Gabor Neural Networks with Proven Approximation Properties

In practice, the time variable cannot be negative. The space $$L^2({\mathbb {R}}_+)$$ of square integrable functions defined on the right half real line $${\mathbb {R}}_+$$ models the causal signal space. This paper focuses on a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$. The density theorem for Gabor systems in $$L^2(\mathbb {R})$$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $$L^2({\mathbb {R}})$$ in terms of the index set alone-independently of window functions. The space $$L^2({\mathbb {R}}_+)$$ admits no nontrivial Gabor system since $${\mathbb {R}}_+$$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ and the notion of $$\Theta $$-transform matrix. Using the $$\Theta $$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ under the condition that $$\log _ba$$ is a positive rational number, where a and b are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $$L^2({\mathbb {R}}_+)$$ is that $$\log _ba \le 1$$. Simultaneously, we obtain a $$\Theta $$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $$L^2(\mathbb {R}_+)$$.

Gabor systems play an important role in various practical problems related to signal and image processing.In this paper, we study the computability of Gabor frames in L 2 .Computable version of some existing results related to Gabor systems are obtained.Also, computability of operators associated with Gabor frames is discussed and the computability of Zak transform is defined.Finally, we discuss the existence of the computable dual frames associated with the Gabor Systems.

G?vruta studied atomic systems in terms of frames for range of operators (that is, for subspaces), namely ?-frames, where the lower frame condition is controlled by the Hilbert-adjoint of a bounded linear operator?. For a locally compact abelian groupGand a positive integer n, westudy frames of matrix-valued Gabor systems in the matrix-valued Lebesgue space L2(G,Cn?n) , where a bounded linear operator ? on L2(G,Cn?n) controls not only lower but also the upper frame condition. We term such frames matrix-valued (?,?*)-Gabor frames. Firstly, we discuss frame preserving mapping in terms of hyponormal operators. Secondly, we give necessary and sufficient conditions for the existence of matrix-valued (?,?*)- Gabor frames in terms of hyponormal operators. It is shown that if ? is adjointable hyponormal operator, then L2(G,Cn?n) admits a ?-tight (?,?*)-Gabor frame for every positive real number ?. A characterization of matrix-valued (?,?*)-Gabor frames is given. Finally, we show that matrix-valued (?,?*)-Gabor frames are stable under small perturbation of window functions. Several examples are given to support our study.

We consider the construction of tight Gabor frames (h, a=1, b=1) from Gabor systems (g, a=1, b=1) with g a window having few zeros in the Zak transform domain via the operation h=Z -1(Zg/|Zg|), where Z is the standard Zak transform. We consider this operation with g the Gaussian, the hyperbolic secant, and for g belonging to a class of positive, even, unimodal, rapidly decaying windows of which the two-sided exponential is a typical example. All these windows g have the property that Zg has a single zero, viz. at (1/2,\1/2), in the unit square [0,1)2. The Gaussian and hyperbolic secant yield a frame for any a,b > 0 with ab < 1, and we show that so does the two-sided exponential. For these three windows it holds that S a -1/2 g h as a 1, where S a is the frame operator corresponding to the Gabor frame (g,a,a). It turns out that the h’s corresponding to g’s of the above type look and behave quite similarly when scaling parameters are set appropriately. We give a particular detailed analysis of the h corresponding to the two-sided exponential. We give several representations of this h, and we show that , and is continuous and differentiable everywhere except at the half-integers, etc., and we pay particular attention to the cases that the time constant of the two-sided exponential g tends to . We also consider the cases that the time constants of the Gaussian and of the hyperbolic secant tend to 0 or to . It so turns out that h thus obtained changes from the box function into its Fourier transform when the time constant changes from 0 to

A study has recently been conducted to assess the extent of hydrocarbon impacts to groundwater&lt;br&gt;and soil resources at a petroleum refinery site in Billings, Montana. To accomplish the study,&lt;br&gt;forty-six groundwater monitoring wells were installed at the site. Data collected from the wells&lt;br&gt;included detailed lithologic descriptions from split-spoon samples, cutting returns from air rotary&lt;br&gt;drilling, and suites of geophysical well logs. Because the quality of the lithologic descriptions&lt;br&gt;from the borings was erratic, our approach was to produce lithofacies interpretations based on&lt;br&gt;gamma ray logs input into a neural network classifier system.&lt;br&gt;The type of neural network used was a self-organizing map. This type of network does not&lt;br&gt;require user interpretations, instead, the network categorizes each input vector into a class based&lt;br&gt;on similarity to other input vectors. The number of output classes is determined by the user. The&lt;br&gt;output classifications were then plotted as ‘pseudo-logs’ and correlation performed using these&lt;br&gt;pseudo-logs.&lt;br&gt;Cross sections constructed using conventional well log interpretation and the neural network&lt;br&gt;classifications show good, general agreement. A significant advantage of the neural network&lt;br&gt;approach over a conventional interpretation approach is that all of the well log data are analyzed&lt;br&gt;concurrently preventing inconsistencies that frequently occur with conventional methods.&lt;br&gt;Another major benefit to the neural network approach is the choice of the number of classes&lt;br&gt;which correlates with the level of lithologic detail that can be resolved.

