Showing papers in "IEEE Transactions on Neural Networks in 2002"
TL;DR: Decomposition implementations for two "all-together" multiclass SVM methods are given and it is shown that for large problems methods by considering all data at once in general need fewer support vectors.
Abstract: Support vector machines (SVMs) were originally designed for binary classification. How to effectively extend it for multiclass classification is still an ongoing research issue. Several methods have been proposed where typically we construct a multiclass classifier by combining several binary classifiers. Some authors also proposed methods that consider all classes at once. As it is computationally more expensive to solve multiclass problems, comparisons of these methods using large-scale problems have not been seriously conducted. Especially for methods solving multiclass SVM in one step, a much larger optimization problem is required so up to now experiments are limited to small data sets. In this paper we give decomposition implementations for two such "all-together" methods. We then compare their performance with three methods based on binary classifications: "one-against-all," "one-against-one," and directed acyclic graph SVM (DAGSVM). Our experiments indicate that the "one-against-one" and DAG methods are more suitable for practical use than the other methods. Results also show that for large problems methods by considering all data at once in general need fewer support vectors.
6,562Â citations
TL;DR: Independent component analysis (ICA), a generalization of PCA, was used, using a version of ICA derived from the principle of optimal information transfer through sigmoidal neurons, which was superior to representations based on PCA for recognizing faces across days and changes in expression.
Abstract: A number of current face recognition algorithms use face representations found by unsupervised statistical methods. Typically these methods find a set of basis images and represent faces as a linear combination of those images. Principal component analysis (PCA) is a popular example of such methods. The basis images found by PCA depend only on pairwise relationships between pixels in the image database. In a task such as face recognition, in which important information may be contained in the high-order relationships among pixels, it seems reasonable to expect that better basis images may be found by methods sensitive to these high-order statistics. Independent component analysis (ICA), a generalization of PCA, is one such method. We used a version of ICA derived from the principle of optimal information transfer through sigmoidal neurons. ICA was performed on face images in the FERET database under two different architectures, one which treated the images as random variables and the pixels as outcomes, and a second which treated the pixels as random variables and the images as outcomes. The first architecture found spatially local basis images for the faces. The second architecture produced a factorial face code. Both ICA representations were superior to representations based on PCA for recognizing faces across days and changes in expression. A classifier that combined the two ICA representations gave the best performance.
2,044Â citations
TL;DR: This paper applies a fuzzy membership to each input point and reformulate the SVMs such that different input points can make different contributions to the learning of decision surface.
Abstract: A support vector machine (SVM) learns the decision surface from two distinct classes of the input points. In many applications, each input point may not be fully assigned to one of these two classes. In this paper, we apply a fuzzy membership to each input point and reformulate the SVMs such that different input points can make different contributions to the learning of decision surface. We call the proposed method fuzzy SVMs (FSVMs).
1,374Â citations
TL;DR: It is demonstrated is that the proposed method can provide the performance of the ideal greedy selection algorithm when information is distributed uniformly and should prove to be a useful method in selecting features for classification problems.
Abstract: Feature selection plays an important role in classifying systems such as neural networks (NNs). We use a set of attributes which are relevant, irrelevant or redundant and from the viewpoint of managing a dataset which can be huge, reducing the number of attributes by selecting only the relevant ones is desirable. In doing so, higher performances with lower computational effort is expected. In this paper, we propose two feature selection algorithms. The limitation of mutual information feature selector (MIFS) is analyzed and a method to overcome this limitation is studied. One of the proposed algorithms makes more considered use of mutual information between input attributes and output classes than the MIFS. What is demonstrated is that the proposed method can provide the performance of the ideal greedy selection algorithm when information is distributed uniformly. The computational load for this algorithm is nearly the same as that of MIFS. In addition, another feature selection algorithm using the Taguchi method is proposed. This is advanced as a solution to the question as to how to identify good features with as few experiments as possible. The proposed algorithms are applied to several classification problems and compared with MIFS. These two algorithms can be combined to complement each other's limitations. The combined algorithm performed well in several experiments and should prove to be a useful method in selecting features for classification problems.
