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52666 results in Statistics and Probability

Page 245 of 2634

  • 4 - Spectral Representation of Diffusion Processes

  • Manuel Domínguez de la Iglesia, Universidad Nacional Autónoma de México
  • Book:
    Orthogonal Polynomials in the Spectral Analysis of Markov Processes
    Published online:
    08 October 2021
    Print publication:
    21 October 2021, pp 254-321

  • Summary

    This chapter is devoted to the spectral analysis of one-dimensional diffusion processes, which are the most basic and important continuous-time Markov processes where now the state space is a continuous interval contained in the real line. Diffusion processes are characterized by an infinitesimal operator which is a second-order differential operator with drift and diffusion coefficients. A spectral representation of the transition probability density of the process is obtained in terms of the orthogonal eigenfunctions of the corresponding infinitesimal operator, for which a Sturm–Liouville problem with certain boundary conditions will be solved. An analysis of the behavior of these boundary points will also be made. An extensive collection of examples related to special functions and orthogonal polynomials is provided, including the Brownian motion with drift and scaling, the Orstein–Uhlenbeck process, a population growth model, the Wright–Fisher model, the Jacobi diffusion model and the Bessel process, among others. Finally, the concept of quasi-stationary distributions is studied, for which the spectral representation plays an important role.

  • 3 - Spectral Representation of Birth–Death Processes

  • Manuel Domínguez de la Iglesia, Universidad Nacional Autónoma de México
  • Book:
    Orthogonal Polynomials in the Spectral Analysis of Markov Processes
    Published online:
    08 October 2021
    Print publication:
    21 October 2021, pp 146-253

  • Summary

    This chapter is devoted to the spectral analysis of birth–death processes on nonnegative integers, which are the most basic and important continuous-time Markov chains. These processes will be characterized by an infinitesimal operator which is a tridiagonal matrix whose spectrum is always contained in the negative real line (including 0). The Karlin–McGregor integral representation formula of the transition probability functions of the process is obtained in terms of orthogonal polynomials with respect to a probability measure with support inside a positive real interval. Although many of the results are similar or equivalent to those of discrete-time birth–death chains, the methods and techniques are quite different. The chapter gives an extensive collection of examples related to orthogonal polynomials, including the M/M/k queue for any k servers, the continuous-time Ehrenfest and Bernoulli–Laplace urn models, a genetics model of Moran and linear birth–death processes. As in the case of discrete-time birth–death chains, the Karlin–McGregor formula is applied to the probabilistic aspects of birth–death processes, such as processes with killing, recurrence, absorption, the strong ratio limit property, the limiting conditional distribution, the decay parameter, quasi-stationary distributions and bilateral birth–death processes on the integers.

  • Immune status affects the clinical features and outcomes of adult patients with respiratory adenovirus infection

  • Handan Zhao, Minghan Zhou, Qing Zheng, Guanjing Lang, Lijun Xu
  • Journal:
    Epidemiology & Infection / Volume 149 / 2021
    Published online by Cambridge University Press:
    21 October 2021, e235
    • Article
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  • The differences in the clinical features and outcomes of respiratory adenovirus infection (RAI) between immunocompetent and immunocompromised adult patients remain unclear. Thirty-nine adult RAI patients, including 28 (71.8%) immunocompetent patients and 11 (28.2%) immunocompromised patients were enrolled in this retrospective study. Demographic characteristics, symptoms, laboratory tests, radiographic findings, therapies and clinical outcomes were compared between the two groups. We found fever (94.9%), cough (66.7%) and sputum production (56.4%) were the most frequent symptoms. Compared with immunocompetent RAI patients, the immunocompromised RAI patients were more likely to experience anaemia (g/l; 90.8 ± 24.0 vs 134.3 ± 14.6, P < 0.001), thrombocytopaenia ( × 109/l; 116.9 ± 92.7 vs 178.4 ± 74.6, P = 0.037), hypoalbuminaemia (g/l; 29.6 ± 5.5 vs 36.9 ± 5.2, P < 0.001), hyponatraemia (mmol/l; 134.8 ± 5.6 vs 138.5 ± 3.9, P = 0.026) and low levels of cholinesterase (U/l; 2650.5 ± 1467.4 vs 5892.8 ± 1875.1, P < 0.001). Chest computed tomography (CT) scans indicated that lung infiltrate was the most common finding (64.1%). Immunocompromised patients had a higher likelihood of bilateral lung involvement (72.7%) and lower lobe involvement (81.8%) of both lungs. The hospitalized mortality rate was 27.3% in immunocompromised RAI patients, but no death occurred among immunocompetent RAI patients (P = 0.018). Our data suggested immunocompromised RAI patients had worse laboratory test results, more bilateral lung and lower lobe involvement and higher in-hospital mortality compared with immunocompetent RAI patients.

Page 245 of 2634