Lognormal Density Function Research Articles

Stationary distribution and probability density function of a stochastic COVID-19 infections model with general incidence

With the sudden outbreak of COVID-19 all over the world, vaccination was one of the most effective treatments. In this paper, a stochastic COVID-19 model with vaccination is developed. We first prove that the stochastic model has a unique global solution by constructing the corresponding Lyapunov function. Then, sufficient condition for ergodic stationary distribution υ(⋅) is provided under the condition R0s>1. Moreover, by applying the six-dimensional Fokker–Planck equation, we obtain the explicit expression of the probability density function near quasi-local equilibrium Q∗, which has the form of lognormal density function. Finally, numerical simulations are presented to verify the above conclusions.

Dynamical behavior and density function of a stochastic model of HPV infection and cervical cancer with a case study for Xinjiang, China

In this article, combining the transmission features of HPV infection and secondary cervical cancer in Xinjiang, China, a stochastic dynamical model for the HPV infection and secondary cervical cancer with environmental white noise is proposed. Firstly, the stochastic extinction of disease is investigated. A sufficient criterion for the asymptotic behavior of any positive solution of stochastic model revolving around the disease-free equilibrium of corresponding deterministic model is established. Secondly, a threshold criterion for the existence of unique ergodic stationary distribution is obtained by means of the auxiliary function. Furthermore, a new technique of partitioned matrix for the calculation of probability density function is proposed, the expression of a log-normal density function around the quasi-endemic equilibrium of stochastic model is calculated. Lastly, the best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the cervical cancer data in Xinjiang province, China. The basic reproduction number is estimated as 1.3496 (95% CnI: (1.3458, 1.3716)). Then, to determine the key parameters of the model, the sensitivity analysis is explored. Some possible interventions and control measures are provided to reduce the HPV infection spread and cervical cancer in Xinjiang of China.

A hypothesis to derive the shape of the dose\u2013response curve for teratogenic radiation effects

Reports of adverse pregnancy outcomes after in utero exposure to very low levels of ionizing radiation are inconsistent with a threshold dose of 100 mSv for teratogenic effects in humans. In the present study, it is hypothesized that the shape of the dose–response relationship for teratogenic effects is a cumulative lognormal distribution without threshold. This hypothesis relies on the assumption that both doses and radiosensitivities in human populations exposed to ionizing radiation are random variables, modeled by lognormal density functions. Here, radiosensitivity is defined as the dose limit up to which radiation damage can be repaired by the cellular repair systems, in short, the repair capacity. Monte Carlo simulation is used to generate N pairs of individual doses and repair capacities. Radiation damage occurs whenever the dose exceeds the related repair capacity. The rate of radiation damage is the number of damages, divided by the number N of pairs. Monte Carlo simulation is conducted for a sufficient number of ascending median doses. The shape of the dose–response relationship is determined by regression of damage rates on mean dose. Regression with a cumulative lognormal distribution function yields a perfect fit to the data. Acceptance of the hypothesis means that studies of adverse health effects following in-utero exposure to low doses of ionizing radiation should not be discarded primarily because they contradict the concept of a threshold dose for teratogenic effects.

Effect of cooling rate on the size fluctuation of V-containing phases in Al-V master alloys

To investigate the relationship between the solidification cooling rate and the size fluctuation of different V-containing phases, Al-4 wt. % V master alloys with various V-containing constituents were prepared at different cooling rates carried out by a wedge-shaped water-cooled copper mold, a graphite mold and a refractory mold at 1050°C, respectively. The variance of lognormal density function was used to characterize the size fluctuation of V-containing phases quantitatively. The results show that the largest size difference of both Al10V phases and Al3V phases is simultaneously present in the ingot prepared by the average solidification cooling rate of 2°C·s-1, while the smallest difference of Al10V phases in size is present in the ingot prepared by the average solidification cooling rate of 271°C·s-1. The size fluctuation of the Al3V phases first increases slightly and then decreases with increasing average solidification cooling rate in the range of 30~195°C·s-1.

