Two-parameter Lognormal Distribution Research Articles

Saturation Equation: An Analytical Expression for Partial Saturation during Wicking Flow in Paper Microfluidic Channels.

The design and fabrication of paper-based microfluidic devices is critically dependent on modeling fluid flow through porous paper membranes. A commonly observed phenomenon is partial saturation, i.e., regions of the paper membrane not being filled completely due to pores of different sizes. The most comprehensive model to date of partial saturation during wicking flow in paper is the Richards equation. However, the solution to the Richards equation requires numerical solvers like COMSOL, which makes it largely inaccessible to the paper microfluidics and lateral flow assay community. There is therefore a need for a simple and appropriate model of partial saturation in paper membranes, easily usable by the wider research community. In the current work, we present an approach to model paper membranes as a bundle of parallel capillaries whose radii follow a two-parameter log-normal distribution. Application of the Washburn equation to the bundle provides a distribution of fluid fronts, which can be used to calculate saturation. Using this approach, we developed the "saturation equation"─an explicit analytical expression to calculate saturation as a function of space and time in 1D wicking flow. Experimentally obtained data for spatiotemporal saturation for four different paper materials were fit to this analytical model to obtain parameters for each material. Results obtained from this analytical model match well with both experimental data and numerical results obtained from the Richards equation. The availability of an explicit analytical expression for partial saturation will enable incorporation of the critical phenomenon of partial saturation in the design of paper microfluidic devices.

EMPIRICAL VULNERABILITY ANALYSIS OF RAILWAY BRIDGE SEISMIC DAMAGE BASED ON 2022 MENYUAN EARTHQUAKE

A 6.9 magnitude earthquake at a depth of 10 km struck Menyuan County, Haibei Prefecture, Qinghai Province, China, on January 8, 2022. This earthquake damaged some railway bridges on the Lanzhou-Xinjiang Passenger Dedicated Line. This study combines relevant historical earthquake damage experience, considers the effects of earthquake intensity, site soil classification, superstructure type, foundation failure factor, number of spans, and total bridge length, and develops empirical formulas for seismic damage prediction of railway bridges using ordinal logistic regression model in SPSS software. The seismic damage matrix, as were the anticipated multi-intensity mean damage index and the empirical vulnerability curve based on the two-parameter lognormal distribution function, were generated on this basis. According to the conclusions, although the suggested particular equations and vulnerability curves do not apply to the remainder of the region owing to geographical uniqueness, the technical approach is valid. It may be used as a reference for seismic damage prediction and vulnerability evaluation in other regions. The empirical vulnerability analysis based on the earthquake damage prediction matrix derived from the regression analysis can provide reasonable and fast forecasts before the next earthquake.

Fragility analysis of a multi-span continuous steel roadway bridge in Türkiye: A case study

One of the most widely utilized methods for determining seismic performance and allowing additional study is fragility analysis. The fragility curve, which is typically characterized by two-parameter log-normal distribution functions, depicts the probability of bridge components exceeding a certain damage limit. This study examines the fragility analysis of a multi-span continuous (MSC) steel roadway bridge in Türkiye. The probabilistic seismic demand model (PSDMs) is illustrated by conducting many time history analyses (THA). The nonlinear analyses are conducted for sixty earthquakes. Logarithmic regression analyses and fragility curves were derived for varying intensity measures (IM) in terms of efficiency. Monte Carlo analysis was used to derive the system fragility curve of the bridge. The PGA and ASI are the most proper intensity measure for the fragility curve of the bridge. Moreover, the slight damage can be visualized with a higher probability for the small intensity measure that even mild earthquake motion can cause some slight damage on the bridge but after slight damage bridge has further capacity until the collapse damage is visualized.

