Earthquake Magnitude Probability and Gamma Distribution

The earthquake magnitude probability distribution is one of the underlying input data for certain earthquake analyses, such as probabilistic seismic hazard analysis. Nowadays, the method proposed by McGuire and Arabasz (1990) is commonly used for obtaining the (simulated) earthquake magnitude probability distributions. However, based on the observed earthquake data in 5 regions (Taiwan, Japan, California, Turkey, and Greece), the model did not fit the observation well. Instead, all of the case studies show that using the newly proposed gamma distribution can improve the simulation significantly compared to the conventional method.

One-sided empirical Bayes test for location parameter in Gamma distribution

In this paper, we devote to constructing the one-sided empirical Bayes(EB) test for the location parameter in the Gamma distribution by nonparametric method. Under some mild conditions, we prove that the EB test is asymptotically optimal with the rate of the order O(n^(-δs/(2s+1))), where 1/2 ≤ δ < 1 and s > 1 is a given natural number. An example is also given to illustrate that the conditions of the main theorems are easily satisfied.

Reconstruction of Probability Density Function for Gamma Distribution

The probability distributions of small sample data are difficult to determine,while a large proportion of samples occur in the early failure period,so it is particularly important to make full use of these data in the statistical analysis.Based on gamma distribution,four methods of probability density function(PDF)reconstruction with early failure data are proposed,and then the mean time between failures(MTBF)evaluation expressions are concluded from the reconstructed PDFs.Both theory analysis and an example show that method 2 is the best evaluation method in dealing with early-failure-small-sample data.The reconstruction methods of PDF also have certain guiding significance for other distribution types.

Designing Bayesian Two-Sided Group Chain Sampling Plan for Gamma Prior Distribution

Acceptance sampling is used to decide either the whole lot will be accepted or rejected,based on inspection of randomly sampled items from the same lot.As an alternative to traditional sampling plans,it is possible to use Baye-sian approaches using previous knowledge on process variation.This study pre-sents a Bayesian two-sided group chain sampling plan(BTSGChSP)by using various combinations of design parameters.In BTSGChSP,inspection is based on preceding as well as succeeding lots.Poisson function is used to derive the probability of lot acceptance based on defective and non-defective products.Gamma distribution is considered as a suitable prior for Poisson distribution.Four quality regions are found,namely:(i)quality decision region(QDR),(ii)probabil-istic quality region(PQR),(iii)limiting quality region(LQR)and(iv)indifference quality region(IQR).Producer’s risk and consumer’s risk are considered to esti-mate the quality regions,where acceptable quality level(AQL)is associated with producer’s risk and limiting quality level(LQL)is associated with consumer’s risk.Moreover,AQL and LQL are used in the selection of design parameters for BTSGChSP.The values based on all possible combinations of design parameters for BTSGChSP are presented and inflection points’values are found.Thefinding exposes that BTSGChSP is a better substitute for the existing plan for industrial practitioners.

Tuning mechanical behaviors of highly entangled hydrogels with the random distribution of mobile entanglements

Highly entangled hydrogels exhibit excellent mechanical properties,including high toughness,high stretchability,and low hysteresis.By considering the evolution of randomly distributed entanglements within the polymer network upon mechanical stretches,we develop a constitutive theory to describe the large stretch behaviors of these hydrogels.In the theory,we utilize a representative volume element(RVE)in the shape of a cube,within which there exists an averaged chain segment along each edge and a mobile entanglement at each corner.By employing an explicit method,we decouple the elasticity of the hydrogels from the sliding motion of their entanglements,and derive the stress-stretch relations for these hydrogels.The present theoretical analysis is in agreement with experiment,and highlights the significant influence of the entanglement distribution within the hydrogels on their elasticity.We also implement the present developed constitutive theory into a commercial finite element software,and the subsequent simulations demonstrate that the exact distribution of entanglements strongly affects the mechanical behaviors of the structures of these hydrogels.Overall,the present theory provides valuable insights into the deformation mechanism of highly entangled hydrogels,and can aid in the design of these hydrogels with enhanced performance.