918Â citations
TL;DR: It is shown that the eigenvectors of a kernel matrix which defines the implicit mapping provides a means to estimate the number of clusters inherent within the data and a computationally simple iterative procedure is presented for the subsequent feature space partitioning of the data.
Abstract: The article presents a method for both the unsupervised partitioning of a sample of data and the estimation of the possible number of inherent clusters which generate the data. This work exploits the notion that performing a nonlinear data transformation into some high dimensional feature space increases the probability of the linear separability of the patterns within the transformed space and therefore simplifies the associated data structure. It is shown that the eigenvectors of a kernel matrix which defines the implicit mapping provides a means to estimate the number of clusters inherent within the data and a computationally simple iterative procedure is presented for the subsequent feature space partitioning of the data.
905Â citations
TL;DR: A novel paradigm is proposed whereby data information is encapsulated in determining the structure and initial parameters of the RBF neural classifier before learning takes place, and the dimension of the search space is drastically reduced in the gradient paradigm.
Abstract: A general and efficient design approach using a radial basis function (RBF) neural classifier to cope with small training sets of high dimension, which is a problem frequently encountered in face recognition, is presented. In order to avoid overfitting and reduce the computational burden, face features are first extracted by the principal component analysis (PCA) method. Then, the resulting features are further processed by the Fisher's linear discriminant (FLD) technique to acquire lower-dimensional discriminant patterns. A novel paradigm is proposed whereby data information is encapsulated in determining the structure and initial parameters of the RBF neural classifier before learning takes place. A hybrid learning algorithm is used to train the RBF neural networks so that the dimension of the search space is drastically reduced in the gradient paradigm. Simulation results conducted on the ORL database show that the system achieves excellent performance both in terms of error rates of classification and learning efficiency.
656Â citations
TL;DR: A survey of the available literature on data mining using soft computing based on the different soft computing tools and their hybridizations used, the data mining function implemented, and the preference criterion selected by the model is provided.
Abstract: The present article provides a survey of the available literature on data mining using soft computing. A categorization has been provided based on the different soft computing tools and their hybridizations used, the data mining function implemented, and the preference criterion selected by the model. The utility of the different soft computing methodologies is highlighted. Generally fuzzy sets are suitable for handling the issues related to understandability of patterns, incomplete/noisy data, mixed media information and human interaction, and can provide approximate solutions faster. Neural networks are nonparametric, robust, and exhibit good learning and generalization capabilities in data-rich environments. Genetic algorithms provide efficient search algorithms to select a model, from mixed media data, based on some preference criterion/objective function. Rough sets are suitable for handling different types of uncertainty in data. Some challenges to data mining and the application of soft computing methodologies are indicated. An extensive bibliography is also included.
630Â citations
TL;DR: In this paper, direct adaptive neural-network control is presented for a class of affine nonlinear systems in the strict-feedback form with unknown nonlinearities by utilizing a special property of the affine term to avoid the controller singularity problem completely.
Abstract: In this paper, direct adaptive neural-network (NN) control is presented for a class of affine nonlinear systems in the strict-feedback form with unknown nonlinearities. By utilizing a special property of the affine term, the developed scheme,avoids the controller singularity problem completely. All the signals in the closed loop are guaranteed to be semiglobally uniformly ultimately bounded and the output of the system is proven to converge to a small neighborhood of the desired trajectory. The control performance of the closed-loop system is guaranteed by suitably choosing the design parameters. Simulation results are presented to show the effectiveness of the approach.
545Â citations
TL;DR: The motivation was to provide a model that adapts its architecture during its unsupervised training process according to the particular requirements of the input data, and by providing a global orientation of the independently growing maps in the individual layers of the hierarchy, navigation across branches is facilitated.