Integration of NMR and Conventional Logs for Vuggy Facies Classification in the Arbuckle Formation: A Machine Learning Approach

Summary Diagenetic effects in carbonate rocks can enhance or occlude depositional pore space. Reliable identification of porosity-enhancing diagenetic features (e.g., vugs and fractures) is essential for petrophysical characterization of reservoir properties (e.g., porosity and permeability), construction of geological and reservoir models, reserve estimation, and production forecasting. Challenges remain in characterizing these diagenetic features from well logs as they are often mixed with variations in mineral and fluid concentrations. Herein, we explore a data-driven approach that is based on a comprehensive well log data set from the Arbuckle Formation in Kansas to classify vuggy facies in carbonate rocks. The available well log data include conventional logs (gamma ray (GRTC), resistivity (RT), neutron/density porosity (NPHI/RHOB), photoelectric factor (PE), and acoustic slowness) and nuclear magnetic resonance (NMR) transverse relaxation time (T2) logs. We parameterized the measured T2 distribution using a multimodal lognormal Gaussian density function and combined the resulting Gaussian parameters with conventional logs as inputs into three supervised machine learning (ML) algorithms; namely, support vector machine (SVM), random forest (RF), and artificial neural network (ANN). The facies labeling data used in this study were based on visual examination of vug sizes from core samples, which include five classes; namely, nonvuggy, pinpoint-size, centimeter-size, fist-size, and super-vuggy. In total, 80% of the data set was used as the training set, and a fivefold cross validation was used for hyperparameter tuning. We conducted a detailed comparison of the above three ML algorithms on the basis of different combinations of features. The highest classification accuracy achieved on the holdout testing set is 84% using SVM on a combination of conventional logs and selected NMR Gaussian parameters as inputs. In general, inclusion of conventional log data improves the prediction accuracy compared with using NMR data alone. Feature selection improves the performance for SVM and ANN but is not recommended for RF.

Experimental Investigation of Stochastic Mechanical Behavior of Cement Emulsified Asphalt Mortar under Monotonic Compression

Experimental investigation on cement emulsified asphalt mortar (CA mortar) under uniaxial monotonic compression by taking into account the stochastic properties were investigated. An analytical constitutive model based on the statistic damage approach capable of mimicking the stochastic mechanical responses of CA mortar under uniaxial compression was proposed. The comparison between the experimental results and the predictions demonstrated that the proposed model was able to characterize the salient features for CA mortar under uniaxial monotonic compression. Furthermore, the compressive stochastic evolution (SE) of CA mortar tested in this work and comparative analyses among typical China Railway Track System-I (CRTS-I) type CA mortar and concrete in several aspects were examined and performed; it was revealed that the Lognormal distribution density function can well represent the damage probability density for CA mortar, and its stochastic constitutive relationship can be reflected by a media process of transition from microscale to macroscale.

Simulating multimodal floc size distributions of suspended cohesive sediments with lognormal subordinates: Comparison with mixing jar and settling column experiments

The Floc Size Distributions (FSDs) of suspended fine-grained sediment flocs play a prime role to estimate their own fate and the transport of contaminates attached to the flocs. However, developing an efficient flocculation model that is capable of simulating continuous and multimodal FSDs is still a challenge. Recently, the population balance equation solved by the Quadrature-Based Method of Moments (QBMM) with lognormal kernel density functions has been developed to investigate the aggregation and breakage processes. It coincides with some recent observations which describe a measured FSD in coastal waters with a set of constituted lognormal distributions. The newly developed lognormal QBMM was tested with several ideal flocculation kinetic kernels, none of which, however, was used for interpreting cohesive sediment dynamics. Therefore, it raised our interest to evaluate the model performance for fine-grained sediments in shear turbulence dominated environments. In this study, additional validations against two kaolinite laboratory experiments were tested in the framework of the extended QBMM. It is hypothesized that these subordinate lognormal distributions share the same value of standard deviation. Different from the previous methods, the common standard deviation is determined empirically to reduce the number of tracers and better represent the FSDs. With sediment flocculation kinetics, the predicted FSDs reasonably reproduce the FSDs observed in both the mixing chamber and the settling column experiments. Despite the lacking of explicit descriptions of microbial effects at the current stage, this model has the potential to be implemented into large-scale particle transport models and deserves a more in-depth study in the future.