Probability Distribution of Summer Daily Precipitation in the Huaihe Basin of China Based on Gamma Distribution

The probability distribution of precipitation in the Huaihe basin （HB） is analyzed with the shape and scale parameters of a Gamma distribution. The summer daily precipitation records of 158 meteorological raingauges are applied over the HB during the period of 1980-2007, and the precipitation samples are classified into unconditional rainy days and conditional rainy days which have a dry or wet preceding day over the years. The Kolmogorov-Smirnov （K-S） test and the comparison between the Gamma distribution probability density function and the sample frequency of daily precipitation records of five representative stations are conducted and analyzed. The results show that the HB is a ＂scale-dominated＂ region characterized with large scale parameters of the Gamma distribution, where rainfall is likely to exhibit large variability leading to extreme wet or dry conditions. Fklrther analysis shows that the confluence area of Sha River and Ying River within the stream between Wangjiaba dam and Bengbu station, the eastern branch of the Huaihe River （HR） between Bengbu station and the Hongze Lake, and the downstream area below the Hongze Lake, are all the areas with a high probability of large precipitation under the condition that the rainy day has a dry preceding day. The eastern part of the Yishu River watershed and the region near Wangjiaba dam are the center of a high probability of large precipitation under the condition that the rainy day has a wet preceding day. Moreover, the day following a dry preceding day has a greater probability of small rainfall. The probability distribution of summer daily precipitation of the HB is significantly skewed. The probability distribution could be more applicable if the rainy days are preceded by a dry or wet day.

Uniformly Minimum-Variance Unbiased Estimator (UMVUE) for the Gamma Cumulative Distribution Function with Known and Integer Scale Parameter

Uniformly minimum-variance unbiased estimator (UMVUE) for the gamma cumulative distribution function with known and integer scale parameter. This paper applies Rao-Blackwell and Lehmann-Scheffeé Theorems to deduce the uniformly minimum-variance unbiased estimator (UMVUE) for the gamma cumulative distribution function with known and integer scale parameters. The paper closes with an example comparing the empirical distribution function with the UMVUE estimates.

Statistical Characteristics of Raindrop Size Distribution in the Tibetan Plateau and Southern China

The characteristics of raindrop size distribution （DSD） over the Tibetan Plateau and southern China are studied in this paper, using the DSD data from April to August 2014 collected by HSC-PS32 disdrometers in Nagqu and Yangjiang, com- prising a total of 9430 and 63661-rain raindrop spectra, respectively. The raindrop spectra, characteristics of parameter variations with rainfall rate, and the relationships between reflectivity factor （Z） and rainfall rate （R） are analyzed, as well as their DSD changes with precipitation type and rainfall rate. The results show that the average raindrop spectra appear to be one-peak curves, the number concentration for larger drops increase significantly with rainfall rate, and its value over southern China is much higher, especially in convective rain larger drops, especially for convective rain in southern China. Standardized Gamma distributions better describe DSD for All three Gamma parameters for stratiform precipitation over the Tibetan Plateau are much higher, while its shape parameter （,u） and mass-weighted mean diameter （Dm）, for convective precipitation, are less. In terms of parameter variation with rainfall rate, the normalized intercept parameter （Nw） over the Tibetan Plateau for stratiform rain increases with rainfall rate, which is opposite to the situation in convective rain. The/1 over the Tibetan Plateau for stratiform and convective precipitation types decreases with an increase in rainfall rate, which is opposite to the case for Dm variation. In Z-R relationships, like ＂Z = ARb＇＇, the coefficient A over the Tibetan Plateau is smaller, while its b is higher, when the rain type transfers from stratiform to convective ones. Furthermore, with an increase in rainfall rate, parameters A and b over southern China increase gradually, while A over the Tibetan Plateau decreases sub- stantially, which differs from the findings of previous studies. In terms of geographic location and climate over the Tibetan Plateau and southern China, the precipitation in the pre-flood seasons is dominated by strong convective rain, while weak convective rain occurs frequently in northern Tibet with lower humidity and higher altitude.

A Comparison of Four Precipitation Distribution Models Used in Daily Stochastic Models

Stochastic weather generators are statistical models that produce random numbers that resemble the observed weather data on which they have been fitted; they are widely used in meteorological and hydrologi- cal simulations. For modeling daily precipitation in weather generators, first-order Markov chain–dependent exponential, gamma, mixed-exponential, and lognormal distributions can be used. To examine the perfor- mance of these four distributions for precipitation simulation, they were fitted to observed data collected at 10 stations in the watershed of Yishu River. The parameters of these models were estimated using a maximum-likelihood technique performed using genetic algorithms. Parameters for each calendar month and the Fourier series describing parameters for the whole year were estimated separately. Bayesian infor- mation criterion, simulated monthly mean, maximum daily value, and variance were tested and compared to evaluate the fitness and performance of these models. The results indicate that the lognormal and mixed-exponential distributions give smaller BICs, but their stochastic simulations have overestimation and underestimation respectively, while the gamma and exponential distributions give larger BICs, but their stochastic simulations produced monthly mean precipitation very well. When these distributions were fitted using Fourier series, they all underestimated the above statistics for the months of June, July and August.

