Zur statistischen Auswertung von mikroskopischen Plankton- und Partikelzählungen

The Poisson distribution is required for the evaluation of counting results. Its availability can be checked with the dispersion index and graphically with a probability sheet. It is shown by an example that with increasing particle density the deviation from the Poisson distribution increases and how these anomalies become visible in the dispersion index and the per-cent error of the count. For the calculation of confidence intervals, a simple approximation formula and the asymptotic behaviour with growing random sample size are represented. The phenomena of over- and under-distribution as well as of “contagion” (Neyman distribution) are mentioned, and the high-speed method of counting according to Tippett and the variance-analytical evaluation methods are referred to.     

On the relationship between the index of dispersion and Allan factor and their power for testing the Poisson assumption

Several statistical tests are available for testing the Poisson hypothesis and/or the equidispersion of a point process. The capability to discriminate between the Poissonian behaviour and more complex processes is fundamental in many areas of research including earthquake analysis, hydrology, ecology, biology, signal analysis and sociology. This study investigates the relationship between two indices often used for detecting departures from equidispersion, namely, the index of dispersion (ID) and the Allan factor (AF). Since an approximation of the sampling distribution of AF for Poisson data has been recently proposed in the literature, we perform a detailed analysis of its properties and its relationship with the asymptotic sampling distribution of ID. Moreover, the statistical power of the AF for testing the Poisson hypothesis is assessed by using an extensive Monte Carlo simulation, and the performances of AF and ID are compared. We propose a simplified version of the AF sampling distribution that does not depend on the rate of occurrence keeping the maximum errors of extreme percentiles always smaller than 2–3 %. The power study highlights that ID systematically outperforms AF for discriminating between equidispersed and under/over dispersed data. Both indices show the same lack of power for distinguishing between data drawn from equidispersed non-Poissonian distribution functions. Therefore, even though AF is a useful statistical tool for detecting the possible fractal behaviour of point processes, ID should be preferred when the analysis aims at assessing equidispersion. The lack of power for a small sample size confirms the difficulty of identifying the true nature of the occurrence process of rare events.     

Model-based geostatistical interpolation of the annual number of ozone exceedance days in the Netherlands

This paper discusses two model-based geostatistical methods for spatial interpolation of the number of days that ground level ozone exceeds a threshold level. The first method assumes counts to approximately follow a Poisson distribution, while the second method assumes a log-Normal distribution. First, these methods were compared using an extensive data set covering the Netherlands, Belgium and Germany. Second, the focus was placed on only the Netherlands, where only a small data set was used. Bayesian techniques were used for parameter estimation and interpolation. Parameter estimates are comparable due to the log-link in both models. Incorporating data from adjacent countries improves parameter estimation. The Poisson model predicts more accurately (maximum kriging standard deviation of 2.16 compared to 2.69) but shows smoother surfaces than the log-Normal model. The log-Normal approach ensures a better representation of the observations and gives more realistic patterns (an RMSE of 2.26 compared to 2.44). Model-based geostatistical procedures are useful to interpolate limited data sets of counts of ozone exceedance days. Spatial risk estimates using existing prior information can be made relating health effects to environmental thresholds.     

Theoretical and Practical Investigations on the Heterogeneous Distribution of Plankton in the Main Basin of the Saidenbach Storage Reservoir

To detect spatial heterogeneous distributions it is necessary to delimitate quantitatively the sum of all errors caused by methods from sampling to counting. This is possible by the coefficient of variation, since it constitutes also a measure of the counting error and since its dependence on the counted number or the mean density of the population is clearly defined. In the investigations the error caused by methods (reproducibility of sampling and counting) does not exceed the theoretically permitted maximum counting error (at p = 0.05). Accordingly, all results of investigations for which the coefficient of variation lies above this counting error limit can be interpreted as actual spatial heterogeneous distributions. During the full circulations of autumn and spring as well as the summer stagnation period, in the Saidenbach storage reservoir no significant differences of the phytoplankton were observed within the range of horizontal distances of appr. 100m for 50% of the cases. For zooplankton. 45% of the investigations indicated a homogeneous distribution. The evaluation of data by the patchiness-index according to Lloyd (L), the Morisita index (M) as well as the graphical comparison between variance and mean value (8²/x) do not show any remarkably different results.     

