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32 pages, 2819 KiB

Open AccessArticle

Disentangling Sources of Multifractality in Time Series

by Robert Kluszczyński, Stanisław Drożdż, Jarosław Kwapień, Tomasz Stanisz and Marcin Wątorek

Mathematics 2025, 13(2), 205; https://doi.org/10.3390/math13020205 - 9 Jan 2025

Cited by 1 | Viewed by 690

Abstract

This contribution addresses the question commonly asked in the scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as [...] Read more.

This contribution addresses the question commonly asked in the scientific literature about the sources of multifractality in time series. Two primary sources are typically considered. These are temporal correlations and heavy tails in the distribution of fluctuations. Most often, they are treated as two independent components, while true multifractality cannot occur without temporal correlations. The distributions of fluctuations affect the span of the multifractal spectrum only when correlations are present. These issues are illustrated here using series generated by several model mathematical cascades, which by design build correlations into these series. The thickness of the tails of fluctuations in such series is then governed by an appropriate procedure of adjusting them to q-Gaussian distributions, and q is treated as a variable parameter that, while preserving correlations, allows for tuning these distributions to the desired functional form. Multifractal detrended fluctuation analysis (MFDFA), as the most commonly used practical method for quantifying multifractality, is then used to identify the influence of the thickness of the fluctuation tails in the presence of temporal correlations on the width of multifractal spectra. The obtained results point to the Gaussian distribution, so

q = 1

, as the appropriate reference distribution to evaluate the contribution of fatter tails to the width of multifractal spectra. An appropriate procedure is presented to make such estimates. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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17 pages, 1606 KiB

Open AccessArticle

Patch-Wise-Based Self-Supervised Learning for Anomaly Detection on Multivariate Time Series Data

by Seungmin Oh, Le Hoang Anh, Dang Thanh Vu, Gwang Hyun Yu, Minsoo Hahn and Jinsul Kim

Mathematics 2024, 12(24), 3969; https://doi.org/10.3390/math12243969 - 17 Dec 2024

Viewed by 1494

Abstract

Multivariate time series anomaly detection is a crucial technology to prevent unexpected errors from causing critical impacts. Effective anomaly detection in such data requires accurately capturing temporal patterns and ensuring the availability of adequate data. This study proposes a patch-wise framework for anomaly [...] Read more.

Multivariate time series anomaly detection is a crucial technology to prevent unexpected errors from causing critical impacts. Effective anomaly detection in such data requires accurately capturing temporal patterns and ensuring the availability of adequate data. This study proposes a patch-wise framework for anomaly detection. The proposed approach comprises four key components: (i) maintaining continuous features through patching, (ii) incorporating various temporal information by learning channel dependencies and adding relative positional bias, (iii) achieving feature representation learning through self-supervised learning, and (iv) supervised learning based on anomaly augmentation for downstream tasks. The proposed method demonstrates strong anomaly detection performance by leveraging patching to maintain temporal continuity while effectively learning data representations and handling downstream tasks. Additionally, it mitigates the issue of insufficient anomaly data by supporting the learning of diverse types of anomalies. The experimental results show that our model achieved a 23% to 205% improvement in the F1 score compared to existing methods on datasets such as MSL, which has a relatively small amount of training data. Furthermore, the model also delivered a competitive performance on the SMAP dataset. By systematically learning both local and global dependencies, the proposed method strikes an effective balance between feature representation and anomaly detection accuracy, making it a valuable tool for real-world multivariate time series applications. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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18 pages, 420 KiB

Open AccessFeature PaperArticle

Boosted Whittaker–Henderson Graduation

by Zihan Jin and Hiroshi Yamada

Mathematics 2024, 12(21), 3377; https://doi.org/10.3390/math12213377 - 29 Oct 2024

Viewed by 908

Abstract

The Whittaker–Henderson (WH) graduation is a smoothing method for equally spaced one-dimensional data such as time series. It includes the Bohlmann filter, the Hodrick–Prescott (HP) filter, and the Whittaker graduation as special cases. Among them, the HP filter is the most prominent trend-cycle [...] Read more.

