A statistical study of the interspike-interval distribution of cortical neurons]

Spontaneous activity was recorded extracellularly by glass microelectrodes from 54 neurons of the Gyrus sigmoideus posterior of unnarcotized and gallamine-immobilazed cats, and the sequential and nonsequential interspike-interval histograms were determined using the multi-channel analyzer CAT 400 C. The interval distributions were characterized by graphic criteria, and it was attempted to describe these distributions mathematically by four biparametric distributions, the Weibull, lognormal, gamma and normal distributions. 80% of the frequency distributions of type I (exponential), II (left skew, gamma-similar) and IV (almost symmetrical) could be assigned to these distributions, namely 43% of the lognormal distribution, 32% of the Weibull distribution, and 5% of the gamma distribution. The interval histograms of the type III (left skew, steep) and V (bimodal) could not be described by any of the distributions selected.     

Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distributions

In this article we study some properties of a new family of distributions, namely Exponentiated Exponential distribution, discussed in Gupta , Gupta , and Gupta (1998). The Exponentiated Exponential family has two parameters (scale and shape) similar to a Weibull or a gamma family. It is observed that many properties of this new family are quite similar to those of a Weibull or a gamma family, therefore this distribution can be used as a possible alternative to a Weibull or a gamma distribution. We present two real life data sets, where it is observed that in one data set exponentiated exponential distribution has a better fit compared to Weibull or gamma distribution and in the other data set Weibull has a better fit than exponentiated exponential or gamma distribution. Some numerical experiments are performed to see how the maximum likelihood estimators and their asymptotic results work for finite sample sizes.     

Times from Infection to Disease-Induced Death and their Influence on Final Population Sizes After Epidemic Outbreaks

For epidemic models, it is shown that fatal infectious diseases cannot drive the host population into extinction if the incidence function is upper density-dependent. This finding holds even if a latency period is included and the time from infection to disease-induced death has an arbitrary length distribution. However, if the incidence function is also lower density-dependent, very infectious diseases can lead to a drastic decline of the host population. Further, the final population size after an epidemic outbreak can possibly be substantially affected by the infection-age distribution of the initial infectives if the life expectations of infected individuals are an unbounded function of infection age (time since infection). This is the case for lognormal distributions, which fit data from infection experiments involving tiger salamander larvae and ranavirus better than gamma distributions and Weibull distributions.     

Identification of age-structured models: cell cycle phase transitions

A methodology is developed that determines age-specific transition rates between cell cycle phases during balanced growth by utilizing age-structured population balance equations. Age-distributed models are the simplest way to account for varied behavior of individual cells. However, this simplicity is offset by difficulties in making observations of age distributions, so age-distributed models are difficult to fit to experimental data. Herein, the proposed methodology is implemented to identify an age-structured model for human leukemia cells (Jurkat) based only on measurements of the total number density after the addition of bromodeoxyuridine partitions the total cell population into two subpopulations. Each of the subpopulations will temporarily undergo a period of unbalanced growth, which provides sufficient information to extract age-dependent transition rates, while the total cell population remains in balanced growth. The stipulation of initial balanced growth permits the derivation of age densities based on only age-dependent transition rates. In fitting the experimental data, a flexible transition rate representation, utilizing a series of cubic spline nodes, finds a bimodal G(0)/G(1) transition age probability distribution best fits the experimental data. This resolution may be unnecessary as convex combinations of more restricted transition rates derived from normalized Gaussian, lognormal, or skewed lognormal transition-age probability distributions corroborate the spline predictions, but require fewer parameters. The fit of data with a single log normal distribution is somewhat inferior suggesting the bimodal result as more likely. Regardless of the choice of basis functions, this methodology can identify age distributions, age-specific transition rates, and transition-age distributions during balanced growth conditions.     

Random Phenotypic Variation of Yeast (Saccharomyces cerevisiae) Single-Gene Knockouts Fits a Double Pareto-Lognormal Distribution

Background

Distributed robustness is thought to influence the buffering of random phenotypic variation through the scale-free topology of gene regulatory, metabolic, and protein-protein interaction networks. If this hypothesis is true, then the phenotypic response to the perturbation of particular nodes in such a network should be proportional to the number of links those nodes make with neighboring nodes. This suggests a probability distribution approximating an inverse power-law of random phenotypic variation. Zero phenotypic variation, however, is impossible, because random molecular and cellular processes are essential to normal development. Consequently, a more realistic distribution should have a y-intercept close to zero in the lower tail, a mode greater than zero, and a long (fat) upper tail. The double Pareto-lognormal (DPLN) distribution is an ideal candidate distribution. It consists of a mixture of a lognormal body and upper and lower power-law tails.

