Exponential Densities Research Articles

Simplified interaction tests for non‐normal data in psychological research

This study examined three simple transformations which increase the power of F tests of main effects and interaction in 2 × 2 factorial designs under violation of normality. The study obtained Monte Carlo results from various heavy‐tailed densities, including mixed‐normal, exponential, Cauchy and Laplace densities, which are associated with grossly distorted probabilities of Type I and Type II errors. Transformation of scores to ranks made the F test for interaction robust and comparable in power to the Mann—Whitney‐Wilcoxon test for the same distributions. Transformation to ‘modular ranks’, having one‐fourth the number of values of conventional ranks, was equally effective. Detection and downweighting of outliers before performing the F test was more effective than rank methods for several distributions. Implications of these findings for the role of scales of measurement and nonparametric methods in psychological research are discussed.

Bounds on approximate steepest descent for likelihood maximization in exponential families

An approximate steepest descent strategy is described, converging in families of regular exponential densities to maximum likelihood estimates of density functions. These density estimates are also obtained by an application of the principle of minimum relative entropy subject to empirical constraints. We prove tight bounds on the increase of the log-likelihood at each iteration of our strategy for families of exponential densities whose log-densities are spanned by a set of bounded basis functions. >

On the statistical optimality of locally monotonic regression

Locally monotonic regression is a recently proposed technique for the deterministic smoothing of finite-length discrete signals under the smoothing criterion of local monotonicity. Locally monotonic regression falls within a general framework for the processing of signals that may be characterized in three ways: regressions are given by projections that are determined by semimetrics, the processed signals meet shape constraints that are defined at the local level, and the projections are optimal statistical estimates in the maximum likelihood sense. the authors explore the relationship between the geometric and deterministic concept of projection onto (generally nonconvex) sets and the statistical concept of likelihood, with the object of characterizing projections under the family of the p-semi-metrics as maximum likelihood estimates of signals contaminated with noise from a well-known family of exponential densities. >

Large classes of proper priors for linear models

We consider the problem of a hierarchical linear model with normal likelihood and variance known in the case where there is very little information about the hyperparameters. We address the problem proposing the use of a very wide class of priors that can be expressed as scale mixtures of normals. Two possibilities are considered: a class based on Double Exponential densities and a class based on Cauchy densities. An example is given to illustrate the techniques.

Moments of inverted scale mixtures

We consider characteristics of the distribution of Y = 1/X when X has a density which can be represented as a scale mixture of exponential densities. We show that moments of order 1 or greater will not exist for this class of distributions, although rational moments may exist. We provide a closed form expression for rational moments strictly in terms of the scaling density which defines the mixture. An intuitive explanation of this phenomenon, based on tail characteristics of the densities in this class, is examined.

Growth-promoting effects of progesterone in a human endometrial cancer cell line (Ishikawa-Var I)

Significant growth responses to progesterone of human endometrial adenocarcinoma cells (Ishikawa-Var I) were observed under in vitro culture conditions. Progesterone affected both the rate of exponential proliferation and cell population densities after the exponential phase. In the presence of the hormone, the doubling time of exponentially proliferating cells was reduced from 44 to 35.6 h and cell densities were increased by as much as 2–3 times over those of controls during approx. 2 weeks in culture. The effects of progesterone on cell population growth were dose dependent. Estradiol (10 −8 M) and testosterone (10 −6 M) did not affect cell densities and the effects of dexamethasone (10 −6 M) were small. In contrast, both progesterone and estradiol stimulated colony formation under anchorage-independent conditions in soft agar. These results suggest the possibility that growth of sensitive cell clones in endometrial tumors could be enhanced in some patients during adjuvant progestin therapy.

Asymptotic distributions of apparent open times and shut times in a single channel record allowing for the omission of brief events.

The openings and shuttings of individual ion channel molecules can be described by a Markov process with discrete states in continuous time. The predicted distributions of the durations of open times, shut times, bursts of openings, etc. are all described, in principle, by mixtures of exponential densities. In practice it is usually found that some of the open times, and the shut times, are too short to be detected reliably. If a fixed dead-time tau is assumed then it is possible to define, as an approximation to what is actually observed, an 'extended opening' or e-opening which starts with an opening of duration at least tau followed by any number of openings and shuttings, all the shut times being shorter than tau; the e-opening ends when a shut time longer than tau occurs. A similar definition is used for e-shut times. The probability densities, f(t), of these extended times have previously been obtained as expressions which become progressively more complicated, and numerically unstable to compute, as t-->infinity. In this paper we present, for the two-state model, an alternative representation as an infinite series of which a small number of terms gives a very accurate approximation of f (t) for large t. For the general model we present an asymptotic representation as a mixture of exponentials which is accurate for all except quite small values of t. Some simple model-independent corrections for missed events are discussed in relationship to the exact solutions.

