding the multiple locations for new facilities with respect to the multiple existing facilities in a given environment is known as Multifacility Location Problem (MLP).Every location problem is normally bounded by some sort of area constraint. But the fact that much of the work carried out in the literature has almost neglected the area constraint which has motivated us to work on Multifacility Location Problem taking the area constraint into consideration. The mathematical model of the multifacility location problem with area constraint has been developed and the solution has been obtained using Kuhn-Tucker theory. This mathematical analysis and solution procedure is highly complex and time consuming. Hence, an attempt has been made to get the solution of a complex, constrained multifacility location problem using Scaled Conjugate Gradient Algorithm (SCGA) in Artificial Neural Networks (ANN). With the help of Numerical examples, it has been established that the solution obtained through ANN model compares well within the acceptable limits with those obtained through analytical method. Indexterms Multifacility Location Problem, Area Constraint, Kuhn-Tucker theory, Artificial Neural Networks (ANN), Scaled Conjugate Gradient Algorithm (SCGA). INTRODUCTION A number of real life situations involve the problem of locating new facilities with respect to the existing facilities. The multifacility location problems refer to the process of finding the multiple locations for the new facilities with respect to the existing multiple facilities in the prevailing environment. The past investigations into Multifacility Location problems have not taken the availability of area for locating the new facilities into consideration. In this regard we may mention the work of Cabot, Francis and Stray[1],Pritsker and Ghare[2], Kuhn and Kuenne [3], Eyster and White[4] and McHose[5]. Hence in our study of MLP, we have taken the area constraints into consideration. We have considered a deterministic model with the concept of Euclidean distance between facilities with the requirement that the new facilities must be located in a restricted area of triangular shape. This leads to linear constraints. We have solved the problem by using Kuhn-Tucker condition and illustrated the solution procedure of the problem by using some numerical data. 1.1 Applications of ANN The training and testing are two important phases in the development of ANN. The large number of training data sets containing input data and target output are to be fed to the ANN in the training phase. The neural networks create connections and learn patterns based on these data sets as pointed out by Rumelhart [7].Each pattern creates a unique configuration of network structure with a unique set of connection strengths or weights. Similarly, if the neural network does not match the pattern within the given tolerance, it will adjust and try again. A neural network adapts in changing inputs and learns trends from data. Each connection weight builds on previous decision nodes, propagating down to a final decision. After the neural network reaches a final decision, it compares its answer against an answer provided in the training set as target output. If there is a match, within a predefined tolerance, the neural network stores these connection weights as successful. If the decision outcome is outside the tolerance, then the neural network cycles through the training set again. A neural network may cycle thousands of times to reach an acceptable tolerance. The optimal network performance is quite dependent upon making the proper network structure [8]. We have first solved a set of multifacility location problems using the analytical method and then we have again solved the same set of problems through ANN technique using SCGA training algorithm. It is found that the solutions obtained through prediction by using this training algorithm compare well with those obtained using analytical method. The prediction error percentage vary between 0 − 13 % which is well within the acceptable limits. Further, the graphical representation of the results obtained analytically (Anal) and through ANN techniques have also been made to have visual clarity in comparison. 1.2 II. PROBLEM FORMULATION ANDANALYTICAL SOLUTION The MLP with triangular area constraint can be stated as:

MPID Control Tuning for a Flexible Manipulator Using a Neural Network

This paper studies the use of neural networks as a tuning tool for the gain in Modified Proportional-Integral-Derivative (MPID) control used to control a flexible manipulator. The vibration control gain in the MPID controller has been determined in an empirical way so far. It is a considerable time consuming process because the vibration control performance depends not only on the vibration control gain but also on the other parameters such as the payload, references and PD joint servo gains. Hence, the vibration control gain must be tuned considering the other parameters. In order to find optimal vibration control gain for the MPID controller, a neural network based approach is proposed in this paper. The proposed neural network finds an optimum vibration control gain that minimizes a criteria function. The criteria function is selected to represent the effect of the vibration of the end effector in addition to the speed of response. The scaled conjugate gradient algorithm is used as a learning algorithm for the neural network. Tuned gain response results are compared to results for other types of gains. The effectiveness of using the neural network appears in the reduction of the computational time and the ability to tune the gain with different loading condition.