Application of continuous normal–lognormal bivariate density functions in a sensitivity analysis of municipal solid waste landfill

The variability of untreated municipal solid waste (MSW) shear strength parameters, namely cohesion and shear friction angle, with respect to waste stability problems, is of primary concern due to the strong heterogeneity of MSW. A large number of municipal solid waste (MSW) shear strength parameters (friction angle and cohesion) were collected from published literature and analyzed. The basic statistical analysis has shown that the central tendency of both shear strength parameters fits reasonably well within the ranges of recommended values proposed by different authors. In addition, it was established that the correlation between shear friction angle and cohesion is not strong but it still remained significant. Through use of a distribution fitting method it was found that the shear friction angle could be adjusted to a normal probability density function while cohesion follows the log-normal density function. The continuous normal–lognormal bivariate density function was therefore selected as an adequate model to ascertain rational boundary values (“confidence interval”) for MSW shear strength parameters. It was concluded that a curve with a 70% confidence level generates a “confidence interval” within the reasonable limits. With respect to the decomposition stage of the waste material, three different ranges of appropriate shear strength parameters were indicated. Defined parameters were then used as input parameters for an Alternative Point Estimated Method (APEM) stability analysis on a real case scenario of the Jakusevec landfill. The Jakusevec landfill is the disposal site of the capital of Croatia – Zagreb. The analysis shows that in the case of a dry landfill the most significant factor influencing the safety factor was the shear friction angle of old, decomposed waste material, while in the case of a landfill with significant leachate level the most significant factor influencing the safety factor was the cohesion of old, decomposed waste material. The analysis also showed that a satisfactory level of performance with a small probability of failure was produced for the standard practice design of waste landfills as well as an analysis scenario immediately after the landfill closure.

An extended quadrature-based moment method with log-normal kernel density functions

An extended quadrature method of moments (EQMOM) with log-normal kernel density functions is developed in this work, and applied to the solution of a population balance equation (PBE) for aggregation and breakup, coalescence, and condensation problems. The cases with one and two kernel density functions are studied analytically, and the existence of an analytical solution is shown. A numerical procedure based on the work of Yuan et al. (2012) is adopted to address cases with a larger number of kernel density functions. Results for the reconstructed number density function (NDF), the time evolution of the zero-order moment and of the mean particle size are compared with those obtained from the rigorous solution of the PBE reported by Vanni (2000) for the cases of aggregation and breakup. A problem concerning coalescence and one regarding condensation, both with analytical solution, are also examined. The results obtained with the proposed approach are compared to those provided by EQMOM with gamma kernel densities. Satisfactory results were obtained for the reconstructed distribution. Excellent agreement was observed between the rigorous solution and the approximated one for the time evolution of the total number density and the mean particle size.

Closed‐Form Model for Hydraulic Properties Based on Angular Pores with Lognormal Size Distribution

We previously presented a mathematical model for hysteretic soil hydraulic properties based on bundles of angular pores. Upscaling to sample‐scale hydraulic properties was done analytically, assuming a Gamma density distribution of pore sizes. The required parameter constraint for getting the closed‐form solution limited the model's applicability for soils with wide pore‐size distributions. The objective of this study was to improve the model by developing a new analytical solution based on a lognormal density function of the pore‐size distribution. Closed‐form expressions were derived for liquid retention, saturated–unsaturated hydraulic conductivity, liquid–vapor interfacial area, and specific surface area at the sample scale. The new model was applied to retention and conductivity data for three soils with different pore‐size distributions. The results show that the previous model is suitable only for sandy materials, whereas the new one describes the soil hydraulic functions of a wider range of soil types quite well.