Income distribution and the residential density gradient

W. C. Wheaton ( Journal of Economic Theory 9, 223–237, 1974 ) asserted that an increase in incomes lowers central densities raising suburban densities in closed monocentric cities with identical resident incomes and tastes. We show that with identical Cobb-Douglas tastes and identical incomes, a sufficient condition for densities to fall everywhere when incomes rise is that the land rent bill of the innermost resident be larger than the commuting bill of the outermost resident. Estimating this model using data in E. S. Mills (“Studies in the Structure of the Urban Economy,” Johns Hopkins Press, Baltimore/London, 1972), we show that densities are predicted to fall everywhere in Baltimore, Denver, Milwaukee, Philadelphia, Rochester, and Toledo when income is increased by 10% circa 1960. But, when we disaggregate the model using the 1960 census income distribution (keeping tastes identical), a 10% increase in incomes rotates densities around distances which vary from 3.3 miles in Toledo to 9.8 miles in Philadelphia. In addition to empirically vindicating Wheaton's assertion, the disaggregated model gives approximately negative exponential densities consistent with spatial equilibrium, and without resorting to R. F. Muth's (“Cities and Housing”, University of Chicago Press, 1969) assumption of a unitary compensated price elasticity of demand.

A procedure to estimate the probability distribution of recurrence times in the west-northwestern Hellenic arc

The determination of the probability distribution of recurrence times is the most important problem in the calculation of long-term conditional probabilities for the recurrence of large and great earthquakes. The principle of maximum entropy in conjunction with a goodness-of-fit test (chi-square or Kolmogorov-Smirnov test) may be employed to obtain estimates of these densities using recurrence data for some seismic regions. Four different distributions are characterized by the property of maximum entropy, as possible laws for recurrence times of the largest earthquakes: uniform, exponential, Gaussian and log-normal. To discriminate among these different probability distributions we use the probability theory and the chi-square test to check the goodness-of-fit to the distribution of recurrence time of shocks of magnitude 6.5 and largest occurred in the west-northwestern zone of the Hellenic arc from 1791 to 1983. It is found that the recurrence times data for the west-northwestern zone of the Hellenic arc cannot be represented by the uniform and the Gaussian probability densities. The recurrence times data for the west-northwestern zone of the Hellenic arc can be described accurately by the exponential and log-normal probability densities, which were predicted from the principle of maximum entropy. In other words, the principle of maximum entropy does not necessarily lead to a unique solution. In turn, the mathematical properties of these distributions could be used to derive different physical properties of the earthquake process in the west-northwestern zone of the Hellenic arc.

A Clonal Selection Based Timecourse Model for Antibody Responses to Killed Vaccine, with Applications to Foot and Mouth Disease

Published models for the timecourse of the immune response are reviewed for their applicability to data from animals that have been injected with killed vaccine. Simple models are required so that statistical fitting procedures become straightforward. A class of models that incorporates the concept of clonal selection has been found useful. Immune memory is described in terms of persistence of mean antibody affinity when the overall concentration of antibody has declined towards background. The model is demonstrated on multipoint affinity distributions and initial negative exponential densities, from which conditions for boundedness can be developed. Predictions are made that are consistent with experiments on the evolution of immunoassay data using foot and mouth disease vaccine.

Robust Bayesian Credible Intervals and Prior Ignorance

Summary In this paper we propose, survey and compare some classes of probability densities that may be used to represent partial prior information, to model either prior ignorance or Bayesian sensitivity analysis. We distinguish two types of models appropriate for two different situations: near ignorance models which are suitable in problems where there is little prior information, and neighbourhood models, which can be used to 'robustify' a strict Bayesian analysis in problems where there is substantial prior information about location. We argue that, especially for the first situation, a reasonable class of prior densities is not the same as a class of reasonable prior densities. We discuss various desiderata for a 'reasonable' class, including coherence and sensible dependence of inferences on sample size. The translation invariant models studied here are classes of conjugate priors, classes of double exponential densities and a neighbourhood of the uniform prior. Of the neighbourhood models we examine examples of E-contamination neighbourhoods (previously studied by Huber, Berger and Berliner) and intervals of measures (DeRobertis and Hartigan). We illustrate the models in the simple problem of constructing credible intervals for an unknown normal mean. Of the models studied in detail, a translation-invariant class of double exponential priors is favoured for modelling little prior information, and a type of interval of measures seems most suitable for robust Bayesian analysis.