Abstract: The self-organizing map (SOM) is a very popular unsupervised neural-network model for the analysis of high-dimensional input data as in data mining applications. However, at least two limitations have to be noted, which are related to the static architecture of this model as well as to the limited capabilities for the representation of hierarchical relations of the data. With our novel growing hierarchical SOM (GHSOM) we address both limitations. The GHSOM is an artificial neural-network model with hierarchical architecture composed of independent growing SOMs. The motivation was to provide a model that adapts its architecture during its unsupervised training process according to the particular requirements of the input data. Furthermore, by providing a global orientation of the independently growing maps in the individual layers of the hierarchy, navigation across branches is facilitated. The benefits of this novel neural network are a problem-dependent architecture and the intuitive representation of hierarchical relations in the data. This is especially appealing in explorative data mining applications, allowing the inherent structure of the data to unfold in a highly intuitive fashion.
485Â citations
TL;DR: The recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time-varying solution of a given Sylvester equation.
Abstract: Presents a recurrent neural network for solving the Sylvester equation with time-varying coefficient matrices. The recurrent neural network with implicit dynamics is deliberately developed in the way that its trajectory is guaranteed to converge exponentially to the time-varying solution of a given Sylvester equation. Theoretical results of convergence and sensitivity analysis are presented to show the desirable properties of the recurrent neural network. Simulation results of time-varying matrix inversion and online nonlinear output regulation via pole assignment for the ball and beam system and the inverted pendulum on a cart system are also included to demonstrate the effectiveness and performance of the proposed neural network.
464Â citations
TL;DR: Both exponential stability and periodic oscillatory solution of bidirectional associative memory (BAM) networks with axonal signal transmission delays are considered by constructing suitable Lyapunov functional and some analysis techniques.
Abstract: Both exponential stability and periodic oscillatory solution of bidirectional associative memory (BAM) networks with axonal signal transmission delays are considered by constructing suitable Lyapunov functional and some analysis techniques. Some simple sufficient conditions are given ensuring the global exponential stability and the existence of periodic oscillatory solutions of BAM with delays. These conditions are presented in terms of system parameters and have important leading significance in the design and applications of globally exponentially stable and periodic oscillatory neural circuits for BAM with delays. In addition, two examples are given to illustrate the results.
TL;DR: This work has developed an 11-transistor silicon circuit that uses silicon physics to naturally implement a similarity computation, local adaptation, simultaneous adaptation and computation and nonvolatile storage, and is an ideal building block for constructing competitive-learning networks.
Abstract: Competitive learning is a general technique for training clustering and classification networks. We have developed an 11-transistor silicon circuit, that we term an automaximizing bump circuit, that uses silicon physics to naturally implement a similarity computation, local adaptation, simultaneous adaptation and computation and nonvolatile storage. This circuit is an ideal building block for constructing competitive-learning networks. We illustrate the adaptive nature of the automaximizing bump in two ways. First, we demonstrate a silicon competitive-learning circuit that clusters one-dimensional (1-D) data. We then illustrate a general architecture based on the automaximizing bump circuit; we show the effectiveness of this architecture, via software simulation, on a general clustering task. We corroborate our analysis with experimental data from circuits fabricated in a 0.35-/spl mu/m CMOS process.
TL;DR: The paper summarizes the different characteristics of Web data, the basic components of Web mining and its different types, and the current state of the art.
Abstract: The paper summarizes the different characteristics of Web data, the basic components of Web mining and its different types, and the current state of the art. The reason for considering Web mining, a separate field from data mining, is explained. The limitations of some of the existing Web mining methods and tools are enunciated, and the significance of soft computing (comprising fuzzy logic (FL), artificial neural networks (ANNs), genetic algorithms (GAs), and rough sets (RSs) are highlighted. A survey of the existing literature on "soft Web mining" is provided along with the commercially available systems. The prospective areas of Web mining where the application of soft computing needs immediate attention are outlined with justification. Scope for future research in developing "soft Web mining" systems is explained. An extensive bibliography is also provided.
