Showing papers on "Poisson distribution published in 1971"

Fitting Discrete Probability Distributions to Evolutionary Events

Abstract: The assumptions underlying the use of the Poisson distribution are essentially that the probability of an event is small but nearly identical for all occurrences and that the occurrence of an event does not alter the probability of recurrence of such events. These assumptions do not seem to be met for evolutionary events since (i) the probability of fixing nucleotide codon substitutions is not equal for all substitutions at a codon, and probably varies for the same substitution in different lineages; (ii) the probability of fixing codon substitutions varies among positions of a cistron; and (iii) the fixation of a nucleotide codon substitution at one position in a cistron modifies, and may even promote, the fixation of a codon substitution elsewhere along the cistron. Natural selection presumably is the causative factor that acts to modify the probability of a nucleotide codon substitution9s being fixed in a population. The use of the negative binomial distribution is consistent with the evidence that selective pressure on amino acid or nucleotide codon positions varies both among codon positions of a cistron and at a particular position during evolutionary time. If the number of fixations of nucleotide codon substitutions per position of cistrons encoding cytochromes c are phyletically inferred (phylogeny based on a paleontological record) rather than phenetically inferred (based on paired comparisons of extant species9 differences in the absence of a phylogeny) the distribution of these fixation data cannot be described adequately by a single Poisson distribution. The fit of these same data to a negative binomial distribution is very satisfactory. It has been argued that the fit of phenetically inferred fixation data, which do not take account of parallel or reverse fixations, to the Poisson distribution was supportive evidence for the hypothesis that protein evolution results from the fixation of selectively neutral codon substitutions. This argument now appears to be undercut by the evidence that data on nucleotide codon fixation are more probably distributed according to the negative binomial distribution. The fact that fixation data can be described by a particular discrete probability distribution does not of itself provide insight into the mechanisms of the evolutionary process. However, the facts—(i) that the assumptions underlying the use of the negative binomial distribution adequately deal with the varying probability of fixing amino acid or nucleotide codon substitutions at and among the positions of a cistron and (ii) that the negative binomial distribution provides an excellent fit for the phyletically inferred fixation data—suggest that the negative binomial is a very appropriate discrete probability distribution for describing evolutionary events. Amino acids or their nucleotide codon substitutions may be fixed at a position of a cistron as though selectively neutral relative to the codon being replaced, even though the codon position will not be selectively neutral, since many amino acids cannot function there. The negative binomial distribution treats this situation well whereas a single Poisson distribution could only be satisfactory if all codon positions that could vary were selectively neutral.

Boundary-crossing probabilities for the Brownian motion and poisson processes and techniques for computing the power of the Kolmogorov-Smirnov test

Abstract: Let w(t), 0 ≦ t ≦ ∞, be a Brownian motion process, i.e., a zero-mean separable normal process with Pr{w(0) = 0} = 1, E{w(t 1)w(t 2)}= min (t 1, t 2), and let a, b denote the boundaries defined by y = a(t), y = b(t), where b(0) < 0 < a(0) and b(t) < a(t), 0 ≦ t ≦ T ≦ ∞. A basic problem in many fields such as diffusion theory, gambler's ruin, collective risk, Kolmogorov-Smirnov statistics, cumulative-sum methods, sequential analysis and optional stopping is that of calculating the probability that a sample path of w(t) crosses a or b before t = T. This paper shows how this probability may be computed for sufficiently smooth boundaries by numerical solution of integral equations for the first-passage distribution functions. The technique used is to approximate the integral equations by linear recursions whose coefficients are estimated by linearising the boundaries within subintervals. The results are extended to cover the tied-down process subject to the condition w(1) = 0. Some related results for the Poisson process and the sample distribution function are given. The procedures suggested are exemplified numerically, first by computing the probability that the tied-down Brownian motion process crosses a particular curved boundary for which the true probability is known, and secondly by computing the finite-sample and asymptotic powers of the Kolmogorov-Smirnov test against a shift in mean of the exponential distribution.