Objective and Methods

If our assumptions are true, the DPLN distribution should provide a better fit to random phenotypic variation in a large series of single-gene knockout lines than other skewed or symmetrical distributions. We fit a large published data set of single-gene knockout lines in Saccharomyces cerevisiae to seven different probability distributions: DPLN, right Pareto-lognormal (RPLN), left Pareto-lognormal (LPLN), normal, lognormal, exponential, and Pareto. The best model was judged by the Akaike Information Criterion (AIC).

Results

Phenotypic variation among gene knockouts in S. cerevisiae fits a double Pareto-lognormal (DPLN) distribution better than any of the alternative distributions, including the right Pareto-lognormal and lognormal distributions.

Conclusions and Significance

A DPLN distribution is consistent with the hypothesis that developmental stability is mediated, in part, by distributed robustness, the resilience of gene regulatory, metabolic, and protein-protein interaction networks. Alternatively, multiplicative cell growth, and the mixing of lognormal distributions having different variances, may generate a DPLN distribution.     

Representing Cumulative Germination.: 2. The Use of the Weibull Function and Other Empirically Derived Curves

The Weibull, Morgan–Mercer–Flodin, Richards, Mitscherlich,Gompertz and logistic functions were each fitted to a wide rangeof cumulative germinations of non-dormant seed. The Weibullproved the most suitable for describing cumulative germinationas it provided a consistently close fit to the data and wasinsensitive to choice of starting values, thus making it fairlyeasy to fit. The others provided either an inferior fit or elsea similar fit but with a greater sensitivity to starting values. The four parameters of the Weibull function reflect maximumgermination, germination rate, the lag in the onset of germinationand the shape of the cumulative distribution. A comparison between non-linear and linear fits of the Mitscherlich,Gompertz and logistic functions showed the clear superiorityof non-linear methods. Cumulative germination, Weibull function, Richards function, Morgan–Mercer–Flodin function, Mitscherlich (monomolecular) function, Gompertz function, logistic (autocatalytic) function, modelling, simulation     

What distribution function do life cycle inventories follow?

Purpose

Life cycle inventory (LCI) results are often assumed to follow a lognormal distribution, while a systematic study that identifies the distribution function that best describes LCIs has been lacking. This paper aims to find the distribution function that best describes LCIs using Ecoinvent v3.1 database using a statistical approach, called overlapping coefficient analysis.

Methods

Monte Carlo simulation is applied to characterize the distribution of aggregate LCIs. One thousand times of simulated LCI results are generated based on the unit process-level parametric uncertainty information, from each of which 1000 randomly chosen data points are extracted. The 1 million data points extracted undergo statistical analyses including Shapiro-Wilk normality test and the overlapping coefficient analysis. The overlapping coefficient is a measure used to determine the shared area between the distribution of the simulated LCI results and three possible distribution functions that can potentially be used to describe them including lognormal, gamma, and Weibull distributions.

Results and discussion

Shapiro-Wilk normality test for 1000 samples shows that average p value of log-transformed LCI results is 0.18 at 95 % confidence level, accepting the null hypothesis that LCI results are lognormally distributed. The overlapping coefficient analysis shows that lognormal distribution best describes the distribution of LCI results. The average of overlapping coefficient (OVL) for lognormal distribution is 95 %, while that for gamma and Weibull distributions are 92 and 86 %, respectively.

Conclusions

This study represents the first attempt to calculate the stochastic distributions of the aggregate LCIs covering the entire Ecoinvent 3.1 database. This study empirically shows that LCIs of Ecoinvent 3.1 database indeed follow a lognormal distribution. This finding can facilitate more efficient storage and use of uncertainty information in LCIs and can reduce the demand for computational power to run Monte Carlo simulation, which currently relies on unit process-level uncertainty data.