TL;DR: This work introduces a new technique for estimating the optical flow field, starting from image sequences, and tracks contours of constant phase over time, since these are more robust to variations in lighting conditions and deviations from pure translation than contouring of constant amplitude.
Abstract: We introduce a new technique for estimating the optical flow field, starting from image sequences. As suggested by Fleet and Jepson (1990), we track contours of constant phase over time, since these are more robust to variations in lighting conditions and deviations from pure translation than contours of constant amplitude. Our phase-based approach proceeds in three stages. First, the image sequence is spatially filtered using a bank of quadrature pairs of Gabor filters, and the temporal phase gradient is computed, yielding estimates of the velocity component in directions orthogonal to the filter pairs' orientations. Second, a component velocity is rejected if the corresponding filter pair's phase information is not linear over a given time span. Third, the remaining component velocities at a single spatial location are combined and a recurrent neural network is used to derive the full velocity. We test our approach on several image sequences, both synthetic and realistic.
TL;DR: The paper discusses implementation issues related to the tuning of the hyperparameters of a support vector machine (SVM) with L/sub 2/ soft margin, for which the radius/margin bound is taken as the index to be minimized, and iterative techniques are employed for computing radius and margin.
Abstract: The paper discusses implementation issues related to the tuning of the hyperparameters of a support vector machine (SVM) with L/sub 2/ soft margin, for which the radius/margin bound is taken as the index to be minimized, and iterative techniques are employed for computing radius and margin. The implementation is shown to be feasible and efficient, even for large problems having more than 10000 support vectors.
TL;DR: It is argued that it is sufficient to build an observer for the output tracking error of uncertain nonlinear systems to ensureUltimate boundedness of the error signals is shown through Lyapunov's direct method.
Abstract: We consider adaptive output feedback control of uncertain nonlinear systems, in which both the dynamics and the dimension of the regulated system may be unknown. However, the relative degree of the regulated output is assumed to be known. Given a smooth reference trajectory, the problem is to design a controller that forces the system measurement to track it with bounded errors. The classical approach requires a state observer. Finding a good observer for an uncertain nonlinear system is not an obvious task. We argue that it is sufficient to build an observer for the output tracking error. Ultimate boundedness of the error signals is shown through Lyapunov's direct method. The theoretical results are illustrated in the design of a controller for a fourth-order nonlinear system of relative degree two and a high-bandwidth attitude command system for a model R-50 helicopter.
TL;DR: A new sufficient condition is given for the uniqueness and global asymptotic stability of the equilibrium point for delayed cellular neural networks (DCNN) and imposes constraints on the feedback and delayed feedback matrices of a DCNN independently of the delay parameter.
Abstract: In this paper, a new sufficient condition is given for the uniqueness and global asymptotic stability of the equilibrium point for delayed cellular neural networks (DCNNs). This condition imposes constraints on the feedback and delayed feedback matrices of a DCNN independently of the delay parameter. This result is also compared with the previous results derived in the literature.
TL;DR: It is proved that the proposed robust adaptive scheme can guarantee the uniform ultimate boundedness of the closed-loop system signals.
Abstract: This paper presents a robust adaptive neural control design for a class of perturbed strict feedback nonlinear system with both completely unknown virtual control coefficients and unknown nonlinearities. The unknown nonlinearities comprise two types of nonlinear functions: one naturally satisfies the "triangularity condition" and can be approximated by linearly parameterized neural networks, while the other is assumed to be partially known and consists of parametric uncertainties and known "bounding functions." With the utilization of iterative Lyapunov design and neural networks, the proposed design procedure expands the class of nonlinear systems for which robust adaptive control approaches have been studied. The design method does not require a priori knowledge of the signs of the unknown virtual control coefficients. Leakage terms are incorporated into the adaptive laws to prevent parameter drifts due to the inherent neural-network approximation errors. It is proved that the proposed robust adaptive scheme can guarantee the uniform ultimate boundedness of the closed-loop system signals.. The control performance can be guaranteed by an appropriate choice of the design parameters. Simulation studies are included to illustrate the effectiveness of the proposed approach.
