Normal Distribution Research Articles

Effects of Different Crystallization Protocols on Marginal Gap of Lithium Disilicate Single Crowns: SEM Analysis

Objective: In everyday dentistry, lithium disilicate is a valid option for single-fix partial dentures, and this material crystallization process is available with two protocols: long and short. This study’s aim was to assess the effects of these two different crystallization protocols, long and short, on the marginal gap of lithium disilicate single crowns. Methods: A total of 24 abutment plastic teeth were scanned using an intra-oral scanner. For each plastic tooth, an identical pair of lithium disilicate crowns was milled (a total of 48 crowns) by a four-axis machine. Each paired sample was categorized into two groups: long crystallization (24 crowns) and short crystallization (24 crowns). To assess precision, each unit’s marginal gap (including abutments and crowns) was meticulously measured at four specified regions using a scanning electron microscope. A Kolmogorov–Smirnov test performed on the study variables indicated a normal distribution (p > 0.05), and it was followed by independent t-tests (α = 0.05). Results: For the long crystallization group, the mean total marginal gap values were 42.91 ± 9.67 μm, and for the short crystallization group, the values were 43.25 ± 8.14 μm, with no significant difference between the groups (p = 0.894). In addition, no significant differences were found between the groups regarding the mean marginal gap measurements for all four surfaces (distal (p = 0.310), mesial (p = 0.732), palatal (p = 0.655), and buccal (p = 0.535)). Conclusions: Both the long and short crystallization methods used for lithium disilicate single crowns demonstrated marginal gap values of less than 120 μm, which are within the clinically acceptable range, with no significant differences across any parameters between the two groups. Regarding the marginal gap value, it is recommended to use the short crystallization protocol as it is more time-efficient.

Assessing the professional social capital of psychiatrists: development of the Resource Generator for Psychiatrists (RG-Psy)

BackgroundIn the field of psychiatry, the dissemination of clinical innovations greatly depends on the social capital of clinicians. An instrument specifically aimed at measuring their professional social capital therefore needs to be developed.MethodsThis survey was conducted to develop and validate the Resource Generator for Psychiatrists, an 11-item questionnaire measuring the social capital of psychiatrists. The online questionnaire was administered through a link sent by e-mail to all psychiatrists and residents in psychiatry licensed to work in Belgium, after excluding ineligible psychiatrists. A total of 1618 psychiatrists or residents were reached. An exploratory factor analysis was conducted. Internal consistency was assessed using Pearson’s correlation, item-total correlation and Cronbach’s alpha. Test-retest reliability was also measured. Multivariable linear regression analysis assessed the association between the total score of the social capital and psychiatrist demographics.ResultsA total of 196 psychiatrists responded to the survey (response rate: 12.1%). The Resource Generator for Psychiatrists showed a normal distribution with a mean of 23.6 (SD = 15.5), good test-retest reliability (ICC = 0.81) and a good total Cronbach’s alpha (0.74). Exploratory factor analysis revealed two main subtypes in psychiatrists’ social capital: “Resources for the clinician” and “Resources for the professional”, with a Cronbach’s alpha of 0.62 and 0.7 respectively. Clinicians attending institutional seminars (β = 5.52, SE = 2.2, p = .013) and working in multidisciplinary settings, such as hospitals (β = 4.75, SE = 2.06, p = .023) or a mobile team (β = 8.75, SE = 3.52, p = .014) were more likely to have higher social capital.ConclusionPsychiatrists’ access to professional resources can be reliably measured by using an 11-item questionnaire.

An unbiased rank-based estimator of the Mann–Whitney variance including the case of ties

Many estimators of the variance of the well-known unbiased and uniform most powerful estimator of the Mann–Whitney effect, are considered in the literature. Some of these estimators are only valid in cases of no ties or are biased in small sample sizes where the amount of bias is not discussed. Here, we derive an unbiased estimator based on different rankings, the so-called ’placements’ (Orban and Wolfe in Commun Stat Theory Methods 9:883–904, 1980), which is therefore easy to compute. This estimator does not require the assumption of continuous distribution functions and is also valid in the case of ties. Moreover, it is shown that this estimator is non-negative and has a sharp upper bound, which may be considered an empirical version of the well-known Birnbaum–Klose inequality. The derivation of this estimator provides an option to compute the biases of some commonly used estimators in the literature. Simulations demonstrate that, for small sample sizes, the biases of these estimators depend on the underlying distribution functions and thus are not under control. This means that in the case of a biased estimator, simulation results for the type-I error of a test or the coverage probability of a confidence interval do not only depend on the quality of the approximation of by a normal distribution but also an additional unknown bias caused by the variance estimator. Finally, it is shown that this estimator is L2-consistent.

