Search results for: Cascade Correlation Neural Network

Authors: Saman M. Abdulla, Najla B. Al-Dabagh, Omar Zakaria

Abstract:

The aim of this article is to explain how features of attacks could be extracted from the packets. It also explains how vectors could be built and then applied to the input of any analysis stage. For analyzing, the work deploys the Feedforward-Back propagation neural network to act as misuse intrusion detection system. It uses ten types if attacks as example for training and testing the neural network. It explains how the packets are analyzed to extract features. The work shows how selecting the right features, building correct vectors and how correct identification of the training methods with nodes- number in hidden layer of any neural network affecting the accuracy of system. In addition, the work shows how to get values of optimal weights and use them to initialize the Artificial Neural Network.

Keywords: Artificial Neural Network, Attack Features, MisuseIntrusion Detection System, Training Parameters.

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Authors: Raquel M. de Sousa, Sofiane Labidi, Allan Kardec D. Barros, Alex O. Barradas Filho, Aldalea L. B. Marques

Abstract:

Several parameters are established in order to measure biodiesel quality. One of them is the iodine value, which is an important parameter that measures the total unsaturation within a mixture of fatty acids. Limitation of unsaturated fatty acids is necessary since warming of higher quantity of these ones ends in either formation of deposits inside the motor or damage of lubricant. Determination of iodine value by official procedure tends to be very laborious, with high costs and toxicity of the reagents, this study uses artificial neural network (ANN) in order to predict the iodine value property as an alternative to these problems. The methodology of development of networks used 13 esters of fatty acids in the input with convergence algorithms of back propagation of back propagation type were optimized in order to get an architecture of prediction of iodine value. This study allowed us to demonstrate the neural networks’ ability to learn the correlation between biodiesel quality properties, in this caseiodine value, and the molecular structures that make it up. The model developed in the study reached a correlation coefficient (R) of 0.99 for both network validation and network simulation, with Levenberg-Maquardt algorithm.

Keywords: Artificial Neural Networks, Biodiesel, Iodine Value, Prediction.

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Authors: Anupama Pande, Vishik Goel

Abstract:

A complex valued neural network is a neural network, which consists of complex valued input and/or weights and/or thresholds and/or activation functions. Complex-valued neural networks have been widening the scope of applications not only in electronics and informatics, but also in social systems. One of the most important applications of the complex valued neural network is in image and vision processing. In Neural networks, radial basis functions are often used for interpolation in multidimensional space. A Radial Basis function is a function, which has built into it a distance criterion with respect to a centre. Radial basis functions have often been applied in the area of neural networks where they may be used as a replacement for the sigmoid hidden layer transfer characteristic in multi-layer perceptron. This paper aims to present exhaustive results of using RBF units in a complex-valued neural network model that uses the back-propagation algorithm (called 'Complex-BP') for learning. Our experiments results demonstrate the effectiveness of a Radial basis function in a complex valued neural network in image recognition over a real valued neural network. We have studied and stated various observations like effect of learning rates, ranges of the initial weights randomly selected, error functions used and number of iterations for the convergence of error on a neural network model with RBF units. Some inherent properties of this complex back propagation algorithm are also studied and discussed.

Keywords: Complex valued neural network, Radial BasisFunction, Image recognition.

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Authors: Defu Zhang, Qingshan Jiang, Xin Li

Abstract:

This paper deals with the application of a well-known neural network technique, multilayer back-propagation (BP) neural network, in financial data mining. A modified neural network forecasting model is presented, and an intelligent mining system is developed. The system can forecast the buying and selling signs according to the prediction of future trends to stock market, and provide decision-making for stock investors. The simulation result of seven years to Shanghai Composite Index shows that the return achieved by this mining system is about three times as large as that achieved by the buy and hold strategy, so it is advantageous to apply neural networks to forecast financial time series, the different investors could benefit from it.

Keywords: Data mining, neural network, stock forecasting.

