ABSTRACT Most deep learning techniques are designed for point‐valued data, while interval‐valued data contain richer information over the same time period, especially during extreme events. To leverage this, we propose DKformer, a novel transformer architecture that models interval data as an inseparable set using the loss function. By combining transformers and loss, DKformer learns complex nonstationary and nonlinear patterns in interval data while accounting for uncertainties and dependencies. Compared to existing interval and point‐based benchmark models, DKformer significantly enhances crude oil price forecast accuracy. Remarkably, it demonstrates robust out‐of‐sample prediction performance across various forecast horizons. Furthermore, the effectiveness of DKformer is preserved even under extreme conditions, including Black Swan events. Our work highlights the potential for DKformer to enhance decision‐making in energy and finance, where robustness to uncertainty and the ability to handle extreme events hold paramount importance.

The skill to understand mathematical concepts is closely related to the comprehensive and functional mastery of mathematical ideas. In learning, an important aspect of material comprehension apart from the skill of understanding mathematical concepts is students' learning independence. This study aims to examine the relationship between mathematical concept comprehension skills and students' learning independence. A correlational method with a quantitative approach was employed in this study. The research was conducted at a junior secondary school in Karawang Regency, namely SMPN 1 Cibuaya, involving 35 students from class VIII-F as the research subjects. The research instruments used were a mathematical concept comprehension skill test and a learning independence questionnaire. The comprehension test consisted of five essay questions, while the questionnaire comprised 28 items. The study involved four stages of data analysis. Specifically, for the learning independence data, the ordinal data were first transformed into interval data using the Successive Interval Method (MSI). The results showed that the correlation test, conducted to determine the strength of the relationship between the two variables, produced a coefficient of 0.606. According to the interval coefficient table, this value falls into the "very strong" category, as it is close to 1. This indicates a positive correlation between the two variables. The influence of mathematical concept comprehension skills and learning independence was found to be 36.72%, with the remaining percentage influenced by other factors. In other words, the higher the students’ learning independence, the higher their mathematical concept comprehension skills and vice versa.

This research addresses the increasing need for innovative digital learning resources that support mathematical understanding in vocational higher education. As learning environments continue shifting toward technology-enhanced formats, e-modules offer a medium for delivering structured, interactive, and student-centered instructional materials. The purpose of this research was to develop a flipbook-based e-module for Mathematics and Statistics course in the Department of Nautical Studies in Politeknik Maririm Negeri Indonesia, and determine its feasibility, quality, and student acceptance. This research used Research and Development (R&D) method and ADDIE (Analysis, Design, Development, Implementation, and Evaluation) model to develop the e-module. But in this research was limited to the Analysis, Design, and Development only. Data were collected through questionnaires distributed to material experts, media experts, and a number of students from Department of Nautical Studies. Quantitative data were converted into interval data using Likert scale. The validation results indicated that the Mathematics and Statistics e-module achieved a highly feasible category. The material expert validation reached 91.25%, indicating excellent content accuracy and pedagogical suitability, while the media expert validation reached 98.75%, indicating the high quality of interface design, navigation, and usability. The responses from students showed the high level of satisfaction and ease of use, with an average score of 84.96%. These results conclude that the developed flipbook-based e-module is a viable and effective interactive learning tool that enhances students’ conceptual understanding of mathematics and statistics, delivers more interesting learning experiences, and supports independent learning.

