Shifted Gamma Distributions Research Articles

Objective Bayesian Approach for Binary Hypothesis Testing of Multivariate Gaussian Observations

This paper proposes an objective Bayesian detector for the binary hypothesis testing problem in which the observation vectors are IID and follow multivariate Gaussian distribution under both the null and alternative hypotheses. The mean vector and covariance matrix under each hypothesis are not known, they also do not have special structure other than the covariance matrices are partially ordered in the positive definite cone with their difference positive semi-definite. An example of such a problem is the detection of Gaussian random signal in Gaussian noise. Non-informative priors are imposed on the unknown parameters for computing the marginals and evaluating the Bayes factor test statistic, where the prior for the covariance matrix is applied to the Cholesky factorization of the precision matrix. For the null hypothesis, we propose conjugate priors for the unknowns. For the alternative hypothesis, we propose uniform prior on the mean vector, and a class of objective priors that encompasses the Jeffreys, Independence Jeffreys, Geisser and Cornfield, Haar Measure and Reference priors for the precision matrix. The proposed priors enable closed-form expressions for the marginals and lead to an objective Bayes factor having the posteriors mainly governed by the data. The proposed detector is analyzed for justifying the priors used, deriving the theoretical moments, and approximating the false alarm and detection probability densities by the shifted Gamma distributions. The developed detector exhibits improvement in detection performance over the energy detector (ED) and the generalized likelihood ratio test (GLRT).

Distributions of hook lengths divisible by two or three

For fixed t=2 or 3, we investigate the statistical properties of {Yˆt(n)}, the sequence of random variables corresponding to the number of hook lengths divisible by t among the partitions of n. We characterize the support of Yˆt(n) and show, in accordance with empirical observations, that the support is vanishingly small for large n. Moreover, we demonstrate that the nonzero values of the mass functions of Yˆ2(n) and Yˆ3(n) approximate continuous functions. Finally, we prove that although the mass functions fail to converge, the cumulative distribution functions of {Yˆ2(n)} and {Yˆ3(n)} converge pointwise to shifted Gamma distributions, completing a characterization initiated by Griffin–Ono–Tsai for t≥4.

Multiplicity moments using Tsallis statistics in high-energy hadron–nucleus interactions

The study of higher-order moments of a distribution and its cumulants constitute a sensitive tool to investigate the correlations between the particle produced in high-energy interactions. In our previous work, we have used the Tsallis [Formula: see text] statistics, NBD, Gamma and shifted Gamma distributions to describe the multiplicity distributions in [Formula: see text]-nucleus and [Formula: see text]-nucleus fixed target interactions at various energies ranging from [Formula: see text][Formula: see text]GeV to 800[Formula: see text]GeV. In this study, we have extended our analysis by calculating the moments using the Tsallis model at these fixed target experiment data. By using the Tsallis model, we have also calculated the average charged multiplicity and its dependence on energy. It is found that the average charged multiplicity and moments predicted by the Tsallis statistics are in much agreement with the experimental values and indicates the success of the Tsallis model on data from visual detectors. The study of moments also illustrates that KNO scaling hypothesis holds good at these energies.

Probabilistic Forecasting of Snowfall Amounts Using a Hybrid between a Parametric and an Analog Approach

Abstract Forecast uncertainty associated with the prediction of snowfall amounts is a complex superposition of the uncertainty about precipitation amounts and the uncertainty about weather variables like temperature that influence the snow-forming process. In situations with heavy precipitation, parametric, regression-based postprocessing approaches often perform very well since they can extrapolate relations between forecast and observed precipitation amounts established with data from more common events. The complexity of the relation between temperature and snowfall amounts, on the other hand, makes nonparametric techniques like the analog method an attractive choice. In this article we show how these two different methodologies can be combined in a way that leverages the respective advantages. Predictive distributions of precipitation amounts are obtained using a heteroscedastic regression approach based on censored, shifted gamma distributions, and quantile forecasts derived from them are used together with ensemble forecasts of temperature to find analog dates where both quantities were similar. The observed snowfall amounts on these dates are then used to compose an ensemble that represents the uncertainty about future snowfall. We demonstrate this approach with reforecast data from the Global Ensemble Forecast System (GEFS) and snowfall analyses from the National Operational Hydrologic Remote Sensing Center (NOHRSC) over an area within the northeastern United States and an area within the U.S. mountain states.