Nonparametric probabilistic seismic demand model and fragility analysis of subway stations using deep learning techniques

A novel and computationally efficient method for developing a nonparametric probabilistic seismic demand model (PSDM) is proposed to conduct the fragility analysis of subway stations accurately and efficiently. The probability density evolution method (PDEM) is used to calculate the evolutionary probability density function of demand measure (DM) without resort to any assumptions of the distribution pattern of DM. To reduce the computational cost of a large amount of nonlinear time history analyses (NLTHAs) in the PDEM, the one-dimensional convolutional neural network (1D-CNN) is used as a surrogate model to predict the time history of structural seismic responses in a data-driven fashion. The proposed nonparametric PSDM is adopted to conduct the fragility analysis of a two-story and three-span subway station, and the results are compared with those from two existing parametric PSDMs, i.e., two-parameter lognormal distribution model and probabilistic neural network (PNN)-based PSDM. The results show that the PDEM-based PSDM has the best performance in describing the probability distribution of seismic responses of underground structures. Different from the fragility curves, the time-dependent fragility surface of the subway station shows how the exceedance probability of damage state changes over time. It can be used to estimate the escape time and thus the number of casualties in an earthquake, which are important indexes when conducting the resilience-based seismic evaluation.

Fragility Study of Buried Pipeline Networks Based on Copula Function

The seismic fragility of water supply networks is an important research component for lifeline engineering in earthquake-prone regions. An accurate computational model was established to rapidly and efficiently assess pipeline damage under seismic excitation. In this study, the fragility of the pipeline network is deduced based on Copula function technology and the series-parallel model. This solves the low accuracy problem caused by simplifying elbow, tee, and other nodes and ignores the correlation between connected components in the fragility calculation of the buried pipeline network. Then, the shaking table tests of the elbow pipe, tee pipe, and scale pipeline network are performed, and the prototype pipeline network’s finite element comparative calculation model is also established. Then, the optimal Copula function for the pipeline network, the Gumbel Copula function, is obtained by calculating the correlation between each component’s seismic demand. It was found that the seismic demand of pipe and elbow components in the sand is higher than that in the clay, while the opposite is valid for the tee component. After that, a two-parameter log-normal distribution is employed to obtain the seismic fragility of three types of components in different soil properties. It was also found that the maximum failure probability can be obtained from the elbow leakage. The fragility of the elbow pipe is greater than that of the tee pipe. The fragility of components is then applied to the fragility calculation of the scale pipeline network by a series-parallel model, and the obtained results are compared with numerical simulations. It is demonstrated that the series-parallel combination calculation-based results have a high safety reserve for the pipeline design.

Development of an analytical probabilistic model to estimate runoff event volumes in South Korea

• A lognormal distribution best represents the rainfall characteristics of South Korea. • An analytical probabilistic model (APM) for lognormal distribution is developed. • The return period of APM is overestimated due to the seasonal rainfall distribution. • The APM adjustment magnitudes and the seasonal rainfall distributions are correlated. This study showed that the complex rainfall characteristics of South Korea, where rainfall is concentrated during a specific period, are poorly represented with a simple single parameter distribution. Thus, a two-parameter lognormal distribution that best represents the rainfall characteristics of South Korea was proposed by comparing four different distributions. Using this distribution, this study developed a suitable analytical probabilistic model (APM) to simulate runoff event volumes in South Korea and suggested an APM parameter-adjustment method based on rainfall characteristics. Thirty-year rainfall data recorded at ten stations were grouped based on the interevent time definition (IETD) of each station. With the grouped rainfall events, the probability density function (PDF) of the lognormal distribution was confirmed to best fit the rainfall event characteristic histograms in South Korea, including those of the rainfall event volume, duration, and interevent time. The APM was developed by deriving the PDF of the runoff event volume with a lognormal distribution, and a frequency analysis was performed based on the runoff event volume obtained by the APM and the Storm Water Management Model (SWMM) to verify the suitability of the derived model. The runoff event volumes output by the APM were compared with the runoff event volumes simulated by the SWMM. The exceedance probabilities and return periods output by the developed APM with a lognormal distribution were very close to the SWMM results. However, the return periods output by the APM were overestimated, and adjustment parameters were needed to match the outputs to the SWMM results. This study found a strong correlation between the adjustment parameters and the coefficients of variation in the average monthly rainfall. This result suggested that model parameter adjustments are necessary before applying an APM to rainfall datasets with large coefficients of variation in average monthly rainfall data.

