Gamma-like Distribution Research Articles

Asymptotic results on tail moment for light-tailed risks

In this paper, we focus on the asymptotic behavior of a recently popular risk measure called the tail moment (TM), which has been extensively applied in the field of risk theory. We conduct the study under the framework in which the individual risks of a financial or insurance system follow convolution equivalent or Gamma-like distributions. Precise asymptotic results are obtained for the TM when the individual risks are mutually independent or have a dependence structure of the Farlie-Gumbel-Morgenstern type. Moreover, based on some specific scenarios, we give an asymptotic analysis on the relative errors between our asymptotic results and the corresponding exact values. Since the model settings in this paper are not covered by traditional ones, our work fills in some gaps of the asymptotic study of the TM for light-tailed risks.

History-dependent changes to distribution of dominance phases in multistable perception.

Multistability - spontaneous switches of perception when viewing a stimulus compatible with several percepts - is often characterized by the distribution of durations of dominance phases. For continuous viewing conditions, these distributions are similar for various multistable displays and share two characteristic features: a Gamma-like distribution shape and dependence of dominance durations on the perceptual history. Both properties depend on a balance between self-adaptation (also conceptualized as a weakening stability prior) and noise. Prior experimental work and simulations that systematically manipulated displays showed that faster self-adaptation leads to a more "normal-like" distribution and, typically, to more regular dominance durations. We used a leaky integrator approach to estimate accumulated differences in self-adaptation between competing representations and used it as a predictor when fitting two parameters of a Gamma distribution independently. We confirmed earlier work showing that larger differences in self-adaptation led to a more "normal-like" distribution suggesting similar mechanisms that rely on the balance between self-adaptation and noise. However, these larger differences led to less regular dominance phases suggesting that longer times required for recovery from adaptation give noise more chances to induce a spontaneous switch. Our results also remind us that individual dominance phases are not "independent and identically distributed."

Computation of Dynamic Operating Balancing Reserve for Wind Power Integration for the Time-Horizon 1–48 Hours

A challenge now facing utilities is how to adjust reserves in the operations-planning horizon of 0 to 48 hours ahead to mitigate the effects of wind variability and forecast uncertainties, in addition to those of load uncertainties and unavailability of generation. Reserves are maintained to ensure a high level of reliability and security to the system. They are subdivided into two groups: those responding within an intrahourly time horizon to regulate power imbalances, and those responding over a 1-48 hours ahead time horizon addressing the net forecast uncertainties. In this paper, we present a methodology for calculating reserves in the latter category, referred to as balancing reserves (BRs), following the integration of wind generation in a power system. Their computation is based on maintaining a predefined level of risk. The novelty here is that wind forecast error distributions are adjusted as a function of wind generation forecast levels. Gamma-like distributions with time-varying parameters, estimated from real data, were chosen to approximate the wind generation forecast errors. It is shown that this improved modeling significantly modifies the values of required balancing reserves and associated risk. The methodology developed is based on a clear criterion, namely risk, and it demonstrates the imperativeness of considering dynamic balancing reserves as a function of the imminent wind generation forecast.

The influence of composition on the local structure around yttrium in quenched silicate melts — Insights from EXAFS

The structural environment around Y in silicate and aluminosilicate glasses containing 5000ppm Y was investigated as a function of melt composition and polymerization using Extended X-ray Absorption Fine Structure (EXAFS) spectroscopy. The used glass compositions were taken from Prowatke and Klemme (2005) varying in the aluminum saturation index (ASI, molar ratio of Al2O3/(Na2O+K2O+CaO)) from 0.115 to 0.768. Furthermore, a set of glass compositions from the system CaO–Al2O3–SiO2 (CAS) was used, for which structural data from computer simulations are available (Haigis et al., 2013--this issue). Structural parameters of the Y–O pair correlation of the first coordination shell were determined from the EXAFS based on a gamma-like distribution function that accounts for the large static disorder and non-Gaussian pair distributions. The analysis shows an increase in the coordination number from 6 to 8, along with an increase of the average Y–O distance by 0.13Å for the composition of Prowatke and Klemme (2005). For the CAS-composition an increase of the coordination number from 6 to 7, along with an increase of the average Y–O distance by 0.06Å is obtained. The change of these parameters is associated with a considerable increase in the asymmetry and width of the Y–O pair distribution. Due to its size and charge, 6-fold coordinated Y will preferentially bond to non-bridging oxygens of the polymeric melt network to form a stable configuration, as is the case for the less polymerized melts with low ASI. In highly polymerized melts with ASI values close to one, 6-fold coordination of Y is not possible because almost only bridging oxygens are available. Consequently, over-bonding of bridging oxygens around Y is counterbalanced by an increase of coordination number and Y–O distance to satisfy local charge balance requirements.

