Random Process Research Articles

Randomized limit theorems for stationary ergodic random processes and fields

Using the randomization approach, introduced by A. Tempelman in Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89–105, we prove: (a) a randomized version of the invariance principle (the functional CLT); (b) a version the Glivenko–Cantelli theorem; (c) a randomized theorem about convergence of empirical processes to the Brownian bridge. We also weaken the moment condition in the randomized CLTs, proved in the mentioned article. The main point of our work is that most of our theorems are valid for all ergodic homogeneous random fields on Zm and Rm,m≥1.

JPSSL: SAR Terrain Classification Based on Jigsaw Puzzles and FC-CRF

Effective features play an important role in synthetic aperture radar (SAR) image interpretation. However, since SAR images contain a variety of terrain types, it is not easy to extract effective features of different terrains from SAR images. Deep learning methods require a large amount of labeled data, but the difficulty of SAR image annotation limits the performance of deep learning models. SAR images have inevitable geometric distortion and coherence speckle noise, which makes it difficult to extract effective features from SAR images. If effective semantic context features cannot be learned for SAR images, the extracted features struggle to distinguish different terrain categories. Some existing terrain classification methods are very limited and can only be applied to some specified SAR images. To solve these problems, a jigsaw puzzle self-supervised learning (JPSSL) framework is proposed. The framework comprises a jigsaw puzzle pretext task and a terrain classification downstream task. In the pretext task, the information in the SAR image is learned by completing the SAR image jigsaw puzzle to extract effective features. The terrain classification downstream task is trained using only a small number of labeled data. Finally, fully connected conditional random field processing is performed to eliminate noise points and obtain a high-quality terrain classification result. Experimental results on three large-scene high-resolution SAR images confirm the effectiveness and generalization of our method. Compared with the supervised methods, the features learned in JPSSL are highly discriminative, and the JPSSL achieves good classification accuracy when using only a small amount of labeled data.

Abstract PO5-21-02: Breast cancer screening based on physical breast examination: ITABERAÍ randomized trial

Abstract Introduction: Approximately 70% of the Brazilian population depends on the public healthcare system (Sistema Único de Saúde - SUS). However, only 16% of SUS users have access to mammography for breast cancer screening, which necessitates the evaluation of new strategies to reduce mortality from the disease, including expanding physical breast examination (PBE) by non-medical professionals. Within the primary healthcare network, the Community Health Agent (CHA) is a technical-level professional who performs activities through home visits. The primary objective of this study is to evaluate the effectiveness of screening actions based on PBE performed by CHA in reducing advanced tumors (Stage III and IV) and breast cancer mortality rates. Secondary objectives include comparing the number of breast cancer diagnoses between the groups, establishing the overdiagnosis rate, comparing cancer-specific survival, comparing overall survival, comparing the number of diagnostic procedures, assessing the interval between the first symptom and diagnosis, evaluating the interval between diagnosis and the start of treatment, and comparing the financial cost of diagnosis and treatment for breast cancer patients. Trial design: Multicenter, phase III, prospective, randomized clinical trial. The target population comprises women aged 40 years or older and users of SUS. The randomization process will be carried out based on the spatial analysis of each participating center, using sociodemographic information on women from each macroregion to verify the homogeneity of data between groups. Intervention: After randomization, CHA in each group underwent training at different times. CHA in the CG received guidance on breast cancer screening and prevention according to Ministry of Health recommendations, while CHA in the IG received the same guidance and participated in theoretical and practical training for PBE (only female professionals). Variables and data collection: An application was developed to be installed on mobile devices, which the CHA used during home visits. The graphical interface of the application was developed in Dart/Flutter. For the backend, Python programming language was used along with the Flask framework to create the Application Programming Interface (API) responsible for the communication between the database and the application. The MySQL software was used for the relational database. The variables used for building the application included sociodemographic data, lifestyle habits, medical history, information on previous examinations, and physical breast examination. Statistical analysis: For the analyses, the database will be exported to the REDCap (Research Electronic Data Capture) platform. The calculations were defined to detect a reduction in severe cases and mortality (10% and 20%, respectively). For this purpose, a sample power of 80% and a Type I error of 5% were adopted after adjusting for intraclass correlation and sampling effect (0.032 and 1.892, respectively). The calculations were estimated using nQuery software (version 9.1). For the city of Itaberaí, a population of 1,894 women was estimated in each group, resulting in a total of 3,788. Present and target accrual: The trial was activated in December 2022, and as of June 30th, 2023, 2,467 patients had been accrued. The study will last for 16 years, and the first data analyses will be conducted in 2024. Registration and ethical aspects: The trial is registered in ReBEC (Brazilian register): RBR-39vm2nd. Contact information: Prof. Dr. Ruffo Freitas-Junior, Federal University of Goiás, ruffojr@terra.com.br. Citation Format: Ruffo Freitas-Junior, Danielle Rodrigues, Rosangela Corrêa, Leonardo Soares. Breast cancer screening based on physical breast examination: ITABERAÍ randomized trial [abstract]. In: Proceedings of the 2023 San Antonio Breast Cancer Symposium; 2023 Dec 5-9; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2024;84(9 Suppl):Abstract nr PO5-21-02.