A THREE-LAYER ADAPTIVE NETWORK FOR PATTERN DENSITY ESTIMATION AND CLASSIFICATION

Based on the assumption that most probability densities in real life can be approximated by a mixture of Gaussian densities, we propose here a three-layer adaptive network with each neuron in the lower hidden layer representing a Gaussian basis function (covariance matrix equal to [Formula: see text] where I is a unit matrix) to estimate various probability densities and serve as a Bayes classifier. The width of the basis function [Formula: see text] may be the same for all neurons in this layer or it may vary from one neuron to another. This paper investigates the effectiveness of the network for both cases and presents a localized learning algorithm to adjust the network parameters. The network was trained with artificial data derived from known mixtures of memoryless Gaussian sources as well as exponential and Gamma densities. The performance of the network as a pattern density estimator was measured in terms of the relative difference between the target probability density function (p.d.f.) which generates the training and testing data and the network output representing the estimation. Samples from two mixtures corresponding to two classes were used to test the network capability as a classifier by comparing its error rate against that of a Bayes classifier. Both one- and two-dimensional cases were explored. The successfulness of the network depended on how well the target p.d.f.’s were represented by the training samples, the number of hidden neurons employed in the network and how thoroughly the network was trained. It was also found that allowing each basis function to have an independent width had a predominant effect on the network performance.

Lower bounds on scintillation detector timing performance

Fundamental method-independent limits on the timing performance of scintillation detectors are useful for identifying regimes in which either present timing methods are nearly optimal or where a considerable performance gain might be realized using better pulse processing techniques. Several types of lower bounds on mean-squared timing error (MSE) performance have been developed and applied to scintillation detectors. The simple Cramèr-Rao (CR) bound can be useful in determining the limiting MSE for scintillators having a relatively high rate of photon production such as BaF 2 and NaI(Tl); however, it tends to overestimate the achievable performance for scintillators with lower rates such as BGO. For this reason, alternative bounds have been developed using rate-distortion theory [1,2] or by assuming that the conversion of energy to scintillation light must pass through excited states which have exponential lifetime densities [3]. The bounds are functions of the mean scintillation pulse shape, the scintillation intensity, and photodetector characteristics; they are simple to evaluate and can be used to conveniently assess the limiting timing performance of scintillation detectors.

The distributions of the apparent open times and shut times in a single channel record when brief events cannot be detected

The openings and shuttings of individual ion channel molecules can be described by a Markov process with discrete states in continuous time. The predicted distributions of the durations of open times, shut times, bursts of openings, etc., are all described, in principle, by mixtures of exponential densities. In practice it is usually found that some of the open times, and/or shut times, are too short to be detected reliably. If a fixed dead-time r is assumed then it is possible to define, as an approximation to what is actually observed, an ‘ extended opening ’ or e-opening which starts with an opening of duration at least r followed by any number of openings and shuttings, all the shut times being shorter than r ; the e-opening ends when a shut time longer than r occurs. A similar definition is used for e-shut times. Several authors have derived approximations to the distribution of durations of e-openings and e-shuttings. In this paper the exact distributions are derived. They are defined piecewise over the intervals r to 2r, 2r to 3 r ,..., etc., the distribution in each interval being a sum of products of polynomials in t with exponential terms. The number of terms is finite, but increases as intervals get further from t = r. An asymptotic form for large t (for which the exact solution becomes difficult to compute) is given for the two state case. The exact solution is compared with several approximations, some of which are shown to be good enough for use in most practical applications.

On the use of time constants as estimates of mean lifetimes in single channel data analysis

For Markov models of single channel kinetics, a sojourn time in a class of states has a density function which is usually a linear combination of exponential densities. There are many instances in the single channel literature where the time constants of exponentials fitted to sojourn time data have been used as estimated mean sojourn times in individual states, though the two may be very different. In the present study the nature and magnitude of this difference in the case of a two state class is illustrated analytically and numerically. The time constants should be viewed at best as approximations, possibly poor, to the estimated mean sojourn times. Estimates of kinetic parameters cannot in general be obtained explicitly from the fitted parameters of the density alone. However, this is shown to be possible in some special cases and enables direct estimation of, for example, the channel opening rate constant β (or an upper limit to the estimate of β in the case of multiple channels) in standard sequential three or four state models of nicotinic receptor kinetics, using only the fitted parameters of the closed-time density.