Bioelectrical Impedance Vector Analysis in Extremely Low-Birth-Weight Infants to Assess Nutritional Status: Breakthroughs and Insights

Background/Objectives: To obtain bioelectrical data to assess nutritional status for extremely low-birth-weight (ELBW) infants upon reaching term-corrected age. Methods: A descriptive, observational, prospective, and single-center study, which included ELBW preterm infants was performed. The study variables collected were gestational age, sex, and anthropometry at birth and at term-corrected age. Bioelectrical impedance vector analysis (BIVA) was performed by a phase-sensitive device (BIA 101 BIVA PRO AKERN srl, Pisa, Italy). The components of the impedance vector—resistance (R) and reactance (Xc)—were normalized for body height (H). For each subject, the measurement was taken between the 36th and 44th weeks of postmenstrual age (PMA). A semi-quantitative analysis of body composition was performed using the vector modality of the BIA. Using the RXc graph method, the bivariate 95% confidence intervals of the mean vectors were constructed. From the bivariate normal distribution of R/H and Xc/H, the bivariate 95%, 75%, and 50% tolerance intervals for this cohort were drawn. The individual impedance vectors were compared with the distribution of the vectors from other populations. Results: 85 ELBW infants (40 male, 45 female) were included, with a mean gestational age at birth of 26 + 6 weeks (±1.76). Mean R/H was 870.33 (±143.21) Ohm/m and Xc/H was 86.84 (±19.05) Ohm/m. We found differences in the bioelectrical data with regard to gender, with resistance values being significantly higher in females. Our ellipses align closely with those from other term neonatal cohorts. Conclusions: Bioelectrical data and the confidence and tolerance ellipses of an ELBW infant cohort are presented and can be used as a reference standard for nutritional assessment at discharge.

Body weight and body surface area of adult patients with selected cancers: An Italian multicenter study.

Although body weight (BW) and body surface area (BSA) are utilized to establish the appropriate dosage of anticancer drugs, their distribution in cancer patients is poorly studied, making it challenging to predict the amount of drug use and related costs of BW or BAS-dosed regimens. This study investigates the distribution of BW and BSA in adults with selected cancers who initiated systemic anticancer treatment in the eastern Emilia-Romagna region hospitals between 2011 and 2021. BW and BSA were collected at the first cycle of each new treatment line, with multiple measurements for patients receiving various treatments or treating for other primary malignancies. Results were grouped by sex, tumor site and treatment setting, and the normal distribution hypothesis was tested for each group. Both linear regression model and quantile regression at the 50th, 25th and 75th percentiles were run to explore the factors influencing BSA. The analysis included 20,634 treatment lines and the corresponding BW and BSA measures from a sample of 13,036 patients. The average BW was 68.05kg (64.20kg for females and 75.07kg for males) and the average BSA was 1.76m2 (1.66m2 for females and 1.87m2 for males). In women, the highest BW was in breast and colon groups, while in men, it was associated with prostate and rectum cancers. The model indicated significant association between BSA, age, sex and tumor localization. Notably, stomach and lung cancers were linked to lower BSA for both sexes (for females -0.081 and -0.041m2 respectively compared to those with breast cancer). Advanced settings were related to lower BSA than neoadjuvant treatment, especially for stomach cancer patients, who experienced a weight loss of 3 to 6kg as therapy progressed. The regression models for predicting BSA can assist regulatory bodies in determining reimbursement for new chemotherapy drugs and help hospitals forecast drug utilization and expenditure more accurately.