When planning statistical studies, it is often necessary to determine the required sample size. The rules for finding the required sample size are quite widely discussed in the works of specialists in various fields of science and practice — engineering and technology, economics and management, medicine, sociology, etc. However, usually only individual specific statements of the problem are considered, and certain recipes for calculating the required quantity are given without justification. sample size, and often erroneous. The article is devoted to the strict formulation of methods for calculating the required sample size, developed to date, as well as the justification of the corresponding algorithms from the standpoint of modern applied mathematical statistics. Four approaches are considered. In the first of them, the sample size is determined by the available resources (material, financial, personnel, etc.) that can be used to obtain sample elements (measurements, observations, analyses, experiments, tests, surveys). In the second, problems of estimating the theoretical characteristics of a random variable are considered. The required sample size is determined by the specified estimation accuracy, by which we mean the half-width of the confidence interval. The third approach is based on the theory of statistical hypothesis testing. We are talking about choosing between the null and alternative hypotheses, based on the level of significance and power of the criterion. One-sided alternatives are discussed in detail. The fourth is developed within the framework of interval data statistics. It developed a general approach based on the principle of equalizing statistical and metrological errors. Based on the known accuracy of the initial measurements, we determine the rational sample size, which is necessary from the point of view under consideration. As basic examples, the rules for calculating the required sample size when studying probability values and mathematical expectations are considered. Limit theorems of probability theory and mathematical statistics are used as the basis of the mathematical apparatus. In further research, it is advisable to study the rate of convergence in the obtained asymptotic expressions and consider finite sample sizes.

Interpretation of laboratory test results is a comparative process that requires reference data. Such data are derived for each analyte separately, without accounting for, the interrelationships among analytes. Physicians use test panels containing multiple analytes to enhance clinical significance and improve the accuracy of decision-making. However, current interpretation practices apply reference intervals and reference change values in a univariate manner-that is, each analyte in the panel is interpreted independently and no reference data are available to interpret the panel as a whole. Yet, metabolism is a network of biomolecules, each of which is related to others. Therefore, a multivariate approach-based on the correlations among biomolecules-can provide a more informative reference than univariate approaches and can be used more effectively in the interpretation of laboratory data. This concept can be summarized by a motto: Combine single tests into meaningful groups, but interpret the group as a single clinical entity. In this opinion paper, we present a practical approach for obtaining reference data for both reference intervals and reference change values to interpret laboratory test panels composed of related analytes.

Interval time series (ITS) analysis has important significance econometric analysis, as it contains information about the range of change and the level or trend of economic processes. More importantly, the rich information of interval data can be used for more accurate quantitative estimation and inference. Considering the possible nonlinear characteristics of ITS data, this article introduces a class of smooth transition interval autoregressive (STIAR) models, which includes the logistic STIAR (LSTIAR) model and the exponential STIAR (ESTIAR) model as special cases. The minimum distance estimation method is proposed to estimate the model parameters and the asymptotic theory of the estimator is established. The nonlinearity test of the model is also well solved. Finally, some numerical simulation results and a practical data example are given.

Accurate estimation of peak water demand remains a challenge in institutional settings with floating populations, particularly in regions where design standards may require revision and validation to accommodate evolving consumption patterns. This pilot study assesses the potential of a probabilistic adaptation of Hunter’s method, calibrated through high-resolution flow and pressure monitoring, for peak flow estimation in five academic buildings in Loja, Ecuador. Over 62 days, usage parameters, duration (t), frequency (i), and peak period (h), were disaggregated from 1 min interval data to derive building-specific binomial probability distributions. The adapted model was compared against three benchmarks: the Neyman–Scott Rectangular Pulse Model (NSRPM), the Water Demand Calculator (WDC), and Ecuador’s Hydro-Sanitary Standard (NHE 2011). Results indicate the proposed approach estimates peak flows within −11.6% to +20.0% of observed values, outperforming WDC (systematic underestimation up to −81.5%) and NHE 2011 (average underestimation of −31.3%), though NSRPM achieved high accuracy for one site (−1.1%) with high inter-building variability (average −38.4%). While limited to a small sample in a single climatic context, this pilot demonstrates that temporal disaggregation of stochastic demand enables a context-sensitive recalibration of Hunter’s method, offering a methodologically sound basis for future validation across diverse institutional settings in the Global South.