POST-COVID-19 RANDOMIZED CONTROLLED TRIALS COMPARING INTRAMEDULLARY NAILS AND SLIDING HIP SCREW CONSTRUCTS FOR INTERTROCHANTERIC FRACTURES DO NOT REPRESENT TYPICAL PATIENT DEMOGRAPHICS: A SYSTEMATIC REVIEW

Extracapsular Hip Fractures (EHF's) are a significant health burden on healthcare services. Optimal treatment is controversial with conflicting evidence being reported. Currently treatment is undertaken with Intramedullary Nail (IMN) or Dynamic Hip Screw (DHS) constructs with a recent increase in IMN use (1). This study aims to conduct a systematic review of Randomised Control Trials published between 2020 and 2023 with particular focus on patient demographics and holistic patient outcomes.Using a unified search-protocol, RCT's published between 2020 and 2023 were collected from CENTRAL, PubMed, MEDLINE and EMBASE. Rayyan software screened duplicates. Using the CASP and Cochrane Risk of Bias Tool papers were critically examined twice, and Blood Loss, Infection and Mobility described the patient journey. Patient demographics were recorded and were contrasted with geographically diverse cohort studies to compare population differences. Parametric tests were used to determine significance levels between population demographics, namely Age and Sex.Eleven papers were included, representing 908 patients (436 Male). The mean age for patients was 64.39. There was considerable risk of bias in 7/11 studies owing to the randomization process and the recording of data. Four Cohort studies were selected for comparison representing 14314 patients. Mean age was significantly different between Cohort Studies and RCT's (Independent T-Test, df 13, t=7.8, p = <0.001, mean difference = 19.251, 95% CI = 13.888, 24.613). This was also true for sex ratios included in the studies (df 13, t = -2.268, p = 0.024, Mean Difference = -0.4884, 95% CI = -0.9702, -0.0066).To conclude, RCT's published in the post COVID-19 era are not representative of patient demographics. This has the potential to provide inaccurate information for implant selection. Additionally further research must be conducted in how to better improve RCT patient inclusion so as to be more representative of patients whilst balancing the risks of operations.

