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Automatic seizure detection is significant for the diagnosis of epilepsy and reducing the massive workload of reviewing continuous EEGs. In this work, a novel approach, combining Stockwell transform (S-transform) with deep Convolutional Neural Networks (CNN), is proposed to detect seizure onsets in long-term intracranial EEG recordings. Primarily, raw EEG data is filtered with wavelet decomposition. Then, S-transform is used to obtain a proper time-frequency representation of each EEG segment. After that, a 15-layer deep CNN using dropout and batch normalization serves as a robust feature extractor and classifier. Finally, smoothing and collar technique are applied to the outputs of CNN to improve the detection accuracy and reduce the false detection rate (FDR). The segment-based and event-based evaluation assessments and receiver operating characteristic (ROC) curves are employed for the performance evaluation on a public EEG database containing 21 patients. A segment-based sensitivity of 97.01% and a specificity of 98.12% are yielded. For the event-based assessment, this method achieves a sensitivity of 95.45% with an FDR of 0.36/h.

We introduce a time-frequency transform based on Gabor functions whose parameters are given by the Fourier transform of the analyzed signal. At any given frequency, the width and the phase of the Gabor function are obtained, respectively, from the magnitude and the phase of the signal's corresponding Fourier component, yielding an analyzing kernel which is a representation of the signal's content at that particular frequency. The resulting Gabor transform tunes itself to the input signal, allowing the accurate detection of time and frequency events, even in situations where the traditional Gabor and S-transform approaches tend to fail. This is the case, for instance, when considering the time-frequency representation of electroencephalogram traces (EEG) of epileptic subjects, as illustrated by the experimental study presented here.

This paper presents an S-transform-based Electroencephalogram channel optimization and feature extraction methodology for monitoring mental vigilance level of humans. Vigilance level detection methodology consists of four steps. In the first stage, two types of Electroencephalogram signals (alert and drowsy) are acquired from 30 healthy subjects and decomposed into sub-bands using the S-transform. In the second stage, permutation entropy of the S-transform coefficients is calculated and Electroencephalogram channel optimization is performed. S-transform-based statistical features are computed from the optimized Electroencephalogram channels, in the third stage. In the fourth stage, artificial intelligence techniques such as Least Square-Support Vector Machine, Artificial Neural Network and Naive Bayes Classifier are used for the classification of Electroencephalogram signals using extracted features. The performance of the feature extraction methodology is tested on the Electroencephalogram data of 30 healthy subjects. Experimental results ensured the effectiveness of proposed methodology for the estimation of mental vigilance level by using Electroencephalogram signals. It is observed that the Artificial Neural Network classifier is a good candidate for pre-emptive automatic vigilance level detection system for Brain-Computer Interface applications.

Highly dispersive surface waves resulting from the combination of strong lateral seafloor heterogeneities, shallow water depths and hard sea bottom severely degrade seismic reflection data quality. Considering that seismic signals are nonstationary and surface waves have various spectra over time, we proposed S-transform-based time frequency wavenumber analysis technique which allows the dynamic analysis of spectrum over time. The data is first transformed from the time-space domain to the time-wavenumber domain through one-dimensional Fourier transform over the spatial variable, then the variable-factor S-transform is applied over time. Nonstationary filtering is then designed to identify and separate surface waves in complex wavefields, based on its low frequency and low velocity properties, in the time-frequency-wavenumber (TFK) domain. Application to field data illustrates that with this technique, not only is the surface wave effectively suppressed, but also the reflective signals are enhanced, which confirm the validity of the method.

We consider stochastic evolution equations in the framework of white noise analysis. Contraction operators on inductive limits of Banach spaces arise naturally in this context and we first extend Banach's fixed point theorem to this type of spaces. In order to apply the fixed point theorem to evolution equations, we construct a topological isomorphism between spaces of generalized random fields and the corresponding spaces of U-functionals. As an application we show that the solutions of some nonlinear stochastic heat equations depend continuously on their initial data. This method also applies to stochastic Volterra equations, stochastic reaction–diffusion equations and to anticipating stochastic differential equations.

Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.