TL;DR: It is shown that, by performing an LM step on a single neighborhood at each training iteration, not only significant savings in memory occupation and computing effort are obtained, but also, the overall performance of the LM method can be increased.
Abstract: Although the Levenberg-Marquardt (LM) algorithm has been extensively applied as a neural-network training method, it suffers from being very expensive, both in memory and number of operations required, when the network to be trained has a significant number of adaptive weights. In this paper, the behavior of a recently proposed variation of this algorithm is studied. This new method is based on the application of the concept of neural neighborhoods to the LM algorithm. It is shown that, by performing an LM step on a single neighborhood at each training iteration, not only significant savings in memory occupation and computing effort are obtained, but also, the overall performance of the LM method can be increased.
TL;DR: A generalization of the error entropy criterion that enables the use of any order of Renyi's entropy and any suitable kernel function in density estimation is proposed and shown that the proposed entropy estimator preserves the global minimum of actual entropy.
Abstract: We have previously proposed the quadratic Renyi's error entropy as an alternative cost function for supervised adaptive system training. An entropy criterion instructs the minimization of the average information content of the error signal rather than merely trying to minimize its energy. In this paper, we propose a generalization of the error entropy criterion that enables the use of any order of Renyi's entropy and any suitable kernel function in density estimation. It is shown that the proposed entropy estimator preserves the global minimum of actual entropy. The equivalence between global optimization by convolution smoothing and the convolution by the kernel in Parzen windowing is also discussed. Simulation results are presented for time-series prediction and classification where experimental demonstration of all the theoretical concepts is presented.
TL;DR: Experimental results demonstrate the usefulness and effectiveness of the proposed similarity measure for image retrieval, which not only takes into consideration the perceptual similarity between images, but also significantly improves the retrieval performance of the Euclidean distance measure.
Abstract: A new scheme of learning similarity measure is proposed for content-based image retrieval (CBIR). It learns a boundary that separates the images in the database into two clusters. Images inside the boundary are ranked by their Euclidean distances to the query. The scheme is called constrained similarity measure (CSM), which not only takes into consideration the perceptual similarity between images, but also significantly improves the retrieval performance of the Euclidean distance measure. Two techniques, support vector machine (SVM) and AdaBoost from machine learning, are utilized to learn the boundary. They are compared to see their differences in boundary learning. The positive and negative examples used to learn the boundary are provided by the user with relevance feedback. The CSM metric is evaluated in a large database of 10009 natural images with an accurate ground truth. Experimental results demonstrate the usefulness and effectiveness of the proposed similarity measure for image retrieval.
TL;DR: The results of the experiments show that the MPEG-7-defined content descriptors can be used as such in thePicSOM system even though Euclidean distance calculation, inherently used in the PicSom system, is not optimal for all of them.
Abstract: Development of content-based image retrieval (CBIR) techniques has suffered from the lack of standardized ways for describing visual image content. Luckily, the MPEG-7 international standard is now emerging as both a general framework for content description and a collection of specific agreed-upon content descriptors. We have developed a neural, self-organizing technique for CBIR. Our system is named PicSOM and it is based on pictorial examples and relevance feedback (RF). The name stems from "picture" and the self-organizing map (SOM). The PicSOM system is implemented by using tree structured SOMs. In this paper, we apply the visual content descriptors provided by MPEG-7 in the PicSOM system and compare our own image indexing technique with a reference system based on vector quantization (VQ). The results of our experiments show that the MPEG-7-defined content descriptors can be used as such in the PicSOM system even though Euclidean distance calculation, inherently used in the PicSOM system, is not optimal for all of them. Also, the results indicate that the PicSOM technique is a bit slower than the reference system in starting to find relevant images. However, when the strong RF mechanism of PicSOM begins to function, its retrieval precision exceeds that of the reference system.
