Irregular Sets Research Articles

On the completely irregular set of maps with the shadowing property

We prove that the completely irregular set is Baire generic for every non-uniquely ergodic transitive continuous map which satisfies the shadowing property and acts on a compact metric space without isolated points. We also show that, under the previous assumptions, the orbit of every completely irregular point is dense. Afterwards, we analyze the connection between transitivity and the shadowing property, draw a few consequences of their joint action within the family of expansive homeomorphisms, and discuss several examples to test the scope of our results.

Response prediction and probabilistic analysis of the vehicle-ballasted track system considering track irregularity based on long-short term memory neural network

It is necessary to efficiently and accurately predict the dynamic behavior of the train-track system subjected to massive random track geometric excitations for system evaluation and maintenance. This paper proposed an engineering practical approach for the train-track system behavior prediction by integrating the deep learning surrogate model-based long-short term memory (LSTM) and physical model of the train-ballasted track interaction (TBTI) system, and then the generalized probability density evolution method (GPDEM) is introduced to achieve the probabilistic assessment of the TBTI system under limited track irregularities, where the track irregularity probability model (TIPM) coupled with generalized-F discrepancy points sets is employed to compress the track irregularity sets to reduce the number of system responses (as training target) but without loss of the responses completeness, therefore, it can promote the efficiency and accuracy of training model. The prediction performance of the surrogate model is illustrated by comparing the time-frequency domain information, mean and standard deviation of the responses obtained by the physical model and surrogate model. The results indicated the proposed method can accurately predict the system response, especially the low and medium frequency response below 50 Hz, and improve the efficiency of stochastic analysis for the TBTI system by approximately 20 times. Furthermore, the applicability of the proposed method is further examined by investigating the influence of probability levels of track irregularity and train speeds on train performance. Its extended model is also implemented to predict the response of systems induced by any measured track irregularity in real-time, which promotes the generalization capability of the proposed surrogate model.

Emergence Behavior Versus Physical-Like Behavior

In this paper, we study the dynamical complexity of points with emergence behavior but without weak face behavior, especially for points without physical-like behavior in certain dynamical systems such as transitive Anosov systems. We use the tools of saturated sets to prove that these points show strong dynamical complexity in the sense of entropy, density and distributional chaos. We obtain some observations of those results related to irregular sets and level sets. These results strengthen the previous results of [Catsigeras et al., 2019; Hou et al., 2023].

Testing many constraints in possibly irregular models using incomplete U-statistics

Abstract We consider the problem of testing a null hypothesis defined by equality and inequality constraints on a statistical parameter. Testing such hypotheses can be challenging because the number of relevant constraints may be on the same order or even larger than the number of observed samples. Moreover, standard distributional approximations may be invalid due to irregularities in the null hypothesis. We propose a general testing methodology that aims to circumvent these difficulties. The constraints are estimated by incomplete U-statistics, and we derive critical values by Gaussian multiplier bootstrap. We show that the bootstrap approximation of incomplete U-statistics is valid for kernels that we call mixed degenerate when the number of combinations used to compute the incomplete U-statistic is of the same order as the sample size. It follows that our test controls type I error even in irregular settings. Furthermore, the bootstrap approximation covers high-dimensional settings making our testing strategy applicable for problems with many constraints. The methodology is applicable, in particular, when the constraints to be tested are polynomials in U-estimable parameters. As an application, we consider goodness-of-fit tests of latent-tree models for multivariate data.

Metric mean dimension of irregular sets for maps with shadowing

We study the metric mean dimension of Φ-irregular set I Φ ( f ) in dynamical systems with the shadowing property. In particular, we prove that for dynamical systems with the shadowing property containing a chain recurrent class Y, the values of topological entropy together with the values of lower and upper metric mean dimension of the set I Φ ( f ) ∩ B ( Y , ε ) ∩ CR ( f ) are bounded from below by the respective values for class Y.

A Minimized Measurement Scheme for Predicting HbA1c Using Discrete Self-Monitoring Blood Glucose Data Within 4 Weeks for People with Type 2 Diabetes.