Abstract 439: Mathematical modeling of cellular dose-response for radiation and radiation-drug combinations including cell cycle effects.

Abstract Thirty-six in vitro radiation dose-response data sets having 7 or more points, covering a variety of cell lines, were used to test four proposed models as alternatives to the linear-quadratic equation: the single-mechanism lognormal, the two-mechanism lognormal, the single-mechanism Hill and the two-mechanism Hill. All four models are forms of our general “damage model,” developed originally for drug dose-response. The lognormal models relate survival to damage using the cumulative lognormal density function, whereas the Hill models use a Hill equation. The models were ranked according to the Akaike Information Criterion, with the single-mechanism lognormal performing the best overall. A subset of the data showing 2-mechanism behavior was identified using the F-test method; for this subset, the two-mechanism lognormal model was superior. The lognormal model is therefore proposed as an alternative to the linear-quadratic. The lognormal damage model is then extended to the case of radiochemotherapy using the concept of additive damage, which we previously developed for drugs with two mechanisms, and then applied to combination exposure. Previously, it was found that fixed-schedule, varying-dose exposures to drug and radiation could be described by the additive damage model. Here, the case of varying schedule is considered, for gemcitabine, which showed schedule-dependent radiosensitization in studies by Pauwels et al (2003, 2009). Cell cycle effects are included through a kinetic equation describing the evolution of the population distribution over the cell cycle. In contrast to some previous models where discrete phases of the cell cycle are treated as compartments, here the distribution is a continuous density function. Gemcitabine is assumed to cause a block at a single point, with dose-dependent duration. The cell cycle phase population density distribution then enters into the radiation damage term. The model thus includes radiation-drug interaction in two ways: through additive damage and through cell cycle effects. The case of fractionated doses of gemcitabine alone is also considered. Overall, the model provides a framework for understanding the complex effects of schedule and dose, as well as a method to potentially predict combination or fractionated dose-response from knowledge of a drug's perturbations to the cell cycle phase distribution. It is expected that these models will have application in preclinical optimization of combination therapy, as well as in computational simulation of anticancer therapy. Citation Format: Katherine S. Williams, Ardith W. El-Kareh, Timothy W. Secomb. Mathematical modeling of cellular dose-response for radiation and radiation-drug combinations including cell cycle effects. [abstract]. In: Proceedings of the 104th Annual Meeting of the American Association for Cancer Research; 2013 Apr 6-10; Washington, DC. Philadelphia (PA): AACR; Cancer Res 2013;73(8 Suppl):Abstract nr 439. doi:10.1158/1538-7445.AM2013-439

A Simple Discrete Approximation for the Renewal Function

Abstract Background: The renewal function is widely useful in the areas of reliability, maintenance and spare component inventory planning. Its calculation relies on the type of the probability density function of component failure times which can be, regarding the region of the component lifetime, modelled either by the exponential or by one of the peak-shaped density functions. For most peak-shaped distribution families the closed form of the renewal function is not available. Many approximate solutions can be found in the literature, but calculations are often tedious. Simple formulas are usually obtained for a limited range of functions only. Objectives: We propose a new approach for evaluation of the renewal function by the use of a simple discrete approximation method, applicable to any probability density function. Methods/Approach: The approximation is based on the well known renewal equation. Results: The usefulness is proved through some numerical results using the normal, lognormal, Weibull and gamma density functions. The accuracy is analysed using the normal density function. Conclusions: The approximation proposed enables simple and fairly accurate calculation of the renewal function irrespective of the type of the probability density function. It is especially applicable to the peak-shaped density functions when the analytical solution hardly ever exists.