We revisit a CKS-space from the viewpoint of standard setup of white noise calculus and prove a general characterization theorem for white noise operators from a CKS-space into another CKS-space. The usual characterizations so far obtained for white noise distributions, white noise test functions and white noise operators are all consequences of the unified characterization theorem.

A linear bounded operator on a test space of white noise functionals can be characterized in terms of growth conditions imposed on the operator's symbol. This paper extends such characterization to operators on the Kondratiev's test space of random variables over a general non-Gaussian probability space.

We follow the biorthogonal approach of Yu. Daletsky and his colleagues. In particular, the test space under consideration is generated by the system of multivariate Appell polynomials defined on the underlying infinite dimensional co-nuclear probability space.

We further extend this biorthogonal approach to operators by providing a biorthogonal chaos decomposition for operators and by giving a biorthogonal construction for an operator's symbol.

In White Noise Analysis (WNA), various random quantities are analyzed as elements of (S)*, the space of Hida distributions.¹ Hida distributions are generalized functions of white noise, which is to be naturally viewed as the derivative of the Brownian motion. On (S)*, the Wick product is defined in terms of the -transform. We have found such a remarkable property that the Wick product has no zero divisors among Hida distributions. This result is a WNA version of Titchmarsh's theorem and is expected to play fundamental roles in developing the "operational calculus" in WNA along the line of Mikusiński's version for solving differential equations.

Multiple stochastic integrals with respect to the Gaussian white noise indexed by R^d are used for the construction of reflection positive non-Gaussian valued random variables (random fields). This construction is then extended to a class of Hida distribution to consider several interesting examples. In particular, some new fields (without full Euclidean invariance but satisfying all other axioms, including space rotation invariance) are constructed through this probabilistic (and purely Euclidean) procedure.

We derive some characteristic properties of the convolution operator acting on white noise functions and prove that the convolution product of white noise distributions coincides with their Wick product. Moreover, we show that the S-transform and the Laplace transform coincide on Fock space.

We revisit the analytic characterization theorem for S-transform of infinite dimensional distributions. Then we prove that the nuclearity of the space of test white noise functionals is a necessary condition for the characterization of the S-transform in terms of analytic and growth conditions.

We study the analytic characterization of $S$-transform in a general setting of white noise functionals. Then, the measurability condition of the norms generating the underlining locally convex space is a necessary and sufficient condition for the analytic characterization of the $S$-transform in terms of analytic and growth conditions.

This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts, the problem reduces to stochastic Schrödinger equation with space–time white noise of Gaussian or Poisson or Lévy type. We identify their mathematical resolution via stochastic quantization. Nonlinear phenomena such as Kerr effect can be modeled by a stochastic nonlinear Schrödinger equation in the focusing case with space–time white noise. A treatment of stochastic transport equation, the Korteweg–De Vries equation as well as a number of other nonlinear wave equations with space–time white noise is also given. The main technique is the S-transform (we will actually use the closely related Hermite transform) which converts the stochastic partial differential equation (PDE) with space–time white noise to a deterministic PDE defined on the Hida–Kondratiev white noise distribution space. We then utilize the inverse S-transform/Hermite transform known as the characterization theorem combined with the infinite-dimensional implicit function theorem for analytic maps to establish local existence and uniqueness theorems for path-wise solutions of this class of problems. The particular focus of this paper on singular white noise distributions is motivated by practical situations where the refractive index fluctuations in the propagation medium in space and time are intense due to turbulence, ionospheric plasma turbulence, marine-layer fluctuations, etc. Since a large class of PDEs, that arise in nonlinear wave propagation, have polynomial-type nonlinearities, white noise distribution theory is an effective tool in studying these problems subject to different types of white noises.