The Whittaker–Henderson (WH) graduation is a smoothing method for equally spaced one-dimensional data such as time series. It includes the Bohlmann filter, the Hodrick–Prescott (HP) filter, and the Whittaker graduation as special cases. Among them, the HP filter is the most prominent trend-cycle decomposition method for macroeconomic time series such as real gross domestic product. Recently, a modification of the HP filter, the boosted HP (bHP) filter, has been developed, and several studies have been conducted. The basic idea of the modification is to achieve more desirable smoothing by extracting long-term fluctuations remaining in the smoothing residuals. Inspired by the modification, this paper develops the boosted version of the WH graduation, which includes the bHP filter as a special case. Then, we establish its properties that are fundamental for applied work. To investigate the properties, we use a spectral decomposition of the penalty matrix of the WH graduation Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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15 pages, 1956 KiB

Open AccessArticle

Information–Theoretic Analysis of Visibility Graph Properties of Extremes in Time Series Generated by a Nonlinear Langevin Equation

by Luciano Telesca and Zbigniew Czechowski

Mathematics 2024, 12(20), 3197; https://doi.org/10.3390/math12203197 - 12 Oct 2024

Viewed by 767

Abstract

In this study, we examined how the nonlinearity

α

of the Langevin equation influences the behavior of extremes in a generated time series. The extremes, defined according to run theory, result in two types of series, run lengths and surplus magnitudes, whose complex [...] Read more.

In this study, we examined how the nonlinearity

α

of the Langevin equation influences the behavior of extremes in a generated time series. The extremes, defined according to run theory, result in two types of series, run lengths and surplus magnitudes, whose complex structure was investigated using the visibility graph (VG) method. For both types of series, the information measures of the Shannon entropy measure and Fisher Information Measure were utilized for illustrating the influence of the nonlinearity

α

on the distribution of the degree, which is the main topological parameter describing the graph constructed by the VG method. The main finding of our study was that the Shannon entropy of the degree of the run lengths and the surplus magnitudes of the extremes is mostly influenced by the nonlinearity, which decreases with with an increase in

α

. This result suggests that the run lengths and surplus magnitudes of extremes are characterized by a sort of order that increases with increases in nonlinearity. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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14 pages, 3429 KiB

Open AccessArticle

Anomaly Detection in Fractal Time Series with LSTM Autoencoders

by Lyudmyla Kirichenko, Yulia Koval, Sergiy Yakovlev and Dmytro Chumachenko

Mathematics 2024, 12(19), 3079; https://doi.org/10.3390/math12193079 - 1 Oct 2024

Cited by 2 | Viewed by 2355

Abstract

This study explores the application of neural networks for anomaly detection in time series data exhibiting fractal properties, with a particular focus on changes in the Hurst exponent. The objective is to investigate whether changes in fractal properties can be identified by transitioning [...] Read more.

This study explores the application of neural networks for anomaly detection in time series data exhibiting fractal properties, with a particular focus on changes in the Hurst exponent. The objective is to investigate whether changes in fractal properties can be identified by transitioning from the analysis of the original time series to the analysis of the sequence of Hurst exponent estimates. To this end, we employ an LSTM autoencoder neural network, demonstrating its effectiveness in detecting anomalies within synthetic fractal time series and real EEG signals by identifying deviations in the sequence of estimates. Whittle’s method was utilized for the precise estimation of the Hurst exponent, thereby enhancing the model’s ability to differentiate between normal and anomalous data. The findings underscore the potential of machine learning techniques for robust anomaly detection in complex datasets. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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27 pages, 36491 KiB

Open AccessArticle

The Time Series Classification of Discrete-Time Chaotic Systems Using Deep Learning Approaches

by Ömer Faruk Akmeşe, Berkay Emin, Yusuf Alaca, Yeliz Karaca and Akif Akgül

Mathematics 2024, 12(19), 3052; https://doi.org/10.3390/math12193052 - 29 Sep 2024

Cited by 1 | Viewed by 1526

Abstract

Discrete-time chaotic systems exhibit nonlinear and unpredictable dynamic behavior, making them very difficult to classify. They have dynamic properties such as the stability of equilibrium points, symmetric behaviors, and a transition to chaos. This study aims to classify the time series images of [...] Read more.