The first-order time-variant reliability expansion method

Time-variant reliability problems are frequently encountered in engineering due to factors like material degradation or random loading modeled as random processes. The PHI2 method, which employs the First Order Reliability Method (FORM), is commonly used to solve such problems. However, it requires repeated searches for Most Probable Points (MPPs), making it computationally expensive. To improve efficiency with little sacrifice of accuracy, this study proposes a First Order Time-variant Reliability Expansion (FOTRE) method, which provides an efficient explicit formulation for MPP regarding time, in contrast to the expensive optimization approach of the PHI2 method. It requires only a single accurate search for the so-called “worst MPP” over the whole lifespan and offers the “adaptive accuracy of outcrossing rate”, which avoids the repeated search for MPPs ensuring computational accuracy. The inspiration behind the FOTRE method stems from the observation that the outcrossing rate tends to be small at time points with relatively large reliability indexes compared to the minimum reliability index βmin, which has a negligible impact on the subsequent structural failure probability over the entire lifespan. This innovative approach significantly improves the efficiency of solving time-variant reliability problems without compromising much of the numerical accuracy. The effectiveness and accuracy of the FOTRE method are demonstrated through several numerical examples.

Dynamics of karyotype evolution.

In the evolution of species, the karyotype changes with a timescale of tens to hundreds of thousand years. In the development of cancer, the karyotype often is modified in cancerous cells over the lifetime of an individual. Characterizing these changes and understanding the mechanisms leading to them has been of interest in a broad range of disciplines including evolution, cytogenetics, and cancer genetics. A central issue relates to the relative roles of random vs deterministic mechanisms in shaping the changes. Although it is possible that all changes result from random events followed by selection, many results point to other non-random factors that play a role in karyotype evolution. In cancer, chromosomal instability leads to characteristic changes in the karyotype, in which different individuals with a specific type of cancer display similar changes in karyotype structure over time. Statistical analyses of chromosome lengths in different species indicate that the length distribution of chromosomes is not consistent with models in which the lengths of chromosomes are random or evolve solely by simple random processes. A better understanding of the mechanisms underlying karyotype evolution should enable the development of quantitative theoretical models that combine the random and deterministic processes that can be compared to experimental determinations of the karyotype in diverse settings.

Artificial Broadcasts as Galactic Populations. I. A Point Process Formalism for Extraterrestrial Intelligences and Their Broadcasts

Artificial broadcasts from extraterrestrial intelligences are a hypothetical class of celestial phenomena. Unlike known astrophysical objects, the societies that generate them may be able to replicate on galactic scales through interstellar travel. Different galaxies could thus have drastically different populations, with variations in abundance of many orders of magnitude. I present a probabilistic formalism to treat this shared history, in which societies and their broadcasts are described by distributions over basic properties like lifespan and energy released. The framework contains a hierarchy of objects related by a tree structure. Discrete societies, the sources of broadcasts, are organized into potentially interstellar metasocieties. The population of each type of object is represented by a random point process in an abstract parameter hyperspace, a haystack. When a selection like an observation draws a sample, the point process is thinned. Given assumptions of interchangeability and independence, observables are modeled with compound Poisson random variables. I present an example of how selection bias can favor sampling longer-lived objects. I rederive the Drake equation for societies in the limit of no expansion. When interstellar replication is present, however, the mean number of detected broadcasts can depend quadratically on stellar mass, suggesting a search strategy favoring large galaxies.

The burden and management competency of cardiomyopathies in China: a nationwide survey study