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Authors: Anupama Pande, Ashok Kumar Thakur, Swapnoneel Roy

Abstract:

A complex valued neural network is a neural network which consists of complex valued input and/or weights and/or thresholds and/or activation functions. Complex-valued neural networks have been widening the scope of applications not only in electronics and informatics, but also in social systems. One of the most important applications of the complex valued neural network is in signal processing. In Neural networks, generalized mean neuron model (GMN) is often discussed and studied. The GMN includes a new aggregation function based on the concept of generalized mean of all the inputs to the neuron. This paper aims to present exhaustive results of using Generalized Mean Neuron model in a complex-valued neural network model that uses the back-propagation algorithm (called -Complex-BP-) for learning. Our experiments results demonstrate the effectiveness of a Generalized Mean Neuron Model in a complex plane for signal processing over a real valued neural network. We have studied and stated various observations like effect of learning rates, ranges of the initial weights randomly selected, error functions used and number of iterations for the convergence of error required on a Generalized Mean neural network model. Some inherent properties of this complex back propagation algorithm are also studied and discussed.

Keywords: Complex valued neural network, Generalized Meanneuron model, Signal processing.

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Authors: Thomas Behan, Zaiyi Liao, Lian Zhao, Chunting Yang

Abstract:

In this paper, low end Digital Signal Processors (DSPs) are applied to accelerate integer neural networks. The use of DSPs to accelerate neural networks has been a topic of study for some time, and has demonstrated significant performance improvements. Recently, work has been done on integer only neural networks, which greatly reduces hardware requirements, and thus allows for cheaper hardware implementation. DSPs with Arithmetic Logic Units (ALUs) that support floating or fixed point arithmetic are generally more expensive than their integer only counterparts due to increased circuit complexity. However if the need for floating or fixed point math operation can be removed, then simpler, lower cost DSPs can be used. To achieve this, an integer only neural network is created in this paper, which is then accelerated by using DSP instructions to improve performance.

Keywords: Digital Signal Processor (DSP), Integer Neural Network(INN), Low Cost Neural Network, Integer Neural Network DSPImplementation.

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Authors: I. Oloyede

Abstract:

The study investigates the accuracy and loss of deep learning algorithms with the set of coronavirus (COVID-19) images dataset by comparing Bayesian convolutional neural network and traditional convolutional neural network in low dimensional dataset. 50 sets of X-ray images out of which 25 were COVID-19 and the remaining 20 were normal, twenty images were set as training while five were set as validation that were used to ascertained the accuracy of the model. The study found out that Bayesian convolution neural network outperformed conventional neural network at low dimensional dataset that could have exhibited under fitting. The study therefore recommended Bayesian Convolutional neural network (BCNN) for android apps in computer vision for image detection.

Keywords: BCNN, CNN, Images, COVID-19, Deep Learning.

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Authors: Abdelhadi Lotfi, Abdelkader Benyettou

Abstract:

In this work, a training algorithm for probabilistic neural networks (PNN) is presented. The algorithm addresses one of the major drawbacks of PNN, which is the size of the hidden layer in the network. By using a cross-validation training algorithm, the number of hidden neurons is shrunk to a smaller number consisting of the most representative samples of the training set. This is done without affecting the overall architecture of the network. Performance of the network is compared against performance of standard PNN for different databases from the UCI database repository. Results show an important gain in network size and performance.

Keywords: Classification, probabilistic neural networks, network optimization, pattern recognition.

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Authors: Rui Antunes, Fernando V. Coito

Abstract:

This article presents the development of a neural network cognitive model for the classification and detection of different frequency signals. The basic structure of the implemented neural network was inspired on the perception process that humans generally make in order to visually distinguish between high and low frequency signals. It is based on the dynamic neural network concept, with delays. A special two-layer feedforward neural net structure was successfully implemented, trained and validated, to achieve minimum target error. Training confirmed that this neural net structure descents and converges to a human perception classification solution, even when far away from the target.

Keywords: Neural Networks, Signal Classification, Adaptative Filters, Cognitive Neuroscience

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Authors: T. Hacib, M. R. Mekideche, N. Ferkha

Abstract:

This paper presents an approach which is based on the use of supervised feed forward neural network, namely multilayer perceptron (MLP) neural network and finite element method (FEM) to solve the inverse problem of parameters identification. The approach is used to identify unknown parameters of ferromagnetic materials. The methodology used in this study consists in the simulation of a large number of parameters in a material under test, using the finite element method (FEM). Both variations in relative magnetic permeability and electrical conductivity of the material under test are considered. Then, the obtained results are used to generate a set of vectors for the training of MLP neural network. Finally, the obtained neural network is used to evaluate a group of new materials, simulated by the FEM, but not belonging to the original dataset. Noisy data, added to the probe measurements is used to enhance the robustness of the method. The reached results demonstrate the efficiency of the proposed approach, and encourage future works on this subject.

