International Journal of Wavelets, Multiresolution and Information Processing

We discuss a volatility functional model and show that jumps asymptotically impact the volatility estimate. This result is useful because our model shows that significant variations affect the estimation of the volatility and historically price series have structures with this type of behavior. We also discuss a method for detecting and locating jumps at different levels and show that the jumps tend to be detected by wavelet coefficients at lower resolution levels accurately. By checking the wavelet coefficients on the different levels, we can find dyadic intervals in some levels, whose corresponding absolute value of the wavelet coefficient exceeds a threshold, and is significantly higher than the others. We applied the procedure in a simulation study and to a series of Google stocks.

In this paper, we improve the broad learning system (BLS) by speeding up the incremental learning for added inputs. We propose an efficient implementation for a step that is in the pseudoinverse computation of a partitioned matrix, to reduce the computational complexity. The proposed efficient implementation has two different forms for the cases of $q>k$ and $q\le k$, respectively, where $q$ and $k$ denote the number of additional training samples and the total number of nodes, respectively. The proposed implementation for $q>k$ utilizes the inverse of a sum of matrices to compute only a $k\times k$ matrix inverse, instead of a $q\times q$ matrix inverse in the original implementation, and the corresponding speedup for the matrix inversion operation in the number of floating-point operations is $\frac{1}{2}{(q/k)}^3$. Moreover, it also speeds up two relevant matrix multiplication operations in the original implementation. On the other hand, the proposed implementation for $q\le k$ speeds up one matrix multiplication operation in the original implementation. The numerical simulations show that both the proposed and original implementations always achieve the same testing accuracy. On the Modified National Institute of Standards and Technology dataset, the speedups of the proposed efficient BLS implementation over the original BLS implementation in total training time are 1.35–1.57 when $q>k$ and 1.09–1.13 when $q<k$, while on the NYU object recognition benchmark dataset, the speedups are 1.10–1.35 when $q>k$ and 1.07–1.09 when $q<k$.

Incremental learning is one of the effective methods of learning from the accumulated training samples and the large-scale dataset. The main advantages of incremental learning consist of making full use of historical information, reducing the training scale greatly and saving space and time consumption. Despite extensive research on incremental support vector machine (SVM) learning algorithms, most of them are based on independent and identically distributed samples (i.i.d.). Not only that, there has been no theoretical analysis of incremental SVM learning algorithms. In this paper, we mainly study the generalization bounds of this incremental SVM learning algorithm whose samples are based on uniformly geometric Markov chains, and exponentially strongly mixing sequence. As a special case, we also obtain the generalization bounds of i.i.d. samples.

Let $\omega$ be a positive, Lebesgue integrable and exponential decay function defined on an infinite interval $I$ and let $L_{1,\omega }(I)$ be the space of weighted Lebesgue integrable functions on $I$. In this paper, we give the relations of the best one-sided approximation and the optimal Hermite–Fejér interpolation by the set of algebraic polynomials of degree not exceeding a given number for the smooth function classes $B{C}^{2r,+}(I)$, $r\in {\mathbb{N}}_{+}$, in the metric of the space $L_{1,\omega }(I),$ and prove that the Hermite–Fejér interpolation based on the set of the zeros of some orthogonal polynomials is optimal in $L_{1,\omega }(I)$. In addition, we show that the approximation error of the optimal Hermite–Fejér interpolation and quadrature errors of the weighted Gaussian quadrature formula are equal, and give the exact constant of the errors.

The robustness of sampling scheme ensures that the exact samples are not crucial to recover the signal under considered and some errors can be tolerated. This paper addresses that two kinds of nonuniform average sampling schemes are robust for finitely generated shift-invariant signals in mixed Lebesgue spaces. The results show that the mixed shift-invariant signals can be reconstructed stably from two classes of nonuniform average samples, respectively, when the generators and sampling functionals are disturbed slightly.

Functional linear regression is one of the main modeling tools for working with functional data. Since functional data are usually stream data essentially and there are some noises in functional data. Many numerical research studies of machine learning indicate that the noise samples not only increase the amount of storage space, but also affect the performance of algorithm. Therefore, in this paper we consider a new learning strategy by introducing incremental learning, Markov sampling for functional linear regression and propose a novel functional incremental linear square regression algorithm based on Markov sampling (FILSR-MS). To have a better understanding of the proposed FILSR-MS, we not only estimate the generalization bound of the proposed algorithm and establish the fast learning rate, but also present some useful discussions. The performance of the proposed algorithm is validated by the numerical experiments for benchmark repository.