The Influence of Rate Heterogeneity among Sites on the Time Dependence of Molecular Rates

Molecular evolutionary rate estimates have been shown to depend on the time period over which they are estimated. Factors such as demographic processes, calibration errors, purifying selection, and the heterogeneity of substitution rates among sites (RHAS) are known to affect the accuracy with which rates of evolution are estimated. We use mathematical modeling and Bayesian analyses of simulated sequence alignments to explore how mutational hotspots can lead to time-dependent rate estimates. Mathematical modeling shows that underestimation of molecular rates over increasing time scales is inevitable when RHAS is ignored. Although a gamma distribution is commonly used to model RHAS, we show that when the actual RHAS deviates from a gamma-like distribution, rates can either be under- or overestimated in a time-dependent manner. Simulations performed under different scenarios of RHAS confirm the mathematical modeling and demonstrate the impacts of time-dependent rates on estimates of divergence times. Most notably, erroneous rate estimates can have narrow credibility intervals, leading to false confidence in biased estimates of rates, and node ages. Surprisingly, large errors in estimates of overall molecular rate do not necessarily generate large errors in divergence time estimates. Finally, we illustrate the correlation between time-dependent rate patterns and differential saturation between quickly and slowly evolving sites. Our results suggest that data partitioning or simple nonparametric mixture models of RHAS significantly improve the accuracy with which node ages and substitution rates can be estimated.

Evaluation of the NASA GISS Single-Column Model Simulated Clouds Using Combined Surface and Satellite Observations

Abstract Three years of surface and Geostationary Operational Environmental Satellite (GOES) data from the Department of Energy Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) site are used to evaluate the NASA GISS Single Column Model (SCM) simulated clouds from January 1999 to December 2001. The GOES-derived total cloud fractions for both 0.5° and 2.5° grid boxes are in excellent agreement with surface observations, suggesting that ARM point observations can represent large areal observations. Low (<2 km), middle (2–6 km), and high (>6 km) levels of cloud fractions, however, have negative biases as compared to the ARM results due to multilayer cloud scenes that can either mask lower cloud layers or cause misidentifications of cloud tops. Compared to the ARM observations, the SCM simulated most midlevel clouds, overestimated low clouds (4%), and underestimated total and high clouds by 7% and 15%, respectively. To examine the dependence of the modeled high and low clouds on the large-scale synoptic patterns, variables such as relative humidity (RH) and vertical pressure velocity (omega) from North American Regional Reanalysis (NARR) data are included. The successfully modeled and missed high clouds are primarily associated with a trough and ridge upstream of the ARM SGP, respectively. The PDFs of observed high and low occurrence as a function of RH reveal that high clouds have a Gaussian-like distribution with mode RH values of ∼40%–50%, whereas low clouds have a gammalike distribution with the highest cloud probability occurring at RH ∼75%–85%. The PDFs of modeled low clouds are similar to those observed; however, for high clouds the PDFs are shifted toward higher values of RH. This results in a negative bias for the modeled high clouds because many of the observed clouds occur at RH values below the SCM-specified stratiform parameterization threshold RH of 60%. Despite many similarities between PDFs derived from the NARR and ARM forcing datasets for RH and omega, differences do exist. This warrants further investigation of the forcing and reanalysis datasets.