Speeding up the scaled conjugate gradient algorithm and its application in neuro-fuzzy classifier training

The aim of this study is to speed up the scaled conjugate gradient (SCG) algorithm by shortening the training time per iteration. The SCG algorithm, which is a supervised learning algorithm for network-based methods, is generally used to solve large-scale problems. It is well known that SCG computes the second-order information from the two first-order gradients of the parameters by using all the training datasets. In this case, the computation cost of the SCG algorithm per iteration is more expensive for large-scale problems. In this study, one of the first-order gradients is estimated from the previously calculated gradients without using the training dataset. To estimate this gradient, a least square error estimator is applied. The estimation complexity of the gradient is much smaller than the computation complexity of the gradient for large-scale problems, because the gradient estimation is independent of the size of dataset. The proposed algorithm is applied to the neuro-fuzzy classifier and the neural network training. The theoretical basis for the algorithm is provided, and its performance is illustrated by its application to several examples in which it is compared with several training algorithms and well-known datasets. The empirical results indicate that the proposed algorithm is quicker per iteration time than the SCG. The algorithm decreases the training time by 20–50% compared to SCG; moreover, the convergence rate of the proposed algorithm is similar to SCG.

Creep feed grinding optimization by an integrated GA-NN system

The present work is aimed to optimize creep feed grinding (CFG) process by an approach using integrated Genetic Algorithm-Neural Network (GA-NN) system. The aim of this creep feed grinding optimization is obtain the maximal metal removal rate (MRR) and the minimum of the surface roughness (R a ). For optimization, metal removal rate is calculated with a mathematic formula and a Back Propagation (BP) artificial neural-network have been used to prediction of R a . The parameters used in the optimization process were reduced to three grinding conditions which consist of wheel speed, workpiece speed and depth of cut. All of other parameters such as workpiece length, workpiece material, wheel diameter, wheel material and width of grinding were taken as constant. The BP neural network was trained using the scaled conjugate gradient algorithm (SCGA). The results of the neural network were compared with experimental values. It shows that the BP model can predict the surface roughness satisfactorily. For optimization of creep feed grinding process, an M-file program has been written in `Matlab' software to integrate GA and NN. After generation of each population by GA, firstly, the BP network predicts R a and then MRR has been calculated with mathematic formula. In this study, the importance of R a and MRR is equal in the optimization process. By using this integrated GA-NN system optimal parameters of creep feed grinding process have been achieved. The obtained results show that, the integrated GA-NN system was successful in determining the optimal process parameters.

Evolutionary artificial neural networks for hydrological systems forecasting

The conventional ways of constructing artificial neural network (ANN) for a problem generally presume a specific architecture and do not automatically discover network modules appropriate for specific training data. Evolutionary algorithms are used to automatically adapt the network architecture and connection weights according to the problem environment without substantial human intervention. To improve on the drawbacks of the conventional optimal process, this study presents a novel evolutionary artificial neural network (EANN) for time series forecasting. The EANN has a hybrid procedure, including the genetic algorithm and the scaled conjugate gradient algorithm, where the feedforward ANN architecture and its connection weights of neurons are simultaneously identified and optimized. We first explored the performance of the proposed EANN for the Mackey–Glass chaotic time series. The performance of the different networks was evaluated. The excellent performance in forecasting of the chaotic series shows that the proposed algorithm concurrently possesses efficiency, effectiveness, and robustness. We further explored the applicability and reliability of the EANN in a real hydrological time series. Again, the results indicate the EANN can effectively and efficiently construct a viable forecast module for the 10-day reservoir inflow, and its accuracy is superior to that of the AR and ARMAX models.

Accelerated conjugate gradient algorithm with finite difference Hessian/vector product approximation for unconstrained optimization

In this paper we propose a fundamentally different conjugate gradient method, in which the well-known parameter β k is computed by an approximation of the Hessian/vector product through finite differences. For search direction computation, the method uses a forward difference approximation to the Hessian/vector product in combination with a careful choice of the finite difference interval. For the step length computation we suggest an acceleration scheme able to improve the efficiency of the algorithm. Under common assumptions, the method is proved to be globally convergent. It is shown that for uniformly convex functions the convergence of the accelerated algorithm is still linear, but the reduction in function values is significantly improved. Numerical comparisons with conjugate gradient algorithms including CONMIN by Shanno and Phua [D.F. Shanno, K.H. Phua, Algorithm 500, minimization of unconstrained multivariate functions, ACM Trans. Math. Softw. 2 (1976) 87–94], SCALCG by Andrei [N. Andrei, Scaled conjugate gradient algorithms for unconstrained optimization, Comput. Optim. Appl. 38 (2007) 401–416; N. Andrei, Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Optim. Methods Softw. 22 (2007) 561–571; N. Andrei, A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization, Appl. Math. Lett. 20 (2007) 645–650], and new conjugacy condition and related new conjugate gradient by Li, Tang and Wei [G. Li, C. Tang, Z. Wei, New conjugacy condition and related new conjugate gradient methods for unconstrained optimization, J. Comput. Appl. Math. 202 (2007) 523–539] or truncated Newton TN by Nash [S.G. Nash, Preconditioning of truncated-Newton methods, SIAM J. on Scientific and Statistical Computing 6 (1985) 599–616] using a set of 750 unconstrained optimization test problems show that the suggested algorithm outperforms these conjugate gradient algorithms as well as TN.