A model to consider the spatial variability of rainfall partitioning within deciduous canopy. I. Model description

In this first paper of two, a numerical simulation model capable of simulating the spatial variability of rainfall partitioning within a canopy is presented. The second paper details how the model is parameterized, and some testing of its capabilities. The first stage of the model is to derive the mature canopy structure. This is achieved through simplified individual tree structures and a random placement routine based on a modified Poisson distribution. Following this the spatial discretization for throughfall is attained as a series of layers of triangles with a tree trunk at every apex. The number of layers is derived from the leaf area index through a modified Poisson distribution. Seasonal variation in the deciduous canopy is simulated through a time vector routine. The rainfall partitioning section of the model is based upon the Rutter model which has been modified to each individual triangle layer. The main feature of this model is that it offers a method of simulating rainfall partitioning at a scale that is a function of the size of the elementary physical components (tree and leaf scale). This can be used to investigate soil moisture variations under a canopy, or to study the variations within the forest hydrological processes themselves. © 1997 John Wiley & Sons, Ltd.     

Derivation of streamflow statistics based on a filtered point process

This work presents the derivation of general streamflow cumulants from daily rainfall time series. The general streamflow cumulants can be used to compute basic streamflow statistics such as mean, variance, coefficient of skewness, and correlation coefficient. Streamflow is considered as a filtered point process where the input is a daily rainfall time series assumed to be a marked point process. The marks of the process are the daily rainfall amounts which are assumed independent and identically distributed. The number of rainfall occurrences is a counting process represented by either the binomial, the Poisson, or the negative binomial probability distribution depending on its ratio of mean to variance. The first three cumulants and the covariance function of J-day averaged streamflows are deduced based on the characteristic function of a filtered point process. These cumulants are functions of the stochastic properties of the daily rainfall process and the basin-response function representing the causal relationship between rainfall and runoff.     

辽宁地区5级以上地震复发间隔及地震概率预测模型研究

详细分析了辽宁地区三个震级档的地震复发间隔分布;基于Poisson分布,构建了辽宁地区5．0、5．5和6．0级三个震级档的Poisson概率预测模型及具有概率水平指标的三级预测预警指标。     

The first and the second e of the extreme value distribution,EV1

In the peak over threshold model resulting in the Extreme-value distribution, type I, (EV1) the firste of the distribution function is based on the Poisson number of exceedances, and the seconde arises from the Exponentially distributed magnitudes.This paper, on the one hand, generalises the Poisson model to the (positive and negative) Binomial distribution, and, on the other hand, the Exponential distribution is generalised to the Generalised Pareto distribution. Lack of fit with respect to the Poisson and Exponential distribution is measured by statistics derived from those which would be locally most powerful if the estimates of the location and scale parameter were equal to the true parameter values. Ways of combining both statistics are discussed.     

利用远震P波接收函数研究中国福建地区地壳厚度和泊松比

地壳厚度和泊松比是反映地壳结构和内部物质组成的重要参数,能够为区域构造和动力学研究提供重要依据.本文基于福建地区分布相对均匀的88个测震台2014—2017年的远震波形数据提取P波接收函数,采用H-κ叠加获得台站下方的地壳厚度和泊松比,并与该地区已有的研究结果进行对比、分析及整合,最终获得了研究区117个观测台站下方的...     

Mixed model for interoccurrence times of earthquakes based on the expectation-maximization algorithm

In this paper, the goodness-of-fit test based on a convex combination of Akaike and Bayesian information criteria is used to explain the features of interoccurrence times of earthquakes. By analyzing the seismic catalog of Iran for different tectonic settings, we have found that the probability distributions of time intervals between successive earthquakes can be described by the generalized normal distribution. This indicates that the sequence of successive earthquakes is not a Poisson process. It is found that by decreasing the threshold magnitude, the interoccurrence time distribution changes from the generalized normal distribution to the gamma distribution in some seismotectonic regions. As a new insight, the probability distribution of time intervals between earthquakes is described as a mixture distribution via the expectation-maximization algorithm.     