A Nonlinear Subspace Approach for Parametric Estimation of PDFs From Short Data Records With Application to Rayleigh Fading

This paper tackles the issue of real-time parametric estimation of a wide class of probability density functions from limited datasets. This type of estimation addresses recent applications that require joint sensing and actuation. The suggested estimator operates in the nonlinear subspace that the parameter space of the distribution creates in the measurement sample space. This enables the estimator to embed a priori available information about the distribution in the computations to produce parameter estimates that are induced by signal components belonging only to the correct class of density functions being considered. It also enables it to nullify the effect of those components that do not belong to this class on the estimation process. The estimator can, with high accuracy, compute quickly the parameters of a wide class of probability density functions from short data records. The approach is developed and basic proofs of correctness are carried-out for the Rayleigh distribution, which is used to characterize wireless communication channels experiencing fast fading in heavily cluttered environments. Simulation results demonstrate the capabilities of the suggested procedure and the clear advantages it has over conventional norm-based estimation techniques. The results also show the ability of the suggested approach to estimate other density functions including the two-parameter lognormal distribution used to characterize shadowing in wireless communication.

Seismic fragility curves for stainless-steel wine storage tanks

Wine storage tanks in seismically active parts of the world sustain significant earthquake-related damage with major financial loss implications. Consequently the development of fragility curves for such tanks using empirical damage data offers an opportunity to better understand real risk exposures and to plan for and mitigate earthquake related damage in the future. The seismic fragility curves can be used for running an earthquake scenario for other wine regions to estimate the probability of earthquake damage and further financial loss estimation. Fragility curves for wine storage tanks were developed utilising damage data collected from approximately 3400 tanks that were subjected to three major earthquakes in New Zealand in 2013 and 2016. Two-parameter (median and log-standard deviation) log-normal distribution variables were used to represent the fragility curves, and peak ground acceleration (PGA) was utilised to represent the intensity measure of the seismic ground motion. Damage states for Flat-Based (FB) and Legged (LG) wine tanks were introduced in accordance with the repair that was recommended by the structural engineers at the time of post-earthquake damage assessment of the affected wineries. A parametric study was also conducted, considering fragility curves for tanks with different capacities and different percentage of wine contents. Based on the analysis, complete damage of FB and LG wine tanks was forecast at median PGAs of 1.0 g and 0.88 g respectively in the 2013 and 2016 earthquakes, while the median PGA for minor damage to FB tanks was 0.18 g and the median PGA for minor damage to LG tanks was 0.23 g. The research presented herein highlighted the importance of evaluating and improving the seismic resilience of storage tanks within wineries located in seismicity active regions.

Estimation of pO2 distribution in EPR oximetry

Electron paramagnetic resonance (EPR) oximetry, using oxygen-sensing implant such as OxyChip, is capable of measuring oxygen concentration in vivo – a critical tissue information required for successful medical treatment such as cancer, wound healing and diabetes. Typically, EPR oximetry produces one value of the oxygen concentration, expressed as pO2 at the site of implant. However, it is well recognized that in vivo one deals with a distribution of oxygen concentration and therefore reporting just one number is not representative_a long-standing critique of EPR oximetry. Indeed, when it comes to the assessment of radiation efficacy one should be guided not by the mean or median but the proportion of oxygenated cancer cells which can be estimated only when the whole oxygen distribution in the tumor is known. Although there is a handful of papers attempting estimation of the oxygen distribution they suffer from the problem of negative frequencies and no theoretical justification and no biomedical interpretation. The goal of this work is to suggest a novel method using the empirical Bayesian approach realized via nonlinear mixed modeling with a priori distribution of oxygen following a two-parameter lognormal distribution with parameters estimated from the multi-implant single component EPR scan. Unlike previous work, the result of our estimation is the distribution with positive values for the frequency and the associated pO2 value. Our algorithm based on nonlinear regression is illustrated with EPR measurements on OxyChips equilibrated with gas mixtures containing four values of pO2 and computation of the proportion of volume with pO2 greater than any given threshold. This approach may become crucial for application of the EPR oximetry in clinical setting when the sucsess of the treatment depends of the proportion of tissue oxygenated.