The public health burden of cardiomyopathies and competency in their management by health agencies in China are not well understood. This study adopted a multi-stage sampling method for hospital selection. In the first stage, nationwide tertiary hospital recruitment was performed. As a result, 88 hospitals with the consent of the director of cardiology and access to an established electronic medical records system, were recruited. In the second stage, we sampled 66 hospitals within each geographic-economic stratification through a random sampling process. Data on (1) the outpatient and inpatient visits for cardiomyopathies between 2017 and 2021 and (2) the competency in the management of patients with cardiomyopathies, were collected. The competency of a hospital to provide cardiomyopathy care was evaluated using a specifically devised scale. The outpatient and inpatient visits for cardiomyopathies increased between 2017 and 2021by 38.6% and 33.0%, respectively. Most hospitals had basic facilities for cardiomyopathy assessment. However, access to more complex procedures was limited, and the integrated management pathway needs improvement. Only 4 (6.1%) of the 66 participating hospitals met the criteria for being designated as a comprehensive cardiomyopathy center, and only 29 (43.9%) could be classified as a primary cardiomyopathy center. There were significant variations in competency between hospitals with different administrative and economic levels. The health burden of cardiomyopathies has increased significantly between 2017 and 2021 in China. Although most tertiary hospitals in China can offer basic cardiomyopathy care, more advanced facilities are not yet universally available. Moreover, inconsistencies in the management of cardiomyopathies across hospitals due to differing administrative and economic levels warrants a review of the nation allocation of medical resources. This work was supported by the Chinese Academy of Medical Sciences (CAMS) Innovation Fund for Medical Sciences (2023-I2M-1-001) and the National High Level Hospital Clinical Research Funding (2022-GSP-GG-17).

Probabilistic models applied to concrete corrosion depth prediction under sulfuric acid environment

Accurate prediction of corrosion depth degradation law can provide a basis for the determination of the sacrificial layer thickness and the cover thickness of the concrete structural members in a corrosive environment. This study aims to explore a more accurate and safe method to predict corrosion depth. Six sets of concrete cylinder concrete specimens were adopted to perform an accelerated corrosion test in sulfuric acid solution with pH ≈ 0.95 for six different scheduled durations. 3D coordinates of a large number of points on the corroded cylindrical surface were obtained with the help of the 3D laser scanning technology. Then, a comparative study was carried out for the corrosion depth measured with the laser scanning method and vernier caliper method. The results prove that concrete corrosion depth predicted by the Gaussian random process model established by the laser scanning method was considered to possess higher credibility and better security.

A Method to Assess Granger Causality, Isolation and Autonomy in the Time and Frequency Domains: Theory and Application to Cerebrovascular Variability.

Concepts of Granger causality (GC) and Granger autonomy (GA) are central to assess the dynamics of coupled physiologic processes. While causality measures have been already proposed and applied in time and frequency domains, measures quantifying self-dependencies are still limited to the time-domain formulation and lack of clear spectral representation. We embed into the linear parametric framework for computing GC from a driver X to a target process Y a measure of Granger Isolation (GI) quantifying the part of the dynamics of Y not originating from X, and a new spectral measure of GA assessing frequency-specific patterns of self-dependencies in Y. The measures are illustrated in theoretical simulations and applied to time series of arterial pressure and cerebral blood flow obtained in syncope subjects and healthy controls. Simulations show that GI is complementary to GC but not trivially related to it, while GA reflects the regularity of the internal dynamics of the target process. In the application to cerebrovascular interactions, spectral GA quantified the physiological response to postural stress of slow cerebral blood flow oscillations, while spectral GC and GI detected an altered response to orthostasis in syncope subjects, likely related to impaired cerebral autoregulation. The new spectral measures of GI and GA are useful complements to GC for the analysis of interacting oscillatory processes, and detect pathophysiological responses to postural stress which cannot be traced in the time domain. The thorough assessment of causality, isolation and autonomy opens new perspectives for the analysis of coupled processes in both physiological and clinical investigations.

A Robust Model Order Reduction Scheme for Lithium-Ion Batteries in Control-Oriented Vehicle Models

The role of batteries in electrification of vehicles is eminent; thus, a dynamic model that represents the physics-based phenomena of the battery system at a minimum computational cost is essential in the model-based design of electrified vehicle control systems. Furthermore, robustness of the reduced-order battery model when maintaining the dominant physics-based phenomena governing the dynamic behavior of the battery system is crucial. Characterization of the power signal applied to the lithium-ion battery in the energy management controller of a plug-in hybrid electric vehicle shows that there is a dominant frequency range in the input signal to the battery. This key feature can be considered as a basis to construct a reduced-order model in which the training input is different from the original power signal. The original idea in this paper is to generate the training input by applying a low-pass filter to the white-noise random signal to maintain the same dominant frequency range observed in the original power signal. Response of the reduced-order model, constructed using the proper orthogonal decomposition, compared to the high-fidelity battery model shows promising results; a maximum relative error of 1% was obtained for the battery state of charge while simulation time was reduced by 42.9%.