Simple Stochastic Model for Planning the Inventory of Spare Components Subject to Wear-out

Simple Stochastic Model for Planning the Inventory of Spare Components Subject to Wear-out We treat an industrial system which comprises of a number of identical components subject to wear-out. To support the system maintenance an appropriate inventory of spare components is needed. In order to plan the sufficient inventory of spare components, two variants of a simple stochastic model are developed. In both variants, the aim is to determine how many spare components are needed at the beginning of a planning interval to meet demand for corrective replacements during this interval. Under the first variant the acceptable probability of spare shortage during the planning interval is chosen as a decision variable. While in the second variant the adequate spare inventory level is assessed by taking into account the expected number of component failures within the planning interval. A comparison of both variants of the model shows that calculations involved in the second variant are simpler. However, it can only be used when the inventory of spare components can be planned for a relatively long period of time. The determination of an adequate number of spare components according to both variants of our model depends on the form of the probability density function of component failure times. Since the components are subject to wear-out, this function exhibits a peak-shaped form that can be described by different statistical density functions. Advantages and disadvantages of using the normal, lognormal, Weibull, and Gamma density function in our model are discussed. Among the probability density functions studied, the normal density function is found to be the most appropriate for calculations in our model. The applicability of both variants of the model is given through numerical examples using field data on electric locomotives of Slovenian Railways.

Pore‐Scale Simulations to Determine the Applied Hydrodynamic Torque and Colloid Immobilization

Values of the applied hydrodynamic torque (Tapplied) and the resisting adhesive torque (Tadhesion) will determine whether a colloid will be immobilized (Tapplied ≤ Tadhesion) or roll (Tapplied > Tadhesion) on a solid water interface. Previous literature has demonstrated in 1–2 collector (grain) systems that the influence of Tapplied on colloid retention can be significant under unfavorable attachment conditions and that only a fraction of the solid surface may contribute to retention. However, many questions remain on how to obtain, analyze, and upscale information on the forces and torques that act on colloids near solid surfaces in porous media. To address some of these gaps in knowledge, high resolution pore‐scale water flow simulations were conducted for sphere packs (25 spheres) over a range of Darcy velocities, grain sizes and distributions, and porosities. The spatial variability of Tapplied was calculated from this information, and successfully described using a lognormal cumulative density function (CDF). Linear interpolation and scaling techniques were subsequently used to predict the lognormal CDF of Tapplied for various colloid sizes, grain sizes and distributions, and water velocities. The lognormal CDF of Tapplied was then evaluated at select values of Tadhesion (i.e, interaction energy) to quantify the fraction and locations on the solid surface that contributes to colloid retention (Sf), and the theoretical maximum solid phase concentration of retained colloids (Smax).

Empirical Study of Six Tests for Equality of Populations with Zero-Inflated Continuous Distributions

We evaluated the properties of six statistical methods for testing equality among populations with zero-inflated continuous distributions. These tests are based on likelihood ratio (LR), Wald, central limit theorem (CLT), modified CLT (MCLT), parametric jackknife (PJ), and nonparametric jackknife (NPJ) statistics. We investigated their statistical properties using simulated data from mixed distributions with an unknown portion of non zero observations that have an underlying gamma, exponential, or log-normal density function and the remaining portion that are excessive zeros. The 6 statistical tests are compared in terms of their empirical Type I errors and powers estimated through 10,000 repeated simulated samples for carefully selected configurations of parameters. The LR, Wald, and PJ tests are preferred tests since their empirical Type I errors were close to the preset nominal 0.05 level and each demonstrated good power for rejecting null hypotheses when the sample sizes are at least 125 in each group. The NPJ test had unacceptable empirical Type I errors because it rejected far too often while the CLT and MCLT tests had low testing powers in some cases. Therefore, these three tests are not recommended for general use but the LR, Wald, and PJ tests all performed well in large sample applications.