Prediction and control of surface roughness in CNC lathe using artificial neural network

In this study, a neural network approach is presented for the prediction and control of surface roughness in a computer numerically controlled (CNC) lathe. Experiments have been performed on the CNC lathe to obtain the data used for the training and testing of a neural network. The parameters used in the experiment were reduced to three cutting parameters which consisted of depth of cutting, cutting speed, and feed rate. Each of the other parameters such as tool nose radius, tool overhang, approach angle, workpiece length, workpiece diameter and workpiece material was taken as constant. A feed forward multi-layered neural network was developed and the network model was trained using the scaled conjugate gradient algorithm (SCGA), which is a type of back-propagation. The adaptive learning rate was used. Therefore, the learning rate was not selected before training and it was adjusted during training to minimize training time. The number of iterations was 8000 and no smoothing factor was used. R a, R z and R max were modeled and were evaluated individually. One hidden layer was used for all models while the numbers of neurons in the hidden layer of the R a model were five and the numbers of neurons in the hidden layers of the R z and R max models were ten. The results of the neural network approach were compared with actual values. In addition, inasmuch as the control of surface roughness is proposed, a control algorithm was developed in the present investigation. The desired surface roughness was entered into the control system as a reference value and the controller determined the cutting parameters for these surface roughness values. A new surface roughness value was determined by sending the cutting parameters to the observer (ANN block). The obtained surface roughness was fed back to the comparison unit and was compared with the reference value and the difference surface roughness was then sent to the controller. The iteration was continued until the difference was reduced to a certain value of surface roughness which could be permitted for machining accuracy. When the surface roughness reached the permitted value, these cutting parameters were sent to the CNC turning system as input values. In conclusion, both the surface roughness values corresponding to the cutting parameters and suitable cutting parameters for a certain surface roughness can be determined prior to a machining operation using the ANN and control algorithm.

Prediction Models and Generalization Performance Study in Electrical Discharge Machining

In the past decade, artificial neural network(ANN) has been applied in Electrical discharge machining(EDM). However, most of them only discuss parameter prediction or optimization result, few tell how to improve generalization performance. In this study, machining process models have been established based on different training algorithms of ANN, namely Levenberg-Marquardt algorithm (LM), Resilient algorithm (RP), Scaled Conjugate Gradient algorithm (SCG) and Quasi-Newton algorithm(BFGS). All models have been trained by same experimental data, checked by another group data, their generalization performance are compared. Take LM as the example, some main factors that may influence generalization performance are discussed.

Scaled conjugate gradient algorithms for unconstrained optimization

In this work we present and analyze a new scaled conjugate gradient algorithm and its implementation, based on an interpretation of the secant equation and on the inexact Wolfe line search conditio...

Scaled memoryless BFGS preconditioned conjugate gradient algorithm for unconstrained optimization

A scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving unconstrained optimization problems is presented. The basic idea is to combine the scaled memoryless BFGS method and the preconditioning technique in the frame of the conjugate gradient method. The preconditioner, which is also a scaled memoryless BFGS matrix, is reset when the Beale–Powell restart criterion holds. The parameter scaling the gradient is selected as the spectral gradient. In very mild conditions, it is shown that, for strongly convex functions, the algorithm is globally convergent. Computational results for a set consisting of 750 unconstrained optimization test problems show that this new scaled conjugate gradient algorithm substantially outperforms the known conjugate gradient methods including the spectral conjugate gradient by Birgin and Martínez [Birgin, E. and Martínez, J.M., 2001, A spectral conjugate gradient method for unconstrained optimization. Applied Mathematics and Optimization, 43, 117–128], the conjugate gradient by Polak and Ribière [Polak, E. and Ribière, G., 1969, Note sur la convergence de méthodes de directions conjuguées. Revue Francaise Informat. Reserche Opérationnelle, 16, 35–43], as well as the most recent conjugate gradient method with guaranteed descent by Hager and Zhang [Hager, W.W. and Zhang, H., 2005, A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM Journal on Optimization, 16, 170–192; Hager, W.W. and Zhang, H., 2004, CG-DESCENT, A conjugate gradient method with guaranteed descent ACM Transactions on Mathematical Software, 32, 113–137].