Implications of the log-normal distribution for updating estimates of the time remaining until ready for phase I post-anesthesia care unit discharge

Background: Multiple managerial decisions on the day of surgery depend on estimating how much longer each patient will remain in the phase I post-anesthesia care unit (PACU). We apply statistical methods developed for the time remaining in long-running surgical cases to the PACU. Methods: We use a previously published historical cohort of N=765 patients from one large teaching hospital who underwent gynecological surgery with time from operating room entrance to end of surgery of at least 4 hours and were admitted to the PACU. Results: The times until patients were ready for discharge followed a two-parameter log-normal distribution (e.g., Lilliefors’ test P = 0.18 and best three-parameter fit with no shift). Because the estimated coefficient of variation was 42%, there was an approximately constant percentage of the patients ready for discharge for each period longer than average. For example, half the patients stayed longer than 2.0 hours. Twenty minutes later, approximately 1/3rd of the patients were ready for discharge, and 2/3rd were not. Twenty minutes later, among the 2/3rd originally remaining, there were 1/3rd since ready for discharge, and so forth. The expected time remaining in the PACU for patients staying longer than the median time was relatively unchanging, independent of the amount of time that they had already been in the PACU. Conclusions: It is imprudent to rely on a certain expected duration of time until PACU discharge because any one time is unreliable. What is dependable is the probabilistic interpretation that, during any one period, there is a relatively unchanging percentage of the patients who will be ready for discharge during the period. Use this mathematical relationship for contacting hospital wards in advance of transfer, optimizing the use of porters, answering queries from families, and forecasting PACU occupancy in real-time.

Assessment of trends in hydrological extremes using regional magnification factors

Detection and attribution of trends in individual at-site series of hydrological extremes is routinely undertaken using simple linear regression-based models. However, the available records are often too short to allow a consistent assessment of trends across different stations in a region. The theoretical developments presented in this paper propose a new method for estimating a regional regression slope parameter across a region, or pooling group, of catchment considered hydrologically similar, and where annual maximum events at different sites are cross-correlated. Assuming annual maximum events to follow a two-parameter log-normal distribution, a series of Monte Carlo simulations demonstrate the ability of the new framework to accurately identify the regional slope, and provide estimates with a reduced sampling variability as compared to the equivalent at-site estimates, thereby enhancing the statistical power of the trend test. This regionally-based trend estimates would allow for a clear characterization of changes across several stations in a region. Finally, the new method is applied to national dataset of annual maximum series of peak flow from 662 gauging sites located across the United Kingdom. The results show that the regional slope estimates are significantly positive (p < 0.05) consistently in the west and north of the country, while mostly not significant in the east and south. This translate into a corresponding increase in design flood (as measured by regional magnification factors) of up-to 50% for time horizon of 50-years into the future.

Percentages of Cases in Operating Rooms of Sufficient Duration to Accommodate a 30-Minute Breast Milk Pumping Session by Anesthesia Residents or Nurse Anesthetists.

Introduction: Accommodating breast milk pumping sessions is required by US federal statute, but fulfillment is challenging for US anesthesia providers (e.g., anesthesia residents and nurse anesthetists). Considerations of good anesthesia practices (e.g., being present for critical portions of cases, including induction and emergence) create limits on which procedures are suitable for such relief. Our objective was to quantify the minimum percentages of cases for which there could reliably (≥ 95%) be at least 30 minutes during the surgical time when the anesthesia provider could receive such breaks.Methods: We studied all surgical cases performed at an anesthesia department over four years, including its inpatient surgical suite, pediatric hospital, and ambulatory surgery center. The 5% lower prediction bounds of surgical times (surgery or procedure start to end) were calculated from three years of historical data (October 1, 2016, to September 30, 2019) based on two-parameter lognormal distributions. The prediction bounds were compared to actual surgical start times during the next one year (October 1, 2019, to September 30, 2020). We considered the interval available for a breast milk pumping session during a case to be from 15 minutes after the start of the surgical time (to allow completion of initial documentation, other activities, and hand-off to the relieving anesthesia provider) until the end of the surgical time.Results: The lower prediction bounds were accurate, with 4.9% (4.6% - 5.2%) of future cases’ surgical times being briefer, matching the nominal 5.0% rate. Applying these bounds, approximately 39% of cases (99% confidence interval 39% - 40%) were reliably of sufficient duration for the anesthesia provider delivering care in that one operating room to receive a 30-minute break for breast milk pumping session between 15 minutes after the start of surgery and procedure end. This percentage (39%) was substantially less than the 72% of the surgical times that were observed, in retrospect, to be sufficiently long because the lower 5% prediction bounds accounted correctly for the uncertainty in the duration of each case. The observed 39% prevalence was significantly fewer than half the cases (P < 0.0001 vs. 50%) suitable for such relief.Conclusions: Individuals making operating room assignments for anesthesia providers need to consider the 5% lower prediction bounds of surgical times for cases in the room when making such assignments for women who require time for breast milk pumping sessions. Such considerations will generally result in assignments to rooms with one or more long-duration cases. Such a strategy may involve changes in preferred assignments for the anesthesia providers and alteration in the order of rotations for anesthesia residents (e.g., palliative care rotation rather than transition to practice at a pediatric ambulatory surgery center). When making room assignments for anesthesia providers who are breastfeeding, our results show that the 5% lower prediction bounds of surgical times need to be calculated; relying on the average surgical times for procedures is insufficient. Our paper also shows how to perform the mathematics using a spreadsheet program or equivalent.

