Single Parametric Distribution Research Articles

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

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

An Inferential Aptness of a Weibull Generated Distribution and Application

In this article an attempt has been made to develop a flexible single parameter continuous distribution using Weibull distribution. The Weibull distribution is most widely used lifetime distributions in both medical and engineering sectors. The exponential and Rayleigh distribution is particular case of Weibull distribution. Here in this study we use these two distributions for developing a new distribution. Important statistical properties of the proposed distribution is discussed such as moments, moment generating and characteristic function. Various entropy measures like Rényi, Shannon and cumulative entropy are also derived. The kthkt⁢h order statistics of pdf and cdf also obtained. The properties of hazard function and their limiting behavior is discussed. The maximum likelihood estimate of the parameter is obtained that is not in closed form, thus iteration procedure is used to obtain the estimate. Simulation study has been done for different sample size and MLE, MSE, Bias for the parameter λλ has been observed. Some real data sets are used to check the suitability of model over some other competent distributions for some data sets from medical and engineering science. In the tail area, the proposed model works better. Various model selection criterion such as -2LL, AIC, AICc, BIC, K-S and A-D test suggests that the proposed distribution perform better than other competent distributions and thus considered this as an alternative distribution. The proposed single parameter distribution is found more flexible as compare to some other two parameter complicated distributions for the data sets considered in the present study.

An Inferential Aptness of a Weibull Generated Distribution and Application

In this article an attempt has been made to develop a flexible single parameter continuous distribution using Weibull distribution. The Weibull distribution is most widely used lifetime distributions in both medical and engineering sectors. The exponential and Rayleigh distribution is particular case of Weibull distribution. Here in this study we use these two distributions for developing a new distribution. Important statistical properties of the proposed distribution is discussed such as moments, moment generating and characteristic function. Various entropy measures like Rényi, Shannon and cumulative entropy are also derived. The kth order statistics of pdf and cdf also obtained. The properties of hazard function and their limiting behavior is discussed. The maximum likelihood estimate of the parameter is obtained that is not in closed form, thus iteration procedure is used to obtain the estimate. Simulation study has been done for different sample size and MLE, MSE, Bias for the parameter λ has been observed. Some real data sets are used to check the suitability of model over some other competent distributions for some data sets from medical and engineering science. In the tail area, the proposed model works better. Various model selection criterion such as -2LL, AIC, AICc, BIC, K-S and A-D test suggests that the proposed distribution perform better than other competent distributions and thus considered this as an alternative distribution. The proposed single parameter distribution is found more flexible as compare to some other two parameter complicated distributions for the data sets considered in the present study.

Identifying density-dependent interactions in collective cell behaviour.

Scratch assays are routinely used to study collective cell behaviour in vitro. Typical experimental protocols do not vary the initial density of cells, and typical mathematical modelling approaches describe cell motility and proliferation based on assumptions of linear diffusion and logistic growth. Jin et al. (Jin et al. 2016 J. Theor. Biol.390, 136-145 (doi:10.1016/j.jtbi.2015.10.040)) find that the behaviour of cells in scratch assays is density-dependent, and show that standard modelling approaches cannot simultaneously describe data initiated across a range of initial densities. To address this limitation, we calibrate an individual-based model to scratch assay data across a large range of initial densities. Our model allows proliferation, motility, and a direction bias to depend on interactions between neighbouring cells. By considering a hierarchy of models where we systematically and sequentially remove interactions, we perform model selection analysis to identify the minimum interactions required for the model to simultaneously describe data across all initial densities. The calibrated model is able to match the experimental data across all densities using a single parameter distribution, and captures details about the spatial structure of cells. Our results provide strong evidence to suggest that motility is density-dependent in these experiments. On the other hand, we do not see the effect of crowding on proliferation in these experiments. These results are significant as they are precisely the opposite of the assumptions in standard continuum models, such as the Fisher-Kolmogorov equation and its generalizations.

