PMC:1647278 / 18832-31181 JSONTXT

{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/1647278","sourcedb":"PMC","sourceid":"1647278","source_url":"https://www.ncbi.nlm.nih.gov/pmc/1647278","text":"Single pattern\nThe exact expression of F+ is computable through many different methods [1-4] but is too much complicated to derive explicitly ∇F+. To overcome this problem, we propose to consider an approximation of F+. As said in introduction, many kind of approximations are available (Gaussian, binomial, compound Poisson or large deviations). In this paper, we have chosen to use a binomial approximation as it provides an expression which is analytically differentiable and is known to be a good heuristic to the problem [8].\nFor a single non-degenerate pattern (i.e. a simple word) W = w1 ... wh (wi ∈ A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@) with h ≥ m - 1 we first denote by\nP(N) = μN (w1 ... wm) × πN (w1 ... wm, wm+1) × ... × πN (wh-m ... wh-1, wh)     (13)\nthe probability for W to occur at a given position in the sequence and then we get\nF + ( N ) ≃ ℙ ( ℬ ( ℓ h , P ( N ) ) ≥ N o b s ) = β ( P ( N ) , N obs , ℓ h − N o b s + 1 ) β ( N o b s , ℓ h − N o b s + 1 )       ( 14 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaacqWGgbGrdaahaaWcbeqaaiabgUcaRaaakiabcIcaOGqabiab=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@8E81@\nwhere ℬ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFSeIqaaa@377E@ denotes the binomial distribution, with ℓh = ℓ - h + 1 and where the β functions (complete and incomplete) and their relation to the binomial cumulative distribution function are described in appendix B.\nNote that if we consider non-overlapping occurrences instead of overlapping ones, we can still use a binomial approximation for the distribution of N, but the expression of P(N) is more complicated as it involves the auto-correlation polynome of the pattern [14]. This point is not developed in this paper.\nReplacing μN and πN by their expression easily gives\nP ( N ) = 1 n − m + 1 ∏ w ∈ A m ∏ a ∈ A N 1 ( w a ) A 1 ( w a ) N 0 ( w ) A 0 ( w )       ( 15 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6E23@\nwhere A1(wa) counts occurrences of the word wa in W = w1 ... wh and A0 (w) counts occurrences of the word w in w2 ... wh-1. Note that in the particular case where h = m - 1, all A0 (w) are null and we simply get (n - m + l) × P (N) = N1 (W).\nUsing the derivative properties of the incomplete beta function (see appendix B for more details) we hence get\n∇ F + ( N ) ≃ P ( N ) N obs − 1 ( 1 − P ( N ) ) ℓ h − N obs β ( N obs , ℓ h − N obs + 1 ) × ∇ P ( N )       ( 16 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHhis0cqWGgbGrdaahaaWcbeqaaiabgUcaRaaakiabcIcaOGqabiab=5eaojabcMcaPiabloKi7maalaaabaGaemiuaaLaeiikaGIae8Nta4KaeiykaKYaaWbaaSqabeaacqWGobGtdaWgaaadbaGaee4Ba8MaeeOyaiMaee4CamhabeaaliabgkHiTiabigdaXaaakiabcIcaOiabigdaXiabgkHiTiabdcfaqjabcIcaOiab=5eaojabcMcaPiabcMcaPmaaCaaaleqabaGaeS4eHW2aaSbaaWqaaiabdIgaObqabaWccqGHsislcqWGobGtdaWgaaadbaGaee4Ba8MaeeOyaiMaee4CamhabeaaaaaakeaaiiGacqGFYoGycqGGOaakcqWGobGtdaWgaaWcbaGaee4Ba8MaeeOyaiMaee4CamhabeaakiabcYcaSiabloriSnaaBaaaleaacqWGObaAaeqaaOGaeyOeI0IaemOta40aaSbaaSqaaiabb+gaVjabbkgaIjabbohaZbqabaGccqGHRaWkcqaIXaqmcqGGPaqkaaGaey41aqRaey4bIeTaemiuaaLaeiikaGIae8Nta4KaeiykaKIaaCzcaiaaxMaadaqadaqaaiabigdaXiabiAda2aGaayjkaiaawMcaaaaa@71E0@\nso all we need is to compute ∇P(N).