Xiaoping Shen

A sampling expansion for nonbandlimited signals in chromatic derivatives

In this paper, we construct infinite-band filterbanks for perfect reconstruction (PR) using Hermite polynomials and Hermite functions. The analysis filters are linear combinations of derivative operators based on these polynomials-the so-called chromatic derivative filters. Together with the synthesis filterbanks, they give PR for a large class of signals that may have infinite bandwidth. Several other related filterbanks are discussed as well. The error in reconstruction for finite-channel chromatic derivative filterbanks is calculated. Examples are given to demonstrate the use of these chromatic filterbanks.

Integration of Self-Organizing Map (SOM) and Kernel Density Estimation (KDE) for network intrusion detection

This paper proposes an approach to integrate the self-organizing map (SOM) and kernel density estimation (KDE) techniques for the anomaly-based network intrusion detection (ABNID) system to monitor the network traffic and capture potential abnormal behaviors. With the continuous development of network technology, information security has become a major concern for the cyber system research. In the modern net-centric and tactical warfare networks, the situation is more critical to provide real-time protection for the availability, confidentiality, and integrity of the networked information. To this end, in this work we propose to explore the learning capabilities of SOM, and integrate it with KDE for the network intrusion detection. KDE is used to estimate the distributions of the observed random variables that describe the network system and determine whether the network traffic is normal or abnormal. Meanwhile, the learning and clustering capabilities of SOM are employed to obtain well-defined data clusters to reduce the computational cost of the KDE. The principle of learning in SOM is to self-organize the network of neurons to seek similar properties for certain input patterns. Therefore, SOM can form an approximation of the distribution of input space in a compact fashion, reduce the number of terms in a kernel density estimator, and thus improve the efficiency for the intrusion detection. We test the proposed algorithm over the real-world data sets obtained from the Integrated Network Based Ohio Universitys Network Detective Service (INBOUNDS) system to show the effectiveness and efficiency of this method.

Bootstrap Methods for Foreign Currency Exchange Rates Prediction

This paper presents the research of using bootstrap methods for time-series prediction. Unlike the traditional single model (neural network, support vector machine, or any other types of learning algorithms) based time-series prediction, we propose to use bootstrap methods to construct multiple learning models, and then use a combination function to combine the output of each model for the final predicted output. In this paper, we use the neural network model as the base learning algorithm and applied this approach to the foreign currency exchange rate predictions. Six major foreign currency exchange rates including Australia dollars (AUD), British pounds (GBP), Canadian dollars (CAD), European euros (EUR), Japanese yen (JPY) and Swiss francs (CHF) are used for prediction (base currency is US Dollar). Simulations on the most recently available exchange rate data (January 01, 2003 to December 27, 2006) on both daily prediction and weekly prediction indicate that the proposed method can significantly improve the forecasting performance compared to the traditional single neural network based approach.

Construction of Periodic Prolate Spheroidal Wavelets Using Interpolation

Periodic prolate spheroidal wavelets (periodic PS wavelets), based on the periodizaton of the first prolate spheroidal wave function (PSWF), were recently introduced by the authors. Because of localization and other properties, these periodic PS wavelets could serve as an alternative to Fourier series for applications in modeling periodic signals. In this paper, we continue our work with periodic PS wavelets and direct our attention to their construction via interpolation. We show that they have a representation in terms of interpolation with the modified Dirichlet kernel. We then derive a group of formulas of interpolation type based on this representation. These formulas enable one to obtain a simple procedure for the calculation of the periodic PS wavelets and finding expansion coefficients. In particular, they are used to compute filter coefficients for the periodic PS wavelets. This is done for a number of concrete cases.

A substitute for summability in wavelet expansions

Gibbs’phenomenon almost always appears in the expansions using classical orthogonal systems. Various summability methods are used to get rid of unwanted properties of these expansions. Similar problems arise in wavelet expansions, but cannot be solved by the same methods. In a previous work [10], two alternative procedures for wavelet expansions were introduced for dealing with this problem. In this article, we are concerned with the details of the implementation of one of the procedures, which works for the wavelets with compact support in the time domain. Estimates based on this method remove the excess oscillations. We show that the dilation equations which arise, though they contain an infinite number of terms, have coefficients which decrease exponnetially. In addition, an iteration relation for the positive estimation function is derived to reduce the amount of calculation in the approximation. Numerical experiments are given to illustrate the theoretical results.