Objective: To establish an accurate and robust calculation model for predicting hemoglobin A1c (HbA1c) for people with type 2 diabetes (T2D) by using the fewest discrete blood glucose values according to an irregular data set and propose an appropriate cost-effective and scientific scheme for routine blood glucose monitoring. Methods: By using two data sets obtained from 2017 to 2022, which involved 2432 people with T2D, ∼420,000 irregular blood glucose values, and 10,000 HbA1c values, multiple blood glucose monitoring schemes were designed and compared to find the optimal one. The data were structured and then fitted using a regularized extreme learning machine, and the results were evaluated on the basis of indicators such as mean absolute error (MAE), root mean square error, and the relevance analysis (R) value; the optimal scheme for routine blood glucose monitoring was determined by combining the accuracy and the cost and was compared with previous studies in terms of accuracy and stability. Results: Data fitting results for the chosen scheme: R = 0.8029 (P < 0.001), MAE = 0.3181% (95% confidence interval, 0.2666-0.3695%). Within the last 4 weeks before the prediction of HbA1c, a minimum of only seven fasting and seven postprandial blood glucose values are needed, of which are one fasting and one postprandial blood glucose values per 4 days. Compared with previous studies, the prediction model shows better accuracy and stability (P < 0.05), especially under the great glucose fluctuation group. Conclusion: A minimized calculation model for accurately and robustly predicting HbA1c using discrete self-monitoring of blood glucose data within 4 weeks for people with T2D has been established and provides a new reference for the design of a scheme for blood glucose monitoring. The diabetes care clinic of Peking University First Hospital (Registration Number: ChiCTR2300068139).

Cubic Bézier-Like Triangular Patches for Rainfall Scattered Data Interpolation and Visualization

Scattered data interpolation plays an important role in computer graphics and scientific visualization. This method can be used to interpolate any regular or irregular data sets. For instance, rainfall distribution is an example of an irregular data that usually appear in statistics. In this study, cubic Bézier-Like triangular patches defined on triangular domain is used to interpolate rainfall data sets. From graphical results, we obtain a very smooth surface. Therefore, the proposed scheme can be used to interpolate and provide a good visualization to the given irregular data sets.

Deep Geometric Learning With Monotonicity Constraints for Alzheimer's Disease Progression.

Alzheimer's disease (AD) is a devastating neurodegenerative condition that precedes progressive and irreversible dementia; thus, predicting its progression over time is vital for clinical diagnosis and treatment. For this, numerous studies have implemented structural magnetic resonance imaging (MRI) to model AD progression, focusing on three integral aspects: 1) temporal variability; 2) incomplete observations; and 3) temporal geometric characteristics. However, many pioneer deep learning-based approaches addressing data variability and sparsity have yet to consider inherent geometrical properties sufficiently. These properties are integral to modeling as they correlate with brain region size, thickness, volume, and shape in AD progression. The ordinary differential equation-based geometric modeling method (ODE-RGRU) has recently emerged as a promising strategy for modeling time-series data by intertwining a recurrent neural network (RNN) and an ODE in Riemannian space. Despite its achievements, ODE-RGRU encounters limitations when extrapolating positive definite symmetric matrices from incomplete samples, leading to feature reverse occurrences that are particularly problematic, especially within the clinical facet. Therefore, this study proposes a novel geometric learning approach that models longitudinal MRI biomarkers and cognitive scores by combining three modules: topological space shift, ODE-RGRU, and trajectory estimation. We have also developed a training algorithm that integrates the manifold mapping with monotonicity constraints to reflect measurement transition irreversibility. We verify our proposed method's efficacy by predicting clinical labels and cognitive scores over time in regular and irregular settings. Furthermore, we thoroughly analyze our proposed framework through an ablation study.

Energy Efficient Clustering Mechanism for Malicious Sensor Nodes in IOT Based MWSN

A mobile wireless sensor network (MWSN) is a collection of wireless nodes that may be set up anywhere and at any time without the need for a pre-existing network structure. Network performance is challenging factor due to nature of mobility and misbehaving nodes. Data loss and misbehaving in sensor nodes degrades performance of network. Some circumstances some malevolent sensor nodes predisposed to eradicate capability of network. The goal of this study is to use an irregular set approach to distinguish hostile nodes. Broadcasting information in route entry table identifies malicious node. Each sensor node in network preserves entry table and broadcast record regarding nearby nodes. To estimate broadcasting record metrics are premeditated as data deliverance proportion, throughput, delay, packet drop, fault rate. Mobile nodes with diverse velocity are simulated in NS2 environment which contains variable node clusters. Based broadcasting records of mobile nodes with diverse velocity are considered to create information table. Fine and malicious nodes are recognized based on rules derived in table of irregular set tactic. Shortest path without malicious node is elected to broadcast packets in the environment. Proposed tactic consequences reveal irregular set tactic amplifies network capability data deliverance percentage, throughput and dwindled end-to-end delay in mobile sensors.

The irregular set for maps with almost weak specification property has full metric mean dimension

Let (X,d) be a compact metric space, f:X→X be a continuous transformation with the almost weak specification property and φ:X→R be a continuous function. We consider the set (called the irregular set for φ) of points for which the Birkhoff average of φ does not exist and show that this set is either empty or carries full Bowen upper and lower metric mean dimension.

