In this paper, we introduce and study fractional versions of the Bell–Touchard process, Poisson-logarithmic process generalized Pólya–Aeppli process. The state probabilities these compound Poisson processes solve a system differential equations that involves Caputo derivative order 0<β<1. It is shown are limiting cases recently introduced namely, counting We obtain mean, variance, covaria...

We study the distribution of a statistic useful in calculating the significance of the number of k-tuple matches detected in biological sequence homology algorithms. The statistic is Rn,k, the total number of heads in head runs of length k or more in a sequence of iid Bernoulli trials of length n. Calculation of the mean is straightforward. Poisson approximation formulas have been used for the ...

BACKGROUND The nature of spike count distributions is of great practical concern for the analysis of neural data. These distributions often have a tendency for 'failures' and a long tail of large counts, and may show a strong dependence of variance on the mean. Furthermore, spike count distributions often show multiplicative rather than additive effects of covariates. We analyzed the responses ...

We discuss the estimation of random errors due to shot noise in backscatter lidar observations that use either photomultiplier tube (PMT) or avalanche photodiode (APD) detectors. The statistical characteristics of photodetection are reviewed, and photon count distributions of solar background signals and laser backscatter signals are examined using airborne lidar observations at 532 nm using a ...

This thesis deals with the application of multiple directional transforms to image restoration, and discusses two cases of image restoration, image denoising and image fusion. In Chapter 1, the background of image restoration is described. First, image denoising problems are addressed. Image denoising is a principal problem of image processing and the purpose is to obtain an original picture as...

Most work on the human fertility transition has focused on declines in mean fertility. However, understanding changes in the variance of reproductive outcomes can be equally important for evolutionary questions about the heritability of fertility, individual determinants of fertility and changing patterns of reproductive skew. Here, we document how variance in completed fertility among women (4...

The repeated presentation of an identical visual stimulus in the receptive field of a neuron may evoke different spiking patterns at each trial. Probabilistic methods are essential to understand the functional role of this variance within the neural activity. In that case, a Poisson process is the most common model of trial-to-trial variability. For a Poisson process, the variance of the spike ...

The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards the variance of the sample mean. In addition, a central limit theorem in the general d–dimensional case is also established. Math. Subje...

The Large Area Telescope (LAT), the main instrument of the Fermi Gamma-Ray Space Telescope, detects high energy gamma rays with energies from 20 MeV to more than 300 GeV. The two main scientific objectives, the study of the Milky Way diffuse background and the detection of point sources, are complicated by the lack of photons. That is why we need a powerful Poisson noise removal method on the s...

We describe a set of new estimators for the N-point correlation functions of point processes. The variance of these estimators is calculated for the Poisson and binomial cases. It is shown that the variance of the unbiased estimator converges to the continuum value much faster than with any previously used alternative, all terms with slower convergence exactly cancel. We compare our estimators ...