Stochastic finite element-based reliability of corroded pipelines with interacting corrosion clusters

The performance of corroded carbon steel pipelines over the course of their design life is generally assessed by probabilistic variables with explicit limit state functions, rather than the realistic representation with stochastic spatial variability and implicit failure considerations. This could be due to the complexities associated with the uncertainty quantification and performance estimation approaches. The consideration of random process representation of corrosion defect propagation and material properties, along with computationally effective implicit formulation is expected to lead to accurate reliability outcomes. This paper proposes a stochastic-based reliability framework considering suitable failure modes represented with surrogate models, that lead to time-variant reliability estimation. The approach combines surrogate computational model with scalar random variables and random field discretisation of underlying characteristics to generate experimental designs and corresponding surrogate models over a time period, which are used to derive reliability estimates. The outcomes of this approach are compared with the results of explicit time-dependent functions. It was observed that reliability estimates of the corroded pipeline change rapidly after the fifth year, providing a much lesser probability of failure (of 3.08×10−3 at 30th-year) compared to the existing models (of 1.80×10−2 at 30th-year), thereby providing an effective pathway for risk-based maintenance and management.

Using Recurrent Neural Networks for the Performance Analysis and Optimization of Stochastic Milkrun-Supplied Flow Lines

AbstractLong-term throughput, as a key performance indicator of a stochastic flow line, is affected by numerous parameters describing the features of the flow line, such as processing time and buffer size. Fast and accurate evaluation methods for a given set of values for those parameters are a prerequisite to systematically optimize such a flow line. In this paper, we consider the case of a flow line with random processing times, limited buffer capacities and so-called milkruns that supply the machines with material parts that are required to perform, e.g., assembly operations on workpieces. In such a system, shortages in the supply of material parts can limit the performance of the flow line. Up to now, there are no accurate analytical approaches to quantify the complex interactions in such milkrun-supplied flow lines for realistic problem sizes. We propose to use recurrent neural networks to determine the long-term throughput of such flow lines enabling us to evaluate production systems of flexible size. Our results show that the throughput can be determined accurately and quickly via recurrent neural networks. Furthermore, we use this new evaluation procedure as a building block to optimize this type of flow line using gradient and local search techniques.

The Influence of Probiotic Lactobacilli on COVID-19 and the Microbiota.

This comprehensive review explores the potential of using lactobacilli as a probiotic in the management of COVID-19. Our findings suggest that lactobacilli show promise in reducing the risk of death, gastrointestinal and overall symptoms, and respiratory failure, as well as in lowering cytokines and inflammatory markers associated with the disease. The molecular mechanisms by which lactobacilli protect against COVID-19 and other viral infections may be related to the reduction in inflammation, modulation of the immune response, and direct interaction with viruses to produce antiviral substances. However, the selected studies demonstrate the presence of mixed findings for various clinical, biochemical, hematological, and immunological parameters, which may be attributed to methodological differences among studies. We highlight the importance of clearly describing randomization processes to minimize bias and caution against small sample sizes and inappropriate statistical tests that could lead to errors. This review offers valuable insights into the therapeutic potential of lactobacilli in the context of COVID-19 and identifies avenues for further research and applications. These findings hold promise for the development of novel approaches to managing COVID-19 and warrant further investigation into the potential benefits of lactobacilli in combating the disease.