Efficient and flexible model-based clustering of jumps in diffusion processes

Jump–diffusion processes involving diffusion processes with discontinuous movements, called jumps, are widely used to model time-series data that commonly exhibit discontinuity in their sample paths. The existing jump–diffusion models have been recently extended to multivariate time-series data. The models are, however, still limited by a single parametric jump-size distribution that is common across different subjects. Such strong parametric assumptions for the shape and structure of a jump-size distribution may be too restrictive and unrealistic for multiple subjects with different characteristics. This paper thus proposes an efficient Bayesian nonparametric method to flexibly model a jump-size distribution while borrowing information across subjects in a clustering procedure using a nested Dirichlet process. For efficient posterior computation, a partially collapsed Gibbs sampler is devised to fit the proposed model. The proposed methodology is illustrated through a simulation study and an application to daily stock price data for companies in the S&P 100 index from June 2007 to June 2017.

Likelihood Confidence Intervals When Only Ranges are Available

Research papers represent an important and rich source of comparative data. The change is to extract the information of interest. Herein, we look at the possibilities to construct confidence intervals for sample averages when only ranges are available with maximum likelihood estimation with order statistics (MLEOS). Using Monte Carlo simulation, we looked at the confidence interval coverage characteristics for likelihood ratio and Wald-type approximate 95% confidence intervals. We saw indication that the likelihood ratio interval had better coverage and narrower intervals. For single parameter distributions, MLEOS is directly applicable. For location-scale distribution is recommended that the variance (or combination of it) to be estimated using standard formulas and used as a plug-in.

Accelerated Evaluation of Automated Vehicles Using Piecewise Mixture Models

The process to certify highly automated vehicles has not yet been defined by any country in the world. Currently, companies test automated vehicles on public roads, which is time-consuming and inefficient. We proposed the accelerated evaluation concept, which uses a modified statistics of the surrounding vehicles and the importance sampling theory to reduce the evaluation time by several orders of magnitude, while ensuring the evaluation results are statistically accurate. In this paper, we further improve the accelerated evaluation concept by using piecewise mixture distribution models, instead of single parametric distribution models. We developed and applied this idea to forward collision control system reacting to vehicles making cutin lane changes. The behavior of the cutin vehicles was modeled based on more than 403,581 lane changes collected by the University of Michigan Safety Pilot Model Deployment Program. Simulation results confirm that the accuracy and efficiency of the piecewise mixture distribution method outperformed single parametric distribution methods in accuracy and efficiency, and accelerated the evaluation process by almost four orders of magnitude.

A finite mixture approach to univariate data simulation with moment matching

Environmental data simulation is carried out for various purposes and is most simply achieved when recorded data can be regarded as a sequence of independent random variables. Simulating from such data involves generating random variables from some fitted probability distribution, preserving the main statistical characteristics. The particular case of data simulation with moment matching is revisited, with a proposal that data be simulated from many-component finite mixture distributions. The simulation procedure matches data moments while also giving greater flexibility for data approximation. In contrast, a single parametric distribution fitted to irregular data will result in data simulations more representative of the fitted distribution than the original data. The flexible finite mixture approach therefore has potential to replace parametric univariate data simulation generally. The method is illustrated with simulations from finite mixtures of generalised beta distributions with matching of data mean, variance, skewness and kurtosis.

Similarity-Based Semilocal Estimation of Post-Processing Models

SummaryWeather forecasts are typically given in the form of forecast ensembles obtained from multiple runs of numerical weather prediction models with varying initial conditions and physics parameterizations. Such ensemble predictions tend to be biased and underdispersive and thus require statistical post-processing. In the ensemble model output statistics approach, a probabilistic forecast is given by a single parametric distribution with parameters depending on the ensemble members. The paper proposes two semilocal methods for estimating the ensemble model output statistics coefficients where the training data for a specific observation station are augmented with corresponding forecast cases from stations with similar characteristics. Similarities between stations are determined by using either distance functions or clustering based on various features of the climatology, forecast errors and locations of the observation stations. In a case-study on wind speed over Europe with forecasts from the ‘Grand limited area model ensemble prediction system’, the similarity-based semilocal models proposed show significant improvement in predictive performance compared with standard regional and local estimation methods. They further allow for estimating complex models without numerical stability issues and are computationally more efficient than local parameter estimation.