Distribution of driving trajectory of passenger car in highway horizontal curves

In this paper the track behavior of passenger car was studied. The vehicle driving trajectory and driving direction were defined, and a classification of the type of vehicle trajectories along the curves was developed. The statistical parameters of vehicle trajectory samples in free flow and their frequency curves and cumulative frequency curves were achieved, K-S test and chi-square test were used to test normal distribution and gamma distribution for collected sample data, and the probabili- ty density functions were given. At last, dispersion degree between vehicle trajectory random varia- ble and the characteristic value of cumulative frequency curve in each key cross section in curves was analyzied. The proposed conclusion can provide theoretical support for the reasonable optimization of widen curve, design of alignment and the management of counter flow conflicts.

EXPLICIT EXPRESSIONS FOR SOME DISTRIBUTIONS RELATED TO RUIN PROBLEMS

The classical risk process that is perturbed by diffusion is studied. The explicit expressions for the ruin probability and the surplus distribution of the risk process at the time of ruin are obtained when the claim amount distribution is a finite mixture of exponential distributions or a Gamma (2, α) distribution.

Non-Maxwellian Kinetic Modelling of City Size Distribution

To extend the kinetic formulation of city size distribution introduced in [1], the non-Maxwellian kinetic modeling is introduced in the present study, in which a variable collision kernel is used in the underlying kinetic equation of Boltzmann type. By resorting to the well-known grazing asymptotic, a kinetic Fokker-Planck counterpart is obtained. The equilibrium of the Fokker-Planck equation belongs to the class of generalized Gamma distributions. Numerical test shows good fit of the generalized Gamma distribution with the city size distribution of China.

IMPULSIVE DIFFERENTIAL EQUATIONS WITH GAMMA DISTRIBUTED MOMENTS OF IMPULSES AND P-MOMENT EXPONENTIAL STABILITY

Differential tigated. We study the properties of solutions sufficient conditions for equations with impulses at random moments are set up and invescase of Gamma distributed random moments of impulses. Several are studied based on properties of Gammma distributions. Some p-moment exponential stability of the solutions are given.

More Erratic and More Extreme: Trends in Precipitation in the State of São Paulo, Brazil

The study assessed changes in rainfall variability and the frequency of extreme events (very wet and very dry) in the state of S?o Paulo, Brazil, for a 40-year period that divided into two sub-groups: 1973-1992 (P1) and 1993-2012 (P2). Data of 79 rain gauge stations were selected to represent the different climatic and geomorphological domains of the state. The annual pattern was evaluated through the scale and the shape parameters of the gamma distribution and the 95th and the 5th percentiles thresholds, the latter also employed to evaluate the seasonal spatial patterns (rainy season, Oct.-Mar. and sub-humid to dry season, Apr.-Sep.). Results showed that the average precipitation was similar in P1 and P2, but S?o Paulo evolved to a pattern of increased irregularity in the rainfall distribution, with a rise of approximately 10% in the number of extremes between 1973 and 2012, especially in the very dry occurrences, and in the north and west of the state, which are the least rainy regions. Moreover, while 55% of the evaluated rain gauges recorded more extreme wet episodes in P2, 76% registered more dry extreme episodes in the same period. Some very dry or very wet events recorded after the 40-year period evaluated were discussed in terms of the associated weather patterns and their impacts on society and attested to the validity of the results found in the quantitative assessment. The qualitative analysis indicates that if the trends of more irregular distribution of rain and increase in extreme events persist, as pointed out by the gamma and percentile analyses, they would continue to bring serious effects on the natural and social systems in the state, which is the most populous and has the strongest and most diversified economy in Brazil.

中国天气发生器的降水模拟

A stochastic model for daily precipitation simulation in China was developedbased on the framework of a 'Richardson-type' weather generator that is an important tool instudying impacts of weather/climate on a variety of systems including ecosystem and risk assessment.The purpose of this work is to develop a weather generator for applications in China. The focus ison precipitation simulation since determination of other weather variables such as temperature isdependent on precipitation simulation. A framework of first order Markov Chain with GammaDistribution for daily precipitation is adopted in this work. Based on this framework, fourparameters of precipitation simulation for each month at 672 stations all over China were determinedusing daily precipitation data from 1961 to 2000. Compared with previous works, our estimation forthe parameters was made for more stations and longer observations, which makes the weather generatormore applicable and reliable. Spatial distributions of the four parameters are analyzed in aregional climate context. The seasonal variations of these parameters at five stations representingregional differences are discussed. Based on the estimated monthly parameters at 672 stations, dailyprecipitations for any period can be simulated. A 30-year simulation was made and compared withobservations during 1971-2000 in terms of annual and monthly statistics. The results aresatisfactory, which demonstrates the usefulness of the weather generator.

