Complexity of Simple, Switched and Skipped Chaotic Maps in Finite Precision

Abstract

In this paper we investigate the degradation of the statistic properties of chaotic maps as consequence of their implementation in a digital media such as Digital Signal Processors (DSP), Field Programmable Gate Arrays (FPGA) or Application-Specific Integrated Circuits (ASIC). In these systems, binary floating-and fixed-point are the numerical representations available. Fixed-point representation is preferred over floating-point when speed, low power and/or small circuit area are necessary. Then, in this paper we compare the degradation of fixed-point binary precision version of chaotic maps with the one obtained by using floating point 754-IEEE standard, to evaluate the feasibility of their FPGA implementation. The specific period that every fixed-point precision produces was investigated in previous reports. Statistical characteristics are also relevant, it has been recently shown that it is convenient to describe the statistical characteristic using both, causal and non-causal quantifiers. In this paper we complement the period analysis by characterizing the behavior of these maps from an statistical point of view using cuantifiers from information theory. Here, rather than reproducing an exact replica of the real system, the aim is to meet certain conditions related to the statistics of systems.

A Smartphone Lightweight Method for Human Activity Recognition Based on Information Theory

“…This work proposes another solution that introduces information theory quantifiers as a new feature extracted from a frequency distribution. These quantifiers of information theory can be defined as measures that characterize properties of a distribution, allowing the extraction of information from time series, where determinism and stochastic are two extremes of the process [27]. The statistical complexity measure of a probability distribution can be used as a feature by machine learning algorithms.…”

“…The HAR-SR maps the implicit features of sensor's signals to a discrete domain as a frequency distribution. From this distribution, this research presents a new feature set for classification models, derived from information theory, such as Shannon's Entropy, Jensen-Shannon Divergence, and the Statistical Complexity Measure [27,41]. By the end, a supervised learning algorithm is used to create a model for activity recognition.…”

“…The Shannon entropy and the divergence of Jensen-Shannon characterize properties of a probability distribution, which can be obtained using the BOP-SAX model, for example. Choosing a proper method for obtaining probability distribution from time series is not trivial since each method uses different techniques that result in different representations that must be chosen according to problem domain [27,44]. The Shannon's entropy [45] is commonly adopted in dealing with information content.…”

“…So, if we have an ordered sequence, such as a simple oscillation or trend, the statistical complexity would be low, and so would an unordered sequence, such as uncorrelated white noise (a random signal with equal intensity at different frequencies). The characterization of the data is more difficult for series located between the two given extremes and, therefore, the complexity would be greater [27]. Thus, the statistical complexity is defined according to Equation (6).…”

“…Unlike the approaches that use the Term Frequency -Inverse Document Frequency (tf-idf) model [25], e.g., SAX-SVM model, BOSS model and MBOSS model, which generally have a larger feature set, the advantage of our method is that the feature set is significantly smaller. The data representation in the discrete domain [19,23] is combined with information theory techniques [26,27] which arrange the input signals in a much smaller space than the original one. This generates a low-cost classification model in terms of computational performance.…”