Generalised storage-yield-reliability modelling: Independent validation of the Vogel–Stedinger (V–S) model using a Monte Carlo simulation approach

Summary The generalised storage-yield-reliability model by Vogel and Stedinger (V–S) is widely used in the literature but there has never been a complete validation of the model with measured river runoff data. This study investigated the problem using three different river runoff data records within a Monte Carlo simulation framework. The V–S model is predicated on the assumption that the probability distribution of reservoir capacity can be described by the 3-parameter log-normal density function. The results showed that the V–S model over-predicted reservoir capacity quantiles for most yield (or demand) ratios but especially so for very high (>0.8) yield ratios. Further investigations revealed that the poor performance of the V–S model was caused by the huge errors associated with its expression for two of the three parameters of the 3-parameter log-normal distribution, i.e. the variance and the lower limit, although the mean of the distribution was relatively well predicted.

Determination of Adhesion between Single Aspergillus niger Spores in Aqueous Solutions Using an Atomic Force Microscope

The interaction force between single cells in contact is of high interest in various interdisciplinary fields of biotechnology, for instance, in cultivation or biofilm formation. A method for the determination of adhesion forces between two single Aspergillus niger spores in different aqueous solutions was established in this study. Adhesion force distributions were determined at three different sodium chloride concentrations and two different pH values using an atomic force microscope (AFM). It was pointed out that adhesion data can be described by log-normal density functions, of which corresponding parameters have been estimated. Using the knowledge of distribution shape, the influence of the environmental condition on the mean values of adhesion force could be studied quantitatively. The highest value of 0.95 nN was observed at pH 2.5 and an ionic strength of 0.5 mol L(-1). Decreasing the ionic strength to 0.05 mol L(-1) decreases the adhesion force mean for about 25%. Increasing the pH value to pH 5 at a sodium chloride concentration of 0.154 mol L(-1) entails a decrease of adhesion from 0.88 to 0.56 nN. These results qualitatively agree with the absolute value of the expected surface potential of Aspergillus niger spores, which is much higher at pH 5 and should take more effect at lower concentrations of counterions.

海氷の一軸圧縮強度と荷重ピーク値の確率分布について

The strength of natural sea ice has a lot of scatter due to variations in its growing process. For engineering purposes, it is necessary to understand this strength randomness. The uniaxial compressive strength of natural sea ice at Lake Saroma was analyzed statistically to find an appropriate probability density function that would fit the strength data. Furthermore, peaks occurring in subsequent ice load data obtained in medium scale field indentation tests carried out by the JOIA project were analyzed statistically, since it had been found that each peak load was due to flaking failure of ice in an independent ice failure zone. From a Chi square test for goodness of fit, it was found that both uniaxial compressive strength and peak load fit a log-normal density function with 3 parameters. From this, a method for generating strength and peaks of ice load based on ice failure in the brittle condition can be developed.

Geometry and Topology of 3D Voronoi Partitions

A new algorithm to determine Voronoi partitions of space using periodic boundary conditions has been implemented in GAMGI free software package. This algorithm was used to study the topological and geometric properties of Voronoi polyhedra obtained for Poisson and Laguerre seeding distributions. The Random Sequential Adsorption (RSA) method was used to generate a Laguerre seeding distribution with a volume fraction of spheres φ ≈ 0.36. The resulting normalised cell volume distributions were compared with the gamma and lognormal density functions using χ2 goodness of fit testing, and in both cases the gamma function provides a better description. The Lewis and Aboav empirical relations approximately apply to both partitions, with least square correlation factors R2 equal to 0.98 for the Lewis law and 0.95 for the Aboav law.

The functional form of the lognormal distribution as a sum of decaying exponential and sinusoidal terms applied to the isometric pinch force of human fingers

The transient data of the pinch force produced between the human forefinger and thumb have been shown to fit the functional form of the well-known lognormal density function. Isometeric force generation is achieved by the stochastic recruitment of individual motor units, which sum together. Evidence from animal and human experiments demonstrates that the force generation can be modelled by underdamped terms. It is shown that a lognormal time series (distribution) can be fitted to a sum of exponential decaying sinusoidal terms.