Robust flood statistics: comparison of peak over threshold approaches based on monthly maxima and TL-moments

ABSTRACT

Flood quantile estimation based on partial duration series (peak over threshold, POT) represents a noteworthy alternative to the classical annual maximum approach since it enlarges the available information spectrum. Here the POT approach is discussed with reference to its benefits in increasing the robustness of flood quantile estimations. The classical POT approach is based on a Poisson distribution for the annual number of exceedences, although this can be questionable in some cases. Therefore, the Poisson distribution is compared with two other distributions (binomial and Gumbel-Schelling). The results show that only rarely is there a difference from the Poisson distribution. In the second part we investigate the robustness of flood quantiles derived from different approaches in the sense of their temporal stability against the occurrence of extreme events. Besides the classical approach using annual maxima series (AMS) with the generalized extreme value distribution and different parameter estimation methods, two different applications of POT are tested. Both are based on monthly maxima above a threshold, but one also uses trimmed L-moments (TL-moments). It is shown how quantile estimations based on this “robust” POT approach (rPOT) become more robust than AMS-based methods, even in the case of occasional extraordinary extreme events.

Editor M.C. Acreman Associate editor A. Viglione     

中国大陆地震空间分布模型检验

本文以中国大陆地震目录为基础资料,以泊松模型为零假设模型,并将Neyman-Scott空间丛集过程的各子模型设为检验模型,采用K-function点过程分析法和最大似然估计法计算各模型参数,并以AIC准侧判定模型的拟合优度,来检验中国大陆地震空间分布模型.检验结果表明:泊松模型的拟合优度最差,说明地震在空间的分布不是完全随机的;广义Thomas模型的拟合优度最好,说明地震的空间分布是丛集的,可用由两个高斯核组成的广义Thomas模型较好地描述.研究结果还表明,同一研究区内,采用不同时段具有不同最小完整起始震级的地震目录计算得到的地震空间分布的丛集尺度几乎不变,这意味着地震空间丛集尺度不受小地震的控制,且可能与研究区的断层规模有关.     

A method for solving relative dispersion of the cloud droplet spectra

The relative dispersion of the cloud droplet spectra or the shape parameter is usually assumed to be a constant in the two-parameter cloud microphysical scheme, or is derived through statistical analysis. However, observations have revealed that the use of such methods is not applicable for all actual cases. In this study, formulas were derived based on cloud microphysics and the properties of gamma function to solve the average cloud droplet radius and the cloud droplet spectral shape parameter. The gamma distribution shape parameter, relative dispersion, and cloud droplet spectral distribution can be derived through solving the droplet spectral shape parameter equation using the average droplet radius, volume radius, and their ratio, thereby deriving an analytic solution. We further examined the equation for the droplet spectral shape parameter using the observational droplet spectral data, and results revealed the feasibility of the method. In addition, when the method was applied to the two-parameter cloud microphysical scheme of the Weather Research and Forecast(WRF) model to further examine its feasibility, the modeling results showed that it improved precipitation simulation performance, thereby indicating that it can be utilized in two-parameter cloud microphysical schemes.     

Random patterns of occurrence of explosive eruptions at Colima Volcano, Mexico

Analysis of the patterns of eruption occurrences may improve our understanding of volcanic processes. In this paper, the available historical data of an individual volcano, Colima in México, are used to classify its eruptions by size using the Volcanic Explosivity Index (VEI). The data shows that, if eruptions are only taken into account above a certain VEI level, the stochastic process associated with the explosive volcanic events can be represented by a non-stationary Poisson point process, which can be reduced to a homogeneous Poisson process through a transformation of the time axis. When eruptions are separated by VEI values, the occurrence patterns of each magnitude category can also be represented by a Poisson distribution. Analysis of the rate of occurrence of all eruptions with VEI greater than 1 permits the recognition of three distinct regimes or rates of volcanic activity during the last 430 years. A double stochastic Poisson model is suggested to describe this non-stationary eruptive pattern of Colima volcano and a Bayesian approach permits an estimation the present hazard.     