Geochemical Characteristics of Modern River-Sand and Its Bearing on the Mineral Exploration in the Manufahi Area, Timor-Leste

Geochemical mapping of river sands in the Manufahi area of Timor-Leste revealed potential areas for future mineral exploration. River sand samples from the study area were collected and geochemically analyzed to identify anomalous concentration distributions of several valuable elements and locate potential target areas and geological formations that may host mineral deposits. The 26 major and minor elements were identified using wavelength-dispersive X-ray fluorescence. The river sands exhibited varying elemental concentrations, with Cr, Cu, Zn, and Ba showing deviations from the normal distribution patterns. Identification of geochemical anomalies is an important task in mineral exploration geochemistry. The mean+2 standard deviations (mean+2STD), median+2 median absolute deviations (median+2MAD), and Tukey’s inner fence (TIF) methods were used to determine the geochemical thresholds. This study shows that TIF and principal component analysis (PCA) methods are highly effective in calculating appropriate threshold values and identifying relevant elemental associations. These approaches have proven useful for delineating target areas for mineral deposits, resulting in reliable outcomes. Four predicted target areas with high potential for deposits and mineralization anomalies of Cr, Cu, Ni, and Ba were delineated in the study area.

Estimating the Generalizations of Process Capability Index Cpmk$C_{pmk}$ in the Presence of Outliers

ABSTRACTProcess capability indices (PCIs) play a fundamental role in assessing and quantifying the capability of manufacturing processes to meet customer specifications. To date, the majority of PCIs that are now in use have been examined through the use of novel generalizations of PCIs with symmetric tolerances to check the performance of an industrial manufacturing process. In this paper, we proposed new estimations of some indices with asymmetric tolerances in the presence of outliers. New robust and parametric estimators of some PCIs are introduced to estimate and compare these indices for each normal distribution with the presence of different outliers. Meanwhile, this paper discusses how well the proposed method can be used for non‐normal data. For illustration purpose, the application example is presented.

Bounded Multivariate Contaminated Normal Mixture Model with an Application in Skin Cancer Detection

Introduction: In real-world datasets, outliers are a common occurrence that can have a significant impact on the accuracy and reliability of statistical analyses. Detecting these outliers and developing robust models to handle their presence is a crucial challenge in data analysis. For instance, natural images may have complex distributions of values due to environmental factors like noise and illumination, resulting in objects with overlapping regions and non-trivial contours that cannot be accurately described by Gaussian mixture models. In many real life applications, observed data always fall in bounded support regions. This leads to the idea of bounded support mixture models. Motivated by the aforementioned observations, we introduce a bounded multivariate cntaminated normal distribution for fitting data with non-Gaussian distributions, asymmetry, and bounded support which makes finite mixture models more robust to fitting, since rare observations are given less importance in calculations. Methods: A family of finite mixtures of bounded multivariate contaminated normal distributions is introduced. The model is well-suited for computer vision and pattern recognition problems due to its heavily-tailed and bounded nature, providing flexibility in modeling data in the presence of outliers. A feasible expectation- maximization algorithm is developed to compute the maximum likelihood estimates of the model parameters using a selection mechanism. Results: The proposed methodology is validated by conducting experiments on both simulated data and two real natural skin cancer images. We estimate the parameters by the proposed expectation-maximization algorithm. The obtained results shown that the proposed model has successfully enhanced accuracy in segmenting skin lesions. Conclusion: The reliable model-based clustering using finite mixtures of bounded multivariate contaminated normal distributions is introduced. An expectation-maximization algorithm was created to estimate parameters, with closed-form expressions utilized at the E-step. Practical tests on images for skin cancer detection showed enhanced accuracy in delineating skin lesions.

Hardness-Dependent Freshwater Quality Criteria for the Protection of Aquatic Organisms for Cadmium in China