Statistical analysis of monthly rainfall in Central West Brazil using probability distributions

In this study, eight probability distributions such as two-parameter Weibull, three-parameter Weibull, two-parameter Rayleigh, two-parameter Gamma, three-parameter Gamma, two-parameter Lognormal, three-parameter lognormal and maximum Gumbel were applied for modelling the monthly rainfall data of the pluviometric station of Campo Grande, MS, Brazil, during the period from 1975 to 2013. The rainfall data were obtained from National Water Agency (ANA) ( https://www.ana.gov.br ). Parameters of these distributions were estimated using the maximum likelihood estimation method. Six goodness of fit tools such as chi-squared test, Kolmogorov–Smirnov test, Akaike information criterion, Bayesian information criterion, root mean square error and coefficient of determination were used to identify the best fitted probability distribution. The goodness of fit tools indicated that although no distribution provides the best fit to the rainfall data for all months, the three-parameter lognormal distribution shows generally better fit than the other distributions. The two-parameter lognormal distribution has the worst fit among the distributions.

Influence of polyester microfiber reinforcement on flexural fatigue characteristics of concrete

The concrete with polyester microfiber is being investigated for the behaviour of flexural strength under static loading. But their behaviour under dynamic loading is unclear and hence there is a need for a comprehensive study. The fatigue performance of concrete with polyester microfiber (0.0, 0.9, 2.0, and 3.0 kg/m3 by volume of concrete) under flexural loading is studied. The cyclic loads are applied at constant maximum stress levels of 80, 85, and 90%, whereas the minimum stress level is fixed at 2% of the static flexural strength of the concrete. The stress level vs. fatigue life (S-N) curves are formulated to observe the effectiveness of microfiber on fatigue performance, follow the logarithmic relationship. Under all the stress levels, concrete with fibre dosage of 0.9 kg/m3 is shown pronouncing effect on fatigue lives compared to concrete with other fibre dosages. The fatigue results are in better agreement with two-parameter Weibull and lognormal distributions. The analysis of covariance (ANCOVA) indicated that the initial stiffness and stress levels are significant in affecting fatigue lives. The unified model is proposed for damage evolution, which can be satisfactorily used to describe the damage of concrete.

Modelling Shot Lengths of Hollywood Motion Pictures with the Dagum Distribution

This paper demonstrates the three-parameter Dagum distribution provides a good fit for shot lengths in Hollywood films due to its ability to model a wide range of skewness and kurtosis values and a variety of tail behaviours by virtue of its two shape parameters. The fit of this distribution is better across films in the sample than the two-parameter lognormal distribution, though animated films are an important exception to this. These results can be applied to more closely replicate the editing practice of film editors when generating film sequences using automated editing software.

Automatic seismic damage identification of reinforced concrete columns from images by a region-based deep convolutional neural network