Estimation of mixing probabilities in multiclass finite mixtures

The problem of estimating prior probabilities in a mixture of M classes with known class conditional distributions is studied. The observation is a sequence of n independent, identically distributed mixture random variables. The first moments of appropriately formulated functions of observations are used to facilitate estimation. The complexity of these functions may vary from linear functions of the observations (in some cases) to complex functions of class conditional density functions of observations, depending on the desired balance between computational simplicity and theoretical properties. A closed-form, recursive, unbiased, convergent estimator using the density function is presented: the result is valid for any problem in which prior probabilities are identifiable. Discrete and mixed densities require a minor modification. Three application examples are described. The class conditional expectations of density functions, required for the initialization of the estimator algorithm, are analytically evaluated for Gaussian and exponential densities. >

Forskolin has biphasic effects on osteoprogenitor cell differentiation in vitro

Cells isolated from fetal rat calvaria (RC) and maintained in vitro in medium containing ascorbic acid and B-glycerophosphate form three-dimensional, mineralized nodules having the histological, immunohistological, and ultrastructural characteristics of woven bone. We have studied the effects of forskolin (FSK), a diterpene that activates adenylate cyclase, in this system. While 10(-7)-10(-5) M FSK significantly stimulated cAMP levels in RC cells, lower concentrations did not. cAMP levels with 10(-5) M FSK reached a maximum by 30 min at 37 degrees C and returned to basal level in 2-3 hr. Changes in cAMP levels correlated with changes in cellular shape: cells treated with 10(-5) M FSK assumed a stellate morphology, lost microfilament bundles, and reduced their substrate adhesiveness, while cells treated with 10(-9) M were not affected. Exponential growth and saturation densities of FSK-treated cultures were similar to untreated cultures, indicating that FSK was neither toxic nor stimulatory to the population. The effect on bone nodule formation of FSK present continuously depended on concentration: 10(-5) M FSK significantly inhibited the number of nodules formed, while 10(-9) M FSK significantly stimulated bone nodule formation. Single short treatments with either 10(-5) M or 10(-9) M FSK had no effect on nodule formation, but repeated short duration treatments (1 hr every 2 days for 21 days) gave results similar to continuous exposure. These results indicate that intermittent elevations in intracellular cAMP have an inhibitory effect on bone formation. In addition, our work indicates that low concentrations of FSK stimulate differentiation of osteoprogenitor cells possibly through a non-cAMP-dependent process.

Band Tails and Thermal Disorder in Doped and Undoped Hydrogenated Amorphous Silicon and Silicon-Germanium Alloys

ABSTRACTThe energy distribution and temperature dependence of the conduction and valence band tail density of states in a-Si:H and a-Si,Ge:H alloys is determined via total yield photoelectron spectroscopy. All films are observed to possess purely exponential conduction and valence band tail densities of states; however, the characteristic energy of the conduction band tail increases much more rapidly with temperature in the range of 300K to 550K than that of the valence band tail. This indicates that over that temperature range the conduction band tail is considerably more susceptible to thermal disorder than to structural disorder whereas the reverse holds for the valence band tail.

Linear relaxation: Distributions, thermal activation, structure, and ambiguity

The equations governing the small-signal response of relaxing, nonresonant systems which may be described by a distribution of relaxation times (DRT) and/or a distribution of activation energies (DAE) are summarized and generalized and their implications discussed for several popular distributions. Much past work, both experimental and theoretical, associated with these distributions is discussed. A distinction is made between physically realistic distributions, which involve finite shortest and longest relaxation times, and the usual mathematical approaches which involve limiting zero and infinite relaxation times. The Lévy DRT, which is of the latter character and which leads to the popular stretched exponential (SE) time and Williams–Watts (WW) frequency responses, is inconsistent with a temperature-independent DAE, reducing its range of applicability for a thermally activated situation. The SE-WW response has been termed universal; it is not, both because of the above facts and also because it does not lead to the often found symmetrical log-frequency response. Both Gaussian and exponential DAEs can lead to both symmetrical and skewed results, and can involve either temperature-dependent or temperature-independent DAEs. However, the Gaussian DAE does not yield fractional power-law time or frequency response over a finite, nonzero range, behavior found in nearly all distributed data. However, all DAEs involving exponential probability densities do lead to such behavior and provide, as well, an explanation of the temperature dependence of power-law exponents. In addition, it appears that the response of systems involving an exponential DAE can fit that of virtually all previous models, including the SE-WW, and thus can fit all data for thermally activated systems which have been fitted by these models. Problems in data fitting and many sources and types of ambiguity and their resolution are discussed. Special attention is devoted to the distinction between parallel, sequential, and hierarchical microscopic-model structure and response, and the various different, but, surprisingly, equivalent ways the overall response can be represented mathematically or by means of equivalent circuits of different connectivity.

Reduction of the moments of intensity fluctuations caused by amplifier saturation for both the K and the log-normally modulated exponential probability densities

Equations are derived for the effect of amplifier saturation on the moments of intensity fluctuations in atmospheric laser propagation. The intensity is assumed to have the recently proposed log-normally modulated exponential probability density. The effect is also calculated for the K probability density.