Cadmium poses a significant threat to freshwater aquatic organisms and ecosystems, making it essential to establish regional freshwater quality criteria (FWQC) in China to safeguard these organisms. The toxicity database for cadmium covered 249 acute toxicity data from 52 species (seven phyla and 27 families) and 62 chronic toxicity data from 21 species (four phyla and 12 families). During short-term exposure, Morone saxatilis displayed the most sensitivity to cadmium, whereas Daphnia magna showed the most sensitivity in long-term exposure scenarios. Significant correlations were identified between water hardness and the toxicity data for cadmium, with the acute toxicity coefficient (KATD) at 1.0227 (n = 52, p < 0.05) and the chronic toxicity coefficient (KCTD) at 0.4983 (n = 21, p < 0.05). With the species sensitivity distribution method, the short-term freshwater quality criteria (S-FWQC) were derived with a normal distribution as the best fit (R2 0.9793), while the long-term freshwater quality criteria (L-FWQC) were calculated using a logistic distribution as the best fit (R2 0.9686). The formulas for the S-FWQC and L-FWQC were represented as 10(1.0227×lg(H)−1.5444) and 10(0.4983×lg(H)−1.7549), respectively, with water hardness serving as an independent variable. This study offers valuable insights for improving the management of cadmium to protect freshwater aquatic organisms in China.

Toxicological Effects of Pendimethalin: Behavioral Changes and LC50 Analysis in Cyprinus carpio

The pre-emergent herbicide pendimethalin is a selective dinitroaniline compound that interferes with the production of microtubules, potentially contaminating groundwater. The current investigation sought to ascertain the acute toxicity levels and behavioral effects of pendimethalin (30% EC) on Cyprinus carpio. Finney's probit bioassay method estimated the lethal toxicity LC50 values for commercial-grade herbicide pendimethalin on C. carpio. This method entails fitting the data collected from experiments to a probit model, which converts the results obtained (such as mortality rates) into a normal distribution scale. It is a useful tool in toxicological research since it enables more precise estimations of the LC50. The lethal toxicity (LC50) values of pendimethalin to C. carpio exposed to 24, 48, 72 and 96-hr were determined to be 2.83µL/L, 2.59µL/L, 2.39µL/L and 2.20µL/L based on the mortality estimations. Pendimethalin-induced behavioral abnormalities in C. carpio were investigated using sublethal concentrations (i.e., 1/15th, 1/10th, and 1/5th of the 96-hour LC50 value) at intervals of 0, 7, 14, 21 and 28 days. Our research observed behavioral shifts such as reduced general activity, reduced foraging behavior, diminished schooling behavior, reduced reflex stimuli, increased opercula movements, irregular movements, mucus secretion, enhanced breathing rates and increased surface air gulping. Our studies showed that pendimethalin can produce both mortality and behavioral alterations in Cyprinus carpio, offering important insights into the pendimethalin fatal and sub-lethal effects. These results highlight how crucial it is to keep an eye on herbicide poisoning in aquatic habitats since even sub-lethal levels can have an effect on fish behavior and the general health of the ecosystem.

Research on Vehicle Path Planning Method with Time Windows in Uncertain Environments

With the growing complexity of logistics and the demand for sustainability, the vehicle routing problem (VRP) has become a key research area. Classical VRPs now incorporate practical challenges such as time window constraints and carbon emissions. In uncertain environments, where many factors are stochastic or fuzzy, optimization models based on uncertainty theory have gained increasing attention. A single-objective optimization model is proposed in this paper to minimize the total cost of VRP in uncertain environments, including fixed costs, transportation costs, and carbon emission costs. Practical constraints like time windows and load capacity are incorporated, and uncertain variables, such as carbon emission factors, are modeled using normal distributions. Two uncertainty models, based on the expected value and chance-constrained criteria, are developed, and their deterministic forms are derived using the inverse distribution method. To solve the problem effectively, a hybrid ant colony–zebra optimization algorithm is proposed, integrating ant colony optimization, zebra optimization, and the 3-opt algorithm to enhance global search and local optimization. Numerical experiments demonstrate the superior performance of the hybrid algorithm, achieving lower total costs compared to standalone ant colony, zebra optimization, genetic algorithm, and particle swarm optimization algorithms. The results highlight its robustness and efficiency in addressing complex constraints.

Establishment of sex-specific reference intervals for PIVKA-II in Southwest China: A real-world data analysis.