     

Size distribution of the quasi-circular vegetation patches in the Yellow River Delta,China

Recent empirical studies have suggested that the patch-size distribution of vegetation can be fitted by a power law, truncated power law, or lognormal model to provide explanatory mechanisms for vegetation pattern formation in arid and semiarid regions. However, contradictory results have been reported. Therefore, additional empirical studies are necessary to test the patch-size distribution of vegetation over several regions before it can be considered as an indicator for assessing the discontinuous transition of ecosystems and understanding the mechanisms of vegetation pattern formation. Analogous to arid and semiarid regions of the world, vegetation patterns are characterized by a two-phase mosaic composed of dense vegetation patches interspersed with areas of bare soil, referred to as quasi-circular vegetation patches (QVPs), in the Yellow River Delta (YRD), China. However, research on the patch-size distribution of the QVPs reflecting vegetation patterns and ecosystem functioning is lacking. To fill this gap, for the first time, we examined the patch-size distribution of the QVPs using the fused IKONOS high-spatial-resolution image and evaluated the statistical distributions that better fit the patch size data of the QVPs in the YRD. We found that a power law, truncated power law, or lognormal distribution was not supported in the study area, whereas gamma distribution reasonably fits the size data of QVPs, implying that micro-depressions, combined with the water-limited and salinization environments had considerable effects on vegetation pattern formation. Our results provide helpful insights and suggest that further studies are needed to classify different types of QVPs. Additionally, more efficient approaches need to be used to fit the statistical distributions for elucidating the spatial vegetation patterns in the YRD.     

Are there differences in structure between marine and terrestrial assemblages?

Recently, interest in species abundance (SAD) distributions has been revived by introduction of a new model, the zero-sum multinomial (ZSM). Yet detailed statistical analyses show that the model does not differ from the lognormal distribution proposed in the 1940s. These analyses were based on data from tropical trees where all individuals in a defined area were identified to species. For many ecological data sets it is not possible to identify and count all individuals in a given area. Here we compare data on marine benthos and fish assemblages with data on terrestrial microfauna and ants. We show that these assemblages show similar SAD patterns and that the SADs are best described by a two-group lognormal model. Whereas the 2-group model fitted all data sets the single group model fitted all except the tropical rainforest ants. However, tests comparing the fits to the 2-group versus the single lognormal model showed that the 2-group model was a significantly better fit to the fish and insect data. The two groups are of rare and common species and the rare group dominates in all four data sets. We suggest that the reason for this is that rare species are continuously immigrating from outside the sampled area. Data on tropical tree assemblages where complete accounts were made do not show such high dominance of rare species where the sampled area is large. We conclude that SAD patterns are similar in marine and terrestrial systems that are open to immigration and that the lognormal distribution is still a valid model for SADs.     

Normal and Extreme Wind Conditions for Power at Coastal Locations in China

In this paper, the normal and extreme wind conditions for power at 12 coastal locations along China’s coastline were investigated. For this purpose, the daily meteorological data measured at the standard 10-m height above ground for periods of 40–62 years are statistically analyzed. The East Asian Monsoon that affects almost China’s entire coastal region is considered as the leading factor determining wind energy resources. For most stations, the mean wind speed is higher in winter and lower in summer. Meanwhile, the wind direction analysis indicates that the prevalent winds in summer are southerly, while those in winter are northerly. The air densities at different coastal locations differ significantly, resulting in the difference in wind power density. The Weibull and lognormal distributions are applied to fit the yearly wind speeds. The lognormal distribution performs better than the Weibull distribution at 8 coastal stations according to two judgement criteria, the Kolmogorov–Smirnov test and absolute error (AE). Regarding the annual maximum extreme wind speed, the generalized extreme value (GEV) distribution performs better than the commonly-used Gumbel distribution. At these southeastern coastal locations, strong winds usually occur in typhoon season. These 4 coastal provinces, that is, Guangdong, Fujian, Hainan, and Zhejiang, which have abundant wind resources, are also prone to typhoon disasters.     

Correlation Goodness‐of‐fit Test for the Logarithmically‐Decreasing Survival Distribution

In this paper, we use the correlation‐type goodness‐of‐fit test for the logarithmically‐decreasing survival distribution. This model was intoduced by Sultan , Balakrishnan and Childs (2001) as a special case of Type‐I truncated logistic distribution. The power of the test based on normal, Weibull and gamma distributions is also calculated. We also give application to real example.     

