Nature and geometry furnish us with many such patterns, such as coastlines, snowflakes, the Cantor set and the Kuch curve.

Understanding signal behavior across scales is vital in areas such as natural phenomena analysis and financial modeling. A key property is self-similarity, quantified by the Hurst exponent (H), which reveals long-term dependencies. Wavelet-based methods are effective for estimating H due to their multi-scale analysis capability, but additive noise in real-world measurements often degrades accuracy. We propose Noise-Controlled ALPHEE (NC-ALPHEE), an enhancement of the Average Level-Pairwise Hurst Exponent Estimator (ALPHEE), incorporating noise mitigation and generating multiple level-pairwise estimates from signal energy pairs. A neural network (NN) combines these estimates, replacing traditional averaging. This adaptive learning maintains ALPHEE's behavior in noise-free cases while improving performance in noisy conditions. Extensive simulations show that in noise-free data, NC-ALPHEE matches ALPHEE's accuracy using both averaging and NN-based methods. Under noise, however, traditional averaging deteriorates and requires impractical level restrictions, while NC-ALPHEE consistently outperforms existing techniques without such constraints. NC-ALPHEE offers a robust, adaptive approach for H estimation, significantly enhancing the reliability of wavelet-based methods in noisy environments.

Differential equations play important roles in modeling complex physical systems.

This paper presents $whittlehurst$, a Python package implementing Whittle's likelihood method for estimating the Hurst exponent in fractional Brownian motion (fBm). While the theoretical foundations of Whittle's estimator are well-established, practical and computational considerations are critical for effective use. We focus explicitly on assessing our implementation's performance across several numerical approximations of the fractional Gaussian noise (fGn) spectral density, comparing their computational efficiency, accuracy, and consistency across varying input sequence lengths. Extensive empirical evaluations show that our implementation achieves state-of-the-art estimation accuracy and computational speed. Additionally, we benchmark our method against other popular Hurst exponent estimation techniques on synthetic and real-world data, emphasizing practical considerations that arise when applying these estimators to financial and biomedical data.

Combustion is the primary process in gas turbine engines, where there is a need for efficient air-fuel mixing to enhance performance. High-shear swirl injectors are commonly used to improve fuel atomization and mixing, which are key factors in determining combustion efficiency and emissions. However, under certain conditions, combustors can experience thermoacoustic instability. In this study, a decision tree-based machine learning algorithm is used to classify combustion conditions by analyzing acoustic pressure and high-speed flame imaging from a counter-rotating high-shear swirl injector of a single can combustor fueled by methane. With a constant Reynolds number and varying equivalence ratios, the combustor exhibits both stable and unstable states. Characteristic features are extracted from the data using time series analysis, providing insight into combustion dynamics. The trained supervised machine learning model accurately classifies stable and unstable operations, demonstrating effective prediction of combustion conditions within the studied parameter range.

Self-similar processes were introduced by Kolmogorov in 1940 (Kolmogorov, 1940). The notion garnered We study the fractal structure of language, aiming considerable attention during the late 1960s, thanks to to provide a precise formalism for quantifying the extensive works of Mandelbrot and his peers (Embrechts properties that may have been previously suspected & Maejima, 2000). Broadly speaking, an object is called but not formally shown. We establish that "self-similar" if it is invariant across scales, meaning its statistical language is: (1) self-similar, exhibiting complexities or geometric properties stay consistent irrespective at all levels of granularity, with no particular of the magnification applied to it (see Figure 1). Nature characteristic context length, and (2) longrange and geometry furnish us with many such patterns, such as dependent (LRD), with a Hurst parameter coastlines, snowflakes, the Cantor set and the Kuch curve. of approximately H = 0.70 0.09. Based Despite the distinction, self-similarity is often discussed on these findings, we argue that short-term patterns/dependencies in the context of "fractals," another term popularized by in language, such as in paragraphs, Mandelbrot in his seminal book The Fractal Geometry of mirror the patterns/dependencies over Nature (Mandelbrot, 1982). However, the two concepts are larger scopes, like entire documents.

This paper presents a novel approach for analysing EEG data from drivers in a simulated driving test. We focused on the Hurst exponent, Shannon entropy, and fractal dimension as markers of the nonlinear dynamics of the brain. The results show significant trends: Shannon Entropy and Fractal Dimension exhibit variations during driving condition transitions, whereas the Hurst exponent reflects memory retention portraying learning patterns. These findings suggest that the tools of Non-linear Dynamical (NLD) Theory as indicators of cognitive state and driving memory changes for assessing driver performance, and advancing the understanding of non-linear dynamics of human cognition in the context of driving and beyond. Our study reveals the potential of NLD tools to elucidate brain state and system variances, enabling their integration into current Deep Learning and Machine Learning models. This integration can extend beyond driving applications and be harnessed for cognitive learning, thereby improving overall productivity and accuracy levels.

In the present research, we delve into the intricate realm of electroencephalogram (EEG) data analysis, focusing on sequence-to-sequence prediction of data across 32 EEG channels. The study harmoniously fuses the principles of applied chaos theory and dynamical systems theory to engender a novel feature set, enriching the representational capacity of our deep learning model. The endeavour's cornerstone is a transformer-based sequence-to-sequence architecture, calibrated meticulously to capture the non-linear and high-dimensional temporal dependencies inherent in EEG sequences. Through judicious architecture design, parameter initialisation strategies, and optimisation techniques, we have navigated the intricate balance between computational expediency and predictive performance. Our model stands as a vanguard in EEG data sequence prediction, demonstrating remarkable generalisability and robustness. The findings not only extend our understanding of EEG data dynamics but also unveil a potent analytical framework that can be adapted to diverse temporal sequence prediction tasks in neuroscience and beyond.