TL;DR: In this paper, a spiking neural network based on spike-time coding and Hebbian learning is proposed for unsupervised clustering on real-world data, and temporal synchrony in a multilayer network can induce hierarchical clustering.
Abstract: We demonstrate that spiking neural networks encoding information in the timing of single spikes are capable of computing and learning clusters from realistic data. We show how a spiking neural network based on spike-time coding and Hebbian learning can successfully perform unsupervised clustering on real-world data, and we demonstrate how temporal synchrony in a multilayer network can induce hierarchical clustering. We develop a temporal encoding of continuously valued data to obtain adjustable clustering capacity and precision with an efficient use of neurons: input variables are encoded in a population code by neurons with graded and overlapping sensitivity profiles. We also discuss methods for enhancing scale-sensitivity of the network and show how the induced synchronization of neurons within early RBF layers allows for the subsequent detection of complex clusters.
TL;DR: An approach for extracting rules from trained NNs for regression by identifying a subregion of the input space and a linear function involving the relevant input attributes of the data approximates the network output for all data samples in this subregion.
Abstract: Neural networks (NNs) have been successfully applied to solve a variety of application problems including classification and function approximation. They are especially useful as function approximators because they do not require prior knowledge of the input data distribution and they have been shown to be universal approximators. In many applications, it is desirable to extract knowledge that can explain how Me problems are solved by the networks. Most existing approaches have focused on extracting symbolic rules for classification. Few methods have been devised to extract rules from trained NNs for regression. This article presents an approach for extracting rules from trained NNs for regression. Each rule in the extracted rule set corresponds to a subregion of the input space and a linear function involving the relevant input attributes of the data approximates the network output for all data samples in this subregion. Extensive experimental results on 32 benchmark data sets demonstrate the effectiveness of the proposed approach in generating accurate regression rules.
TL;DR: A visualization-induced SOM (ViSOM) is proposed to overcome shortcomings of the self-organizing map (SOM), which can accommodate both training data and new arrivals and is much simpler in computational complexity.
Abstract: When used for visualization of high-dimensional data, the self-organizing map (SOM) requires a coloring scheme, such as the U-matrix, to mark the distances between neurons. Even so, the structures of the data clusters may not be apparent and their shapes are often distorted. In this paper, a visualization-induced SOM (ViSOM) is proposed to overcome these shortcomings. The algorithm constrains and regularizes the inter-neuron distance with a parameter that controls the resolution of the map. The mapping preserves the inter-point distances of the input data on the map as well as the topology. It produces a graded mesh in the data space such that the distances between mapped data points on the map resemble those in the original space, like in the Sammon mapping. However, unlike the Sammon mapping, the ViSOM can accommodate both training data and new arrivals and is much simpler in computational complexity. Several experimental results and comparisons with other methods are presented.
TL;DR: In this paper, an optimal neurocontroller that replaces the conventional automatic voltage regulator (AVR) and the turbine governor for a turbogenerator connected to the power grid is presented.
Abstract: This paper presents the design of an optimal neurocontroller that replaces the conventional automatic voltage regulator (AVR) and the turbine governor for a turbogenerator connected to the power grid. The neurocontroller design uses a novel technique based on the adaptive critic designs (ACDs), specifically on heuristic dynamic programming (HDP) and dual heuristic programming (DHP). Results show that both neurocontrollers are robust, but that DHP outperforms HDP or conventional controllers, especially when the system conditions and configuration change. This paper also shows how to design optimal neurocontrollers for nonlinear systems, such as turbogenerators, without having to do continually online training of the neural networks, thus avoiding risks of instability.
TL;DR: Analysis of learned EEG patterns confirms that for a subject to operate satisfactorily a brain interface, the latter must fit the individual features of the former.