Coalescent Time Distributions in Trees of Arbitrary Size

The relationship between speciation times and the corresponding times of gene divergence is of interest in phylogenetic inference as a means of understanding the past evolutionary dynamics of populations and of estimating the timing of speciation events. It has long been recognized that gene divergence times might substantially pre-date speciation events. Although the distribution of the difference between these has previously been studied for the case of two populations, this distribution has not been explicitly computed for larger species phylogenies. Here we derive a simple method for computing this distribution for trees of arbitrary size. A two-stage procedure is proposed which (i) considers the probability distribution of the time from the speciation event at the root of the species tree to the gene coalescent time conditionally on the number of gene lineages available at the root; and (ii) calculates the probability mass function for the number of gene lineages at the root. This two-stage approach dramatically simplifies numerical analysis, because in the first step the conditional distribution does not depend on an underlying species tree, while in the second step the pattern of gene coalescence prior to the species tree root is irrelevant. In addition, the algorithm provides intuition concerning the properties of the distribution with respect to the various features of the underlying species tree. The methodology is complemented by developing probabilistic formulae and software, written in R. The method and software are tested on five-taxon species trees with varying levels of symmetry. The examples demonstrate that more symmetric species trees tend to have larger mean coalescent times and are more likely to have a unimodal gamma-like distribution with a long right tail, while asymmetric trees tend to have smaller mean coalescent times with an exponential-like distribution. In addition, species trees with longer branches generally have shorter mean coalescent times, with branches closest to the root of the tree being most influential.

Spontaneous switching among multiple spatio-temporal patterns in three-oscillator systems constructed with oscillatory cells of true slime mold

Three-oscillator systems with plasmodia of true slime mold, Physarum polycephalum, which is an oscillatory amoeba-like unicellular organism, were experimentally constructed and their spatio-temporal patterns were investigated. Three typical spatio-temporal patterns were found: rotation ( R ), partial in-phase ( P I ), and partial anti-phase with double frequency ( P A ). In pattern R , phase differences between adjacent oscillators were almost 120 ∘. In pattern P I , two oscillators were in-phase and the third oscillator showed anti-phase against the two oscillators. In pattern P A , two oscillators showed anti-phase and the third oscillator showed frequency doubling oscillation with small amplitude. Actually each pattern is not perfectly stable but quasi-stable. Interestingly, the system shows spontaneous switching among the multiple quasi-stable patterns. Statistical analyses revealed a characteristic in the residence time of each pattern: the histograms seem to have Gamma-like distribution form but with a sharp peak and a tail on the side of long period. That suggests the attractor of this system has complex structure composed of at least three types of sub-attractors: a “Gamma attractor”—involved with several Poisson processes, a “deterministic attractor”—the residence time is deterministic, and a “stable attractor”—each pattern is stable. When the coupling strength was small, only the Gamma attractor was observed and switching behavior among patterns R , P I , and P A almost always via an asynchronous pattern named O . A conjecture is as follows: Internal/external noise exposes each pattern of R , P I , and P A coexisting around bifurcation points: That is observed as the Gamma attractor. As coupling strength increases, the deterministic attractor appears then followed by the stable attractor, always accompanied with the Gamma attractor. Switching behavior could be caused by regular existence of the Gamma attractor.

Compartment Volume Influences Microtubule Dynamic Instability: A Model Study

Microtubules (MTs) are cytoskeletal polymers that exhibit dynamic instability, the random alternation between growth and shrinkage. MT dynamic instability plays an essential role in cell development, division, and motility. To investigate dynamic instability, simulation models have been widely used. However, conditions under which the concentration of free tubulin fluctuates as a result of growing or shrinking MTs have not been studied before. Such conditions can arise, for example, in small compartments, such as neuronal growth cones. Here we investigate by means of computational modeling how concentration fluctuations caused by growing and shrinking MTs affect dynamic instability. We show that these fluctuations shorten MT growth and shrinkage times and change their distributions from exponential to non-exponential, gamma-like. Gamma-like distributions of MT growth and shrinkage times, which allow optimal stochastic searching by MTs, have been observed in various cell types and are believed to require structural changes in the MT during growth or shrinkage. Our results, however, show that these distributions can already arise as a result of fluctuations in the concentration of free tubulin due to growing and shrinking MTs. Such fluctuations are possible not only in small compartments but also when tubulin diffusion is slow or when many MTs (de)polymerize synchronously. Volume and all other factors that influence these fluctuations can affect MT dynamic instability and, consequently, the processes that depend on it, such as neuronal growth cone behavior and cell motility in general.

COBE satellite measurement, Cantorian space and cosmic strings

This note draws attention to the possibility of a deep connection between COBE measurement, the recent results of Cantorian-fractal spacetime and cosmic strings. Our conclusion is that all three lines of research seem to confirm the big bang scenario and relate to a gamma-like distribution function revealing a link to a common statistical and thermodynamical origin.