NAION is a severe form of vision loss and the onset of anamorphic, painless vision loss, and in most cases, it is linked to risk factors that are systemic including diabetes and hypertension. The objective of this study was to evaluate the risk of NAION as a result of semaglutide use in a systematic review of literature that supports the claim by examining the incidence, the risk factors and potential mechanisms of the relationship between the two. A literature review of the publications on the topic published in January 2024-June 2025 was carried out. The included studies were those that evaluate the risk of NAION in patients under semaglutide treatment, which comprise studies in the cohort category, clinical trials, and case reports. Hazard ratios (HR) data, 95% confidence interval (CI) data and incidence rate data were extracted and analyzed. Five studies were analyzed and reported different findings on the correlation between semaglutide and NAION. There was a report of substantial increase in the hazard ratio (HR of 2.58 up to 4.28) by certain studies whereas there was no significant risk increase in others. Most studies had confidence intervals that were wide indicating variation in results. Small Bruchs membrane opening (BMO) and crowded optic discs were risk factors that were found to predispose NAION in semaglutide users. Although risk of NAION in people taking semaglutide seems to be low in the general population, there are some subgroups which may face this risk especially those that have predisposing ocular risk factors. The general therapeutic advantages of semaglutide in the treatment of type 2 diabetes and obesity offset the risks that the drug poses to the eyes of the majority of patients..

Introduction: Prolonged QT interval has been associated with first stroke of any type in the general population. Whether QT prolongation predicts recurrent stroke is unknown. Hypothesis: We hypothesized that QT prolongation would be associated with increased risk of recurrent stroke in individuals who recently suffered a cryptogenic stroke. Methods: The Atrial Cardiopathy and Antithrombotic Drugs in Prevention After Cryptogenic Stroke (ARCADIA) trial randomized individuals with cryptogenic stroke and atrial cardiopathy to apixaban versus aspirin to assess the prevention of recurrent strokes. We excluded participants with missing QT interval data and those who had ventricular conduction defects or pacemakers. The baseline electrocardiogram (ECG) was the first technically adequate ECG to be done after stroke onset. QT interval was corrected for heart rate (QTc) using a cohort-specific correction formula, as well as the Framingham, Hodges, Bazett, and Fridericia formulae. For each QTc variable, prolonged QTc was defined as >450 ms in men or >460 ms in women. Cox proportional hazards models were used to assess the association between QTc and recurrent stroke. Models were adjusted for randomization arm, age, sex, race, smoking, hypertension, diabetes, prior stroke, and N-terminal pro-B-type natriuretic peptide. Results: Among 881 participants, 139 (15.8%) had a prolonged cohort-specific QTc. Over a mean of 1.8 years, 62 recurrent strokes of any type occurred (crude rate 7.0%, annualized rate 3.9% per year). After multivariable adjustment, prolongation of cohort-specific QTc was associated with decreased risk of stroke (hazard ratio [95% confidence interval] = 0.72 [0.54-0.95] per standard deviation and 0.16 [0.04-0.64] for prolonged vs. normal QTc). These findings were consistent across all methods of QT correction ( Table) . Accounting for QRS duration, incident atrial fibrillation, and the competing risk of death did not change the results. Conclusions: In a population with recent cryptogenic stroke and atrial cardiopathy, QTc prolongation was associated with a lower risk of recurrent stroke. These findings contrast with the association observed between QTc prolongation and stroke in the general population, which may reflect selection bias or bias from ECGs obtained in the acute phase of stroke. If replicated, these findings suggest that repolarization markers like QTc may play different roles in stroke risk stratification for first versus recurrent stroke.