We aim to establish the sex-related reference intervals (RIs) of PIVKA-II in southwest China by indirect method with the real-world data. Between 29 July 2016 and 5 February 2024, PIVKA-II test data were collected from 120,780 healthy adult participants (aged 18 to 97 years) in the Laboratory Information System (LIS) of West China Hospital to establish reference intervals. Additionally, a validation group comprised of 2068 healthy adults was evaluated using the same detection algorithm and platform. Following the CLSI EP28-A3 guideline, Box-Cox transformation was applied for normal transformation, and outliers were identified using the Tukey method. Furthermore, we employed the standard normal deviate test (z-test) recommended by Harris and Boyd to determine whether to stratify reference intervals by age and sex subclasses. We successfully established population-specific RI for PIVKA-II in southwest China using an indirect method. By utilizing a robust dataset and conducting rigorous statistical analyses, we delineated sex-specific RIs, with values of 0-35 mAU/mL for males and 0-29 mAU/mL for females according to the normal distribution method, and 0-32 mAU/mL for males and 0-28 mAU/mL for females using the non-parametric method. These intervals are more suitable for the local population than those derived from manual methods. These RIs provide valuable guidance for the accurate interpretation of PIVKA-II levels in the local population.

Dynamic Process Dispersion Monitoring Through Bayesian‐Driven Smoothing Adjustments

ABSTRACTThis study introduces a Bayesian adaptive exponentially weighted moving average (AEWMA) control chart designed to monitor process dispersion in case of normal distribution by integrating various loss functions. It exhibits robust performance in identifying shifts in process dispersion across different scales. Evaluation involves Monte Carlo simulations to calculate run length characteristics and a comprehensive comparative analysis against existing charts. Our findings emphasize the increased sensitivity of the Bayesian AEWMA control chart to shifts of different magnitudes. Furthermore, an experiment was performed in the context of another field, specifically semiconductor manufacturing, to compare the performance of the proposed Bayesian control chart using different loss functions. It showed that the suggested chart was much better than the existing control charts in terms of observing out‐of‐control signals. In summary, this article develops an innovative approach with various loss function approaches and improves the accuracy and efficiency depending on the dispersion change in the Phase II process, thus it is a valuable contribution to the further development of the quality control and monitoring process.

The Critical Influence of Wire Diameter and Bending for Orthodontic Wire Integration—New Insights for Maxillary Movements (In Vitro Study)

Background: Traditional methods for palatal expansion using fixed appliances often face limitations in comfort and aesthetics. In comparison, aligner therapy has limitations, particularly regarding maxillary expansion. The aim of this study is to examine the biomechanical properties regarding the wire diameter and bending of different stainless steel wires to evaluate their potential for incorporation into maxillary aligner therapy. Materials and Methods: Three rectangular stainless steel wires (0.016″ × 0.022″, 0.017″ × 0.025″, and 0.019″ × 0.025″) were tested for mechanical expansion forces in the intermolar region, comparing non-tooth-shaped bent wires (A groups) and tooth-shaped bent wires (B groups). Using a Z010 testing machine (ZwickRoell GmbH and Co. KG, Ulm, Germany), expansion forces were measured at 1 mm intervals over a 5 mm distance, with 15 samples analyzed per group. Statistical analyses included the Shapiro–Wilk test for normal distribution, the Mann–Whitney U test, which revealed significant results (U = 225, p < 0.001), and the Kruskal–Wallis test, which indicated significance (H = 39.130; df = 2; p < 0.001). Results: Tooth-shaped bent wires exhibited significantly lower expansion forces than non-tooth-shaped bent wires for all tested wire types. This difference was most notable in wires with larger transverse profiles (0.019″ × 0.025″), where the tooth-shaped bent wires displayed a marked reduction in mechanical load capacity. Specific force measurements for non-tooth-shaped wires ranged from 760.61 ± 79.51 mN at 1 mm of deformation to 2468.46 ± 66.27 mN at 5 mm of deformation, while tooth-shaped wires ranged from 116.80 ± 3.74 mN to 1979.49 ± 23.23 mN. Conclusions: These findings suggest that non-tooth-shaped bent wires offer a more efficient and uniform expansion potential for maxillary movements due to their stable elastic properties. Clinically, integrating non-tooth-shaped stainless steel wires into aligner therapy may provide a viable method for maxillary expansion, supporting both first- and second-order movements in orthodontic treatment. Further research is needed to explore the integration of such wires for effective maxillary expansion in aligner therapy.