Care home staff interventions to optimise pain assessment and management in people with advanced dementia in long-term care settings: a systematic review

Abstract Introduction As a person’s dementia symptoms worsen, they become increasingly frail and dependent on others for support,[1] which often necessitates moving into long-term care (LTC). Pain is often poorly recognised and undertreated in people with advanced dementia.[2] Aim To identify and evaluate the effectiveness of care home staff interventions involving pain assessment and management in people with advanced dementia in LTC settings. Methods This systematic review was conducted and reported in accordance with the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) guidelines and registered with the International Prospective Register of Systematic Reviews (PROSPERO; CRD42022355487). Nine databases (Medline, Scopus, The Cochrane Library, EMBASE, Web of Science, International Pharmaceutical Abstracts, PsycINFO, CINAHL, World Health Organisation International Clinical Trials Registry Platform) were searched. The search strategy was developed in Medline and adapted as appropriate for the other databases; it comprised the following search terms: ‘dementia’, ‘Alzheimer’s disease’, ‘pain assessment’, ‘pain management’, ‘pain intervention’, ‘care home’, and ‘nursing home’. The selection of studies was limited to randomised controlled trials (RCTs) in which ≥50% of participants had advanced (moderate-severe) dementia and published in English. Electronic databases were searched from date of inception to May 2023. Two reviewers independently assessed potentially relevant articles for inclusion, completed data extraction and assessed risk of bias using the Cochrane Risk of Bias (RoB) 2.0 tool. A third reviewer was consulted when consensus could not be reached. A narrative analysis was undertaken due to the heterogeneity of outcome measures. Results In total, 3504 articles were identified, with 998 records remaining after duplicate removal. This left 2456 studies for title/abstract screening, and 22 full-text studies were assessed for eligibility; five studies met the inclusion criteria. A total of 1363 participants (mean age 83-88 years) and 123 care homes were included in the studies. The studies reported that these interventions were shown to be effective compared to control groups: the Pain recognition and Treatment (PRT) protocol in China, regular use of the PACSLAC pain assessment tool in Canada, the use of a Comprehensive Observational Pain Management Protocol in Hong Kong, the COSMOS intervention in Norway, and the STA OP stepwise multidisciplinary intervention in the Netherlands. Primary outcomes concerning pain scores measured by PACSLAC (n=2 studies), PAINAD (n=2 studies), and MOBID-2 (n=1 study) showed a statistically significant decrease. Secondary outcomes measured included several clinical outcomes associated with pain, depression, presence of neuropsychiatric symptoms and nursing staff stress. Four of the included studies were judged to be at ‘some concerns of bias’ according to RoB 2.0, due to unclear information in two or more domains (randomisation process, deviations from intended interventions, and measurement of the outcomes); one study was judged as having a ‘low risk of bias’. Conclusion The use of pain assessment tools decreased pain scores in people with advanced dementia in LTC. A limitation of this review is the exclusion of non-English language studies. Further, the clinical and methodological heterogeneity of included studies made a quantitative comparison difficult, thus, the findings are based on narrative analysis.

Improving awareness of foreign domestic workers during the COVID-19 pandemic using infographics: An experience during the crisis

The COVID-19 epidemic is a significant humanitarian crisis witnessed in contemporary times, necessitating the exploration of techniques and approaches that might augment the capabilities of individuals within society to mitigate the adverse consequences of this pandemic. Domestic workers have always been an important part of societal structure, and it is essential to ensure their awareness about the perils of the pandemic as well as implement preventive measures. This is crucial in order to effectively address the hazards associated with the ongoing epidemic. The primary objective of this study was to assess the efficacy of infographic technology in augmenting the level of knowledge among foreign domestic workers on the COVID-19 epidemic. The present study employed a quasi-experimental design to examine and compare the effects of infographic technology on the experimental group, while the control group received only public awareness messages disseminated to all citizens and residents. The study sample comprised (16) foreign domestic workers, who were assigned to the two research groups through a random allocation process. The development of the COVID-19 pandemic awareness scale was conducted. The measurement scale consisted of three dimensions: the cognitive awareness dimension, including a set of (12) statements; the performative awareness dimension, comprising a set of (8) statements; and the psychological awareness dimension, comprising a set of (10) statements. The findings demonstrated the superiority of the experimental group, which used the infographic approach, over the control group in fostering awareness of the COVID-19 epidemic throughout its three dimensions among the participants of the study. The study suggested the need of expanding the utilization of infographics in an attempt to enhance knowledge and awareness of the COVID-19 pandemic, alongside the establishment of varied databases for migrant domestic workers. The objective is also to develop an extensive digital awareness strategy about the impacts of the COVID-19 epidemic, so as to include an increased number of informative infographics.