Generating Calibrated Ensembles of Physically Realistic, High-Resolution Precipitation Forecast Fields Based on GEFS Model Output

Abstract Enhancements of multivariate postprocessing approaches are presented that generate statistically calibrated ensembles of high-resolution precipitation forecast fields with physically realistic spatial and temporal structures based on precipitation forecasts from the Global Ensemble Forecast System (GEFS). Calibrated marginal distributions are obtained with a heteroscedastic regression approach using censored, shifted gamma distributions. To generate spatiotemporal forecast fields, a new variant of the recently proposed minimum divergence Schaake shuffle technique, which selects a set of historic dates in such a way that the associated analysis fields have marginal distributions that resemble the calibrated forecast distributions, is proposed. This variant performs univariate postprocessing at the forecast grid scale and disaggregates these coarse-scale precipitation amounts to the analysis grid by deriving a multiplicative adjustment function and using it to modify the historic analysis fields such that they match the calibrated coarse-scale precipitation forecasts. In addition, an extension of the ensemble copula coupling (ECC) technique is proposed. A mapping function is constructed that maps each raw ensemble forecast field to a high-resolution forecast field such that the resulting downscaled ensemble has the prescribed marginal distributions. A case study over an area that covers the Russian River watershed in California is presented, which shows that the forecast fields generated by the two new techniques have a physically realistic spatial structure. Quantitative verification shows that they also represent the distribution of subgrid-scale precipitation amounts better than the forecast fields generated by the standard Schaake shuffle or the ECC-Q reordering approaches.

Satellite Precipitation Characterization, Error Modeling, and Error Correction Using Censored Shifted Gamma Distributions.

Satellite multisensor precipitation products (SMPPs) have a variety of potential uses, but suffer from relatively poor accuracy due to systematic biases and random errors in precipitation occurrence and magnitude. We use the Censored Shifted Gamma Distribution (CSGD) to characterize the Tropical Rainfall Measurement Mission Multi-Satellite Precipitation Analysis (TMPA), a commonly-used SMPP, and to compare it against the rain gage-based North American Land Data Assimilation System Phase 2 (NLDAS-2) reference precipitation dataset across the conterminous United States. The CSGD describes both the occurrence and the magnitude of precipitation. Climatological CSGD characterization reveals significant regional differences between TMPA and NLDAS-2 in terms of magnitude and probability of occurrence. We also use a flexible CSGD-based error modeling framework to quantify errors in TMPA relative to NLDAS-2. The framework can model conditional bias as either a linear or nonlinear function of satellite precipitation rate and can produce a "conditional CSGD" of describing the distribution of "true" precipitation based on a satellite observation. The framework is also used to "merge" TMPA with atmospheric variables from Modern-Era Retrospective analysis for Research and Applications (MERRA-2) to reduce SMPP errors. Despite the coarse resolution of MERRA-2, this merging offers robust reductions in random error due to the better performance of numerical models in resolving stratiform precipitation. Improvements in the near-realtime version of TMPA are relatively greater than for the higher-latency research version.