Probability distributions for hurricane effects

Abstract: This investigation treats indirect development of probability functions for hurricane effects for specified time intervals. Methods outlined here provide a way of obtaining magnitude-recurrence interval relationships for hurricane effects. Methods are suitable for sites where insufficient data does not permit getting effect magnitude-recurrence interval information directly from historic records. A stochastic model for hurricane occurrences around sites is developed; a periodic Poisson distribution best described hurricanes on the Texas Coast. Data imply that Texas hurricanes tend to follow a cyclic trend with a period of about 33 yr. A hurricane occurrence model is then combined with distribution for hurricane wind, given a hurricane occurrence. If hurricane occurrences follow a periodic Poisson law, then exceedances of given magnitudes of hurricane effects are also shown to obey a periodic Poisson law.

Recent results in the statistical analysis of univariate point processes

Abstract: : Results on the statistical analyses of series of events published subsequent to the monograph by Cox and Lewis on this subject are surveyed. Special emphasis is given to tests for renewal processes, a considerable amount being now known about the distributions of some of the test statistics involved, and to testing the functional form of a trend in a nonhomogeneous Poisson process, as well as the point process model itself. A survey of work in special processes such as cluster processes and doubly stochastic Poisson processes is also given.

Poisson flats in euclidean spaces part ii: homogeneous poisson flats and the complementary theorem

Abstract: Part I [21] treated the case of a finite number of independent random uniform s-flats in an 'admissible' subset of Ed (s = 0, ,d 1). In this second part, the natural and fruitful 'Poisson extension' to a 'countable number of independent random uniform s-flats in Ed itself' is considered. It is worth mentioning at the outset that to have read Part I is not a prerequisite for reading the present paper. Although results of that part are often applied here, they serve only in an auxiliary capacity, thereby allowing the main thread of the theory to be developed without interruption.

Investigation of magnetization and density of a north atlantic seamount using poisson’s theorem

Abstract: The relationship between the gravitational and magnetic potentials caused by a uniform distribution of mass and magnetization may be used to obtain independent information about these physical properties. The general relationship in the frequency domain between the Fourier transforms of the gravity and magnetic anomaly fields is established through the Poisson theorem. The discrete Fourier transforms of the sampled continuous functions are used in an analysis which leads to a system of linear equations involving terms in density, magnetization, and calculated finite Fourier‐series coefficients. A least squares solution of the system yields the three components of the total magnetization vector divided by the density. From these results, the direction of total magnetization and the minimum of the Koenigsberger ratio Q can be determined uniquely. The remanent magnetization direction and certain other information can be derived for special cases in which the value of one or more of the physical property term...

Estimation of the Mean by Shrinkage to a Point

Abstract: The problem of shrinking the maximum likelihood estimator (MLE) of the mean μ of various populations towards a natural origin μ0 is studied. Estimators of the general form, are proposed for the cases of Normal, Gamma, Poisson, and Binomial distributions. The behavior of these estimators is compared with that of the ones suggested by Thompson [3]. Our estimators are shown to be better in the sense of having a smaller mean square error in an interval around μ0.

On the quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals

Abstract: We consider a single server queueing system M/G/1 in which customers arrive in a Poisson process with mean λt , and the service time has distribution dB ( t ), 0 t W ( t ) be the virtual waiting time process, i.e., the time that a potential customer arriving at the queueing system at time t would have to wait before beginning his service. We also let the random variable denote the first busy period initiated by a waiting time u at time t = 0.

The determination of poisson's ratio by compression tests of cylindrical specimens.

Abstract: Theoretical basis is presented for determining Poisson's ratio in uniaxial compression tests of cylindrical specimens. A simplified measurement technique and instrumentation for obtaining the required parameters are described in detail. Experimental data for Centennial sweet potato flesh at 72°F and four moisture contents are provided and analyzed to give the time dependent master curve and the time-moisture shift factor; thus, the time and moisture influenced viscoelastic property of Poisson's ratio is completely specified.

