Search Results (7)

In this article, new properties of the Poisson distribution of order k with parameter $\lambda$ are found. Based on them, the modes of the Poisson distributions of order $k=3$ and 4 are derived for $\lambda$ in $(0,1)$. They are 0, 3, 5, and 0, 4, 7, 8, respectively, for $\lambda$ in specified subintervals of (0, 1). In addition, using Mathematica, computational results for the modes of the Poisson distributions of order $k=2,3$, and 4 are presented for $\lambda$ in specified subintervals of $(0,2)$. Full article

This paper proposes a new fractional Poisson process through a recursive fractional differential governing equation. Unlike the homogeneous Poison process, the Caputo derivative on the probability distribution of k jumps with respect to time is linked to all probability distribution functions of j jumps, where j is a non-negative integer less than or equal to k. The distribution functions of arrival times are derived, while the inter-arrival times are no longer independent and identically distributed. Further, this new fractional Poisson process can be interpreted as a homogeneous Poisson process whose natural time flow has been randomized, and the underlying time randomizing process has been studied. Finally, the conditional distribution of the kth order statistic from random number samples, counted by this fractional Poisson process, is also discussed. Full article

We show that the most important measures of quantum chaos, such as frame potentials, scrambling, Loschmidt echo and out-of-time-order correlators (OTOCs), can be described by the unified framework of the isospectral twirling, namely the Haar average of a k-fold unitary channel. We show that such measures can then always be cast in the form of an expectation value of the isospectral twirling. In literature, quantum chaos is investigated sometimes through the spectrum and some other times through the eigenvectors of the Hamiltonian generating the dynamics. We show that thanks to this technique, we can interpolate smoothly between integrable Hamiltonians and quantum chaotic Hamiltonians. The isospectral twirling of Hamiltonians with eigenvector stabilizer states does not possess chaotic features, unlike those Hamiltonians whose eigenvectors are taken from the Haar measure. As an example, OTOCs obtained with Clifford resources decay to higher values compared with universal resources. By doping Hamiltonians with non-Clifford resources, we show a crossover in the OTOC behavior between a class of integrable models and quantum chaos. Moreover, exploiting random matrix theory, we show that these measures of quantum chaos clearly distinguish the finite time behavior of probes to quantum chaos corresponding to chaotic spectra given by the Gaussian Unitary Ensemble (GUE) from the integrable spectra given by Poisson distribution and the Gaussian Diagonal Ensemble (GDE). Full article

The generalized cumulative residual entropy is a recently defined dispersion measure. In this paper, we obtain some further results for such a measure, in relation to the generalized cumulative residual entropy and the variance of random lifetimes. We show that it has an intimate connection with the non-homogeneous Poisson process. We also get new expressions, bounds and stochastic comparisons involving such measures. Moreover, the dynamic version of the mentioned notions is studied through the residual lifetimes and suitable aging notions. In this framework we achieve some findings of interest in reliability theory, such as a characterization for the exponential distribution, various results on k-out-of-n systems, and a connection to the excess wealth order. We also obtain similar results for the generalized cumulative entropy, which is a dual measure to the generalized cumulative residual entropy. Full article

When modelling insurance claim count data, the actuary often observes overdispersion and an excess of zeros that may be caused by unobserved heterogeneity. A common approach to accounting for overdispersion is to consider models with some overdispersed distribution as opposed to Poisson models. Zero-inflated, hurdle and compound frequency models are typically applied to insurance data to account for such a feature of the data. However, a natural way to deal with unobserved heterogeneity is to consider mixtures of a simpler models. In this paper, we consider k-finite mixtures of some typical regression models. This approach has interesting features: first, it allows for overdispersion and the zero-inflated model represents a special case, and second, it allows for an elegant interpretation based on the typical clustering application of finite mixture models. k-finite mixture models are applied to a car insurance claim dataset in order to analyse whether the problem of unobserved heterogeneity requires a richer structure for risk classification. Our results show that the data consist of two subpopulations for which the regression structure is different. Full article

The analysis of spatial arrangement of incompletely developed fruits (IDF) in capitula could be used to understand the nature and the relative arrangement of these fruits at maturity, previously unexplained by current models. The objective of this work was to quantify and define the distribution pattern of visible IDF (IDF_vis) at physiological maturity in the capitulum of the cultivated sunflower, in two genotypes with different self-compatibility expression grown in three different environments. Spatial characteristics and the possibility of randomness of IDF_vis pattern generation were also evaluated. We were able to define four IDF_vis patterns: Type I, where the distribution of the IDF_vis was located mainly at the capitulum center, Type II, where the distribution remained grouped at its center but spreads towards the periphery, Type III, where the distribution was more homogeneous over the entire capitulum surface and Type IV with a homogeneous but very dispersed distribution over the entire capitulum surface. Second order spatial point pattern analysis techniques for a plane (Ripley’s K) were applied to the distribution of IDF_vis in the four predefined IDF_vis patterns. Using the ADE-4 software, spatial distribution patterns contained in a circular surface and corrected for edge effects were analyzed. By grouping the different types of IDF_vis patterns by environment and genotype, a tendency was observed to generate preferably two types of patterns, Type I and Type IV, directly related to the genotype and not to the environment. The K index obtained for each type of pattern showed that, for the scales analyzed, Types I, II and III can be defined as grouped, since they laid outside the Poisson confidence limits. The Type IV pattern presented results consistent with a completely randomized distribution. It was observed a low-frequency appearance of the IV (random) pattern and only for one genotype in the different environments studied, while in the rest of the genotype x environment combinations there was always a greater degree of grouping (non random; Type; I, II and III patterns). Proved that mostly of the IDF_vis patterns presented in the sunflower capitulum were mainly non random, the results shown here suggest that, to the intrinsic characteristics of the plant to express this character, mainly physiological, intra-receptacle physical factors could be added in the post-pollination stage, capable of altering the normal development of the embryos. Full article

Location modeling, both inductive and deductive, is widely used in archaeology to predict or investigate the spatial distribution of sites. The commonality among these approaches is their consideration of only spatial effects of the first order (i.e., the interaction of the locations with the site characteristics). Second-order effects (i.e., the interaction of locations with each other) are rarely considered. We introduce a deductive approach to investigating such second-order effects using linguistic hypotheses about settling behavior in the Final Palaeolithic. A Poisson process was used to simulate a point distribution using expert knowledge of two distinct hunter–gatherer groups, namely, reindeer hunters and elk hunters. The modeled points and point densities were compared with the actual finds. The G-, F-, and K-function, which allow for the identification of second-order effects of varying intensity for different periods, were applied. The results reveal differences between the two investigated groups, with the reindeer hunters showing location-related interaction patterns, indicating a spatial memory of the preferred locations over an extended period of time. Overall, this paper shows that second-order effects occur in the geographical modeling of archaeological finds and should be taken into account by using approaches such as the one presented in this paper. Full article