How does reach‐scale stream‐hyporheic transport vary with discharge? Insights from rSAS analysis of sequential tracer injections in a headwater mountain stream

AbstractThe models of stream reach hyporheic exchange that are typically used to interpret tracer data assume steady‐flow conditions and impose further assumptions about transport processes on the interpretation of the data. Here we show how rank Storage Selection (rSAS) functions can be used to extract “process‐agnostic” information from tracer breakthrough curves about the time‐varying turnover of reach storage. A sequence of seven slug injections was introduced to a small stream at base flow over the course of a diel fluctuation in stream discharge, providing breakthrough curves at discharges ranging from 0.7 to 1.2 L/s. Shifted gamma distributions, each with three parameters varying stepwise in time, were used to model the rSAS function and calibrated to reproduce each breakthrough curve with Nash‐Sutcliffe efficiencies in excess of 0.99. Variations in the fitted parameters over time suggested that storage within the reach does not uniformly increase its turnover rate when discharge increases. Rather, changes in transit time are driven by both changes in the average rate of turnover (external variability) and changes in the relative rate that younger and older water contribute to discharge (internal variability). Specifically, at higher discharge, the turnover rate increased for the youngest part of the storage (corresponding to approximately 5 times the volume of the channel), while discharge from the older part of the storage remained steady, or declined slightly. The method is shown to be extensible as a new approach to modeling reach‐scale solute transport that accounts for the time‐varying, discharge‐dependent turnover of reach storage.

Statistical Postprocessing of Ensemble Precipitation Forecasts by Fitting Censored, Shifted Gamma Distributions*

Abstract A parametric statistical postprocessing method is presented that transforms raw (and frequently biased) ensemble forecasts from the Global Ensemble Forecast System (GEFS) into reliable predictive probability distributions for precipitation accumulations. Exploratory analysis based on 12 years of reforecast data and ⅛° climatology-calibrated precipitation analyses shows that censored, shifted gamma distributions can well approximate the conditional distribution of observed precipitation accumulations given the ensemble forecasts. A nonhomogeneous regression model is set up to link the parameters of this distribution to ensemble statistics that summarize the mean and spread of predicted precipitation amounts within a certain neighborhood of the location of interest, and in addition the predicted mean of precipitable water. The proposed method is demonstrated with precipitation reforecasts over the conterminous United States using common metrics such as Brier skill scores and reliability diagrams. It yields probabilistic forecasts that are reliable, highly skillful, and sharper than the previously demonstrated analog procedure. In situations with limited predictability, increasing the size of the neighborhood within which ensemble forecasts are considered as predictors can further improve forecast skill. It is found, however, that even a parametric postprocessing approach crucially relies on the availability of a sufficiently large training dataset.

Different rates of DNA replication at early versus late S-phase sections: multiscale modeling of stochastic events related to DNA content/EdU (5-ethynyl-2'deoxyuridine) incorporation distributions.

Mathematical modeling allows relating molecular events to single-cell characteristics assessed by multiparameter cytometry. In the present study we labeled newly synthesized DNA in A549 human lung carcinoma cells with 15-120 min pulses of EdU. All DNA was stained with DAPI and cellular fluorescence was measured by laser scanning cytometry. The frequency of cells in the ascending (left) side of the "horseshoe"-shaped EdU/DAPI bivariate distributions reports the rate of DNA replication at the time of entrance to S phase while their frequency in the descending (right) side is a marker of DNA replication rate at the time of transition from S to G2 phase. To understand the connection between molecular-scale events and scatterplot asymmetry, we developed a multiscale stochastic model, which simulates DNA replication and cell cycle progression of individual cells and produces in silico EdU/DAPI scatterplots. For each S-phase cell the time points at which replication origins are fired are modeled by a non-homogeneous Poisson Process (NHPP). Shifted gamma distributions are assumed for durations of cell cycle phases (G1, S and G2 M), Depending on the rate of DNA synthesis being an increasing or decreasing function, simulated EdU/DAPI bivariate graphs show predominance of cells in left (early-S) or right (late-S) side of the horseshoe distribution. Assuming NHPP rate estimated from independent experiments, simulated EdU/DAPI graphs are nearly indistinguishable from those experimentally observed. This finding proves consistency between the S-phase DNA-replication rate based on molecular-scale analyses, and cell population kinetics ascertained from EdU/DAPI scatterplots and demonstrates that DNA replication rate at entrance to S is relatively slow compared with its rather abrupt termination during S to G2 transition. Our approach opens a possibility of similar modeling to study the effect of anticancer drugs on DNA replication/cell cycle progression and also to quantify other kinetic events that can be measured during S-phase.