A connection between the Lagrangian stochastic–convective and cumulant expansion approaches for describing solute transport in heterogeneous porous media

The equation describing the ensemble-average solute concentration in a heterogeneous porous media can be developed from the Lagrangian (stochastic–convective) approach and from a method that uses a renormalized cumulant expansion. These two approaches are compared for the case of steady flow, and it is shown that they are related. The cumulant expansion approach can be interpreted as a series expansion of the convolution path integral that defines the ensemble-average concentration in the Lagrangian approach. The two methods can be used independently to develop the classical form for the convection–dispersion equation, and are shown to lead to identical transport equations under certain simplifying assumptions. In the development of such transport equations, the cumulant expansion does not require a priori the assumption of any particular distribution for the Lagrangian displacements or velocity field, and does not require one to approximate trajectories with their ensemble-average. In order to obtain a second-order equation, the cumulant expansion method does require truncation of a series, but this truncation is done rationally by the development of a constraint in terms of parameters of the transport field. This constraint is less demanding than requiring that the distribution for the Lagrangian displacements be strictly Gaussian, and it indicates under what velocity field conditions a second-order transport equation is a reasonable approximation.     

A hierarchical generalized linear model with variable selection: studying the response of a representative fish assemblage for large European rivers in a multi-pressure context

Global change calls for an understanding of how temperature and flow regimes influence aquatic ecosystems. Fish assemblages are a major component of river ecosystems and are thought to exhibit more integrative informative responses than single species to environmental variations, whether rare and sudden or gradual and continuous. The use of long-term datasets is thus of primary importance, allied to statistical modeling. For each of three previously identified species clusters, we performed Bayesian variable selection and inference within a hierarchical log Poisson Generalized Linear Model using a spike and slab normal prior to pinpoint which subset of environmental variables is of importance for each fish assemblage. Fish counts from electrofishing experiments are known to provide overdispersed data and, not surprisingly, the contribution of recorded environmental effects is found to be weak compared with those of other intra-assemblage sources of variation. The posterior distribution of the regression parameters is in coherence with what was expected from biological knowledge of the three species clusters. In particular, thermophilic species tend to benefit from warmer waters, whereas the recruitment of cold water species decreases due to global warming effects. Our study provides an example of the advantages of hierarchical modeling for quantifying interspecies ecological effects and selecting common environmental variables of importance.     

水平层状介质中基阶瑞利面波椭圆极化特征数值分析与研究

近年来的工程实践表明,以速度频散特性为理论基础的传统瑞利面波法的实际应用存在场地限制的瓶颈问题.为克服该困难,本文从面波的基础理论着手,在前人建立的水平层状介质中瑞利面波速度频散方程的基础上,推导了水平层状介质中瑞利面波质点位移的解析解公式.以工程中常见的几种典型的水平层状地层模型为例,结合地脉动单点谱比法(HVSR,Horizontal to Vertical Spectral Ratio),对基阶瑞利面波的椭圆极化特征进行了数值模拟研究.研究结果表明:与瑞利面波的速度频散特性类似,其椭圆极化同样具有频散特性,且椭圆极化时的质点位移水平分量与垂直分量的频谱比与地层泊松比结构有关.瑞利面波的这种椭圆极化特性展示了利用单点瑞利面波的多分量评价地层泊松比结构的理论可行性.     

The calculation of the statistical counting error for 222Rn scintillation counting

The calculation of the statistical counting error for alpha-scintillation counting of²²²Rn is complicated by the fact that two of the short-lived daughter products of²²²Rn are alpha emitters. All observed daughter-product counts attributable to parent decays that have already been detected must be excluded in the calculation of this error. An equation is derived which takes this into consideration.