A Proficient Obtrusion Recognition Clustered Mechanism for Malicious Sensor Nodes in a Mobile Wireless Sensor Network

A collection of wireless nodes that may be installed at any location and at any time without requiring an established network structure is called a mobile wireless sensor network. The problem of network performance arises from the mobility of nodes and their misbehaviour. Network performance is negatively impacted by data loss and sensor node misbehaviour. In certain cases, there are malicious sensor nodes that are designed to destroy the network’s capacity. This work aims to identify hostile nodes using an irregular set technique. The route entry table’s broadcasting metadata helps identify rogue nodes. Every sensor node in the network broadcasts information about adjacent nodes and maintains a sentry table. Premeditated data delivery proportion, throughput, delay, packet drop, and fault rate are used to estimate broadcasting record parameters. In the NS2 environment, mobile nodes with varying velocities are simulated. To generate an information table, mobile nodes with varying speeds are examined based on their broadcasting records. On the basis of guidelines taken from the irregular set tactic table, good and bad nodes are distinguished. Packets are disseminated along the shortest path that doesn’t contain any malicious nodes. The results of the proposed technique show that an irregular set tactic increases throughput, network capability, data delivery percentage, and end-to-end delay reduction in mobile sensors.

Performance of machine learning methods for modeling reservoir management based on irregular daily data sets: a case study of Zit Emba dam

Performance of machine learning methods for modeling reservoir management based on irregular daily data sets: a case study of Zit Emba dam

Nonlinear grey Bernoulli model with physics-preserving Cusum operator

The Cusum (cumulative sum) operator is a fundamental prerequisite for the nonlinear grey Bernoulli model. Traditionally, it is believed to visually identify the underlying dynamic pattern of the original time series. This paper presents the misconceptions concerning the Cusum operation and the over-optimization of the initial condition in the classical nonlinear grey Bernoulli model, both of which inspire the proposal of a physics-preserving Cusum operator. Under a state space framework, separable nonlinear least squares and nonlinear least squares are formulated to generate simultaneous estimates of structural parameters and initial condition. By combining the Cusum operators and parameter estimation methods, four modeling paradigms are generated and comprehensively compared. The simulation results show that (i) the physics-preserving Cusum outperforms the traditional Cusum and (ii) nonlinear least squares outperforms separable nonlinear least squares, especially in irregular sampling settings with large time intervals and high noise levels. Finally, the proposed approach is used to identify the underlying dynamics from short-term traffic flow data, and the results validate its effectiveness.

Continuous time recurrent neural networks: Overview and benchmarking at forecasting blood glucose in the intensive care unit

ObjectiveBlood glucose measurements in the intensive care unit (ICU) are typically made at irregular intervals. This presents a challenge in choice of forecasting model. This article gives an overview of continuous time autoregressive recurrent neural networks (CTRNNs) and evaluates how they compare to autoregressive gradient boosted trees (GBT) in forecasting blood glucose in the ICU. MethodsContinuous time autoregressive recurrent neural networks (CTRNNs) are a deep learning model that account for irregular observations through incorporating continuous evolution of the hidden states between observations. This is achieved using a neural ordinary differential equation (ODE) or neural flow layer. In this manuscript, we give an overview of these models, including the varying architectures that have been proposed to account for issues such as ongoing medical interventions. Further, we demonstrate the application of these models to probabilistic forecasting of blood glucose in a critical care setting using electronic medical record and simulated data and compare with GBT and linear models. ResultsThe experiments confirm that addition of a neural ODE or neural flow layer generally improves the performance of autoregressive recurrent neural networks in the irregular measurement setting. However, several CTRNN architecture are outperformed by a GBT model (Catboost), with only a long short-term memory (LSTM) and neural ODE based architecture (ODE-LSTM) achieving comparable performance on probabilistic forecasting metrics such as the continuous ranked probability score (ODE-LSTM: 0.118 ± 0.001; Catboost: 0.118 ± 0.001), ignorance score (0.152 ± 0.008; 0.149 ± 0.002) and interval score (175 ± 1; 176 ± 1). ConclusionThe application of deep learning methods for forecasting in situations with irregularly measured time series such as blood glucose shows promise. However, appropriate benchmarking by methods such as GBT approaches (plus feature transformation) are key in highlighting whether novel methodologies are truly state of the art in tabular data settings.

Lyapunov irregular set of Banach cocycles

Let be a topologically mixing subshift of finite type and let be a Hölder continuous map where is the set of invertible elements of a Banach ring B. We prove that either all ergodic measures have the same maximal Lyapunov exponent, or the irregular set is residual in and has full upper capacity topological entropy of . We also prove for generic fiber-bunched cocycles, similar statements can be strengthened by considering the Bowen topological entropy.