Background: Acute stress has been linked to increased arrhythmic activity, but little is known about how mental disorders modify this relationship in patients with coronary heart disease (CHD). Increased arrhythmia burden substantially elevates the risk of sudden cardiac death; thus, examining the role of mental disorders in mental stress-induced arrhythmia is essential for informing targeted prevention strategies in this high-risk population. We tested the hypothesis that in individuals with recent myocardial infarction (MI) and in normal sinus rhythm, mental disorders are associated with an increased risk of ectopic atrial and ventricular arrhythmias during a standardized speech stressor challenge. Methods: We studied a cohort of 304 post-MI participants with structured clinical interviews to diagnose major depression and post-traumatic stress disorder (PTSD). We measured premature atrial (PAC) and ventricular contractions (PVC) from Holter data in 5-minute intervals before, during, and after a standardized laboratory-based 5-minute speech stressor. We examined the associations between mental disorders and changes in PAC/PVC counts after stress using multivariable logistic regression. Results: The mean age 52 (SD ±7) years, 56% were women, and 52% were black (Table 1). Among participants with MI, depression was associated with greater PAC increases during stress (OR=3.16, 95% CI: 1.22–7.58, p=0.012); this remained significant after adjustment (OR=4.04, 95% CI: 1.46–10.70, p=0.005). PTSD was associated with higher PAC counts during recovery versus rest (OR=4.73, 95% CI: 1.42–13.80, p=0.005), persisting after adjustment (OR=11.80, 95% CI: 2.93–47.50, p < 0.001). Depression associated with increased PVCs during recovery compared to rest (OR=2.27, 95% CI: 0.98–4.93, p=0.046). However, statistical significance was lost after adjusting for demographic and traditional risk factors. Conclusions: In young and middle-aged individuals with prior MI, PTSD and depression were associated with an increased burden of arrhythmia under mental stress. The associations were stronger for atrial arrhythmias than ventricular arrhythmias.

This study investigates the comparative efficacy of Summated Rating Scales (SRS) and traditionalordinal scales (raw Likert-type responses) in measuring critical thinking skills among vocationalstudents, employing Item Response Theory (IRT) to evaluate their psychometric properties.Addressing the limitations of ordinal scales notably inconsistent intervals between responsecategories the research adopts a descriptive quantitative methodology involving 269 students fromstate vocational high schools in Yogyakarta, Indonesia. Data were collected using a five-pointLikert scale instrument, validated for content (Aiken’s V = 0.94), and analyzed through two IRTframeworks: Polytomous IRT for unscaled ordinal data and Continuous Response Model (CRM)IRT for SRS-transformed interval data. Key findings reveal that SRS enhances measurementprecision by normalizing response distributions into proportional intervals (e.g., recalibrated scores:0.00, 0.73, 1.46, 2.07, 2.84), thereby resolving issues of unequal category spacing inherent toordinal scales. Polytomous IRT demonstrated robust item fit (e.g., Partial Credit Model fit for 5/6items) and strong difficulty parameter invariance (r = 0.84), yet exhibited instability in abilityestimates (r = 0.37) due to extreme response patterns. Conversely, CRM IRT applied to scaleddata produced stable ability estimates (r = 0.46) and eliminated infinite values in MaximumLikelihood Estimation, underscoring its superiority in handling continuous metrics. However, ordinalscales retained higher consistency in difficulty calibration across subgroups. The study concludesthat integrating SRS with CRM IRT offers a refined approach for critical thinking assessments,balancing precision and fairness, while ordinal scales remain pragmatic for contexts prioritizingsimplicity. These insights advocate for the adoption of advanced scaling techniques in vocationaleducation to improve the validity of competency evaluations, with recommendations for futureresearch to explore hybrid models and longitudinal applications.

The γ-ray emitting binary system exhibits periodic emission across the electromagnetic spectrum, extending from radio frequencies up to the very-high-energy regime. The most prominent features are three periods with values P_1 = 26.5 d, P_2 = 26.9 d, and P_ ̊m long = 4.6 years. Occasionally, a fourth period of 26.7 d is also detected. Mathematically, these four periods are interrelated via the interference pattern of a beating. Competing scenarios that seek to determine which of these periods are physical and which are secondary are under debate. In addition, the detection of a fifth period, P_3 = 26.3 d, was recently claimed. Our aim is to determine which of these periods are intrinsic (i.e., whether they are likely to be related to physical processes) and which of these are secondary (i.e., resulting from interference). We chose to avoid any assumption concerning the physical scenario and restricted our analysis to the phenomenology of the radio emission variability. We selected intervals from archival radio data and applied the generalized Lomb-Scargle periodogram. We fit the observational data to generate synthetic datasets that only contain specific signals in the form of pure sine waves. We analyzed these synthetic data to assess the impact of these signals and their interference on the light curves and the periodogram. The two-peaked profile, consisting of P_1 and P_2, was detected in the periodogram of the actual data for intervals that are significantly shorter than P_ long, provided that these intervals contain the minimum of the long-term modulation. The characteristics of the observational data and their periodogram could only be reproduced with synthetic data if these explicitly included all of the three periods, P_1, P_2, and P_ long, with the residuals being limited by noise. We have found, for the first time, that all three periods, P_1, P_2, and P_̊m long, could, in fact, correspond to physically real processes occurring in . This should be considered when modeling the physical processes in this binary system.

