Dynamical density delay maps: simple, new method for visualising the behaviour of complex systems

Background. Physiologic signals, such as cardiac interbeat intervals, exhibit complex fluctuations. However, capturing important dynamical properties, including nonstationarities may not be feasible from conventional time series graphical representations. Methods. We introduce a simple-to-implement visualisation method, termed dynamical density delay mapping (``D3-Map'' technique) that provides an animated representation of a system's dynamics. The method is based on a generalization of conventional two-dimensional (2D) Poincar� plots, which are scatter plots where each data point, x(n), in a time series is plotted against the adjacent one, x(n+1). First, we divide the original time series, x(n) (n=1,..., N), into a sequence of segments (windows). Next, for each segment, a three-dimensional (3D) Poincar� surface plot of x(n), x(n+1), hx(n),x(n+1) is generated, in which the third dimension, h, represents the relative frequency of occurrence of each (x(n),x(n+1)) point. This 3D Poincar\'e surface is then chromatised by mapping the relative frequency h values onto a colour scheme. We also generate a colourised 2D contour plot from each time series segment using the same colourmap scheme as for the 3D Poincar\'e surface. Finally, the original time series graph, the colourised 3D Poincar\'e surface plot, and its projection as a colourised 2D contour map for each segment, are animated to create the full ``D3-Map.'' Results. We first exemplify the D3-Map method using the cardiac interbeat interval time series from a healthy subject during sleeping hours. The animations uncover complex dynamical changes, such as transitions between states, and the relative amount of time the system spends in each state. We also illustrate the utility of the method in detecting hidden temporal patterns in the heart rate dynamics of a patient with atrial fibrillation. The videos, as well as the source code, are made publicly available. Conclusions. Animations based on density delay maps provide a new way of visualising dynamical properties of complex systems not apparent in time series graphs or standard Poincar\'e plot representations. Trainees in a variety of fields may find the animations useful as illustrations of fundamental but challenging concepts, such as nonstationarity and multistability. For investigators, the method may facilitate data exploration

Visibility graphs of random scalar fields and spatial data

The family of visibility algorithms were recently introduced as mappings between time series and graphs. Here we extend this method to characterize spatially extended data structures by mapping scalar fields of arbitrary dimension into graphs. After introducing several possible extensions, we provide analytical results on some topological properties of these graphs associated to some types of real-valued matrices, which can be understood as the high and low disorder limits of real-valued scalar fields. In particular, we find a closed expression for the degree distribution of these graphs associated to uncorrelated random fields of generic dimension, extending a well known result in one-dimensional time series. As this result holds independently of the field's marginal distribution, we show that it directly yields a statistical randomness test, applicable in any dimension. We showcase its usefulness by discriminating spatial snapshots of two-dimensional white noise from snapshots of a two-dimensional lattice of diffusively coupled chaotic maps, a system that generates high dimensional spatio-temporal chaos. We finally discuss the range of potential applications of this combinatorial framework, which include image processing in engineering, the description of surface growth in material science, soft matter or medicine and the characterization of potential energy surfaces in chemistry, disordered systems and high energy physics. An illustration on the applicability of this method for the classification of the different stages involved in carcinogenesis is briefly discussed

Leakage Detection Framework using Domain-Informed Neural Networks and Support Vector Machines to Augment Self-Healing in Water Distribution Networks

The reduction of water leakage is essential for ensuring sustainable and resilient water supply systems. Despite recent investments in sensing technologies, pipe leakage remains a significant challenge for the water sector, particularly in developed nations like the UK, which suffer from aging water infrastructure. Conventional models and analytical methods for detecting pipe leakage often face reliability issues and are generally limited to detecting leaks during nighttime hours. Moreover, leakages are frequently detected by the customers rather than the water companies. To achieve substantial reductions in leakage and enhance public confidence in water supply and management, adopting an intelligent detection method is crucial. Such a method should effectively leverage existing sensor data for reliable leakage identification across the network. This not only helps in minimizing water loss and the associated energy costs of water treatment but also aids in steering the water sector towards a more sustainable and resilient future. As a step towards ‘self-healing’ water infrastructure systems, this study presents a novel framework for rapidly identifying potential leakages at the district meter area (DMA) level. The framework involves training a domain-informed variational autoencoder (VAE) for real-time dimensionality reduction of water flow time series data and developing a two-dimensional surrogate latent variable (LV) mapping which sufficiently and efficiently captures the distinct characteristics of leakage and regular (non-leakage) flow. The domain-informed training employs a novel loss function that ensures a distinct but regulated LV space for the two classes of flow groupings (i.e., leakage and non-leakage). Subsquently, a binary SVM classifier is used to provide a hyperplane for separating the two classes of LVs corresponding to the flow groupings. Hence, the proposed framework can be efficiently utilised to classify the incoming flow as leakage or non-leakage based on the encoded surrogates LVs of the flow time series using the trained VAE encoder. The framework is trained and tested on a dataset of over 2000 DMAs in North Yorkshire, UK, containing water flow time series recorded at 15-minute intervals over one year. The framework performs exceptionally well for both regular and leakage water flow groupings with a classification accuracy of over 98 % on the unobserved test datase

Manifold Learning Approach for Chaos in the Dripping Faucet

Dripping water from a faucet is a typical example exhibiting rich nonlinear phenomena. For such a system, the time stamps at which water drops separate from the faucet can be directly observed in real experiments, and the time series of intervals \tau_n between drop separations becomes a subject of analysis. Even if the mass m_n of a drop at the onset of the n-th separation, which cannot be observed directly, exhibits perfectly deterministic dynamics, it sometimes fails to obtain important information from time series of \tau_n. This is because the return plot \tau_n-1 vs. \tau_n may become a multi-valued function, i.e., not a deterministic dynamical system. In this paper, we propose a method to construct a nonlinear coordinate which provides a "surrogate" of the internal state m_n from the time series of \tau_n. Here, a key of the proposed approach is to use ISOMAP, which is a well-known method of manifold learning. We first apply it to the time series of $\tau_n$ generated from the numerical simulation of a phenomenological mass-spring model for the dripping faucet system. It is shown that a clear one-dimensional map is obtained by the proposed approach, whose characteristic quantities such as the Lyapunov exponent, the topological entropy, and the time correlation function coincide with the original dripping faucet system. Furthermore, we also analyze data obtained from real dripping faucet experiments which also provides promising results.Comment: 9 pages, 10 figure

Classification of damage in structural systems using time series analysis and supervised and unsupervised pattern recognition techniques

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