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Fractional negative binomial and Pólya processes

Vellaisamy P., Maheshwari A.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

77--101

In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process (SFPP) is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

Coalescing stochastic processes in retrival from semantic memory

Queau P-Y., Woyczyński W. A., Lerner A. J.

Mathematica Applicanda

2015

Vol. 43, No. 2

189--224

Semantic memory retrieval is one of the most fundamental cognitive functions in humans. It is not fully understood and researchers from various fields of science struggle to find a model that would correlate well with experimental results and help understanding the complex background processes involved. To study such a phenomenon we need a relevant experimental protocol which can isolate the basic cognitive functions of interest from other perturbations. A variety of existing medical tests can provide such information, and the one we analyze is the Category Fluency Test (CFT). It was originally designed to measure frontal brain lobe damages in injured patients, and it tests directly the semantic memory retrieval, which is affected in cases of injury but can be also influenced by dementia, Alzheimer syndrome, or just aging. This paper introduces a new paradigm in analysis of the temporal structure of CFT responses by utilizing coalescent stochastic process model. We believe that this particular model is relevant to how this cognitive function operates and can lead to a better understanding of the background processes. The method turns out to be better at separating the two cognitively different groups studied here than the Weibull model from our previous paper Meyer et al.(2012), and could potentially be used for early diagnostics of dementia or Alzheimer's disease. Two other models, one based on the concept of Levy processes, and one related to the fractional Poisson model, are also explored.

Praca proponuje model procesów koalescencyjnych w celu wyjaśnienia mechanizmów odzysku pojęć i nazw z pamięci semantycznej. Model jest pretestowany używając dobrze znanego eksperymentalnego Testu Biegłości Kategorycznej, który jest standardowym narzędziem neurologów badających pacjentów z objawami demencji. Możliwości modelowania opartego na procesach Lévy’ego i ułamkowych procesach Poissona są również zbadane.