In this paper, we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure ${\pi }_{\sigma }^{\beta }$, $0<\beta \le 1$, on the dual of Schwartz test function space ${{𝒟}}^{\prime }$. The Hilbert space $L^2({\pi }_{\sigma }^{\beta })$ of complex-valued functions is described in terms of a system of generalized Appell polynomials ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$ associated to the measure ${\pi }_{\sigma }^{\beta }$. The kernels $C_n^{\sigma ,\beta }(\cdot )$, $n\in {\mathbb{N}}_0$, of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$, there is a generalized dual Appell system ${\mathbb{Q}}^{\sigma ,\beta ,\alpha }$ that is biorthogonal to ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$. The test and generalized function spaces associated to the measure ${\pi }_{\sigma }^{\beta }$ are completely characterized using an integral transform as entire functions.

In this paper, the conventional Stockwell transform is effectively used to classify the ECG arrhythmias. The performance of ECG classification mainly depends on feature extraction based on an efficient formation of morphological and temporal features and the design of the classifier. Feature extraction is the important component of designing the system based on pattern recognition since even the best classifier will not perform better if the good features are not selected properly. Here, the S-transform (ST) is used to extract the morphological features which is appended with temporal features. This feature set is independently classified using artificial neural network (NN) and support vector machine (SVM). In this work, five classes of ECG beats (normal, ventricular, supra ventricular, fusion and unknown beats) from Massachusetts Institute of Technology-Beth Israel Hospital (MIT-BIH) arrhythmia database are classified according to AAMI EC57 1998 standard (Association for the Advancement of Medical Instrumentation). Performance is evaluated on several normal and abnormal ECG signals of MIT-BIH arrhythmias database using two classifier techniques: ST with NN classifier (ST-NN) and other proposed ST with SVM classifier (ST-SVM). The proposed method achieves accuracy of 98.47%. The performance of the proposed technique is compared with ST-NN and earlier reported technique.

Accurate identification of intended grip actions using the myoelectric signal recorded from the surface of the residual muscles can facilitate natural control of a prosthetic hand for an amputee. However, this is not trivial due to the complexity of the hand muscles. To overcome these shortcomings, there is the need for determining features of the myoelectric recordings that can be used for accurate identification of the grip actions. This study reports the use of S-transform (ST) of the surface myoelectric recordings for recognizing the intent of the user to generate a set of grip patterns. Surface Electromyogram (sEMG) recorded while performing five different hand/finger grip patterns was analyzed. ST of the signal was computed to analyze the signal in a windowed time–frequency domain. The energy and mean amplitude of the transformed signal were classified using a neural network. The method was tested for able-hand and trans-radial amputee subjects. The results show that ST showed improved sensitivity, specificity and accuracy for both healthy and amputee people.

For the Hankel–Stockwell transform, the Price uncertainty principle is proved, we define the Localization operators and we study their boundedness and compactness. We also show that these operators belong to the so-called Schatten–von Neumann class.

Due to the advantages offered by the S-transform (ST) distribution, it has been recently successfully implemented for various applications such as seismic and image processing. The desirable properties of the ST include a globally referenced phase as the case with the short time Fourier transform (STFT) while offering a higher spectral resolution as the wavelet transform (WT). However, this estimator suffers from some inherent disadvantages seen as poor energy concentration with higher frequencies. In order to improve the performance of the distribution, a modification to the existing technique is proposed. Additional parameters are proposed to control the window's width which can greatly enhance the signal representation in the time–frequency plane. The new estimator's performance is evaluated using synthetic signals as well as biomedical data. The required features of the ST which include invertability and phase information are still preserved.

Electric load movement forecast is increasingly importance for the industry. This study addresses the load movement forecast modeling based on complex matrix interpolation of the S-transform (ST). In complex matrix of time-frequency representation of the ST, each row follows conjugate symmetric property and each column appears a certain degree of similarity. Based on these characteristics, a complex matrix interpolation method for the time-frequency representation of the ST is proposed to interpolate each row of the complex matrix based on the conjugate symmetric property, and then to perform nearest-neighbor interpolation on each column. Then with periodic extension for daily and yearly electric load movement, a forecast model employing the complex matrix interpolation of the ST is introduced. The forecast approach is applied to predict daily load movement of the European Network on Intelligent Technologies (EUNITE) load dataset and annual electric load movement of State Gird Corporation of China and its branches in 2005 and 2006. Result analysis indicates workability and effectiveness of the proposed method.