Appendix C
We give here the complete expression of σ for a single pattern in the special case of an order m = 0 homogeneous Markov model of parameter μ.
The MLE of μ is given by
μ N  = N 1   n        ( 76 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF8oqBdaWgaaWcbaacbeGae4Nta4eabeaakiabg2da9maalaaabaGae4Nta40aaSbaaSqaaiabigdaXaqabaaakeaacqWGUbGBaaGaaCzcaiaaxMaacqGGOaakcqaI3aWncqaI2aGncqGGPaqkaaa@3976@
where N1 is the frequency of all letters.
A Gaussian approximation gives
L MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaeHbnf2C0vMCJfMCKbaceiGaa8htaaaa@394B@ (N1) ≃ N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFneVtaaa@383B@ (E1, C1,1)     (77)
with E1 = nμ and, for all a, b ∈ A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@,
C1,1 (a, b) = nμ (a) I MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=Hi8jbaa@3894@a = b - nμ(a) × nμ(b)     (78)
We have also
P ( N ) = 1 n h    ∏ a ∈ A   N 1  ( a )  A 1  ( a )           ( 79 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaacqWGqbaucqGGOaakieqacqWFobGtcqGGPaqkcqGH9aqpdaWcaaqaaiabigdaXaqaaiabd6gaUnaaCaaaleqabaGaemiAaGgaaaaakmaarafabaGae8Nta40aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWGHbqycqGGPaqkdaahaaWcbeqaaiabdgeabnaaBaaameaacqaIXaqmaeqaaSGaeiikaGIaemyyaeMaeiykaKcaaaqaaiabdggaHjabgIGioJWaaiab+bq8bbqab0Gaey4dIunakiaaxMaacaWLjaGaeiikaGIaeG4naCJaeGyoaKJaeiykaKcaaa@5594@
which implies for all a ∈ A MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaWaaeGaeaaakeaaimaacqWFaeFqaaa@3821@ that
∂ P ( N )  ∂ N 1  ( a )   = A 1  ( a )  N 1  ( a )  ︸  G 1  ( a )     × P ( N )       ( 80 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabgkGi2kabdcfaqjabcIcaOGqabiab=5eaojabcMcaPaqaaiabgkGi2kab=5eaonaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIaemyyaeMaeiykaKcaaiabg2da9maalaaabaGaemyqae0aaSbaaSqaaiabigdaXaqabaGccqGGOaakcqWGHbqycqGGPaqkaeaadaagaaqaaiab=5eaonaaBaaaleaacqaIXaqmaeqaaOGaeiikaGIaemyyaeMaeiykaKcaleaacqWFhbWrdaWgaaadbaGaeGymaedabeaaliabcIcaOiabdggaHjabcMcaPaGccaGL44paaaGaey41aqRaemiuaaLaeiikaGIae8Nta4KaeiykaKIaaCzcaiaaxMaacqGGOaakcqaI4aaocqaIWaamcqGGPaqkaaa@5687@
So finally we get
σ ≃ Q   t  G 1  × C 1 , 1   × G 1          ( 81 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCcqWIdjYocqWGrbqudaGcaaqaaiabbccaGmaaCaaaleqabaacbiGae4hDaqhaaGqabOGae03raC0aaSbaaSqaaiabigdaXaqabaGccqGHxdaTcqqFdbWqdaWgaaWcbaGaeGymaeJaeiilaWIaeGymaedabeaakiabgEna0kab9DeahnaaBaaaleaacqaIXaqmaeqaaaqabaGccaWLjaGaaCzcaiabcIcaOiabiIda4iabigdaXiabcMcaPaaa@44E0@
where Q is either defined by equation (24) if the pattern is over-represented or by equation (28) if under-represented.