Keywords: Inverse problem, MLP neural network, parametersidentification, FEM.

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Authors: Hazem M. El-Bakry, Qiangfu Zhao

Abstract:

In recent years, fast neural networks for object/face detection have been introduced based on cross correlation in the frequency domain between the input matrix and the hidden weights of neural networks. In our previous papers [3,4], fast neural networks for certain code detection was introduced. It was proved in [10] that for fast neural networks to give the same correct results as conventional neural networks, both the weights of neural networks and the input matrix must be symmetric. This condition made those fast neural networks slower than conventional neural networks. Another symmetric form for the input matrix was introduced in [1-9] to speed up the operation of these fast neural networks. Here, corrections for the cross correlation equations (given in [13,15,16]) to compensate for the symmetry condition are presented. After these corrections, it is proved mathematically that the number of computation steps required for fast neural networks is less than that needed by classical neural networks. Furthermore, there is no need for converting the input data into symmetric form. Moreover, such new idea is applied to increase the speed of neural networks in case of processing complex values. Simulation results after these corrections using MATLAB confirm the theoretical computations.

Keywords: Fast Code/Data Detection, Neural Networks, Cross Correlation, real/complex values.

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Authors: Magdi M. Nabi, Ding-Li Yu

Abstract:

This paper presents a neural network based model predictive control (MPC) strategy to control a strongly exothermic reaction with complicated nonlinear kinetics given by Chylla-Haase polymerization reactor that requires a very precise temperature control to maintain product uniformity. In the benchmark scenario, the operation of the reactor must be guaranteed under various disturbing influences, e.g., changing ambient temperatures or impurity of the monomer. Such a process usually controlled by conventional cascade control, it provides a robust operation, but often lacks accuracy concerning the required strict temperature tolerances. The predictive control strategy based on the RBF neural model is applied to solve this problem to achieve set-point tracking of the reactor temperature against disturbances. The result shows that the RBF based model predictive control gives reliable result in the presence of some disturbances and keeps the reactor temperature within a tight tolerance range around the desired reaction temperature.

Keywords: Chylla-Haase reactor, RBF neural network modelling, model predictive control.

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Authors: Wajdi Bellil, Chokri Ben Amar, Adel M. Alimi

Abstract:

This paper proposes a comparison between wavelet neural networks (WNN), RBF neural network and polynomial approximation in term of 1-D and 2-D functions approximation. We present a novel wavelet neural network, based on Beta wavelets, for 1-D and 2-D functions approximation. Our purpose is to approximate an unknown function f: Rn - R from scattered samples (xi; y = f(xi)) i=1....n, where first, we have little a priori knowledge on the unknown function f: it lives in some infinite dimensional smooth function space and second the function approximation process is performed iteratively: each new measure on the function (xi; f(xi)) is used to compute a new estimate f as an approximation of the function f. Simulation results are demonstrated to validate the generalization ability and efficiency of the proposed Beta wavelet network.

Keywords: Beta wavelets networks, RBF neural network, training algorithms, MSE, 1-D, 2D function approximation.

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Authors: S. Anna Durai, E. Anna Saro

Abstract:

Image Compression using Artificial Neural Networks is a topic where research is being carried out in various directions towards achieving a generalized and economical network. Feedforward Networks using Back propagation Algorithm adopting the method of steepest descent for error minimization is popular and widely adopted and is directly applied to image compression. Various research works are directed towards achieving quick convergence of the network without loss of quality of the restored image. In general the images used for compression are of different types like dark image, high intensity image etc. When these images are compressed using Back-propagation Network, it takes longer time to converge. The reason for this is, the given image may contain a number of distinct gray levels with narrow difference with their neighborhood pixels. If the gray levels of the pixels in an image and their neighbors are mapped in such a way that the difference in the gray levels of the neighbors with the pixel is minimum, then compression ratio as well as the convergence of the network can be improved. To achieve this, a Cumulative distribution function is estimated for the image and it is used to map the image pixels. When the mapped image pixels are used, the Back-propagation Neural Network yields high compression ratio as well as it converges quickly.

Keywords: Back-propagation Neural Network, Cumulative Distribution Function, Correlation, Convergence.