Research on Production Decision Modelling Based on Sequential Testing and Minimum Objective Planning

In this paper, the decision-making problem encountered by an enterprise in the production process is studied in depth by using the sequential test and the minimum objective planning method. The study begins with a sampling and testing model for spare parts, simplifies the model based on normal distribution, and applies the sequential test technique to reduce the number of necessary samples and effectively optimize the cost of sampling and testing. Immediately after that, we constructed a minimum objective planning model to model the production process of spare parts and finished products and their related costs in detail. We solved the optimal production cost for each of the six real production scenarios faced by the enterprise. The results show that our model can not only effectively reduce production costs, but also significantly improve decision-making efficiency.

Joint Modelling of Longitudinal Measurements and Time-to-Event Outcomes With a Cure Fraction Using Functional Principal Component Analysis.

In studying the association between clinical measurements and time-to-event outcomes within a cure model, utilizing repeated observations rather than solely baseline values may lead to more accurate estimation. However, there are two main challenges in this context. First, longitudinal measurements are usually observed at discrete time points and second, for diseases that respond well to treatment, a high censoring proportion may occur by the end of the trial. In this article, we propose a joint modelling approach to simultaneously study the longitudinal observations and time-to-event outcome with an assumed cure fraction. We employ the functional principal components analysis (FPCA) to model the longitudinal data, offering flexibility by not assuming a specific form for the longitudinal curve. We used a Cox's proportional hazards mixture cure model to study the survival outcome. To investigate the longitudinal binary observations, we adopt a quasi-likelihood method which builds pseudo normal distribution for the binary data and use the E-M algorithm to estimate the parameters. The tuning parameters are selected using the Akaike information criterion. Our proposed method is evaluated through extensive simulation studies and applied to a clinical trial data to study the relationship between the longitudinal prostate specific antigen (PSA) measurements and overall survival in men with metastatic prostate cancer.

On a Corrective Decile-based Confidence Interval Estimator of Mean for Normal and Skewed Distributions

This study addresses an issue with the decile t-confidence interval (dt-CI), which fails to achieve the desired coverage probabilities for large samples or skewed distributions (Mokhtar, Yusof & Sapiri, 2024). The article proposes a new corrective decile t-confidence interval (cdt-CI) that resolves these issues by modifying the decile standard deviation. Simulations using normal, chi-squared, log-normal, and gamma distributions show that the cdt-CI outperforms existing methods, particularly for skewed data, in terms of coverage probability and robustness. The real-life data sets analyzed in this study also support the conclusions of the simulation study.

Experimental Study on the Characteristics of Corrosion-Induced Cracks and Steel Corrosion Depth of Carbonated Recycled Aggregate Concrete Beams

The durability of carbonated recycled aggregate concrete (C-RAC) beams is still unclear at present. In this paper, the characteristics of corrosion-induced cracks and the steel corrosion depth of C-RAC beams were investigated through the accelerated corrosion test. The results showed that when accelerating corrosion to the 40th day, compared to the non-carbonated recycled aggregate concrete (NC-RAC) beam, the corrosion-induced cracking area of the C-RAC beam with a 100% carbonated recycled coarse aggregate (C-RCA) replacement ratio decreased by 40.00%, while the total length of the corrosion-induced cracks (CCs) increased by 51.82%. The type of probability distribution for the width of the CCs on the tension side of the C-RAC beams was a lognormal distribution. Compared with the NC-RAC beam, the mean value of the width of the CCs of the C-RAC beam with a 100% C-RCA replacement ratio decreased by 66.67%, the crack width distribution was more concentrated, and the quartiles and median were all reduced. With an increase in the C-RCA replacement ratio, the fractal dimension and the scale coefficient of CCs on the tension side of the beams showed an approximate trend of first increasing and then decreasing. The distribution of the corrosion depth of longitudinal tensile steel bars in the C-RAC beams was a mainly normal distribution. When the C-RCA replacement ratio increased from 30% to 100%, the mean value of the corrosion depth of the longitudinal tensile steel bars decreased by 33.46%, and the trend of changes in the quartiles and medians was basically the same as the trend of changes in the mean value. The research results can provide some reference for promoting the engineering application of C-RAC beams.