Effect of Porosity on Strength Distribution of Microcrystalline Cellulose

Fracture strength of pharmaceutical compacts varies even for nominally identical samples, which directly affects compaction, comminution, and tablet dosage forms. However, the relationships between porosity and mechanical behavior of compacts are not clear. Here, the effects of porosity on fracture strength and fracture statistics of microcrystalline cellulose compacts were investigated through diametral compression tests. Weibull modulus, a key parameter in Weibull statistics, was observed to decrease with increasing porosity from 17 to 56 vol.%, based on eight sets of compacts at different porosity levels, each set containing ∼50 samples, a total of 407 tests. Normal distribution fits better to fracture data for porosity less than 20 vol.%, whereas Weibull distribution is a better fit in the limit of highest porosity. Weibull moduli from 840 unique finite element simulations of isotropic porous materials were compared to experimental Weibull moduli from this research and results on various pharmaceutical materials. Deviations from Weibull statistics are observed. The effect of porosity on fracture strength can be described by a recently proposed micromechanics-based formula.Key words: diametral compression test, finite element simulations, normal distribution, reliability, Weibull modulus     

Frequent and occasional species and the shape of relative-abundance distributions

Recently, three different models have been proposed to explain the distribution of abundances in natural communities: the self‐similarity model; the zero‐sum ecological drift model; and the occasional–frequent species model of Magurran and Henderson. Here we study patterns of relative abundance in a large community of forest Hymenoptera and show that it is indeed possible to divide the community into a group of frequent species and a group of occasional species. In accordance with the third model, frequent species followed a lognormal distribution. Relative abundances of the occasional species could be described by the self‐similarity model, but did not follow a log‐series as proposed by the occasional–frequent model. The zero‐sum ecological drift model makes no explicit predictions about frequent and occasional species but the abundance distributions of the hymenopteran species did not show the excess of rare species predicted by this model. Separate fits of this model to the frequent and to the occasional species were worse than the respective fits of the lognormal and the self‐similarity model.     

A General and Dynamic Species Abundance Model, Embracing the Lognormal and the Gamma Models

One aspect of community ecology that has been given particular attention is the pattern of species abundances in a community. The species may have a wide range of abundances; some are very common and others rare. When species abundance models are fitted to observations, the lognormal model and one of the gamma models (e.g., the log-series model) are usually applied. The model that gives the best fit according to some goodness-of-fit test is then chosen. By applying a diffusion approximation for each species' dynamics with density regulation of the straight theta-logistic type, we here present a general species abundance model that embraces the two most widely applied species abundance models, the lognormal and the gamma. Our general model will, therefore, provide a better fit than the two special cases, except when it corresponds to one of them. In contrast to the classical models, ours is also dynamic, making it possible to evaluate the fluctuations in species abundance over time through both biotic and abiotic factors. The model is fitted to several species abundance data sets and our results compared to previous attempts to fit a model, usually either the lognormal or the log-series.     

Statistical analyses support power law distributions found in neuronal avalanches

The size distribution of neuronal avalanches in cortical networks has been reported to follow a power law distribution with exponent close to -1.5, which is a reflection of long-range spatial correlations in spontaneous neuronal activity. However, identifying power law scaling in empirical data can be difficult and sometimes controversial. In the present study, we tested the power law hypothesis for neuronal avalanches by using more stringent statistical analyses. In particular, we performed the following steps: (i) analysis of finite-size scaling to identify scale-free dynamics in neuronal avalanches, (ii) model parameter estimation to determine the specific exponent of the power law, and (iii) comparison of the power law to alternative model distributions. Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect). This scale-free dynamics suggests the power law as a model for the distribution of avalanche sizes. Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports. Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test. Both the power law distribution without and with exponential cut-off provided significantly better fits to the cluster size distributions in neuronal avalanches than the exponential, the lognormal and the gamma distribution. In summary, our findings strongly support the power law scaling in neuronal avalanches, providing further evidence for critical state dynamics in superficial layers of cortex.     

Analytical Quantile Solution for the S‐distribution,Random Number Generation and Statistical Data Modeling