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Authors: Han Xiao, Xiaoyan Zhu

Abstract:

Margin-Based Principle has been proposed for a long time, it has been proved that this principle could reduce the structural risk and improve the performance in both theoretical and practical aspects. Meanwhile, feed-forward neural network is a traditional classifier, which is very hot at present with a deeper architecture. However, the training algorithm of feed-forward neural network is developed and generated from Widrow-Hoff Principle that means to minimize the squared error. In this paper, we propose a new training algorithm for feed-forward neural networks based on Margin-Based Principle, which could effectively promote the accuracy and generalization ability of neural network classifiers with less labelled samples and flexible network. We have conducted experiments on four UCI open datasets and achieved good results as expected. In conclusion, our model could handle more sparse labelled and more high-dimension dataset in a high accuracy while modification from old ANN method to our method is easy and almost free of work.

Keywords: Max-Margin Principle, Feed-Forward Neural Network, Classifier.

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Authors: Koray U. Erbas

Abstract:

Neural networks are appealing for many applications since they are able to learn complex non-linear relationships between input and output data. As the number of neurons and layers in a neural network increase, it is possible to represent more complex relationships with automatically extracted features. Nowadays Deep Neural Networks (DNNs) are widely used in Computer Vision problems such as; classification, object detection, segmentation image editing etc. In this work, Facial Emotion Recognition task is performed by proposed Convolutional Neural Network (CNN)-based DNN architecture using FER2013 Dataset. Moreover, the effects of different hyperparameters (activation function, kernel size, initializer, batch size and network size) are investigated and ablation study results for Pooling Layer, Dropout and Batch Normalization are presented.

Keywords: Convolutional Neural Network, Deep Learning, Deep Learning Based FER, Facial Emotion Recognition.

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Authors: El Mahdi Barrah, Said Safi, Abdessamad Malaoui

Abstract:

In this paper, we present an algorithm for reconstruction from incomplete 2D scans for tifinagh characters. This algorithm is based on using correlation between the lost block and its neighbors. This system proposed contains three main parts: pre-processing, features extraction and recognition. In the first step, we construct a database of tifinagh characters. In the second step, we will apply “shape analysis algorithm”. In classification part, we will use Neural Network. The simulation results demonstrate that the proposed method give good results.

Keywords: Tifinagh character recognition, Neural networks, Local cost computation.

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Authors: Jasmin Velagic, Nedim Osmic, Bakir Lacevic

Abstract:

In this paper the neural network-based controller is designed for motion control of a mobile robot. This paper treats the problems of trajectory following and posture stabilization of the mobile robot with nonholonomic constraints. For this purpose the recurrent neural network with one hidden layer is used. It learns relationship between linear velocities and error positions of the mobile robot. This neural network is trained on-line using the backpropagation optimization algorithm with an adaptive learning rate. The optimization algorithm is performed at each sample time to compute the optimal control inputs. The performance of the proposed system is investigated using a kinematic model of the mobile robot.

Keywords: Mobile robot, kinematic model, neural network, motion control, adaptive learning rate.

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Authors: H.Mohammadi Majd, M.Jalali Azizpour, M. Goodarzi

Abstract:

In this paper back-propagation artificial neural network (BPANN) with Levenberg–Marquardt algorithm is employed to predict the limiting drawing ratio (LDR) of the deep drawing process. To prepare a training set for BPANN, some finite element simulations were carried out. die and punch radius, die arc radius, friction coefficient, thickness, yield strength of sheet and strain hardening exponent were used as the input data and the LDR as the specified output used in the training of neural network. As a result of the specified parameters, the program will be able to estimate the LDR for any new given condition. Comparing FEM and BPANN results, an acceptable correlation was found.

Keywords: BPANN, deep drawing, prediction, limiting drawingratio (LDR), Levenberg–Marquardt algorithm

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Authors: P. R. Jayathilaka, K.L. Jayaratne, H.L. Premaratne

Abstract:

This research presented in this paper is an on-going project of an application of neural network and fuzzy models to evaluate the sociological factors which affect the educational performance of the students in Sri Lanka. One of its major goals is to prepare the grounds to device a counseling tool which helps these students for a better performance at their examinations, especially at their G.C.E O/L (General Certificate of Education-Ordinary Level) examination. Closely related sociological factors are collected as raw data and the noise of these data are filtered through the fuzzy interface and the supervised neural network is being utilized to recognize the performance patterns against the chosen social factors.

