Fixed Shape Parameter Research Articles

Effect of heat shock and recovery temperature on variability of single cell lag time of Cronobacter turicensis

The effect of heat stress and subsequent recovery temperature on the individual cellular lag of Cronobacter turicensis was analysed using optical density measurements. Low numbers of cells were obtained through serial dilution and the time to reach an optical density of 0.035 was determined. Assuming the lag of a single cell follows a shifted Gamma distribution with a fixed shape parameter, the effect of recovery temperature on the individual lag of untreated and sublethally heat treated cells of Cr. turicensis were modelled. It was found that the shift parameter (Tshift) increased asymptotically as the temperature decreased while the logarithm of the scale parameter (θ) decreased linearly with recovery temperature. To test the validity of the model in food, growth of low numbers of untreated and heat treated Cr. turicensis in artificially contaminated infant first milk was measured experimentally and compared with predictions obtained by Monte Carlo simulations. Although the model for untreated cells slightly underestimated the actual growth in first milk at low temperatures, the model for heat treated cells was in agreement with the data derived from the challenge tests and provides a basis for reliable quantitative microbiological risk assessments for Cronobacter spp. in infant milk.

Estimating the spot rate curve using the Nelson–Siegel model: A ridge regression approach

The Nelson–Siegel model is widely used in practice for fitting the term structure of interest rates. Due to the ease in linearizing the model, a grid search or an OLS approach using a fixed shape parameter are popular estimation procedures. The estimated grid search parameters, however, have been reported (1) to behave erratically over time, and (2) to have relatively large variances. On the other hand, parameter estimates based on a fixed shape parameter, while avoiding multicollinearity, turn out to be too smooth. We show that the Nelson–Siegel model can become heavily collinear depending on the estimated/fixed shape parameter. A simple procedure based on ridge regression can remedy the reported problems significantly.

Methodology and Equations of Mineral Production Forecast. Part II. The Fundamental Equation. Crude Oil Production in USA

The fundamental equation of mineral production allows to model and design the dynamics of mineral production, however complex they are or could be. It considers not only the case of a constant production to reserves ratio for given intervals of time, but with a piecewise approach, it is also enabled to account the variation on time of this ratio. With a constant production to reserves ratio, the limit expression of the fundamental equation takes the form of an Erlang distribution with a fixed shape parameter. The rate parameter equals the scale factor. The discrete piecewise version, instead of considering the reserves and the production to reserves ratio being constant through certain intervals of time, updates both variables by units of time. This version, using either lineal or non lineal functions for the variables involved, let to model known production profiles or to forecast them by experimental design. The Hubbert’s linearization updated with recent data and the p-box method applied to determine ultimate recovery of U.S. crude oil reserves indicate official accounts underestimate them. The analysis of the ideal model of production based on Hubbert’s linearization and curve, can be made by decomposing it in the distribution with time of the reserves and of the production to reserves ratio. The distribution of reserves with time is synchronized for both the ideal Hubbert’s curve and real profiles, disregarding whether they match or not. The departure of real profiles from the ideal Hubbert’s curve lies on the differences or correspondences of the distribution with time of the production to reserves ratio. The MonteCarlo simulation applied to forecast US crude oil production for the next five years points to a slow decline, with average annual yields presenting a difference lower than 10% between the start and the end of the simulation.

Estimating the Yield Curve Using the Nelson-Siegel Model: A Ridge Regression Approach

The Nelson-Siegel model is widely used in practice for fitting the term structure of interest rates. Due to the ease in linearizing the model, a grid search or an OLS approach using a fixed shape parameter are popular estimation procedures. The estimated parameters, however, have been reported (1) to behave erratically over time, and (2) to have relatively large variances. We show that the Nelson-Siegel model can become heavily collinear depending on the estimated/fixed shape parameter. A simple procedure based on ridge regression can remedy the reported problems significantly.

Statistical analysis of rainfall intermittency at small spatial and temporal scales

[1] Rainfall intermittency is analyzed and quantified at small spatial and temporal scales using 2 years of radar and disdrometer data collected in Switzerland. Analytical models are fitted and used to describe the intermittency for spatial scales between 0 and 30 km and temporal resolutions between 30 s and 6 h, providing climatological parameterizations for efficient and accurate upscaling/downscaling of intermittent rainfall fields. First, the zero rainfall probability is analyzed with respect to the considered spatial resolution. Second, the spatial autocorrelation of rainfall intermittency is quantified with respect to the temporal resolution. Finally, the temporal autocorrelation is analyzed with respect to the spatial resolution. The results show that all these different aspects of rainfall intermittency can be accurately described by a scaled exponential function with a fixed shape parameter and a variable scale parameter. Models describing this variability are provided.

Information geometry in neural spike sequences

An information geometrical method is developed for characterizing or classifying neurons in cortical areas whose spike rates fluctuate in time. The interspike intervals (ISIs) of a spike sequence of a neuron is modeled as a gamma process with a time-variant spike rate, a fixed shape parameter and a fixed absolute refractory period. We formulate the problem of estimating the fixed parameters as semiparametric estimation and apply an information geometrical method to derive the optimal estimators from a statistical viewpoint.

