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  • Cambridge University Press (CUP)  (11)
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  • Cambridge University Press (CUP)  (11)
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  • 1
    Online Resource
    Online Resource
    Finite-size corrections to Poisson approximations of rare events in renewal processes
    Cambridge University Press (CUP) ; 2001
    In:  Journal of Applied Probability Vol. 38, No. 2 ( 2001-06), p. 554-569
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 38, No. 2 ( 2001-06), p. 554-569
    Abstract: Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1239/jap/996986762
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2001
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
    Location Call Number Limitation Availability
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    Online Resource
  • 2
    Online Resource
    Online Resource
    Finite-size corrections to Poisson approximations of rare events in renewal processes
    Cambridge University Press (CUP) ; 2001
    In:  Journal of Applied Probability Vol. 38, No. 02 ( 2001-06), p. 554-569
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 38, No. 02 ( 2001-06), p. 554-569
    Abstract: Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200020039
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2001
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
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    Online Resource
  • 3
    Online Resource
    Online Resource
    An existence theorem for the discrete coagulation-fragmentation equations
    Cambridge University Press (CUP) ; 1984
    In:  Mathematical Proceedings of the Cambridge Philosophical Society Vol. 96, No. 2 ( 1984-09), p. 351-357
    Details
    In: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press (CUP), Vol. 96, No. 2 ( 1984-09), p. 351-357
    Abstract: This paper gives an existence result for the discrete coagulation-fragmentation equations: (If k = 1, the first and last sums are 0.)
    Type of Medium: Online Resource
    ISSN: 0305-0041 , 1469-8064
    URL: Article
    DOI: 10.1017/S0305004100062253
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1984
    detail.hit.zdb_id: 1483586-1
    detail.hit.zdb_id: 209201-3
    SSG: 17,1
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    Online Resource
  • 4
    Online Resource
    Online Resource
    A branching-process solution of the polydisperse coagulation equation
    Cambridge University Press (CUP) ; 1984
    In:  Advances in Applied Probability Vol. 16, No. 01 ( 1984-03), p. 56-69
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 16, No. 01 ( 1984-03), p. 56-69
    Abstract: The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when The critical time t c when diverges corresponds to a critical branching process, while post-critical times t & gt; t c correspond to supercritical branching processes.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.1017/S0001867800022345
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1984
    detail.hit.zdb_id: 1474602-5
    Location Call Number Limitation Availability
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  • 5
    Online Resource
    Online Resource
    A branching-process solution of the polydisperse coagulation equation
    Cambridge University Press (CUP) ; 1984
    In:  Advances in Applied Probability Vol. 16, No. 1 ( 1984-03), p. 56-69
    Details
    In: Advances in Applied Probability, Cambridge University Press (CUP), Vol. 16, No. 1 ( 1984-03), p. 56-69
    Abstract: The polydisperse coagulation equation models irreversible aggregation of particles with varying masses. This paper uses a one-parameter family of discrete-time continuous multitype branching processes to solve the polydisperse coagulation equation when The critical time t c when diverges corresponds to a critical branching process, while post-critical times t 〉 t c correspond to supercritical branching processes.
    Type of Medium: Online Resource
    ISSN: 0001-8678 , 1475-6064
    URL: Article
    DOI: 10.2307/1427224
    RVK:
    SA 1600
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1984
    detail.hit.zdb_id: 1474602-5
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  • 6
    Online Resource
    Online Resource
    Markov Additive Processes and Repeats in Sequences
    Cambridge University Press (CUP) ; 2007
    In:  Journal of Applied Probability Vol. 44, No. 02 ( 2007-06), p. 514-527
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 44, No. 02 ( 2007-06), p. 514-527
    Abstract: Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k ), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200003132
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2007
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
    Location Call Number Limitation Availability
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    Online Resource
  • 7
    Online Resource
    Online Resource
    Markov Additive Processes and Repeats in Sequences
    Cambridge University Press (CUP) ; 2007
    In:  Journal of Applied Probability Vol. 44, No. 02 ( 2007-06), p. 514-527
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 44, No. 02 ( 2007-06), p. 514-527
    Abstract: Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k ), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200117991
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2007
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
    Location Call Number Limitation Availability
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    Online Resource
  • 8
    Online Resource
    Online Resource
    An existence theorem for the discrete coagulation–fragmentation equations
    Cambridge University Press (CUP) ; 1985
    In:  Mathematical Proceedings of the Cambridge Philosophical Society Vol. 98, No. 01 ( 1985-7), p. 183-
    Details
    In: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press (CUP), Vol. 98, No. 01 ( 1985-7), p. 183-
    Type of Medium: Online Resource
    ISSN: 0305-0041 , 1469-8064
    URL: Article
    DOI: 10.1017/S0305004100063362
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 1985
    detail.hit.zdb_id: 1483586-1
    detail.hit.zdb_id: 209201-3
    SSG: 17,1
    Location Call Number Limitation Availability
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    Online Resource
  • 9
    Online Resource
    Online Resource
    Markov Additive Processes and Repeats in Sequences
    Cambridge University Press (CUP) ; 2007
    In:  Journal of Applied Probability Vol. 44, No. 2 ( 2007-06), p. 514-527
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 44, No. 2 ( 2007-06), p. 514-527
    Abstract: Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k ), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1239/jap/1183667418
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2007
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
    Location Call Number Limitation Availability
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    Online Resource
  • 10
    Online Resource
    Online Resource
    Path reversal, islands, and the gapped alignment of random sequences
    Cambridge University Press (CUP) ; 2004
    In:  Journal of Applied Probability Vol. 41, No. 04 ( 2004-12), p. 975-983
    Details
    In: Journal of Applied Probability, Cambridge University Press (CUP), Vol. 41, No. 04 ( 2004-12), p. 975-983
    Abstract: In bioinformatics, the notion of an ‘island’ enhances the efficient simulation of gapped local alignment statistics. This paper generalizes several results relevant to gapless local alignment statistics from one to higher dimensions, with a particular eye to applications in gapped alignment statistics. For example, reversal of paths (rather than of discrete time) generalizes a distributional equality, from queueing theory, between the Lindley (local sum) and maximum processes. Systematic investigation of an ‘ownership’ relationship among vertices in ℤ 2 formalizes the notion of an island as a set of vertices having a common owner. Predictably, islands possess some stochastic ordering and spatial averaging properties. Moreover, however, the average number of vertices in a subcritical stationary island is 1, generalizing a theorem of Kac about stationary point processes. The generalization leads to alternative ways of simulating some island statistics.
    Type of Medium: Online Resource
    ISSN: 0021-9002 , 1475-6072
    URL: Article
    DOI: 10.1017/S0021900200020751
    RVK:
    SA 6100
    Language: English
    Publisher: Cambridge University Press (CUP)
    Publication Date: 2004
    detail.hit.zdb_id: 1474599-9
    detail.hit.zdb_id: 219147-7
    SSG: 3,2
    Location Call Number Limitation Availability
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    Online Resource
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