This paper proposed a modified faster region-based convolutional neural network (faster R-CNN) for the multitype seismic damage identification and localization (i.e., concrete cracking, concrete spalling, rebar exposure, and rebar buckling) of damaged reinforced concrete columns from images. Four hundred raw images containing different damages and complicated background information are taken by a consumer-grade camera in various locations and arbitrary perspectives to simulate the diverse situations where real-world postearthquake damaged structural images are taken by nonprofessionals. Rectangular bounding boxes are obtained to localize multitype structural damages along with the corresponding category labels and classification probabilities. Data augmentation is implemented by rotation at every 90°, vertical and horizontal flipping operations. An interactive labeling process for the ground-truth regions of the aforementioned damages is performed by a semiautomatic MATLAB program. A four-step alternating training procedure is adopted on the basis of the mini-batch stochastic gradient decent algorithm with momentum by backpropagation. Test results show that the trained faster R-CNN can automatically identify and localize the aforementioned multitype seismic damages and the overall average precision reaches 80%. The relative errors of coordinates of the left-top point obey minimum extreme value distributions, and those of width and height obey three-parameter lognormal distributions. The intersection ratio between the identification and ground truth has a mean value of 0.88, and the width–height ratio obeys a two-parameter lognormal distribution. Updated convolutional kernels in the first layer have shown trending, focusing, and line detectors for the feature extraction of multitype damages. Trending and focusing detectors contribute to the recognition of local damage regions, for example, concrete spalling and rebar exposure, whereas line detectors are more sensitive to the segmentation geometry, that is, concrete cracks.

Monte Carlo comparison of estimation methods for the two-parameter lognormal distribution

In the paper we compare performance of estimation methods for the two-parameter lognormal distribution via the Monte Carlo simulation. The comparison of performances is made with respect to their biases, variances, root mean square error. The methods are applied on real data set representing experimentally obtained values of ultimate tensile strength of material.

Damage states of yielding and collapse for elevated water tanks supported on RC frame staging

Elevated water tanks are inverted pendulum type structures where drift limit is an important criterion for seismic design and performance evaluation. Explicit drift criteria for elevated water tanks are not available in the literature. In this study, probabilistic approach is used to determine maximum drift limit for damage state of yielding and damage state of collapse for the elevated water tanks supported on RC frame staging. The two damage states are defined using results of incremental dynamic analysis wherein a total of 2160 nonlinear time history analyses are performed using twelve artificial spectrum compatible ground motions. Analytical fragility curves are developed using two-parameter lognormal distribution. The maximum allowable drifts corresponding to yield and collapse level requirements are estimated for different tank capacities. Finally, a single fragility curve is developed which provides maximum drift values for the different probability of damage. Further, for rational consideration of the uncertainties in design, three confidence levels are selected and corresponding drift limits for damage states of yielding and collapse are proposed. These values of maximum drift can be used in performance-based seismic design for a particular damage state depending on the level of confidence.

Estimation on Reliability Models of Bearing Failure Data

The failure data of bearing products is random and discrete and shows evident uncertainty. Is it accurate and reliable to use Weibull distribution to represent the failure model of product? The Weibull distribution, log-normal distribution, and an improved maximum entropy probability distribution were compared and analyzed to find an optimum and precise reliability analysis model. By utilizing computer simulation technology and k-s hypothesis testing, the feasibility of three models was verified, and the reliability of different models obtained via practical bearing failure data was compared and analyzed. The research indicates that the reliability model of two-parameter Weibull distribution does not apply to all situations, and sometimes, two-parameter log-normal distribution model is more precise and feasible; compared to three-parameter log-normal distribution model, the three-parameter Weibull distribution manifests better accuracy but still does not apply to all cases, while the novel proposed model of improved maximum entropy probability distribution fits not only all kinds of known distributions but also poor information issues with unknown probability distribution, prior information, or trends, so it is an ideal reliability analysis model with least error at present.

Determining the Parameters of the Intermediate Duration - Intensity Equation for Mediterranean Region, Turkey

The aim of this study is to determine the parameter estimates of the general intensity-duration-frequency (IDF) formula for the 20 rainfall stations in the Mediterranean Region of Turkey. The maximum annual rainfall data were statistically analyzed for the durations 1, 2, 3, 4, 5, 6, 8, 12, 18 and 24 hours. Generalized Extreme Value (GEV), Gumbel, Normal, two-parameter Lognormal, three-parameter Lognormal, Gamma, Pearson type-III and Log-Pearson type-III distributions were considered and compared using Chi square (χ2) goodness-fit test. Parameters of these distributions were estimated using both maximum likelihood and moment methods. The parameters of the general IDF equation i = (A* TB)/(tC + D) and coefficient of correlation (R) for different return periods (2, 5, 10, 25, 50, 75 and 100 years) were calculated by using nonlinear estimation method. Coefficient of correlation result (R = 0,875) of general IDF equation showed the suitability for intermediate durations in the Mediterranean Region.