Periodic Prolate Spheroidal Wavelets

ABSTRACT Prolate spheroidal wavelets (PS wavelets) were recently introduced by the authors. They were based on the first prolate spheroidal wave function (PSWF) and had many desirable properties lacking in other wavelets. In particular, the subspaces belonging to the associated multiresolution analysis (MRA) were shown to be closed under differentiation and translation. In this paper, we introduce periodic prolate spheroidal wavelets. These periodic wavelets are shown to possess properties inherited from PS wavelets such as differentiation and translation. They have the potential for applications in modeling periodic phenomena as an alternative to the usual periodic wavelets as well as the Fourier basis.

Wavelet Like Behavior of Slepian Functions and Their Use in Density Estimation

Abstract Slepian functions (Prolate Spheroidal Wave Functions) are obtained by maximizing the energy of a σ-bandlimited function (normalized with total energy 1) on a prescribed interval [−τ, τ]. The solution to this problem leads to an eigenvalue problem λf(t) = {sinσ(t − x)/π(t − x)}f(x)dx, whose solutions, in turn, form an orthogonal sequence {ϕ n }. This sequence is a basis of the Paley-Wiener space B σ of σ-bandlimited functions. For σ = π, integer translates of the Slepian functions of order 0, {ϕ0(t − n)} form a Riesz basis of the same space. Furthermore, by using ϕ0 as a scaling function we can construct a wavelet theory based on them. Two methods of density estimations thus naturally arise; one based on the orthogonal system {ϕ n } and the other on the scaling functions {ϕ0(t − n)}. The former gives more rapid convergence, while the latter avoids Gibbs phenomenon, is locally positive, and allows the use of thresholding methods. Both approaches exhibit a strong localization property.

A Quadrature Formula Based On Sampling In Meyer Wavelet Subspaces

In this paper we discuss a weighted trapezoidal rule based on sampling in Meyer wavelet subspaces. For a wide class of functions, we obtain convergence and error bounds. Some examples are given to construct sampling functions.

DensityRank: A novel feature ranking method based on kernel estimation

This paper proposes a novel feature ranking method, DensityRank, based on kernel estimation on the feature spaces to improve the classification performance. As the availability of raw data in many of todays applications continues to grow at an explosive rate, it is critical to assess the learning capabilities of different features and select the important subset of features to improve learning accuracy as well as reduce computational cost. In our approach, kernel methods are used to estimate the probability density function for each feature across different class labels. Discrepancy analysis based on the mean integrated square error (MISE) between pairs of such density estimations is used to provide the ranking values. Then, the ranked subspace method is adopted to select subsets of important features that are used to develop the learning models. Comparative study of this method with those of traditional ranking methods related to Fishers discrimination ratio and information gain theory, as well as the random subspace algorithm and the bootstrap aggregating (bagging), are presented in this paper. Simulation results on various real-world data sets illustrate the effectiveness of the proposed method.

Recovery of digitized signals using Slepian functions

The continuous prolate spheroidal wave functions (Slepian functions) were found to be useful for analog signal processing several decades ago. But the digital revolution left them in the dust since they did not seem naturally adapted to discrete analysis. Yet they have many desirable, even unique, properties that originally made them fascinating and could lead to some applications in digital and analog-digital signal processing practice. The simplest such applications involve the conversion of a digital signal to an analog signal and recovery of a bandlimited signal from their values at countable many distinct points on the real line. The Shannon sampling theorem, which is based on the sinc expansions, plays the most important theoretical foundation in such approaches. In this work, by using the natural connection between the Slepian functions and sinc function, several new formulae based on integer values of Slepian functions are developed. These formulae are then used to replace the sinc function in sampling theorems for digitizing bandlimited signals. Finally, they are used to construct analysis and synthesis filter banks for sampled values of a bandlimited signal.