Residential Parking Service Model Based on Resource Sharing

Aiming at solving pressing parking issues in the urban environment, a residential parking spaces sharing model was proposed in this study. In this model, firstly, the residential community pattern, the status of idle parking spaces, and the temporal and spatial characteristics of sharing parking had been analyzed. Next,in the convenience of modeling,medical institutions that have the most prominent parking problems were selected as the subject of study. Based on the K-S statistical analysis results and the actual parking sharing situation,it was observed that the residential parking sharing time satisfied the shifted negative exponential distribution( SNED). Finally,a probability model of shared service capacity based on the SNED and critical time condition was established. By applying the statistical analysis method,the time of vehicles passing in and out of parking spaces, the idle time of parking spaces, the shifted distribution parameters, and other important model parameters had been calibrated,which was leading to the algorithm of model. In addition,considering the feasibility of model without sufficient data,the vehicle travel probability,the stagnation rate of parking space,and the status of parking spaces were defined and the reference data were also provided. The results of case studies indicate that it is very promising to solve urban parking issues if the residential community shares its rich parking resources with adjacent commercial buildings.

Intermittency of Wind Wave Group Breaking

Knowledge on intermittency of wave breaking is so far limited to a few summary statistics, while the probability distribution of time interval between breaking events can provide a full view of intermittency. Based on a series of experiments on wind wave breaking, such probability distributions are investigated. Breaking waves within a wave group were taken as a single breaking event according to recent studies. Interval between successive wave groups with breaker is the focus of this paper. For intervals in our experiments with different fetch and wind conditions, their distributions are all skewed and weighted on small intervals. Results of Kolmogorov-Smirnov tests on time series of these intervals indicate that they all follow gamma distribution, and some are even exponential type. Average breaking-group-interval decreases with friction velocity and significant steepness until the wind is strong enough;most of them are more than 10 times the dominant wave period. Group breaking probability proposed by Babanin recently and the average number of breaking waves in wave groups are also discussed, and they are seemingly more reasonable and sensitive than traditional breaking probability defined in terms of single wave.

An Estimation Method for Kill Probability Based on Random Residence Time Series

Relations between statistical residence time series and effective shooting are analyzed in accordance with the properties of the random residence time of maneuver targets crossing shot area in a given time. An estimation method for kill probability is proposed, which solves the probability of number of residence times satisfied effective shooting in given time. Some expressions and their approximate formulae of kill probability are derived, under known the distribution of residence time series. Theoretical analysis and simulation results show that this method is suitable for evaluating the hit ability of fire system for maneuver targets in random shooting.

Characterizing neural activities evoked by manual acupuncture through spiking irregularity measures

The neural system characterizes information in external stimulations by different spiking patterns. In order to examine how neural spiking patterns are related to acupuncture manipulations, experiments are designed in such a way that different types of manual acupuncture （MA） manipulations are taken at the ＇Zusanli＇ point of experimental rats, and the induced electrical signals in the spinal dorsal root ganglion are detected and recorded. The interspike interval （ISI） statistical histogram is fitted by the gamma distribution, which has two parameters： one is the time-dependent firing rate and the other is a shape parameter characterizing the spiking irregularities. The shape parameter is the measure of spiking irregularities and can be used to identify the type of MA manipulations. The coefficient of variation is mostly used to measure the spike time irregularity, but it overestimates the irregularity in the case of pronounced firing rate changes. However, experiments show that each acupuncture manipulation will lead to changes in the firing rate. So we combine four relatively rate- independent measures to study the irregularity of spike trains evoked by different types of MA manipulations. Results suggest that the MA manipulations possess unique spiking statistics and characteristics and can be distinguished according to the spiking irregularity measures. These studies have offered new insights into the coding processes and information transfer of acupuncture.

Stochastic modeling for starting-time of phase evolution of random seismic ground motions

In response to the challenge inherent in classical high-dimensional models of random ground motions, a family of simulation methods for nonstationary seismic ground motions was developed previously through employing a wave-group propagation formulation with phase spectrum model built up on the frequency components’ starting-time of phase evolution. The present paper aims at extending the formulation to the simulation of non-stationary random seismic ground motions. The ground motion records associated with N–S component of Northridge Earthquake at the type-II site are investigated. The frequency components’ starting-time of phase evolution of is identified from the ground motion records, and is proved to admit the Gamma distribution through data fitting. Numerical results indicate that the simulated random ground motion features zeromean, non-stationary, and non-Gaussian behaviors, and the phase spectrum model with only a few starting-times of phase evolution could come up with a sound contribution to the simulation.