Sparse Non-Uniform Linear Array-Based Propagator Method for Direction of Arrival Estimation

A novel approach that does not require the number of sources as a priori is proposed to estimate the direction of arrival (DOA) based on a sparse non-uniform linear antenna array. To ensure the identifiability of the DOA, a specific configuration scheme of sparse array is designed. Based on this specific sparse array, firstly the fourth-order cumulant (FOC) is adopted to eliminate the impact imposed by Gaussian noise. Secondly, to circumvent eigenvalue decomposition or singular value decomposition, a propagator is constructed by using a Hermitian FOC matrix and a hyperparameter. Finally, a projection onto an irregular Toeplitz set is proposed to further improve estimation accuracy.

On multifractal analysis and large deviations of singular hyperbolic attractors

In this paper we study the multifractal analysis and large derivations for singular hyperbolic attractors, including the geometric Lorenz attractors. For each singular hyperbolic homoclinic class whose periodic orbits are all homoclinically related and such that the space of ergodic probability measures is connected, we prove that: (i) level sets associated to continuous observables are dense in the homoclinic class and satisfy a variational principle; (ii) irregular sets are either empty or are Baire generic and carry full topological entropy. The assumptions are satisfied by C 1-generic singular hyperbolic attractors and Cr -generic geometric Lorenz attractors . Finally we prove level-2 large deviations bounds for weak Gibbs measures, which provide a large deviations principle in the special case of Gibbs measures. The main technique we apply is the horseshoe approximation property.

And empirical measures, the irregular set and entropy

AbstractFor integers a and $b\geq 2$ , let $T_a$ and $T_b$ be multiplication by a and b on $\mathbb {T}=\mathbb {R}/\mathbb {Z}$ . The action on $\mathbb {T}$ by $T_a$ and $T_b$ is called $\times a,\times b$ action and it is known that, if a and b are multiplicatively independent, then the only $\times a,\times b$ invariant and ergodic measure with positive entropy of $T_a$ or $T_b$ is the Lebesgue measure. However, it is not known whether there exists a non-trivial $\times a,\times b$ invariant and ergodic measure. In this paper, we study the empirical measures of $x\in \mathbb {T}$ with respect to the $\times a,\times b$ action and show that the set of x such that the empirical measures of x do not converge to any measure has Hausdorff dimension one and the set of x such that the empirical measures can approach a non-trivial $\times a,\times b$ invariant measure has Hausdorff dimension zero. Furthermore, we obtain some equidistribution result about the $\times a,\times b$ orbit of x in the complement of a set of Hausdorff dimension zero.

Social welfare relations and irregular sets

Social welfare relations satisfying Pareto and equity principles on infinite utility streams have revealed a non-constructive nature, specifically by showing that in general they imply the existence of non-Ramsey sets and non-Lebesgue measurable sets. In [4, Problem 11.14], the authors ask whether such a connection holds with non-Baire sets as well. In this paper we answer such a question showing that several versions of Pareto principles acting on different utility domains imply the existence of non-Baire sets. Furthermore, we analyze in more details the needed fragments of AC and we start a systematic investigation of a social welfare diagram in a similar fashion done in the past decades concerning cardinal invariants and regularity properties of the reals. In doing that we use tools from forcing theory, such as specific tree-forcings (in particular variants of Silver and Mathias forcings) and Shelah's amalgamation.

Floods and Heavy Precipitation at the Global Scale: 100‐Year Analysis and 180‐Year Reconstruction

AbstractFloods and heavy precipitation have disruptive impacts worldwide, but their historical variability remains only partially understood at the global scale. This article aims at reducing this knowledge gap by jointly analyzing seasonal maxima of streamflow and precipitation at more than 3,000 stations over a 100‐year period. The analysis is based on Hidden Climate Indices (HCIs). Like standard climate indices (e.g., Nino 3.4, NAO), HCIs are used as covariates explaining the temporal variability of data, but unlike them, HCIs are estimated from the data. In this work, a distinction is made between common HCIs, that affect both heavy precipitation and floods, and specific HCIs, that exclusively affect one or the other. Overall, HCIs do not show noticeable autocorrelation, but some are affected by noticeable trends. In particular, strong and wide‐ranging trends are identified in precipitation‐specific HCIs, while trends affecting flood‐specific HCIs are weaker and have more localized effects. A probabilistic model is then derived to link HCIs and large‐scale atmospheric variables (pressure, wind, temperature) and to reconstruct HCIs since 1836 using the 20CRv3 reanalysis. In turn this allows estimating the probability of occurrence of floods and heavy precipitation at the global scale. This 180‐year reconstruction highlights flood hot‐spots and hot‐moments in the distant past, well before the establishment of perennial monitoring networks. The approach presented in this study is generic and paves the way for an improved characterization of historical variability by making a better use of long but highly irregular station data sets.