Communication: A minimal model for the diffusion-relaxation backbone dynamics of proteins

We present a model for the local diffusion-relaxation dynamics of the C(α)-atoms in proteins describing both the diffusive short-time dynamics and the asymptotic long-time relaxation of the position autocorrelation functions. The relaxation rate spectra of the latter are represented by shifted gamma distributions, where the standard gamma distribution describes anomalous slow relaxation in macromolecular systems of infinite size and the shift accounts for a smallest local relaxation rate in macromolecules of finite size. The resulting autocorrelation functions are analytic for any time t ≥ 0. Using results from a molecular dynamics simulation of lysozyme, we demonstrate that the model fits the position autocorrelation functions of the C(α)-atoms exceptionally well and reveals moreover a strong correlation between the residue's solvent-accessible surface and the fitted model parameters.

The Comparative Analysis of the Term Structure Models of the Affine Yield Class

There are some various methods of construction of models of the short interest rate (SIR) for the analysis of a term structure of the affine yield class. In the paper the analysis of three from them is carried out: V-model (Vasicek, 1977), CIR-model (Cox, Ingersoll and Ross, 1985), MC-model (Medvedev and Cox, 1996). The main attention is given to a research of mutual properties of functions of the affine structure of these models and properties of probability distributions of stochastic processes, which the SIR in these models follows. The unity of models is that the process of the SIR is described by the same scheme with the lower reflecting boundary, which in the V-model is receded at the minus infinity, it is equal to zero in CIR-model, and in a MC-model this boundary takes an intermediate position. It is shown that the conditional distribution of the SIR is compound Poisson, which in case of a V-model converges to normal, for a CIR-model is a mixture of gamma distributions, and in a MC-model is a mixture of shifted gamma distributions.

Effect of micropore size distribution on the surface diffusivity in microporous solids

In this paper, we will theoretically deal with the concentration dependence of the observed surface diffusivity due to the size distribution of the micropore. The mathematical model is derived based on the assumption of a local adsorption isotherm and a local surface diffusive flux for a particular micropore of radius r. The observed quantities—amount adsorbed and the overall flux, and, hence, the observed surface diffusion coefficient—are obtained from the integration over the full range of micropore. Various micropore distributions, such as bimodal, uniform, shifted gamma distributions, have been used in the numerical simulations. It is found that the micropore size distribution per se induces an increase in the observed surface diffusion coefficient with respect to loading, and the rate of increase depends on the various of the micropore size distribution as well as the shape of the distribution. For all distributions, the increase can be much higher, depending very much on the smallest micropore size. This study suggests that the micropore size distribution may be one of the sources for the variation of the observed surface diffusion coefficient with fractional loading, commonly observed in the literature. When dead-end pores are present (i.e. pores do not contribute to the surface flux), the observed surface diffusivity does not increase as drastically as that for the case of no dead-end pores. This is true irrespective of the capacity and affinity of the dead-end pores relative to those of the “flowing” pores.

A note on transit time and reliability for regular-route trucking

This paper describes the development of an empirical model for predicting transit time and reliability for regular-route LTL trucking. It is postulated that the distribution of transit time of a city-pair is a Gamma function with parameters determined by the length of haul. Parameters for a family of shifted Gamma distributions have been estimated using the travel time information from seven regular-route motor common carriers of general freight. Transit time and reliability are two of the most important level of service measures for any mode. The model developed is a level of service model from the “shipper's view” of the carrier's operation. It can be used in many of the partial equilibrium analyses of the transportation market. To model the intercity freight demand is a typical example. The procedures for applying the transit time model developed in the paper are outlined.