Iterative adaptive RBF methods for detection of edges in two-dimensional functions

Radial basis functions have gained popularity for many applications including numerical solution of partial differential equations, image processing, and machine learning. For these applications it is useful to have an algorithm which detects edges or sharp gradients and is based on the underlying basis functions. In our previous research, we proposed an iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities in one-dimensional problems. The iterative edge detection method is based on the observation that the absolute values of the expansion coefficients of multiquadric radial basis function approximation grow exponentially in the presence of a local jump discontinuity with fixed shape parameters but grow only linearly with vanishing shape parameters. The different growth rate allows us to accurately detect edges in the radial basis function approximation. In this work, we extend the one-dimensional iterative edge detection method to two-dimensional problems. We consider two approaches: the dimension-by-dimension technique and the global extension approach. In both cases, we use a rescaling method to avoid ill-conditioning of the interpolation matrix. The global extension approach is less efficient than the dimension-by-dimension approach, but is applicable to truly scattered two-dimensional points, whereas the dimension-by-dimension approach requires tensor product grids. Numerical examples using both approaches demonstrate that the two-dimensional iterative adaptive radial basis function method yields accurate results.

Evaluation of ground-based remotely sensed liquid water cloud properties using shortwave radiation measurements

Water cloud optical and microphysical properties are required for a better understanding of the impact of aerosols on the solar radiation feedback. The introduced retrieval technique provides droplet concentration, effective radius and optical thickness on a basis of ground-based remote sensing observations and a vertical cloud model. Mainly cloud radar, microwave radiometer and ceilometer observations are used in this approach. The model assumptions are related to a sub-adiabatic approach, in which cloud mixing processes are predefined to be homogeneous. A gamma droplet size distribution with a fixed shape parameter is considered to relate the observations with the retrieval products. The technique is applied on a water cloud case and the uncertainties in relation to the model assumptions and errors in the measurements are determined. The greatest uncertainty in the retrieval of droplet concentration is related to the fixed shape parameter, which requires in-situ data for validation processes to quantify the results. The optical parameters are less sensitive to the model assumptions and they are evaluated with a radiative closure experiment. They are used as input for radiative transfer calculations in order to compare the simulations with radiation measurements at the ground. There is a relatively good agreement between the simulated and measured radiation considering the horizontal cloud inhomogeneity, although a bias of around 5% still exists. Therefore the technique might be a suitable approach of retrieving quantitative cloud optical properties independently from radiation observations.

Modeling and predicting inactivation of Escherichia coli under isobaric and dynamic high hydrostatic pressure

Abstract Published experimental isobaric inactivation curves of Escherichia coli cells in peptone solution and raw milk cheese were fitted with the Weibull model (with a fixed shape parameter) and the pressure dependent rate parameter of the Weibull model by log-logistic equation. These were used to predict these microorganisms' nonisobaric survival curves under different pressure profiles. The assumption has been that the inactivation rate in nonisobaric processes is a function of the instantaneous pressure and the corresponding survival ratio. This could be expressed by a difference equation and can be solved incrementally by a software like Microsoft Excel® which is freely available in the web. Although the predictions were not perfectly matched with the data, reasonable estimates were obtained under different pressure profiles. In principle, the same methodology can be used to predict nonisobaric survival curves during pulsed high pressure treatments if the temperature effects are negligible. Industrial relevance There is now enough evidence that a log-linear isothermal survival curve of bacterial cells is an exception rather than a rule and the same can be said about semilogarithmic survival curves of organisms exposed to high hydrostatic pressure (HHP). Therefore accurate description of survival curves with other modeling techniques is essential. Moreover, userfriendly software like Microsoft Excel® is more familiar than the other softwares and could help the people in the food industry to predict the nonisobaric inactivation of microorganisms.

Retrieval of Raindrop Size Distribution from Simulated Dual-Frequency Radar Measurements

Abstract Observations of raindrop size distributions (DSDs) have validated the use of three-parameter distribution functions in representing the observed spectra. However, dual-frequency radar measurements are limited to retrieving two independent parameters of the DSD, thus requiring a constraint on a three-parameter distribution. In this study, disdrometer observations from a variety of climate regions are employed to develop constraints on the gamma distribution that are optimized for dual-frequency radar rainfall retrievals. These observations are composited by reflectivity, and then gamma parameters are fit to the composites. The results show considerable variability in shape parameter between regions and within a region at different reflectivities. Most notable is that oceanic regions exhibit maxima in shape parameter at 13.6-GHz reflectivities between 40 and 50 dBZ, in contrast to continental regions. The shape parameter and slope parameter of all composite DSDs are poorly correlated. Thus, constraints of a constant shape parameter or shape parameter–slope parameter relationship are inadequate to represent the observed variability. However, the shape and slope parameters are highly correlated at a given reflectivity. Constraints of a fixed shape parameter and relationships between a shape parameter m and slope parameter Λ, both of which are given as functions of 13.6-GHz reflectivity, are applied to retrieve rain rate, liquid water content, and mean mass diameter from the composites. The m–Λ relationships perform best at high reflectivity (dBZ13.6 > 35), whereas the fixed shape parameter generally results in lower error at medium and low reflectivities (dBZ13.6 < 35). All calculations have been made under the assumption that the reflectivity measurements have been corrected for attenuation.