Realistic Spiking Neuron Statistics in a Population are Described by a Single Parametric Distribution

The spiking of activity of neurons throughout the cortex is random and complicated. This complicated activity requires theoretical formulations in order to understand the underlying principles of neural processing. A key aspect of theoretical investigations is characterizing the probability distribution of spiking activity. This study aims to better understand the statistics of the time between spikes, or interspike interval, in both real data and a spiking model with many time scales. Exploration of the interspike intervals of neural network activity can provide a better understanding of neural responses to different stimuli. We consider different parametric distribution fitting techniques to characterize the random spike times of a population of neurons in the visual cortex of a mammal. Five different probability distribution functions were considered, including three mixture models, and their goodness of fit was determined through two criteria: maximum likelihood and Akaike Information Criteria. Despite being largely heterogeneous, both criteria indicated that one distribution, although different for each criteria, was the best fitting for all of the neurons in the data set. The Gamma-Gamma mixture distribution was the best according to maximum likelihood and the Exponential distribution was the best according to AIC. The statistical methodology applied to a burst model yielded the same results, and the AIC formula was further investigated to better understand its consistent selection of the same parametric distribution. We find that complicated neural spiking activity can sometimes be described by a single parametric distribution, which is hopefully comforting for theorists.

Scaling theory for anomalous semiclassical quantum transport

Quantum transport through devices coupled to electron reservoirs can be described in terms of the full counting statistics (FCS) of charge transfer. Transport observables, such as conductance and shot-noise power are just cumulants of FCS and can be obtained from the sample’s average density of transmission eigenvalues, which in turn can be obtained from a finite element representation of the saddle-point equation of the Keldysh (or supersymmetric) nonlinear sigma model, known as quantum circuit theory. Normal universal metallic behavior in the semiclassical regime is controlled by the presence of a Fabry–Pérot singularity in the average density of transmission eigenvalues. We present general conditions for the suppression of Fabry–Pérot modes in the semiclassical regime in a sample of arbitrary shape, a disordered conductor or a network of ballistic quantum dots, which leads to an anomalous metallic phase. Through a double-scaling limit, we derive a scaling equation for anomalous metallic transport, in the form of a nonlinear differential equation, which generalizes the ballistic-diffusive scaling equation of a normal metal. The two-parameter stationary solution of our scaling equation generalizes Dorokhov’s universal single-parameter distribution of transmission eigenvalues. We provide a simple interpretation of the stationary solution using a thermodynamic analogy with a spin-glass system. As an application, we consider a system formed by a diffusive wire coupled via a barrier to normal-superconductor reservoirs. We observe anomalous reflectionless tunneling, when all perfectly transmitting channels are suppressed, which cannot be explained by the usual mechanism of disorder-induced opening of tunneling channels.

Multi-Source-Driven Asynchronous Diffusion Model for Video-Sharing in Online Social Networks

Characterizing the video diffusion in online social networks (OSNs) is not only instructive for network traffic engineering, but also provides insights into the information diffusion process. A number of continuous-time diffusion models have been proposed to describe video diffusion under the assumption that the activation latency along social links follows a single parametric distribution. However, such assumption has not been empirically verified. Moreover, a user usually has multiple activated neighbors with different activation times, and it is hard to distinguish the different contributions of these multiple potential sources. To fill this gap, we study the multiple-source-driven asynchronous information diffusion problem based on substantial video diffusion traces. Specifically, we first investigate the latency of information propagation along social links and define the single-source (SS) activation latency for an OSN user. We find that the SS activation latency follows the exponential mixture model. Then we develop an analytical framework which incorporates the temporal factor and the influence of multiple sources to describe the influence propagation process. We show that one's activation probability decreases exponentially with time. We also show that the time shift of the exponential function is only determined by the most recent source (MRS) active user, but the total activation probability is the combination of influence exerted by all active neighbors. Based on these discoveries, we develop a multi-source-driven asynchronous diffusion model (MADM). Using maximum likelihood techniques, we develop an algorithm based on expectation maximization (EM) to learn model parameters, and validate our proposed model with real data. The experimental results show that the MADM obtains better prediction accuracy under various evaluation metrics.