Cycle-time and residence-time density approximations in a stochastic model for circulatory transport.

The concentration of a drug in the circulatory system is studied under two different elimination strategies. The first strategy--geometric elimination--is the classical one which assumes a constant elimination rate per cycle. The second strategy--Poisson elimination--assumes that the elimination rate changes during the process of elimination. The problem studied here is to find a relationship between the residence-time distribution and the cycle-time distribution for a given rule of elimination. While the presented model gives this relationship in terms of Laplace-Stieltjes transform., the aim here is to determine the shapes of the corresponding probability density functions. From experimental data, we expect positively skewed, gamma-like distributions for the residence time of the drug in the body. Also, as some elimination parameter in the model approaches a limit, the exponential distribution often arises. Therefore, we use Laguerre series expansions, which yield a parsimonious approximation of positively skewed probability densities that are close to a gamma distribution. The coefficients in the expansion are determined by the central moments, which can be obtained from experimental data or as a consequence of theoretical assumptions. The examples presented show that gamma-like densities arise for a diverse set of cycle-time distribution and under both elimination rules.

Cycle-time and residence-time density approximations in a stochastic model for circulatory transport

The concentration of a drug in the circulatory system is studied under two different elimination strategies. The first strategy—geometric elimination—is the classical one which assumes a constant elimination rate per cycle. The second strategy—Poisson elimination—assumes that the elimination rate changes during the process of elimination. The problem studied here is to find a relationship between the residence-time distribution and the cycle-time distribution for a given rule of elimination. While the presented model gives this relationship in terms of Laplace-Stieltjes transform, the aim here is to determine the shapes of the corresponding probability density functions. From experimental data, we expect positively skewed, gamma-like distributions for the residence time of the drug in the body. Also, as some elimination parameter in the model approaches a limit, the exponential distribution often arises. Therefore, we use Laguerre series expansions, which yield a parsimonious approximation of positively skewed probability densities that are close to a gamma distribution. The coefficients in the expansion are determined by the central moments, which can be obtained from experimental data or as a consequence of theoretical assumptions. The examples presented show that gamma-like densities arise for a diverse set of cycle-time distributions and under both elimination rules.

The Surplus Theory of Social Stratification and the Size Distribution of Personal Wealth

The Surplus Theory of Social Stratification explains of wealth in terms of (1) the fugitivity of wealth not needed to sustain the production of more wealth, (2) the tendency of wealth to flow into the hands of those who are already disproportionately wealthy, and (3) the ability of workers in an industrial society to retain a greater share of the wealth they produce than workers in societies with more primitive technologies. Size distributions of wealth from societies at different levels of technology can be fitted by a family of gamma distributions, whose shape parameter is related to a society's level of technology. The Surplus Theory implies a stochastic process that generates gamma-like distributions. Analysis of this the Inequality Process, explains many facts about size distributions of personal wealth. This paper presents a general theory of the size distribution of personal wealth. The paper shows that the Surplus Theory of Social Stratification implies a stochastic called here the inequality process, that reproduces the major features of size distributions of personal wealth in societies at various technological levels. 'This paper brings together two literatures without cross-citations. These are (1) the literature on the fitting of probability density functions to size distributions of personal wealth, and (2) the literature in anthropology, social archeology, and sociology on the emergence of as populations of hunter/gatherers developed agriculture, the literature of the Surplus Theory of Social Stratification. This paper assumes that the concept of wealth has very nearly the same meaning in societies as different as those of hunter/gatherers on the one hand and industrial societies on the other. However, no comparisons

A note on seasonal Markov chains with gamma or gamma-like distributions

Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain {Xt} has a Laplace transform (LT) L(Xt+1; θ |Χ t=x) of the ‘exponential' form H(θ) exp{ – G(θ)x}. An algorithm is derived for the computation of the LT of Σat,Χ t for this class, and for a seasonal generalization of it.A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968).A seasonal version of this process is developed, valid for any number of seasons.Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.

A note on seasonal Markov chains with gamma or gamma-like distributions

Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain {Xt } has a Laplace transform (LT) L(Xt +1; θ |Χ t=x) of the ‘exponential' form H(θ) exp{ – G(θ)x}. An algorithm is derived for the computation of the LT of Σat,Χ t for this class, and for a seasonal generalization of it. A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968). A seasonal version of this process is developed, valid for any number of seasons. Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.