Usefulness of the reversible jump Markov chain Monte Carlo model in regional flood frequency analysis

Regional flood frequency analysis is a convenient way to reduce estimation uncertainty when few data are available at the gauging site. In this work, a model that allows a non‐null probability to a regional fixed shape parameter is presented. This methodology is integrated within a Bayesian framework and uses reversible jump techniques. The performance on stochastic data of this new estimator is compared to two other models: a conventional Bayesian analysis and the index flood approach. Results show that the proposed estimator is absolutely suited to regional estimation when only a few data are available at the target site. Moreover, unlike the index flood estimator, target site index flood error estimation seems to have less impact on Bayesian estimators. Some suggestions about configurations of the pooling groups are also presented to increase the performance of each estimator.

A note on derivation of the generating function for the right truncated Rayleigh distribution

An expression is obtained for the probability that a Weibull random variable falls after the truncation and within a finite interval. However small, the truncation in the Weibull distribution (when the value of the shape parameter is two, it is called the Rayleigh distribution) has an impact. An attempt is made to obtain generating functions for two fixed shape parameters.

Information Geometry of Interspike Intervals in Spiking Neurons

An information geometrical method is developed for characterizing or classifying neurons in cortical areas, whose spike rates fluctuate in time. Under the assumption that the interspike intervals of a spike sequence of a neuron obey a gamma process with a time-variant spike rate and a fixed shape parameter, we formulate the problem of characterization as a semiparametric statistical estimation, where the spike rate is a nuisance parameter. We derive optimal criteria from the information geometrical viewpoint when certain assumptions are added to the formulation, and we show that some existing measures, such as the coefficient of variation and the local variation, are expressed as estimators of certain functions under the same assumptions.

On Empirical Bayes Testing for a Location Parameter in a Shifted Gamma Distribution

Consider a location parameter θ in a shifted gamma distribution having a where is the fixed shape parameter. We study the two-action problem of testing against using the empirical Bayes approach. An empirical Bayes test is constructed. Under some mild conditions, is shown to be asymptotically optimal at a rate of order for where n is the number of past data available when the current decision problem is considered. The achieved rate of much improves the existing result in the literature.

Bivariate Distributions with Pearson Type VII Conditionals

In this article the most general class of bivariate distributions such that both conditional densities are Pearson Type VII, with fixed shape parameter, is fully characterized. Some of its properties and relations with other distributions are explored. The estimation of parameters is considered by the methods of maximum likelihood and pseudolikelihood and a method for random variate generation is presented along with a simulation experiment. Bivariate and multivariate extensions of the Pearson Type VII conditionals distribution are also discussed.

Some Remarks on Exponential Families

Abstract The following facts may serve to provide a feeling about how restrictive the assumption of an exponential family is. (a) A one-parameter exponential family in standard form with respect to Lebesgue measure is a location parameter family iff it is normal with fixed variance. (b) It is a scale parameter family iff it is gamma with fixed shape parameter. Both facts are known (see Borges and Pfanzagl 1965; Ferguson 1962; Lindley 1958) but may not have received as much attention as they deserve. Under the assumption of differentiable densities, short and elementary proofs are given.

A differential equations approach to the modal location for a family of bivariate gamma distributions

Analytical and numerical results are given for determining the location of the mode of a class of bivariate gamma densities as a function of the parameters. The model location for a class of bivariate gammas as considered by Kibble (1941, Sankhya A 5 137–150) is shown to satisfy a nonlinear differential equation in ϱ, the correlation coefficient for fixed shape parameter. Qualitative and asymptotic properties of the modal location are also given. Whenever the shape parameters are unequal, analytical and numerical results are used to provide a conjecture for the modal location in the general case.

Estimating pareto quantiles using two order statistics

Asymptotically best linear unbiased estimators of the population quantiles for the location-scale Pareto distribution with fixed shape parameter are obtained using two suitably chosen order statistics. Formulae for the appropriate order statistics, coefficients, variances, and asymptotic relative efficiencies (relative to the usual non-parametric estimator for quantiles) are given

A new life-quality measure for electron tubes

Since the Military Standard MIL-E-1B replaced JAN-1A Specifications, the tube industry and tube users have been searching for a common satisfactory measure for the life quality of electron tubes. This paper proposes a new measure, incorporating the information in the electron tube failure-age distribution for replacement of the obsolete Average Life Percentage originally specified in JAN-1A. A Weibull distribution with a fixed shape parameter was found appropriate for describing the failure age of electron tubes under a wide variety of application. Hence, a single measure, the scale parameter, will characterize the life quality of electron tubes. Secondary life-quality measures, such as mean life, median life and the recently introduced “reliability,” hazard rate, etc., can all be expressed in terms of the proposed measure. Some numerical examples are given at the end of the paper.