Prediction of Performance and Smoke Emission Using Artificial Neural Network in a Diesel Engine

The fuel injection pressure is one of the significant operating parameters affects atomization of fuel and mixture formation; therefore, it determines the performance and emissions of a diesel engine. Increasing the fuel injection pressure decrease the particle diameter and caused the diesel fuel spray to vaporize quickly. However, with decreasing fuel particles their inertia will also decrease and for this reason fuel can not penetrate deeply into the combustion chamber. In this study, artificial neural-networks (ANNs) are used to determine the effects of injection pressure on smoke emissions and engine performance in a diesel engine. Experimental studies were used to obtain training and test data. Injection pressure was changed from 100bar to 300bar in experiment (standard injection pressure of test engine is 150bar). Injection pressure and engine speed have been used as the input layer; smoke emission, engine torque and specific fuel consumption have been used as the output layer. Two different training algorithms were studied. The best results were obtained from Levenberg-Marquardt (LM) and Scaled Conjugate gradient (SCG) algorithms with 11 neurons. However, The LM algorithm is faster than the SCG algorithm, and its error values are smaller than those of the SCGs. For the torque with LM algorithm, fraction of variance (R2) and mean absolute percentage error (MAPE) were found to be 0.9927 and 7.2108%, respectively. Similarly, for the specific fuel consumption (SFC), R2 and MAPE were calculated as 0.9872 and 6.0261%, respectively. For the torque with SCG algorithm, R2 and MAPE were found to be 0.9879 and 9.0026%, respectively. Similarly, for the specific fuel consumption (SFC), R2 and MAPE were calculated as 0.9793 and 8.7974%, respectively. So, these ANN predicted results can be considered within acceptable limits and the results show good agreement between predicted and experimental values.

A scaled BFGS preconditioned conjugate gradient algorithm for unconstrained optimization

This letter presents a scaled memoryless BFGS preconditioned conjugate gradient algorithm for solving unconstrained optimization problems. The basic idea is to combine the scaled memoryless BFGS method and the preconditioning technique in the frame of the conjugate gradient method. The preconditioner, which is also a scaled memoryless BFGS matrix, is reset when the Powell restart criterion holds. The parameter scaling the gradient is selected as the spectral gradient. Computational results for a set consisting of 750 test unconstrained optimization problems show that this new scaled conjugate gradient algorithm substantially outperforms known conjugate gradient methods such as the spectral conjugate gradient SCG of Birgin and Martínez [E. Birgin, J.M. Martínez, A spectral conjugate gradient method for unconstrained optimization, Appl. Math. Optim. 43 (2001) 117–128] and the (classical) conjugate gradient of Polak and Ribière [E. Polak, G. Ribière, Note sur la convergence de méthodes de directions conjuguées, Revue Francaise Informat. Reserche Opérationnelle, 3e Année 16 (1969) 35–43], but subject to the CPU time metric it is outperformed by L-BFGS [D. Liu, J. Nocedal, On the limited memory BFGS method for large scale optimization, Math. Program. B 45 (1989) 503–528; J. Nocedal. http://www.ece.northwestern.edu/~nocedal/lbfgs.html].

Implementing Temporal-Difference Learning with the Scaled Conjugate Gradient Algorithm

This paper investigates the use of the scaled conjugate gradient (SCG) algorithm in temporal-difference (TD) learning for time series prediction. Special emphasis is given on the implementation details, after examining the theoretical background of the algorithm and the learning methodology and how these could be combined. Simple time series (linear, sinusoidal, etc.) as well as more complex ones, coming from real data, are used to examine the behavior of this novel combination of learning algorithm and methodology. Preliminary experimental results indicate that the implementation as presented in this paper indeed works, but the performance (in terms of learning speed and generalization ability) of TD learning using the SCG algorithm is not as good as expected, at least on the representative problems examined. An attempt to rationalize these results is presented.

Soft sensor modeling based on Gaussian processes

In order to meet the demand of online optimal running, a novel soft sensor modeling approach based on Gaussian processes was proposed. The approach is moderately simple to implement and use without loss of performance. It is trained by optimizing the hyperparameters using the scaled conjugate gradient algorithm with the squared exponential covariance function employed. Experimental simulations show that the soft sensor modeling approach has the advantage via a real-world example in a refinery. Meanwhile, the method opens new possibilities for application of kernel methods to potential fields.