Two different types of adaptive networks are considered for solving the centralized and distributed hy-pothesis testing problem. The performance of the two different types of networks is compared under differentperformance indices and training rules. It is shown that training rules based on the Neyman-Pearson criterion outperform error based training rules. Simulations are provided for data that are linearly and nonlinearly separable. I. INTRODUCTION The optimum Bayesian and Neyman-Pearson solution to the distributed decision fusion problem bearsstriking similarities to the structure of a neural network (NN), [28,29]. Moreover, NNs can, in principle learn arbitrary input-output mappings, provided that they are sufficiently smooth. These two facts motivate theuse of NNs for solving the centralized and distributed hypothesis testing problem. In selecting the properNN layout, one could argue that a perceptron-type NN can learn any input-output mapping, thus it can betrained to solve the hypothesis testing problem. However, the ability of a perceptron-type NN to learn an

Objects placed in mammograms and their features have high diagnostic value. The method of extraction and calculation parameters of the images is a key element in the whole process of assessment of individual medical cases. This applies both to detect the location of microcalcifications, as well as the accurate assessment of the degree of malignancy [1, 2, 3, 4, 8, 9]. The analysis of the literature shows that in many cases, to assess them as neural classifiers used neural networks unidirectional scholars algorithm back propagation of error. The large popularity of this type of network stems from the rich knowledge about them supported a wide range of applications [11, 12, 13, 17, 18]. This does not mean that this type of network is the best tool for the identification of cancer. In order to verify the correctness of the research, it was decided to compare the two types of neural networks: one-way multi-layer network MLBP [17, 18] and a network of independently building the architecture i.e. Fahlmana network [5, 17]. Both networks were used for the same task, the task of verification and classification of medical data based on the feature vectors. The study attempted to assess the mammograms for detecting the locus and the evaluation grade of microcalcifications [19, 23]. According to the stated objective, the structure of the neural networks were designed so that it was possible to indicate location and performance evaluation grade localized potential microcalcifications. Way of describing mammogram—a set of characteristics and size of the database are the factors that were taken as those that have a key impact on the quality of the diagnostic computer tools [21, 22, 23].

Long-term sea level variations in the northern South China Sea (SCS) are known to significantly impact coastal ecosystems and socio-economic activities. To improve sea level prediction accuracy, four models—harmonic analysis and three artificial neural networks (ANNs), namely genetic algorithm-optimized back propagation (GA-BP), radial basis function (RBF), and long short-term memory (LSTM)—are developed and compared using 52 years of observational data (1960–2004). Key evaluation metrics are presented to demonstrate the models’ effectiveness: for harmonic analysis, the root mean square error (RMSE) is reported as 14.73, the mean absolute error (MAE) is 12.61, the mean bias error (MBE) is 0.0, and the coefficient of determination (R2) is 0.84; for GA-BP, the RMSE is measured as 29.1371, the MAE is 24.9411, the MBE is 5.6809, and the R2 is 0.4003; for the RBF neural network, the RMSE is calculated as 27.1433, the MAE is 22.7533, the MBE is 2.1322, and the R2 is 0.4690; for LSTM, the RMSE is determined as 23.7929, the MAE is 19.7899, the MBE is 1.3700, and the R2 is 0.5872. The key findings include the following: (1) A significant sea level rise trend at 1.4 mm/year is observed in the northern SCS. (2) Harmonic analysis is shown to outperform all ANN models in both accuracy and robustness, with sea level variations effectively characterized by four principal and six secondary tidal constituents. (3) Despite their complexity, ANN models (including LSTM) are found to fail in surpassing the predictive capability of the traditional harmonic method. These results highlight the continued effectiveness of harmonic analysis for long-term sea level forecasting, offering critical insights for coastal hazard mitigation and sustainable development planning.

For future 4G wireless system, higher spectrum efficiency and access speed are required, hence overlapping the time slot and multi frequency carriers will meet such demand, which forms the original thought of OVTFDM system. Based on Gabor transformation and under wavelet frame work, computation complex can be reduced on both transfer and receiver side. Further more, in OVTFDM system avoid cancellation of LSI, only detect user code sequence, classify it to recover original signal. In this paper we present a fast decoding algorithm by using Gabor neural network, and evaluate its performance.