An iterative construction of confidence intervals for a proportion

Under the criterion of the set inclusion, two-sided admissible 1 con- fidence intervals for the probability of success for a binomial random variable are constructed using a new iterative method that is based on a direct analysis of coverage probability. A refined Clopper-Pearson interval is derived and compared with the Blyth-Still-Casella interval, and is recommended for statistical practice due to its performance and accessibility. A generalization is provided to the case of a discrete sample space with a single parameter distribution. Some details and an R-code that computes the refined Clopper-Pearson interval are given in Supple- mentary Materials.

Validation of theα-μModel of the Power Spectral Density of GPS Ionospheric Amplitude Scintillation

Theα-μmodel has become widely used in statistical analyses of radio channels, due to the flexibility provided by its two degrees of freedom. Among several applications, it has been used in the characterization of low-latitude amplitude scintillation, which frequently occurs during the nighttime of particular seasons of high solar flux years, affecting radio signals that propagate through the ionosphere. Depending on temporal and spatial distributions, ionospheric scintillation may cause availability and precision problems to users of global navigation satellite systems. The present work initially stresses the importance of the flexibility provided byα-μmodel in comparison with the limitations of a single-parameter distribution for the representation of first-order statistics of amplitude scintillation. Next, it focuses on the statistical evaluation of the power spectral density of ionospheric amplitude scintillation. The formulation based on theα-μmodel is developed and validated using experimental data obtained in São José dos Campos (23.1°S; 45.8°W; dip latitude 17.3°S), Brazil, located near the southern crest of the ionospheric equatorial ionization anomaly. These data were collected between December 2001 and January 2002, a period of high solar flux conditions. The results show that the proposed model fits power spectral densities estimated from field data quite well.

Acoustic Emission Signals Characteristics of Q345R at Tensile Progress in the Normal and Low Temperatures

Acoustic emission signals were collected during the tensile process of Q345R at the normal and low temperatures. The signals were analyzed by the single parameter analysis, distribution analysis and correlation analysis to result the acoustic emission characteristics of Q345R in the normal and low temperatures.

Skew mixture models for loss distributions: A Bayesian approach

The derivation of loss distribution from insurance data is a very interesting research topic but at the same time not an easy task. To find an analytic solution to the loss distribution may be misleading although this approach is frequently adopted in the actuarial literature. Moreover, it is well recognized that the loss distribution is strongly skewed with heavy tails and presents small, medium and large size claims which hardly can be fitted by a single analytic and parametric distribution. Here we propose a finite mixture of Skew Normal distributions that provides a better characterization of insurance data. We adopt a Bayesian approach to estimate the model, providing the likelihood and the priors for the all unknown parameters; we implement an adaptive Markov Chain Monte Carlo algorithm to approximate the posterior distribution. We apply our approach to a well known Danish fire loss data and relevant risk measures, such as Value-at-Risk and Expected Shortfall probability, are evaluated as well.

Understanding Animal Group-Size Distributions

One of the most striking aspects of animal groups is their remarkable variation in size, both within and between species. While a number of mechanistic models have been proposed to explain this variation, there are few comprehensive datasets against which these models have been tested. In particular, we only vaguely understand how environmental factors and behavioral activities affect group-size distributions. Here we use observations of House sparrows (Passer domesticus) to investigate the factors determining group-size distribution. Over a wide range of conditions, we observed that animal group sizes followed a single parameter distribution known as the logarithmic distribution. This single parameter is the mean group size experienced by a randomly chosen individual (including the individual itself). For sparrows, the experienced mean group size, and hence the distribution, was affected by four factors: morning temperature, place, behavior and the degree of food spillage. Our results further indicate that the sparrows regulate the mean group size they experience, either by groups splitting more or merging less when local densities are high. We suggest that the mean experienced group size provides a simple but general tool for assessing the ecology and evolution of grouping.