The selection of a specific statistical distribution as a model for describing the population behavior of a given variable is seldom a simple problem. One strategy consists in testing different distributions (normal, lognormal, Weibull, etc.), and selecting the one providing the best fit to the observed data and being the most parsimonious. Alternatively, one can make a choice based on theoretical arguments and simply fit the corresponding parameters to the observed data. In either case, different distributions can give similar results and provide almost equivalent models for a given data set. Model selection can be more complicated when the goal is to describe a trend in the distribution of a given variable. In those cases, changes in shape and skewness are difficult to represent by a single distributional form. As an alternative to the use of complicated families of distributions as models for data, the S‐distribution [Voit, E. O. (1992) Biom. J. 7 , 855–878] provides a highly flexible mathematical form in which the density is defined as a function of the cumulative. S‐distributions can accurately approximate many known continuous and unimodal distributions, preserving the well known limit relationships between them. Besides representing well‐known distributions, S‐distributions provide an infinity of new possibilities that do not correspond with known classical distributions. Although the utility and performance of this general form has been clearly proved in different applications, its definition as a differential equation is a potential drawback for some problems. In this paper we obtain an analytical solution for the quantile equation that highly simplifies the use of S‐distributions. We show the utility of this solution in different applications. After classifying the different qualitative behaviors of the S‐distribution in parameter space, we show how to obtain different S‐distributions that accomplish specific constraints. One of the most interesting cases is the possibility of obtaining distributions that acomplish P(X ≤ X_c) = 0. Then, we demonstrate that the quantile solution facilitates the use of S‐distributions in Monte‐Carlo experiments through the generation of random samples. Finally, we show how to fit an S‐distribution to actual data, so that the resulting distribution can be used as a statistical model for them.     

Modelling rate distributions using character compatibility: implications for morphological evolution among fossil invertebrates

Rate distributions are important considerations when testing hypotheses about morphological evolution or phylogeny. They also have implications about general processes underlying character evolution. Molecular systematists often assume that rates are Poisson processes with gamma distributions. However, morphological change is the product of multiple probabilistic processes and should theoretically be affected by hierarchical integration of characters. Both factors predict lognormal rate distributions. Here, a simple inverse modelling approach assesses the best single-rate, gamma and lognormal models given observed character compatibility for 115 invertebrate groups. Tests reject the single-rate model for nearly all cases. Moreover, the lognormal outperforms the gamma for character change rates and (especially) state derivation rates. The latter in particular is consistent with integration affecting morphological character evolution.     

The size distribution of human hematogenous metastases.

The development of primary cancers and their subsequent metastases occur through a complex sequence of discrete steps. A hypothesis is proposed here whereby the time available for the growth of metastases is normally distributed, presumably as a consequence of the summation of multiple independently distributed time intervals from each of the steps and of the Central Limit Theorem. For exponentially growing metastases, the corresponding size distribution would be lognormal; Gompertzian growth would imply a modified (Gompertz-normal) distribution, where larger metastases would occur less frequently as a consequence of a decreased growth rate. These two size distributions were evaluated against 18 human autopsy cases where precise size measurements had been collected from over 3900 macroscopic hematogenous organ metastases. The lognormal distribution provided an approximate agreement. Its main deficiency was a tendency to over-represent metastases greater than 10 mm diameter. The Gompertz-normal distribution provided more stringent agreement, correcting for this over-representation. These observations supported the hypothesis of normally distributed growth times, and qualified the utility of the lognormal and Gompertz-normal distributions for the size distribution of metastases.     

The distribution of the intervals between neural impulses in the maintained discharges of retinal ganglion cells

Simulated neural impulse trains were generated by a digital realization of the integrate-and-fire model. The variability in these impulse trains had as its origin a random noise of specified distribution. Three different distributions were used: the normal (Gaussian) distribution (no skew, normokurtic), a first-order gamma distribution (positive skew, leptokurtic), and a uniform distribution (no skew, platykurtic). Despite these differences in the distribution of the variability, the distributions of the intervals between impulses were nearly indistinguishable. These inter-impulse distributions were better fit with a hyperbolic gamma distribution than a hyperbolic normal distribution, although one might expect a better approximation for normally distributed inverse intervals. Consideration of why the inter-impulse distribution is independent of the distribution of the causative noise suggests two putative interval distributions that do not depend on the assumed noise distribution: the log normal distribution, which is predicated on the assumption that long intervals occur with the joint probability of small input values, and the random walk equation, which is the diffusion equation applied to a random walk model of the impulse generating process. Either of these equations provides a more satisfactory fit to the simulated impulse trains than the hyperbolic normal or hyperbolic gamma distributions. These equations also provide better fits to impulse trains derived from the maintained discharges of ganglion cells in the retinae of cats or goldfish. It is noted that both equations are free from the constraint that the coefficient of variation (CV) have a maximum of unity. The concluding discussion argues against the random walk equation because it embodies a constraint that is not valid, and because it implies specific parameters that may be spurious.