Gabor frames and directional time–frequency analysis

The existence of tight Gabor duals for Gabor frames and subspace Gabor frames

We consider the difficult problem of deciding whether a triple(g,a,b),with window g ∈ L 2(ℝ) and time shift parameter a and frequency shift parameterbis a Gabor frame from two different points of view. We first identify two classes of non-negative windows g ∈ L 2(ℝ) such that their Zak transforms have no and just one zero per unit square, respectively. The first class consists of all integrable, non-negative windowsg that are supported by and strictly decreasing on [0, ∞). The second class consists of all even, non-negative, continuous, integrable windowsg that satisfy on [0, ∞) a condition slightly stronger than strict convexity (superconvexity). Accordingly, the members of these two classes generate Gabor frames for integer oversampling factor (ab)-1 ≥ 1 and ≥ 2, respectively. When we weaken the condition of superconvexity into strict convexity, the Zak transformsZg may have as many zeros as one wants, but in all cases (g, a, b) is still a Gabor frame when(ab) -1 is an integer ≥ 2. As a second issue we consider the question for which a, b > 0 the triple (g, a, b) is a Gabor frame, where gis the characteristic function of an interval [0, ∞) with c0 > 0 fixed. It turns out that the answer to the latter question is quite complicated, where irrationality or rationality of abgives rise to quite different situations. A pictorial display, in which the various cases are indicated in the positive (a, b)-quadrant, shows a remarkable resemblance to the design of a low-budget tie.

This thesis considers the application of neural networks in cost estimating in project management and whether they lead to more accurate estimates. It strikes two areas of research, namely neural networks and project management; an introductory chapter on both subjects is included. The statistical problem of parametric cost estimating is described and an explanation of the general principles is given. The Multi-Layer Perceptron with the Backpropagation learning algorithm is determined to be the most appropriate network and a selection of available software programs is reviewed. A Multi-Layer Perceptron neural model is used to determine one of the most important cost estimating relationships of the PRICE model. A comparison of the outputs of the neural network and the PRICE model shows that the Backpropagation algorithm is able to find the underlying estimating relationships used by PRJCE. To investigate whether other underlying functions can be found with artificial intelligence methods, other input parameters are selected and the costs generated by the PRICE model and by the neural network are compared with each other. Further experiments were undertaken in order to improve the performance of the neural network. The neural networks were applied to real data. and their output compared with the PRICE model. The processes of achieving better results are analogous to those used for the artificial data. A neural network was created which performs better than the PRICE model in terms of the accuracy of the estimates produced. The results are discussed and the collection of significant and accurate information and then deciding on which type of network is the best network to be used are identified as the major problems in the application of artificial intelligence for cost estimation in project management. The limitations and restrictions of the implementation of neural networks are examined and the scope and topics of further research are suggested.

In this study, the thermoluminescence (TL) glow curve test results performed with eleven different dose values were used as training data, and its attempted to estimate the test results of the curves performed at four different doses using artificial intelligence methods. While the dose values of the data used for training were 10, 20, 50, 100, 150, 220, 400, 500, 600, 700, and 900 Gy, the selected dose values of the data for the testing were 40, 276, 320, and 800 Gy. The success of the experimental and artificial neural network results was determined according to the mean squared error (RMSE), regression error (R2), root squared error (RSE), and mean absolute error (MAE) criteria. Studies have been carried out on seven different neural network types. These networks are adaptive network-based fuzzy inference system (ANFIS), general regression neural network (GRNN), radial basis neural network (RBNN), cascade-forward backprop neural network (CFBNN), Elman backprop neural network (EBNN), feed-forward backprop neural network (FFBNN), and layer recurrent neural network (LRNN). This study concluded that the neural network with the Elman backpropagation network type demonstrated the best network performance. In this network, the training success rate is 80.8%, while the testing success rate is 87.95%.

Rolling resistance is considered as a significant criterion in vehicles energy consumption. In off-road vehicles, soil and tire interaction encompasses more parameters that lead to the thorny problem in its analysis. The typical experimental models are useful remedies in the estimation perspective, but they can not be extended to all conditions. Neural networks as efficient tools can simplify the examination procedure. This research was aimed to raise the Wismer–Luth model accuracy to rely on cone index as soil condition representative by utilizing three neural networks. Three types of networks were investigated, comprising multi-layer perceptron (MLP), radial basis function (RBF) and generalized regression neural network (GRNN). Various feature vectors consisting of vertical load parameters, velocity, traffic, and cone index values were evaluated to estimate the optimal value of the cone index. GRNN had the best performance with the least error rate in comparison with RBF and MLP. Based on the results of feature vector analysis, employing either forward velocity or multi-pass factors had a negligible effect on model improvement, but by incorporating all parameters (velocity, load, and number of passes), the accuracy of the model increased significantly.