Informal contacts in science: a probabilistic model for communication processes.

Abstract: Significant contacts among scientists within research specialties are generally infrequent and are distributed as an essentially random process, the pattern of most contacts conforming to a Poisson distribution. Extremely productive persons in a specialty, however, seem to form a separate distribution; they have a considerably higher number of contacts.

Analysis of Ephemeral Flow in Aridlands

Abstract: An important parameter is found to be the flow duration, which describes the opportunity for natural recharge. Arid-land hydrology is, in effect, characterized by the absence of base flow, special precipitation patterns, and scarcity of data. A stochastic model of flow duration is developed for a stream near Tucson, AZ, using a regional model of rainfall. The number of flow events per summer seems to follow a Poisson distribution and that in winter, a geometric distribution, whereas the flow duration per event seems to follow a negative binomial distribution in both cases. Time series estimates of monthly streamflow of the Rillito Creek, AZ, are given to illustrate the correlation between various months and the difficulty of interpreting such time-lumped data.

Cost-Minimizing Inventory Control of Items Having A Special Type of Erratic Demand Pattern

Abstract: : The paper considers an item with a demand pattern characterized by Poisson occurrences of transactions whose sizes are geometrically distributed. Empirically it has been found that a significant number of inventoried items possess a demand pattern of this type. For a reorder-point (R), order-up-to level (M) system under continuous review and complete backordering a practical procedure is developed for determining the values of R and M as functions of three given parameters. (Author)

On the measurement of Poisson's ratio for modeling clay

Abstract: Resonance testing of Plasticine clay indicates that, for small strains (≤10−5) in the frequency range 100–3000 Hz, the material can be considered to be a linear viscoelastic solid with parameters which depend on temperature, frequency and prior large-strain history. In order to measure Poisson's ratio, it is necessary to take special precautions to eliminate large straining between small-strain tests of different tensorial character. A simple but effective test configuration for measuring Poisson's ratio is described and test results are displayed.

N-fold joint photocounting distribution for modulated laser radiation: Transmission through the turbulent atmosphere

Abstract: In this paper a generalised result for theN-fold joint photoelectron counting distribution for independently modulated radiation is given. We extend the recent results of Diament and Teich, for the one-fold photoelectron counting distribution for light propagated through an atmosphere characterised by log-normal irradiance fluctuations, to theN-fold joint photoelectron counting distribution. An approximate solution for thisN-fold distribution is obtained, for detection intervals {Ti} «τa whereτa is the characteristic time of the atmospheric turbulence. We present specifically the two-fold joint photocounting distribution for amplitude-stabilised laser radiation passing through such an atmosphere for several levels of turbulence and degrees of correlation. Cases including additive, independent, non-interfering Poisson noise are considered. Computer generated plots of the photocounting distribution are presented. For noise-free detection, the otherwise narrow-peaked photocounting distribution is seen to broaden markedly and shift its peak to lower counts as the turbulence level increases. Furthermore, a non-singular counting distribution is obtained for fully correlated detection. In the presence of additive noise and varying only the signal-to-noise ratioγ, the probability surface is intermediate between that of the Poisson and that of the noise-free log-normal fading counting distribution. The peak, however, is observed to decrease and then again increase in magnitude asγ → ∞, for correlated detection only. These results are expected to be of use in the study of atmospheric turbulence, as well as in the evaluation of certain stochastic functionals that occur in optical communication theory for the turbulent atmospheric channel.

Poisson distributions of error in mental test theory

Abstract: An axiomatic model for tests of pure speed is developed, leading to the conclusion that such test scores may, under certain circumstances, be regarded as realizations of a Poisson process. This leads to further study of the situation in terms of strong true score theory with the assumption of a Poisson distribution of scores and true score equal to the parameter of the Poisson distribution. Important conclusions are that (1) the Spearman-Brown formula holds in continuous time, (2) the raw moments of the distribution of true scores are obtainable from a knowledge of the factorial moments of observed scores over a population of examinees, and (3) under reasonable assumptions the distribution of observed scores over examinees ought to be negative binomial. An empirical example is given demonstrating the fit of the model to various predictions based on the model. In particular, the Poisson assumption can be tested independently of any assumption about the distribution of true or observed scores over examinees.

