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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D1: Probability and Statistics".

Deadline for manuscript submissions: closed (28 February 2026) | Viewed by 15443

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Dr. Esam Mahdi

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School of Mathematics and Statistics, Carleton University, Ottawa K1S 5B6, Canada
Interests: time series analysis; spatio-temporal analysis; quantile regression; functional and structural MRI; interdisciplinary research
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Special Issue Information

Dear Colleagues,

Time series analysis, with its diverse applications spanning from finance to climate science and from medicine to marketing, plays a pivotal role in uncovering patterns, forecasting trends, and driving decision-making processes in various domains. We aim to explore innovative dimensions within this domain, delving into theoretical frameworks, potential applications, and the seamless integration with machine learning, neural networks (NNs), as well as recent breakthroughs in multivariate and high-dimensional analyses across various domains.

Dr. Esam Mahdi
Guest Editor

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Keywords

time series models
ARMA model
ARCH model
leverage
long memory
nonlinear time series
forecasting methods
portmanteau tests
autocorrelation function
time-varying parameters
machine learning models
neural networks
temporal data mining
cryptocurrency analysis

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Published Papers (5 papers)

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Research

31 pages, 4763 KB

Open AccessArticle

What Drives Multi-Chain Crypto Forecasting: Model Choice, Feature Selection, and Transferability

by Mingxing Wang, Yufeng Xiao, Pavel Braslavski and Dmitry I. Ignatov

Mathematics 2026, 14(8), 1286; https://doi.org/10.3390/math14081286 - 13 Apr 2026

Viewed by 1656

Abstract

Increasingly shaped by heterogeneous on-chain activity rather than a single shared market process, this study investigates 7-day-ahead forecasting using 147 market and on-chain indicators across eight major blockchain ecosystems from October 2023 to April 2025. We benchmark statistical, deep-learning, and foundation-model baselines under [...] Read more.

0.0428

) in a univariate setting without additional optimized covariates, while TFT remains slightly weaker even under its best feature-input configuration (MAPE:

0.0435

;

p = 0.9359

versus TiRex), suggesting a persistent practical advantage for TiRex. Importantly, TiRex’s zero-shot nature confers a substantial efficiency edge: by bypassing feature selection, it delivers comparable accuracy at a fraction of the computational cost. At the same time, feature selection materially affects many model families, with Boruta chosen in roughly 71.7% of best configurations. Taken together, the evidence supports a selective-feature principle: robust forecasting depends on validated, chain-specific features rather than larger feature sets. Feature-importance and overlap analyses further indicate a mixed structure of transferability, where broad market proxies provide baseline context while chain-specific variables drive marginal gains. Overall, this study highlights that effective multi-chain forecasting is primarily a feature selection problem under statistical uncertainty, while also showing that zero-shot designs like TiRex can achieve state-of-the-art accuracy with unmatched efficiency, offering practical implications for building leaner, more robust trading systems. Full article

(This article belongs to the Special Issue Applications of Time Series Analysis)

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34 pages, 453 KB

Open AccessArticle

Parametric Estimation of a Merton Model Using SOS Flows and Riemannian Optimization

by Luca Di Persio and Paul Bastin

Mathematics 2026, 14(7), 1217; https://doi.org/10.3390/math14071217 - 4 Apr 2026

Viewed by 869

Abstract

We consider the problem of Bayesian parameter inference in the Merton structural credit risk model, where the posterior is induced by a jump-diffusion likelihood and the marginal evidence is not available in closed form. To approximate this posterior, we construct a variational family based on triangular sum-of-squares (SOS) polynomial flows, in which each component map is monotone by construction: its diagonal derivative is a positive definite quadratic form on a monomial basis, yielding a closed-form log-Jacobian and explicit gradients with respect to all flow parameters. The symmetric positive definite matrices parametrizing the flow are optimized by intrinsic Riemannian gradient ascent on the positive definite cone equipped with the affine-invariant metric, which preserves feasibility at every iterate without projection. We show that the rank-one Jacobian gradients produced by the SOS structure have unit norm in the affine-invariant metric, establishing a direct algebraic coupling between the transport family and the optimization geometry and implying a universal 1-Lipschitz bound for the log-Jacobian along geodesics. On the likelihood side, we derive exact score identities for all five structural parameters of the Merton model—drift, volatility, jump intensity, jump mean, and jump volatility—through both the Poisson log-normal mixture and the Fourier inversion representations. Strictly positive parameters are handled via exponential reparametrization, and the resulting gradients propagate end-to-end through the flow. We establish uniform truncation bounds on compact parameter sets for the infinite mixture and its associated score series, providing rigorous control over the finite approximations used in practice. The base distribution is chosen to be uniform on

