Zero-Crossing Analysis and Information Divergence of Lévy Walks for Real-Time Feature Extraction

Zero-Crossing Analysis and Information Divergence of Lévy Walks for Real-Time Feature Extraction

Jesus David Terrazas Gonzalez (University of Manitoba, Department of Electrical and Computer Engineering, Winnipeg, Canada) and Witold Kinsner (University of Manitoba, Department of Electrical and Computer Engineering, Winnipeg, Canada)
Copyright: © 2016 |Pages: 19
DOI: 10.4018/IJHCR.2016100104
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Abstract

A method, based on the Smirnov transform, for generating synthetic data with the statistical properties of Lévy-walks is presented. This method can be utilized for generating arbitrary prescribed probability density functions (pdf). A cybersecurity engineering problem associated with Internet traffic is addressed. The synthetic Lévy-walks process is intertwined with sections of distinct characteristics creating a composite signal that is analyzed through zero-crossing rate (ZCR) within a varying-size window to identify sections. The advantages of the ZCR computation directly in the time-domain are appealing for real-time implementations. Moreover, the characterization of the degree of closeness, via the Kullback-Leibler divergence (KLD), among the pdfs of arbitrary processes (focusing on Lévy walks) and model pdfs is presented. The results obtained from the KLD experiments provide a categorical determination of the closeness degree. These results, a remarkable achievement in this research, are also promising to be used as features for classification of complex signals in real-time.
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Introduction

Biologists have found that mobility patterns of foraging animals (e.g., spider monkey, albatrosses (seabirds), jackals, and marine predators) resemble what physicists have long called Lévy-walks (Atkinson, Rhodes, Macdonald, & Anderson, 2002; Ramos-Fernandez, Morales, Miramontes, Cocho, Larralde, & Ayala-Orozco, 2004; Viswanathan, Afanasyev, Buldyrev, Murphy, Prince, & Stanley, 1996). The term “Lévy-walk” was coined by (Shlesinger, Klafter, & Wong, 1982) to explain atypical particle diffusion not governed by Brownian motion (BM) (Rhee, Shin, Hong, Lee, & Chong, 2008; Rhee, Shin, Hong, Lee, Kim, & Chong, 2011).

Patterns of human mobility have features resembling Lévy walks (e.g., heavy-tail flight and pause-time distributions and the superdiffusive nature of mobility (Rhee et al., 2011)). Some deviations from pure Lévy walks, in human mobility, occur due to various factors specific to humans (e.g., geographical constraints such as roads, buildings, obstacles and traffic) (Rhee et al., 2008). General examples of Lévy-processes are: Poisson processes, compound Poisson processes, BM, Gamma processes, inverse Gaussian processes, and those with stable distributions (Kyprianou, 2014).

Biological systems exhibiting Lévy-walks have been of inspiration for designing adaptive behaviour in robotic applications and artificial agents. The behaviour imitation of biological creatures has contributed to the advancement of adaptive searching (Nurzaman, Matsumoto, Nakamura, Koizumi, & Ishiguro, 2009). Motion of bacteria responding to the presence of chemical concentration gradients, aka bacterial chemotaxis, has been adopted and implemented to realize simple yet effective searching behaviour for gradient-inducing targets in robots or artificial agents (Rhee et al., 2011). Lévy-walks are also applied in complex mobile sensing scenarios related to the Internet-of-Things (IoT), where smartphones are taking a central role. Smartphones allow crowd-sourced sensing in the IoT because of their embedded different types of sensors (e.g., camera, microphone, GPS, thermometer, accelerometer, and communications like Wi-Fi, Bluetooth, and near field communications (NFC)) (Thejaswini, Rajalakshmi, & Desai, 2014). Due to the nature of human mobility, smartphones are immersed in scenarios subjected to Lévy-walks when monitoring environmental factors like temperature, humidity, urban noise pollution, carbon footprint, air pollution, and urban traffic (Thejaswini et al., 2014).

The search problem distinguishes between two kinds of interacting organisms: either a “searcher” (e.g., forager, predator, parasite, pollinator, the active gender in the mating process) or else a “target” (e.g., prey, food, the passive gender for mating) (Viswanathan et al., 1996). When compared to BM, it is known that the probability for returning to previously visited site of a particle making a Lévy-flight is smaller, and therefore advantageous when target sites are sparsely and randomly distributed (Rhee et al., 2011).

The nature of the searching drive can be guided almost entirely by external cues, either by the cognitive (memory) or detective (e.g., olfaction and vision) skills of the searcher. Nevertheless, in certain situations the movement is non-oriented, thus becoming a stochastic process (Stark & Woods, 2002) in essence. Therefore, in such cases (and even when a small deterministic component in the locomotion exists) it is a random search that defines the outcomes of biological activities (e.g., a predator cannot search for so long without finding food, or it will perish). It is known that Lévy-walks search patterns lead to optimal outcomes in computing (e.g., computer databases and networks). These outcomes are possible because Lévy-walks can be applied not only to continuous spaces, but also discrete ones (e.g., inherent discrete nature of searches in Internet or memory search in neural-networks) (Viswanathan, Raposo, & da Luz, 2008).

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