Delta method
Let us start with a simple case. We consider a single pattern which is over-represented (seen more than expected) so we have
S N  = − log ⁡  10   F +  ( N )   with   F +  ( N ) ≜ ℙ μ N  , π N    ( N ≥ N obs   )           ( 8 )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeqadaaabaGaem4uam1aaSbaaSqaaGqabiab=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@681D@
where the function F+ also depends on the sequence length ℓ and the considered pattern.
If F+ is differentiate, the delta-method (a simple first order Taylor expansion around N = E, see [13]) provides the following approximation:
S N  ≃ − log ⁡  10   F +  ( E ) −   t  ( N − E ) ∇ F +  ( E )  ln ⁡ ( 10 ) F +  ( E )         ( 9 )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaWgaaWcbaacbeGae8Nta4eabeaakiabloKi7iabgkHiTiGbcYgaSjabc+gaVjabcEgaNnaaBaaaleaacqaIXaqmcqaIWaamaeqaaOGaemOray0aaWbaaSqabeaacqGHRaWkaaGccqGGOaakcqWFfbqrcqGGPaqkcqGHsisldaWcaaqaaiabbccaGmaaCaaaleqabaGaemiDaqhaaOGaeiikaGIae8Nta4KaeyOeI0Iae8xrauKaeiykaKIaey4bIeTaemOray0aaWbaaSqabeaacqGHRaWkaaGccqGGOaakcqWFfbqrcqGGPaqkaeaacyGGSbaBcqGGUbGBcqGGOaakcqaIXaqmcqaIWaamcqGGPaqkcqWGgbGrdaahaaWcbeqaaiabgUcaRaaakiabcIcaOiab=veafjabcMcaPaaacaWLjaGaaCzcamaabmaabaGaeGyoaKdacaGLOaGaayzkaaaaaa@5A3D@
and hence, using (7) we have
S N  ≃ S −   t  ( N − E ) ∇ F +  ( E )  ln ⁡ ( 10 ) F +  ( E )         ( 10  )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaWgaaWcbaacbeGae8Nta4eabeaakiabloKi7iabdofatjabgkHiTmaalaaabaGaeeiiaaYaaWbaaSqabeaacqWG0baDaaGccqGGOaakcqWFobGtcqGHsislcqWFfbqrcqGGPaqkcqGHhis0cqWGgbGrdaahaaWcbeqaaiabgUcaRaaakiabcIcaOiab=veafjabcMcaPaqaaiGbcYgaSjabc6gaUjabcIcaOiabigdaXiabicdaWiabcMcaPiabdAeagnaaCaaaleqabaGaey4kaScaaOGaeiikaGIae8xrauKaeiykaKcaaiaaxMaacaWLjaWaaeWaaeaacqaIXaqmcqaIWaamaiaawIcacaGLPaaaaaa@503C@
for n large enough. The distribution of S^ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGtbWugaqcaaaa@2DEB@ is therefore approximated by
L MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaeHbnf2C0vMCJfMCKbaceiGaa8htaaaa@394B@ (SN) ≃ N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFneVtaaa@383B@ (S, σ2)     (11)
with
σ =   t  ∇ F +  ( E ) × C × ∇ F +  ( E )    ln ⁡ ( 10 ) F +  ( E )         ( 12  )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCcqGH9aqpdaWcaaqaamaakaaabaGaeeiiaaYaaWbaaSqabeaacqWG0baDaaGccqGHhis0cqWGgbGrdaahaaWcbeqaaiabgUcaRaaakiabcIcaOGqabiab+veafjabcMcaPiabgEna0kab+neadjabgEna0kabgEGirlabdAeagnaaCaaaleqabaGaey4kaScaaOGaeiikaGIae4xrauKaeiykaKcaleqaaaGcbaGagiiBaWMaeiOBa4MaeiikaGIaeGymaeJaeGimaaJaeiykaKIaemOray0aaWbaaSqabeaacqGHRaWkaaGccqGGOaakcqGFfbqrcqGGPaqkaaGaaCzcaiaaxMaadaqadaqaaiabigdaXiabikdaYaGaayjkaiaawMcaaaaa@543D@
In consequence, computing σ requires both to compute C (done in appendix A) and ∇F+ (E).