Discrete-time chaotic systems exhibit nonlinear and unpredictable dynamic behavior, making them very difficult to classify. They have dynamic properties such as the stability of equilibrium points, symmetric behaviors, and a transition to chaos. This study aims to classify the time series images of discrete-time chaotic systems by integrating deep learning methods and classification algorithms. The most important innovation of this study is the use of a unique dataset created using the time series of discrete-time chaotic systems. In this context, a large and unique dataset representing various dynamic behaviors was created for nine discrete-time chaotic systems using different initial conditions, control parameters, and iteration numbers. The dataset was based on existing chaotic system solutions in the literature, but the classification of the images representing the different dynamic structures of these systems was much more complex than ordinary image datasets due to their nonlinear and unpredictable nature. Although there are studies in the literature on the classification of continuous-time chaotic systems, no studies have been found on the classification of discrete-time chaotic systems. The obtained time series images were classified with deep learning models such as DenseNet121, VGG16, VGG19, InceptionV3, MobileNetV2, and Xception. In addition, these models were integrated with classification algorithms such as XGBOOST, k-NN, SVM, and RF, providing a methodological innovation. As the best result, a 95.76% accuracy rate was obtained with the DenseNet121 model and XGBOOST algorithm. This study takes the use of deep learning methods with the graphical representations of chaotic time series to an advanced level and provides a powerful tool for the classification of these systems. In this respect, classifying the dynamic structures of chaotic systems offers an important innovation in adapting deep learning models to complex datasets. The findings are thought to provide new perspectives for future research and further advance deep learning and chaotic system studies. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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13 pages, 6439 KiB

Open AccessArticle

Exploring the Dynamic Behavior of Crude Oil Prices in Times of Crisis: Quantifying the Aftershock Sequence of the COVID-19 Pandemic

by Fotios M. Siokis

Mathematics 2024, 12(17), 2743; https://doi.org/10.3390/math12172743 - 3 Sep 2024

Cited by 1 | Viewed by 1389

Abstract

Crude oil prices crashed and dropped into negative territory at the onset of the COVID-19 pandemic. This extreme event triggered a series of great-magnitude aftershocks. We seek to investigate the cascading dynamics and the characteristics of the series immediately following the oil market [...] Read more.

Crude oil prices crashed and dropped into negative territory at the onset of the COVID-19 pandemic. This extreme event triggered a series of great-magnitude aftershocks. We seek to investigate the cascading dynamics and the characteristics of the series immediately following the oil market crash. Utilizing a robust method named the Omori law, we quantify the correlations of these events. This research presents empirical regularity concerning the number of times that the absolute value of the percentage change in the oil index exceeds a given threshold value. During the COVID-19 crisis, the West Texas Intermediate (WTI) oil prices exhibit greater volatility compared to the Brent oil prices, with higher relaxation values at all threshold levels. This indicates that larger aftershocks decay more rapidly, and the period of turbulence for the WTI is shorter than that of Brent and the stock market indices. We also demonstrate that the power law’s exponent value increases with the threshold value’s magnitude. By proposing this alternative method of modeling extreme events, we add to the current body of literature, and the findings demonstrate its practical use for decision-making authorities—particularly financial traders who model high-volatility products like derivatives. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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Review

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25 pages, 2166 KiB

Open AccessReview

Review of the Natural Time Analysis Method and Its Applications

by Panayiotis A. Varotsos, Efthimios S. Skordas, Nicholas V. Sarlis and Stavros-Richard G. Christopoulos

Mathematics 2024, 12(22), 3582; https://doi.org/10.3390/math12223582 - 15 Nov 2024

Cited by 1 | Viewed by 869

Abstract

A new concept of time, termed natural time, was introduced in 2001. This new concept reveals unique dynamic features hidden behind time-series originating from complex systems. In particular, it was shown that the analysis of natural time enables the study of the dynamical [...] Read more.

A new concept of time, termed natural time, was introduced in 2001. This new concept reveals unique dynamic features hidden behind time-series originating from complex systems. In particular, it was shown that the analysis of natural time enables the study of the dynamical evolution of a complex system and identifies when the system enters a critical stage. Hence, natural time plays a key role in predicting impending catastrophic events in general. Several such examples were published in a monograph in 2011, while more recent applications were compiled in the chapters of a new monograph that appeared in 2023. Here, we summarize the application of natural time analysis in various complex systems, and we review the most recent findings of natural time analysis that were not included in the previously published monographs. Specifically, we present examples of data analysis in this new time domain across diverse fields, including condensed-matter physics, geophysics, earthquakes, volcanology, atmospheric sciences, cardiology, engineering, and economics. Full article

(This article belongs to the Special Issue Recent Advances in Time Series Analysis)

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