On the Distribution of Higher Basidiomycetes in the Sibsagar District, Assam

Abstract: The distribution of the higher basidiomycetes of the Sibsagar district of Assam has been studies with the application of Poisson's probability distribution. The fungi mostly occur in patches or pockets of podzolic soil formations having a rich forest cover, Agaricaceae occurring most commonly. The types of fungi were also classified as soil-inhabiting and wood-inhabiting, the former with sporophores submerged in the soil superficially. The Poisson's probability distribution also explains the mode of distribution to certain extent provided the places of occurrence of these fungi are not disturbed markedly.

Space-time validation of a thunderstorm rainfall model†

Abstract: A probability model for predicting the occurrence and magnitude of thunderstorm rainfall developed in the southwestern United States was tested in the metropolitan Chicago area with reasonable success, especially for the moderate to the extreme runoff-producing events. The model requires the estimation of two parameters, the mean number of events per year and the conditional probability of rain given that an event has occurred. To tie in the data from more than one gage in an area, an event can be defined in several ways, such as the areal mean rainfall exceeding 0.50 inch and at least one gage receiving more than 1.0 inch. This type of definition allows both of the model parameters to be obtained from daily warm-season rainfall records. Regardless of the definition used a Poisson distribution adequately described the number of events per season. A negative binomial distribution was derived as representing the frequency density function for rainfall where several gages are employed in defining a storm. Chicago data fit both distributions very well at events with relatively high return periods. The results indicate the possibility of using the model on a regional basis where limited amount of data may be used to estimate parameters for extensive areas.

Parameter identification for a class of multivariable non-linear processes

Abstract: A new signal-processing technique is presented and applied to the problem of determining the parameters of processes whoso dynamic behaviour is characterized by a set of ordinary first-order non-linear differential equations. The process signals and certain products of the process signals are used to excite Poisson filter chains. The Poisson filter chains provide exponentially smoothed moments of the signals at the filter chain inputs. Simultaneous sample values of these Poisson filtered signals are then used to form linear algebraic equations in the unknown process parameters. The method of instrumental variables is utilized to combat inaccuracies in the measured process signals. Finally, the method is illustrated by an example and the results of simulation studies are presented to demonstrate the feasibility of the method.

Maximum Likelihood Estimation from Grouped Poisson Data

Abstract: In this article, sufficient conditions are given for the existence of a unique solution of the likelihood equation which results from a grouped data sample. The necessary and sufficient conditions for the convergence of a sequence defined by the method of successive approximations to this unique solution are also given. Finally, it is shown that when the groups from an underlying Poisson distribution are connected the method of successive approximations will converge to the unique solution of the resulting likelihood equation regardless of the starting value chosen provided that the sample is not concentrated entirely in either or both the first and last groups.

Selective interaction of a poisson and renewal process: the dependency structure of the intervals between responses

Abstract: This paper studies the dependency structure of the intervals between responses in the renewal inhibited Poisson process, and continues the author's earlier work on this type of process ((1970a), (1970b)). A new approach to the intervals between events in a stationary point process, based on the idea of an average event, is introduced. Average event initial conditions (as opposed to equilibrium initial conditions previously determined) for the renewal inhibited Poisson process are obtained and event stationarity of the resulting response process is established. The joint distribution and correlation between pairs of contiguous synchronous intervals is obtained; further, the joint distribution of non-contiguous pairs of synchronous intervals is derived. Finally, the joint distributions of pairs of contiguous synchronous and asynchronous intervals are related, and a similar but more general stationary point result is conjectured.

Almost periodic solutions of Poisson’s equation

Abstract: It is well known that if /(/) is almost periodic with respect to a real variable t, and if F(,t)=f'f(s)ds is bounded for — «