EXOPLANETARY SPIN-ORBIT ALIGNMENT: RESULTS FROM THE ENSEMBLE OF ROSSITER-MCLAUGHLIN OBSERVATIONS

One possible diagnostic of planet formation, orbital migration, and tidal evolution is the angle ψ between a planet's orbital axis and the spin axis of its parent star. In general, ψ cannot be measured, but for transiting planets one can measure the angle λ between the sky projections of the two axes via the Rossiter–McLaughlin effect. Here, we show how to combine measurements of λ in different systems to derive statistical constraints on ψ. We apply the method to 11 published measurements of λ, using two different single-parameter distributions to describe the ensemble. First, assuming a Rayleigh distribution (or more precisely, a Fisher distribution on a sphere), we find that the peak value is less than 22° with 95% confidence. Second, assuming that a fraction f of the orbits have random orientations relative to the stars, and the remaining fraction (1 − f) are perfectly aligned, we find f < 0.36 with 95% confidence. This latter model fits the data better than the Rayleigh distribution, mainly because the XO-3 system was found to be strongly misaligned while the other 10 systems are consistent with perfect alignment. If the XO-3 result proves robust, then our results may be interpreted as evidence for two distinct modes of planet migration.

Hypothesis Tests for Point-Mass Mixture Data with Application to `Omics Data with Many Zero Values

Data composed of a continuous component plus a point-mass frequently arises in genomic studies. The distribution of this type of data is characterized by the proportion of observations in the point mass and the distribution of the continuous component. Standard statistical methods focus on one of these effects at a time and can fail to detect differences between experimental groups. We propose a novel empirical likelihood ratio test (LRT) statistic for simultaneously testing the null hypothesis of no difference in point-mass proportions and no difference in means of the continuous component. This study evaluates the performance of the empirical LRT and three existing point-mass mixture statistics: 1) Two-part statistic with a t-test for testing mean differences (Two-part t), 2) Two-part statistic with Wilcoxon test for testing mean differences (Two-part W), and 3) parametric LRT. Our investigations begin with an analysis of metabolomics data from Arabidopsis thaliana, which contains many metabolites with a large proportion of observed concentrations in a point-mass at zero. All four point-mass mixture statistics identify more significant differences than standard t-tests and Wilcoxon tests. The empirical LRT appears particularly effective. These findings motivate a large simulation study that assesses Type I and Type II error of the four test statistics with various choices of null distribution. The parametric LRT is frequently the most powerful test, as long as the model assumptions are correct. As is common in 'omics data, the Arabidopsis metabolites have widely varying concentration distributions. A single parametric distribution cannot effectively represent all of these distributions, and individually selecting the optimal parametric distribution to use in the LRT for each metabolite is not practical. The empirical LRT, which does not require parametric assumptions, provides an attractive alternative to parametric and standard methods.

Quantification of variability and uncertainty using mixture distributions: evaluation of sample size, mixing weights, and separation between components.

Variability is the heterogeneity of values within a population. Uncertainty refers to lack of knowledge regarding the true value of a quantity. Mixture distributions have the potential to improve the goodness of fit to data sets not adequately described by a single parametric distribution. Uncertainty due to random sampling error in statistics of interests can be estimated based upon bootstrap simulation. In order to evaluate the robustness of using mixture distribution as a basis for estimating both variability and uncertainty, 108 synthetic data sets generated from selected population mixture log-normal distributions were investigated, and properties of variability and uncertainty estimates were evaluated with respect to variation in sample size, mixing weight, and separation between components of mixtures. Furthermore, mixture distributions were compared with single-component distributions. Findings include: (1). mixing weight influences the stability of variability and uncertainty estimates; (2). bootstrap simulation results tend to be more stable for larger sample sizes; (3). when two components are well separated, the stability of bootstrap simulation is improved; however, a larger degree of uncertainty arises regarding the percentiles coinciding with the separated region; (4). when two components are not well separated, a single distribution may often be a better choice because it has fewer parameters and better numerical stability; and (5). dependencies exist in sampling distributions of parameters of mixtures and are influenced by the amount of separation between the components. An emission factor case study based upon NO(x) emissions from coal-fired tangential boilers is used to illustrate the application of the approach.