Keywords: Education, Fuzzy, neural network, prediction, Sociology

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Authors: N.Vahdatifar, Ss.Mortazavi, R.Kianinezhad

Abstract:

In this paper, a Neural Network based predictive DTC algorithm is proposed .This approach is used as an alternative to classical approaches .An appropriate riate Feed - forward network is chosen and based on its value of derivative electromagnetic torque ; optimal stator voltage vector is determined to be applied to the induction motor (by inverter). Moreover, an appropriate torque and flux observer is proposed.

Keywords: Neural Networks, Predictive DTC

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Authors: Yong-Quan Zhou, Deng-Xu He, Zheng Nong

Abstract:

In this paper two models using a functional network were employed to solving classification problem. Functional networks are generalized neural networks, which permit the specification of their initial topology using knowledge about the problem at hand. In this case, and after analyzing the available data and their relations, we systematically discuss a numerical analysis method used for functional network, and apply two functional network models to solving XOR problem. The XOR problem that cannot be solved with two-layered neural network can be solved by two-layered functional network, which reveals a potent computational power of functional networks, and the performance of the proposed model was validated using classification problems.

Keywords: Functional network, neural network, XOR problem, classification, numerical analysis method.

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Authors: Li Kewen, Su Zhaoxin, Wang Xingmou, Zhu Jian Bing

Abstract:

Conventional reservoir prediction methods ar not sufficient to explore the implicit relation between seismic attributes, and thus data utilization is low. In order to improve the predictive classification accuracy of reservoir lithology, this paper proposes a deep learning lithology prediction method based on ResNet (Residual Neural Network) and SENet (Squeeze-and-Excitation Neural Network). The neural network model is built and trained by using seismic attribute data and lithology data of Shengli oilfield, and the nonlinear mapping relationship between seismic attribute and lithology marker is established. The experimental results show that this method can significantly improve the classification effect of reservoir lithology, and the classification accuracy is close to 70%. This study can effectively predict the lithology of undrilled area and provide support for exploration and development.

Keywords: Convolutional neural network, lithology, prediction of reservoir lithology, seismic attributes.

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Authors: Magdi M. Nabi, Ding-Li Yu

Abstract:

Control of a semi-batch polymerization reactor using an adaptive radial basis function (RBF) neural network method is investigated in this paper. A neural network inverse model is used to estimate the valve position of the reactor; this method can identify the controlled system with the RBF neural network identifier. The weights of the adaptive PID controller are timely adjusted based on the identification of the plant and self-learning capability of RBFNN. A PID controller is used in the feedback control to regulate the actual temperature by compensating the neural network inverse model output. Simulation results show that the proposed control has strong adaptability, robustness and satisfactory control performance and the nonlinear system is achieved.

Keywords: Chylla-Haase polymerization reactor, RBF neural networks, feed-forward and feedback control.

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Authors: Hazem M. El-Bakry, Ahmed Atwan

Abstract:

Computer networks are essential part in computerbased information systems. The performance of these networks has a great influence on the whole information system. Measuring the usability criteria and customers satisfaction on small computer network is very important. In this article, an effective approach for measuring the usability of business network in an information system is introduced. The usability process for networking provides us with a flexible and a cost-effective way to assess the usability of a network and its products. In addition, the proposed approach can be used to certify network product usability late in the development cycle. Furthermore, it can be used to help in developing usable interfaces very early in the cycle and to give a way to measure, track, and improve usability. Moreover, a new approach for fast information processing over computer networks is presented. The entire data are collected together in a long vector and then tested as a one input pattern. Proposed fast time delay neural networks (FTDNNs) use cross correlation in the frequency domain between the tested data and the input weights of neural networks. It is proved mathematically and practically that the number of computation steps required for the presented time delay neural networks is less than that needed by conventional time delay neural networks (CTDNNs). Simulation results using MATLAB confirm the theoretical computations.

Keywords: Usability Criteria, Computer Networks, Fast Information Processing, Cross Correlation, Frequency Domain.

