Deterministic Sampling from Univariate Normal Distributions with Sierpinski Space-Filling Curves

This work addresses the problem of sampling from Gaussian probability distributions by means of uniform samples obtained deterministically and directly from space-filling curves (SFCs), a purely topological concept. To that end, the well-known inverse cumulative distribution function method is used, with the help of the probit function,which is the inverse of the cumulative distribution function of the standard normal distribution. Mainly due to the central limit theorem, the Gaussian distribution plays a fundamental role in probability theory and related areas, and that is why it has been chosen to be studied in the present paper. Numerical distributions (histograms) obtained with the proposed method, and in several levels of granularity, are compared to the theoretical normal PDF, along with other already established sampling methods, all using the cited probit function. Final results are validated with the Kullback-Leibler and two other divergence measures, and it will be possible to draw conclusions about the adequacy of the presented paradigm. As is amply known, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. That said, sequences resulting from this kind of procedure are not truly random. Even so, and to be coherent with the literature, the expression ”random number” will be used along the text to mean ”pseudo-random number”.

A STUDY OF FRACTAL GEOMETRY IN WIRELESS SENSOR NETWORKS

Fractal geometry is a subject that studies non-integer dimensional figures. Most of the fractal geometry figures have a nested or recursive structure. This paper attempts to apply the nested or recursive structure characteristics of fractal geometry to wireless sensor networks. We selected two filling curves, Node-Gosper and Moore, as our research subjects. Node-Gosper Curve is a curve based on node-replacement with a fractal dimension of two. Its first-order graph consists of seven basic line segments. When the hierarchy becomes larger, it can be filled with a hexagonal-like shape. To allow the mobile anchor node of wireless sensor networks to walk along this curve, the number of levels of the Node-Gosper Curve can be adjusted according to parameters such as the sensing area and transmission range. Many space-filling curves have the common shortcoming that they cannot loop on their own, that is, the starting point and the end point are not close, which will cause the mobile anchor node to use extra paths from the end point back to the starting point. The Moore curve has a self-loop, i.e., the starting point and the ending point are almost at the same position. This paper applies Moore curve to the path planning of the mobile anchor node. We can use this path to traverse the entire sensing area and stay in the central point of each square cluster to collect the information of the nodes where the events occurred. The self-loop characteristic of the Moore curve is expected to reach each sensor to collect data faster than other space filling curves, that is, the transmission latency of the sensor traversal will be reduced

The Effect of Space-filling Curves on the Efficiency of Hand Gesture Recognition Based on sEMG Signals

Over the past few years, Deep learning (DL) has revolutionized the field of data analysis. Not only are the algorithmic paradigms changed, but also the performance in various classification and prediction tasks has been significantly improved with respect to the state-of-the-art, especially in the area of computer vision. The progress made in computer vision has produced a spillover in many other domains, such as biomedical engineering. Some recent works are directed towards surface electromyography (sEMG) based hand gesture recognition, often addressed as an image classification problem and solved using tools such as Convolutional Neural Networks (CNN). This paper extends our previous work on the application of the Hilbert space-filling curve for the generation of image representations from multi-electrode sEMG signals, by investigating how the Hilbert curve compares to the Peano- and Z-order space-filling curves. The proposed space-filling mapping methods are evaluated on a variety of network architectures and in some cases yield a classification improvement of at least 3%, when used to structure the inputs before feeding them into the original network architectures.

The Metric Topology

The chapter is an extensive account of the metric topology and is a prerequisite for all the subsequent chapters. The leading sections develop the basic metric properties such as closure and interior, continuity and equivalent metrics, separation properties, product spaces, and countability axioms. This is followed by a detailed study of completeness, compactness, local compactness, and function spaces. Chapter applications include contraction mappings, continuous nowhere differentiable functions, space-filling curves, closed convex subsets of ?n, and a number of approximation results. The chapter concludes with a detailed section on orthogonal polynomials and Fourier series of continuous functions, which, together with section 3.7, provides an excellent background for Hilbert spaces. The study of sequence and function spaces in this chapter leads up gradually into Banach spaces.