{[0, 1]}^{5}

, whose bounded support ensures uniform control of the monomial basis and stabilizes the polynomial calculus. These ingredients are assembled into a fully explicit modified ELBO with implementable gradients, combining Euclidean updates for vector parameters and intrinsic manifold updates for matrix parameters. Full article

(This article belongs to the Special Issue Applications of Time Series Analysis)

19 pages, 1281 KB

Open AccessFeature PaperArticle

A Novel Hybrid Approach Using an Attention-Based Transformer + GRU Model for Predicting Cryptocurrency Prices

by Esam Mahdi, Carlos Martin-Barreiro and Xavier Cabezas

Mathematics 2025, 13(9), 1484; https://doi.org/10.3390/math13091484 - 30 Apr 2025

Cited by 12 | Viewed by 8835

Abstract

In this article, we introduce a novel deep learning hybrid model that integrates attention Transformer and gated recurrent unit (GRU) architectures to improve the accuracy of cryptocurrency price predictions. By combining the Transformer’s strength in capturing long-range patterns with GRU’s ability to model short-term and sequential trends, the hybrid model provides a well-rounded approach to time series forecasting. We apply the model to predict the daily closing prices of Bitcoin and Ethereum based on historical data that include past prices, trading volumes, and the Fear and Greed Index. We evaluate the performance of our proposed model by comparing it with four other machine learning models, two are non-sequential feedforward models: radial basis function network (RBFN) and general regression neural network (GRNN), and two are bidirectional sequential memory-based models: bidirectional long short-term memory (BiLSTM) and bidirectional gated recurrent unit (BiGRU). The model’s performance is assessed using several metrics, including mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE), along with statistical validation through the non-parametric Friedman test followed by a post hoc Wilcoxon signed-rank test. Results demonstrate that our hybrid model consistently achieves superior accuracy, highlighting its effectiveness for financial prediction tasks. These findings provide valuable insights for enhancing real-time decision making in cryptocurrency markets and support the growing use of hybrid deep learning models in financial analytics. Full article

(This article belongs to the Special Issue Applications of Time Series Analysis)

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25 pages, 1580 KB

Open AccessArticle

Online Monitoring of Structural Change Points Based on Ratio-Type Statistics

by Wenjie Li, Hao Jin and Minghua Wu

Mathematics 2025, 13(8), 1315; https://doi.org/10.3390/math13081315 - 17 Apr 2025

Cited by 1 | Viewed by 1023

Abstract

For scenarios where the type of structural break in a time series is unknown, this paper proposes a modified ratio-type test statistic to enable effective online monitoring of structural breaks, while circumventing the estimation of long-term variance. Under specific assumptions, we rigorously derive the asymptotic distribution of the test statistic under the null hypothesis and establish its consistency under the alternative hypothesis. In cases where both variance and mean breaks coexist, we introduce a refined mixed-break monitoring procedure based on the consistent estimation of breakpoints. The proposed method first provides consistent estimations of the mean change points and variance change points separately; then, mean and variance removal are performed on original data; finally, the previously removed trend is added back. Compared to traditional monitoring methods, which have to use two test statistics, this method requires only one to simultaneously monitor both types of change points, resulting in a significantly simplified monitoring process. This approach effectively reduces mutual interference between the two types of breaks, thereby enhancing the power of the test. Extensive numerical simulations confirm that this method can accurately detect the presence of structural breaks and reliably identify their types. Finally, case studies are provided to demonstrate the efficacy and practical applicability of the proposed method. Full article

(This article belongs to the Special Issue Applications of Time Series Analysis)

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23 pages, 6097 KB

Open AccessArticle

Decomposition-Aware Framework for Probabilistic and Flexible Time Series Forecasting in Aerospace Electronic Systems

by Yuanhong Mao, Xin Hu, Yulang Xu, Yilin Zhang, Yunan Li, Zixiang Lu and Qiguang Miao

Mathematics 2025, 13(2), 262; https://doi.org/10.3390/math13020262 - 14 Jan 2025

Cited by 2 | Viewed by 1380

Abstract

Degradation prediction for aerospace electronic systems plays a crucial role in maintenance work. This paper proposes a concise and efficient framework for multivariate time series forecasting that is capable of handling diverse sequence representations through a Channel-Independent (CI) strategy. This framework integrates a decomposition-aware layer to effectively separate and fuse global trends and local variations and a temporal attention module to capture temporal dependencies dynamically. This design enables the model to process multiple distinct sequences independently while maintaining the flexibility to learn shared patterns across channels. Additionally, the framework incorporates probabilistic distribution forecasting using likelihood functions, addressing the dynamic variations and uncertainty in time series data. The experimental results on multiple real-world datasets validate the framework’s effectiveness, demonstrating its robustness and adaptability in handling diverse sequences across various application scenarios. Full article

(This article belongs to the Special Issue Applications of Time Series Analysis)

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