\nFor all (w, a) ∈ A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@m × A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@ we have\n∂ P ( N ) ∂ N 0 ( w ) = − A 0 ( w ) N 0 ( w ) × P ( N )       ( 17 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2kabdcfaqjabcIcaOGqabiab=5eaojabcMcaPaqaaiabgkGi2kab=5eaonaaBaaaleaaieaacqGFWaamaeqaaOGaeiikaGIaem4DaCNaeiykaKcaaiabg2da9iabgkHiTmaalaaabaGaemyqae0aaSbaaSqaaiabicdaWaqabaGccqGGOaakcqWG3bWDcqGGPaqkaeaacqWFobGtdaWgaaWcbaGae4hmaadabeaakiabcIcaOiabdEha3jabcMcaPaaacqGHxdaTcqWGqbaucqGGOaakcqWFobGtcqGGPaqkcaWLjaGaaCzcamaabmaabaGaeGymaeJaeG4naCdacaGLOaGaayzkaaaaaa@5086@\nand\n∂ P ( N ) ∂ N 1 ( w ) = − A 1 ( w a ) N 1 ( w a ) × P ( N )       ( 18 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2kabdcfaqjabcIcaOGqabiab=5eaojabcMcaPaqaaiabgkGi2kab=5eaonaaBaaaleaaieaacqGFXaqmaeqaaOGaeiikaGIaem4DaCNaeiykaKcaaiabg2da9iabgkHiTmaalaaabaGaemyqae0aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWG3bWDcqWGHbqycqGGPaqkaeaacqWFobGtdaWgaaWcbaGae4xmaedabeaakiabcIcaOiabdEha3jabdggaHjabcMcaPaaacqGHxdaTcqWGqbaucqGGOaakcqWFobGtcqGGPaqkcaWLjaGaaCzcamaabmaabaGaeGymaeJaeGioaGdacaGLOaGaayzkaaaaaa@5324@\nIf we denote by\nP = μ (w1 ... wm) × π (w1 ... wm, wm+1) × ... × π (wh-m ... wh-1, wh)     (19)\nthe true probability for W to occur at a given position in the sequence X then we get, using (7) in (13), that\nP ( E ) = p × ( 1 − 1 n − m + 1 ) h − m ≃ p       ( 20 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaucqGGOaakieqacqWFfbqrcqGGPaqkcqGH9aqpcqWGWbaCcqGHxdaTdaqadaqaaiabigdaXiabgkHiTmaalaaabaGaeGymaedabaGaemOBa4MaeyOeI0IaemyBa0Maey4kaSIaeGymaedaaaGaayjkaiaawMcaamaaCaaaleqabaGaemiAaGMaeyOeI0IaemyBa0gaaOGaeS4qISJaemiCaaNaaCzcaiaaxMaadaqadaqaaiabikdaYiabicdaWaGaayjkaiaawMcaaaaa@4A3A@\nfor n large enough. We hence get\n∇ F + ( E ) ≃ p N obs ( 1 − p ) ℓ h − N obs β ( N obs , ℓ h − N obs + 1 ) × G       ( 21 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHhis0cqWGgbGrdaahaaWcbeqaaiabgUcaRaaakiabcIcaOGqabiab=veafjabcMcaPiabloKi7maalaaabaGaemiCaa3aaWbaaSqabeaacqWGobGtdaWgaaadbaGaee4Ba8MaeeOyaiMaee4CamhabeaaaaGccqGGOaakcqaIXaqmcqGHsislcqWGWbaCcqGGPaqkdaahaaWcbeqaaiabloriSnaaBaaameaacqWGObaAaeqaaSGaeyOeI0IaemOta40aaSbaaWqaaiabb+gaVjabbkgaIjabbohaZbqabaaaaaGcbaacciGae4NSdiMaeiikaGIaemOta40aaSbaaSqaaiabb+gaVjabbkgaIjabbohaZbqabaGccqGGSaalcqWItecBdaWgaaWcbaGaemiAaGgabeaakiabgkHiTiabd6eaonaaBaaaleaacqqGVbWBcqqGIbGycqqGZbWCaeqaaOGaey4kaSIaeGymaeJaeiykaKcaaiabgEna0kab=DeahjaaxMaacaWLjaWaaeWaaeaacqaIYaGmcqaIXaqmaiaawIcacaGLPaaaaaa@6649@\nwhere tG = [tG0 tG1] is defined by\nG 0 ( w ) = − A 0 ( w ) E 0 ( w ) and G 1 ( w a ) = − A 1 ( w a ) E 1 ( w a )       ( 22 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@5D85@\nUsing equation (12) we finally get\nσ ≃ Q +   t G × C × G       ( 23 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCcqWIdjYocqWGrbqudaahaaWcbeqaaiabgUcaRaaakmaakaaabaGaeeiiaaYaaWbaaSqabeaacqWG0baDaaacbeGccqGFhbWrcqGHxdaTcqGFdbWqcqGHxdaTcqGFhbWraSqabaGccaWLjaGaaCzcamaabmaabaGaeGOmaiJaeG4mamdacaGLOaGaayzkaaaaaa@4098@\nwhere\nQ + = p N obs ( 1 − p ) ℓ h − N obs ln ⁡ ( 10 ) β ( p , N obs , ℓ h − N obs + 1 )       ( 24 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@675F@\nand then, a computation of σ is possible by plug-in. Without considering the computation of E and C, the complexity of this approach is O(h) (where h is the size of the pattern).\nWhen a degenerate pattern (finite set of words) is considered instead of a single word, it is easy to adapt this method by summing the contribution p of each word belonging to the pattern. 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