Several methods have been proposed for process monitoring with interval-valued data. We examine these methods and propose a simulation-based approach that is much easier to implement and has a much clearer interpretation. We consider both Phase I and Phase II implementation of interval data control charts and assess the loss of information from the use of interval data of various widths.

IntroductionAI is radically changing workplace ecosystems in the midst of the Fourth Industrial Revolution, making human-machine collaboration a need for organizations. The ambidextrous processes by which AI simultaneously encourages and constrains inventive behaviors need systematic examination, even though employee innovation is still essential for maintaining competitive advantage. In order to understand the paradoxical consequences of AI, this study builds a dual-path moderated mediation model based on the Job Demands-Resources (JD-R) paradigm.MethodsUsing a two-wave longitudinal design with a 3-month interval and multi-source data from 250 experts in China, we combined survey measurements with quasi-experimental manipulations. The following findings were obtained using structural equation modeling (SEM) and bootstrapping.Results(1) AI technology adoption is a job resource that increases Felt Obligation for Constructive Change (FOCC), but it also acts as a job demand that inhibits innovation by creating a sense of job insecurity; (2) task crafting is a crucial boundary condition that amplifies the positive mediation path while attenuating the negative pathway.DiscussionBased on the aforementioned findings, this study highlights the importance of considering employees' psychological states and behavioral changes while fostering technological innovation, exposing the intricacy of artificial intelligence technology in HRM from both a subjective and objective standpoint. Job insecurity is a possible drawback of technology use, hence businesses should take appropriate steps to lessen employee uneasiness while using new technologies. Felt Obligation for Constructive Change, on the other hand, is a crucial strategy for encouraging creative behavior. To do this, managers must investigate and enhance employees' intrinsic motivation for their everyday tasks and foster a culture of creativity. Task crafting, as an effective self-management and driving factor, is also very important to reduce the negative effects of technology adoption and increase its positive effects. For this reason, businesses should support and encourage employees to improve their autonomy and flexibility, iterate on their work methods, and stimulate their ability to innovate. This will not only help employees develop their own skills but also give businesses a competitive edge and continuous innovation motivation.

In this study, we propose a novel conceptualization and methodology for extended intervals, defining them as subsets of the Cartesian product, where R × Z 2 , where Z 2 = { 0 , 1 } represents the direction of interval traversal. This new framework simplifies the representation and computation of extended intervals by introducing a unique metric d γ derived from a function γ ( t ) , which we prove to be an adapted measure. This metric not only facilitates efficient calculation but also demonstrates excellent properties for practical implementation, as evidenced by its integration into the R software. Our approach significantly extends traditional interval analysis by incorporating directional information, which is crucial for accurately modeling and analyzing variability in interval data. Furthermore, we explore the application of this framework to extended interval-valued ARMA (AutoRegressive Moving Average) time series, providing a comprehensive theoretical foundation along with the proof of the Wold decomposition theorem for stationary extended interval-valued time series. The practical utility of our method is demonstrated through numerical simulations and empirical analysis of some financial markets time series. Our results show that extended intervals offer a robust and intuitive means to capture both the direction and magnitude of variations, making them highly valuable for time series forecasting and analysis in finance and other fields.