This paper addresses the problem of Bayesian wavelet estimating the mean vector of multivariate normal distribution under a multivariate normal prior distribution based on linear and nonlinear exponential balanced loss functions. The covariance matrix of multivariate normal distribution is considered known. Bayes estimators of mean vector parameter of multivariate normal distribution are achieved. Then two soft shrinkage wavelet threshold estimators based on Stein’s unbiased risk estimate (SURE) and Bayes estimators are provided. Finally, the performance of the soft shrinkage wavelet estimators was checked through simulation study and Electrical Grid Stability Simulated data set. Simulation and real data results showed the better performance of SURE thresholds based on linear and nonlinear exponential balanced loss functions compared to other classical wavelet methods. Also, they showed better performance for SURE threshold based on nonlinear exponential balanced loss function in multivariate normal distribution with small dimensions.

As neuroscience states, the ability of our brain to process visual information is similar to that of neurons. Our cognitive ability carries out this process using the functional magnetic resonance imaging (fMRI) profiles that it collects during the presentation of visual stimuli around us. However, this process poses its own set of challenges due to complexities like structure and noise levels. To address these challenges, high-quality computational approaches have been executed. This work proposes a hybrid computational approach, that combines feature extraction and conditional generative adversarial network techniques. This research suggests the use of the hybrid form incorporating Kohonen Network and Information Maximizing Generative Adversarial Networks (KNInfoGAN) to map the fMRI data to visual stimuli. For feature extraction, the Kohonen network is used, and the visual stimuli are reconstructed using the InfoGAN approach. A series of experiments were conducted on dual synthetic fMRI datasets using a KNInfoGAN model, and the results were obtained with an accuracy rate of 99% for handwritten numbers and 87% for handwritten English characters, approximately. An accuracy rate of 73% has been obtained upon utilizing the ImageNet database to examine the accomplishments of the suggested approach for highly complex natural stimulus. The suggested technique in this study outperforms the previously existing techniques in image reconstruction, with enhanced quality and a lower computational cost. The same has been verified based on the comparative examination of a wide range of tests conducted in this study.

We study the efficiency of the vector greedy algorithms for simultaneous sparse approximation in a Banach space. We employ a unified way of analyzing the approximation properties of the Vector Weak Chebyshev Greedy Algorithm, the Vector Weak Greedy Algorithm with Free Relaxation and the Vector Rescaled Weak Relaxed Greedy Algorithm. We obtain the necessary and sufficient conditions for the convergence and the optimal convergence rate on the basic sparse class for these algorithms. It shows that these vector greedy algorithms are simple but highly efficient in dealing with multi-target element.

This paper presents a non-parametric Bayesian approach for modeling multiple response variables using a Gaussian process regression (GPR) model. The response functions are modeled using a dependent Gaussian process (GP) prior, and the estimation, prediction, and inference issues are discussed within this framework. To establish the information consistency of the dependent GPs prediction strategy, a stretching-restriction method is proposed. The covariance structure is constructed using convolved Gaussian processes (CGPs) to illustrate the results. Simulations and real data analyses show that the proposed dependent GPR yields reasonably good prediction accuracy.

The theory of $n$-Hilbert space is very important in Functional analysis. Based on wavelet theory, the Frame theory is developed, and the classical results of frame theory are established in Hilbert space. There are several new extensions and generalizations on frames in Hilbert space such as weaving frames in Hilbert space. In this paper, the concept of weaving frames in $n$-Hilbert space is introduced. We obtain some new properties of weaving frames in $n$-Hilbert space. First, we give some characterizations of weaving frames in $n$-Hilbert space. Then, we give some stability and perturbation results on weaving frames in $n$-Hilbert space.

In this paper, using standard pairs, we present a method to construct symbols satisfying the sum rules of order $p$ for any given $p$. It is shown that using matrix polynomial theory symbols, symmetric or non-symmetric, satisfying the sum rules of order $p$ can be constructed efficiently. The construction is illustrated using various examples.