The dynamical lineage of isolated, HI-rich ultra-diffuse galaxies

Ultra-diffuse galaxies (UDGs) exhibit morphological similarities with other low-luminosity galaxies, indicating a possible evolutionary connection. We investigated for the common dynamical characteristics of isolated, HI-rich UDGs with other low-luminosity field galaxies, namely the low surface brightness galaxies (LSBs) and the dwarf irregulars (dIrrs). We considered samples of each of the UDGs, LSBs, and the dIrrs. We first obtained scaling relations involving mass and structural parameters for the LSB and the dIrr samples and superposed the UDGs on them. We then carried out a two-sample Anderson-Darling test to analyse whether the UDGs belong to the population of the LSBs or the dIrrs. Thereafter, we constructed distribution function-based stellar-dynamical models of these galaxies to determine their kinematical parameters. We followed up with the Mann-Whitney U-test to determine if our UDG, LSB, and dIrr samples belong to different parent populations so far as kinematics is concerned. Finally, we conducted principal component analyses involving both structural and kinematical parameters to identify the key properties accounting for the variance in the data for the respective galaxy populations. From the galaxy scaling relation studies, we note that the UDGs and the LSBs constitute statistically different populations. However, for the UDGs and the dIrrs, the null hypotheses of these statistical tests cannot be rejected for the following scaling relations: (i) stellar mass versus atomic hydrogen mass, (ii) stellar mass versus dynamical mass, and (iii) dark matter core density versus core radius mass scaling. Interestingly, the dynamical models suggest that the UDGs, LSBs, and the dIrrs constitute different galaxy populations, as reflected by their radial-to-vertical velocity dispersion and the rotational velocity-to-total stellar velocity dispersion. Finally, we observe that the total HI and stellar mass mostly regulate the variance in the structural and kinematical data for both the UDGs and the dIrrs, while the ratio of radial-to-vertical velocity dispersion and the total HI mass dominate the variation in the LSBs. The UDGs and the LSBs represent statistically different galaxy populations with respect to their mass and structural properties. However, the statistical studies do not negate the fact that the structural parameters of the UDGs and the dIrrs follow the same normal distributions. However, the UDGs, LSBs, and the dIrrs constitute very different populations as far as their kinematical parameters are concerned. Finally, we note that the variation in the structural and kinematical data of both the UDGs and the dIrrs is mostly accounted for by their stellar mass and HI mass, whereas for the LSBs, the variance is explained by the ratio of the radial-to-vertical stellar dispersion followed by the HI mass. Thus, we may conclude that the UDGs and dIrrs share a common dynamical lineage.

Stochastic Risk Assessment Framework of Deep Shale Reservoirs by a Deep Learning Method and Random Field Theory

Risk assessment of deep shale reservoirs is very important for subsurface energy development. However, due to complex geological environments and physicochemical interactions, shale reservoir fabric parameters exhibit variability. Moreover, the actual investigation and testing information is very limited, which is a typical small-sample problem. In this paper, the heterogeneity and statistical characteristics of deep shale reservoirs are clarified by the measured mechanical parameters. A deep learning method for deep shale reservoirs with limited survey data information is proposed. The variability of deep shale reservoirs is characterized by random field theory. A variable stiffness method and stochastic analysis method are developed to evaluate the risk of deep shale reservoirs. The detailed workflow of the stochastic risk assessment framework is presented. The frequency distribution and failure risk of deep shale reservoirs are calculated and analyzed. The risk assessment of deep shale reservoirs under different model parameters is discussed. The results show that a stochastic risk assessment framework of deep shale reservoirs, using a deep learning method and random field theory, is scientifically reasonable. The scatter plots of the elasticity modulus (EM), cohesive force (CF), and Poisson ratio (PR) distribute along the 45-degree line. The different distributed variables of EM, CF, and PR have a positive correlation. The statistical properties of the measurement data and deep learning data are approximately the same. The principal stress of deep shale follows the normal distribution with significance level 0.1. Under positive copula conditions, the maximum failure probability is 5.99%. Under negative copula conditions, the maximum failure probability is 4.58%. Different copula functions under positive and negative copula conditions have different failure probabilities. For the exponential correlation structure, the minimum failure probability is 3.46%, while the maximum failure probability is 6.19%. The mean failure probability of the EM, CF, and PR of deep shale reservoirs is 4.85%. Different random field-related structures and parameters have different influences on the failure risk.