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Authors: Anita S. Gangal, P. K. Kalra, D. S. Chauhan

Abstract:

The backpropagation algorithm in general employs quadratic error function. In fact, most of the problems that involve minimization employ the Quadratic error function. With alternative error functions the performance of the optimization scheme can be improved. The new error functions help in suppressing the ill-effects of the outliers and have shown good performance to noise. In this paper we have tried to evaluate and compare the relative performance of complex valued neural network using different error functions. During first simulation for complex XOR gate it is observed that some error functions like Absolute error, Cauchy error function can replace Quadratic error function. In the second simulation it is observed that for some error functions the performance of the complex valued neural network depends on the architecture of the network whereas with few other error functions convergence speed of the network is independent of architecture of the neural network.

Keywords: Complex backpropagation algorithm, complex errorfunctions, complex valued neural network, split activation function.

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Authors: Nahid Ghasemi, Mohammad Goodarzi, Morteza Khosravi

Abstract:

Simultaneous determination of multicomponents of phenol, resorcinol and catechol with a chemometric technique a PCranking artificial neural network (PCranking-ANN) algorithm is reported in this study. Based on the data correlation coefficient method, 3 representative PCs are selected from the scores of original UV spectral data (35 PCs) as the original input patterns for ANN to build a neural network model. The results obtained by iterating 8000 .The RMSEP for phenol, resorcinol and catechol with PCranking- ANN were 0.6680, 0.0766 and 0.1033, respectively. Calibration matrices were 0.50-21.0, 0.50-15.1 and 0.50-20.0 μg ml-1 for phenol, resorcinol and catechol, respectively. The proposed method was successfully applied for the determination of phenol, resorcinol and catechol in synthetic and water samples.

Keywords: Phenol, Resorcinol, Catechol, Principal componentrankingArtificial Neural Network, Chemometrics.

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Authors: A. Chaouachi, R.M. Kamel, R. Ichikawa, H. Hayashi, K. Nagasaka

Abstract:

This paper presents the applicability of artificial neural networks for 24 hour ahead solar power generation forecasting of a 20 kW photovoltaic system, the developed forecasting is suitable for a reliable Microgrid energy management. In total four neural networks were proposed, namely: multi-layred perceptron, radial basis function, recurrent and a neural network ensemble consisting in ensemble of bagged networks. Forecasting reliability of the proposed neural networks was carried out in terms forecasting error performance basing on statistical and graphical methods. The experimental results showed that all the proposed networks achieved an acceptable forecasting accuracy. In term of comparison the neural network ensemble gives the highest precision forecasting comparing to the conventional networks. In fact, each network of the ensemble over-fits to some extent and leads to a diversity which enhances the noise tolerance and the forecasting generalization performance comparing to the conventional networks.

Keywords: Neural network ensemble, Solar power generation, 24 hour forecasting, Comparative study

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Authors: S. Anna Durai, E. Anna Saro

Abstract:

In general the images used for compression are of different types like dark image, high intensity image etc. When these images are compressed using Counter Propagation Neural Network, it takes longer time to converge. The reason for this is that the given image may contain a number of distinct gray levels with narrow difference with their neighborhood pixels. If the gray levels of the pixels in an image and their neighbors are mapped in such a way that the difference in the gray levels of the neighbor with the pixel is minimum, then compression ratio as well as the convergence of the network can be improved. To achieve this, a Cumulative Distribution Function is estimated for the image and it is used to map the image pixels. When the mapped image pixels are used the Counter Propagation Neural Network yield high compression ratio as well as it converges quickly.

Keywords: Correlation, Counter Propagation Neural Networks, Cummulative Distribution Function, Image compression.

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Authors: Mohammad Reza Ghasemi, Amin Ghorbani

Abstract:

The main goal of the present work is to decrease the computational burden for optimum design of steel frames with frequency constraints using a new type of neural networks called Wavelet Neural Network. It is contested to train a suitable neural network for frequency approximation work as the analysis program. The combination of wavelet theory and Neural Networks (NN) has lead to the development of wavelet neural networks. Wavelet neural networks are feed-forward networks using wavelet as activation function. Wavelets are mathematical functions within suitable inner parameters, which help them to approximate arbitrary functions. WNN was used to predict the frequency of the structures. In WNN a RAtional function with Second order Poles (RASP) wavelet was used as a transfer function. It is shown that the convergence speed was faster than other neural networks. Also comparisons of WNN with the embedded Artificial Neural Network (ANN) and with approximate techniques and also with analytical solutions are available in the literature.

Keywords: Weight Minimization, Frequency Constraints, Steel Frames, ANN, WNN, RASP Function.

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