Beta regression models are a class of models used frequently to model response variables in the interval (0, 1). Although there are articles in which these models are used to model clustered and longitudinal data, the prediction of random effects is limited, and residual analysis has not been implemented. In this paper, a random intercept beta regression model is proposed for the complete analysis of this type of data structure. We proposed some types of residuals and formulate a methodology to obtain the best prediction of random effects. This model is developed through the parameterisation of beta distribution in terms of the mean and dispersion parameters. A log-likelihood function is approximated by the Gauss–Hermite quadrature to numerically integrate the distribution of random intercepts. A simulation study is used to investigate the performance of the estimation process and the sampling distributions of the residuals.

It is proposed and substantiated to use a mathematical model for making decisions regarding the credibility of content posted on social networks, based on establishing a relationship between the outcome on which the decision about the credibility or unreliability of the content is made and the factors influencing it. Quantitative factors for assessing the user profile in the network have been justified: the number of posts, shares, or likes made by users within a short time after the content appears; the number of comments or reactions at certain time intervals; the number of participants interacting with the content within half a day after publication; the viral spread coefficient of the content. The resulting indicator of such a model is proposed to be the degree of credibility of specific content, ranging from 0 to 1. It is proposed and justified that methods of interval data analysis should be used to represent and analyze this indicator based on expert assessment of the content. An optimization problem is formulated for the two-stage identification of the model based on interval data analysis: forming the current structure based on candidate models (model structure synthesis), estimating its parameters and verifying the adequacy of the model. A hybrid method for identifying interval models of user profiles in a social network is proposed and substantiated. This method combines a metaheuristic algorithm for synthesizing the model structure based on the behavioral model of a bee colony with gradient methods for identifying the parameters of candidate models. Examples of applying the proposed method and mathematical model for making decisions about the credibility of content are presented.

Background:Extra-articular fractures of the proximal tibia are extremely difficult to treat. These fractures can be treated with intramedullary nailing (IMN), closed reduction, and minimally invasive plating.Objective:The objective of this study was to compare the efficacy of proximal plating versus IMN in the treatment of extra-articular proximal tibial fractures.Materials and Methods:The present randomized prospective clinical study was carried out at the department of orthopedics, Isra University Hospital, Hyderabad, from May 2024 to December 2024 on 62 individuals with fracture of the proximal tibia who underwent proximal tibial plating (PTP) or IMN after taking permission from the ethical board of the institute. Computer-assisted prospective randomization was used to assign participants to groups. Participants in Group A received IMN treatment, whereas those in Group B received PTP. Data were analyzed through the Statistical Package for the Social Sciences version 24. The significance threshold that was used was P < 0.05. Age, sex, hospitalization, infection rate, fracture union duration, and fracture angulation were all compared between the two groups. The interval data (age, operative time, fracture union time, and knee movement) were subjected to a paired-sample t-test.Results:A total of 48 cases were studied, which were divided into two groups. Significantly shorter stay at hospital after surgery (value of P = 0.035), time for union, and time to full bearing of weight (P = 0.004) were experienced by the IMN group as compared to Group B. However, neither technique clearly outperformed the other groups with regard to operative time (P = 0.082), rate of infection (P = 0.738), knee range of motion (P = 0.462), or degrees of malunion or non-union.Conclusion:The current study concluded that there was no noteworthy difference between the two techniques in relation to surgical duration, rate of infection, and frequency of malunion and non-union. However, IMN is better than minimally invasive plating in terms of the duration of hospital stay, speed of union, and early full weight bearing.

Abstract: A key aspect of smart infrastructure for electricity is implementing Meter Data Management (MDM) for the entire portfolio of consumption points. The paper presents the experience of implementing the Meter Data Management solution at Delgaz Grid S.A., outlining the steps for data migration, methods for interfacing with metering devices, data flow optimizations, and performance analysis mechanisms. Specifically, the scientific contribution of this project includes integrating efficient transfer protocols and advanced functions for daily interval data correction, reactive power or loss calculations, and data and process synchronization. The applied solutions leverage a scalable and secure infrastructure capable of managing large volumes of real-time data, ensuring high interoperability between the MDM central platform, metering devices, billing systems, and interconnections with other entities. This flexible, load-resistant structure allows for efficient resource allocation and enhanced operational performance.

Kernel clustering with automatic variable weighting for interval data