Abstract: This paper proposes a novel and simple local neural classifier for the recognition of mental tasks from on-line spontaneous EEG signals The proposed neural classifier recognizes three mental tasks from on-line spontaneous EEG signals Correct recognition is around 70% This modest rate is largely compensated by two properties, namely low percentage of wrong decisions (below 5%) and rapid responses (every 1/2 s) Interestingly, the neural classifier achieves this performance with a few units, normally just one per mental task Also, since the subject and his/her personal interface learn simultaneously from each other, subjects master it rapidly (in a few days of moderate training) Finally, analysis of learned EEG patterns confirms that for a subject to operate satisfactorily a brain interface, the latter must fit the individual features of the former
TL;DR: A system to estimate participants' focus of attention from gaze directions and sound sources is developed and can be used as an index for a multimedia meeting record and for analyzing a meeting.
Abstract: A user's focus of attention plays an important role in human-computer interaction applications, such as a ubiquitous computing environment and intelligent space, where the user's goal and intent have to be continuously monitored. We are interested in modeling people's focus of attention in a meeting situation. We propose to model participants' focus of attention from multiple cues. We have developed a system to estimate participants' focus of attention from gaze directions and sound sources. We employ an omnidirectional camera to simultaneously track participants' faces around a meeting table and use neural networks to estimate their head poses. In addition, we use microphones to detect who is speaking. The system predicts participants' focus of attention from acoustic and visual information separately. The system then combines the output of the audio- and video-based focus of attention predictors. We have evaluated the system using the data from three recorded meetings. The acoustic information has provided 8% relative error reduction on average compared to only using one modality. The focus of attention model can be used as an index for a multimedia meeting record. It can also be used for analyzing a meeting.
TL;DR: A novel regression approach, termed as the robust support vector regression (RSVR) network, is proposed to enhance the robust capability of SVR and shows that even the training lasted for a long period, the testing errors would not go up and the overfitting phenomenon is indeed suppressed.
Abstract: Support vector regression (SVR) employs the support vector machine (SVM) to tackle problems of function approximation and regression estimation. SVR has been shown to have good robust properties against noise. When the parameters used in SVR are improperly selected, overfitting phenomena may still occur. However, the selection of various parameters is not straightforward. Besides, in SVR, outliers may also possibly be taken as support vectors. Such an inclusion of outliers in support vectors may lead to seriously overfitting phenomena. In this paper, a novel regression approach, termed as the robust support vector regression (RSVR) network, is proposed to enhance the robust capability of SVR. In the approach, traditional robust learning approaches are employed to improve the learning performance for any selected parameters. From the simulation results, our RSVR can always improve the performance of the learned systems for all cases. Besides, it can be found that even the training lasted for a long period, the testing errors would not go up. In other words, the overfitting phenomenon is indeed suppressed.
TL;DR: A new method for the estimation of the fractal dimension of a geometrical object using fuzzy logic techniques and it is proposed that this dimension incorporates the concept of a fuzzy set, which can be considered a weaker definition (but more realistic) of the Fractal dimension.
Abstract: In this paper, we describe a new method for the estimation of the fractal dimension of a geometrical object using fuzzy logic techniques. The fractal dimension is a mathematical concept, which measures the geometrical complexity of an object. The algorithms for estimating the fractal dimension calculate a numerical value using as data a time series for the specific problem. This numerical (crisp) value gives an idea of the complexity of the geometrical object (or time series). However, there is an underlying uncertainty in the estimation of the fractal dimension because we use only a sample of points of the object, and also because the numerical algorithms for the fractal dimension are not completely accurate. For this reason, we have proposed a new definition of the fractal dimension that incorporates the concept of a fuzzy set. This new definition can be considered a weaker definition (but more realistic) of the fractal dimension, and we have named this the "fuzzy fractal dimension." We can apply this new definition of the fractal dimension in conjunction with soft computing techniques for the problem of time series prediction. We have developed hybrid intelligent systems combining neural networks, fuzzy logic, and the fractal dimension, for the problem of time series prediction, and we have achieved very good results.