Workload prediction based on improved error correlation logistic regression algorithm and Cross‐TRCN of spatiotemporal neural network

AbstractIn view of the randomness of user network usage behavior in data centers, which leads to a large randomness in power load, and considering that a single randomness processing method is usually difficult to fully characterize the uncertain characteristics of the system, this paper proposes a dual fusion prediction analysis model based on an improved error correlation logic regression algorithm and a novel spatiotemporal neural network structure called Cross‐TRCN. Two weight coefficients λ1 and λ2 are introduced to fuse the prediction results with different long‐term sequence prediction performance, thereby further eliminating the influence of random errors. The results show that it is feasible to predict the workload of data centers based on the improved error correlation logic regression algorithm and the innovative spatiotemporal neural network structure Cross‐TRCN.

Asymptotic normality of spectral means of Hilbert space valued random processes

A variety of statistics for functional time series allows for a representation as weighted average of corresponding periodogram operators over the frequency domain. We study consistency and asymptotic normality of such spectral mean estimators under mild assumptions. We show that weak convergence of these estimators can be deduced from the (joint) weak convergence of the sample autocovariance operators. The latter is established for a large class of weakly dependent functional time series, which admit expansions as Bernoulli shifts and the weak dependence is quantified by the condition of L4-m-approximability.

Stochastic processes and mean square calculus on fractal curves

Abstract In this paper, random and stochastic processes are defined on fractal curves. Fractal calculus is used to define the cumulative distribution function, probability density function, moments, variance, and correlation function of stochastic processes on fractal curves. A new framework, which is a generalization of mean square calculus, is formulated. The sequence of random variables on the fractal curve, fractal mean square continuity, mean square F α {F^{\alpha}} -derivative, and fractal mean square integral are discussed. The mean square solution of a fractal stochastic equation is derived and plotted to illustrate the details.

МЕТОДИ ОЦІНКИ ВІДПОВІДНОСТІ ПСЕВДОВИПАДКОВОЇ ПОСЛІДОВНОСТІ ДО ВИПАДКОВОГО РОЗПОДІЛУ В ЗАДАЧАХ АВТОМАТИЗАЦІЇ, КІБЕРБЕЗПЕЦІ, КОМП’ЮТЕРНО-ІНТЕГРОВАНИХ ТЕХНОЛОГІЯХ ТА НЕРУЙНІВНОМУ КОНТРОЛІ

Creation of methods for determining the degree of closeness of a given pseudorandom sequence to a random distribution is a very relevant task in many fields where pseudorandom numbers are used. For example, in cryptography it is important that pseudorandom sequences are unpredictable and have the properties of random numbers. Methods for determining proximity to a random distribution help in assessing the quality of such sequences and identifying flaws in them. In scientific research and engineering projects, pseudorandom numbers are used to model complex systems and simulate random phenomena. It is important to have a means of determining how well such numbers reproduce a random process. In statistics, it is important that pseudorandom sequences meet the requirements of various statistical properties. Methods of determining proximity to a random distribution help in evaluating the properties of these sequences. In software engineering and other fields, it is important to test the quality of pseudorandom number generators to avoid vulnerabilities and flaws in programs. Thus, the development and improvement of methods for determining the degree of closeness of pseudorandom sequences to a random distribution is an urgent task with important practical applications.