Two distinct patterns
We consider now two patterns V and W instead of one and want to study the joint distribution of SN (V) and SN (W) their corresponding pattern statistics.
With a similar argument as in section "delta method", it is easy to show that
ℒ ( [ S N  ( V )    S N  ( W )      ]   )  ≃ N ( [ S ( V )    S ( W )      ]  , [ σ V 2    σ V , W      σ V , W     σ W 2       ]   )        ( 29 )   MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamrtHrhAL1wy0L2yHvtyaeHbnfgDOvwBHrxAJfwnaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@6F1A@
where σV (resp. σW) is the standard deviation σ for the pattern V (resp. W) and where
σ V , W   =   t  ∇ F V ε  ( E ) × C × ∇ F W η  ( E )  ln ⁡ ( 10 ) F V ε  ( E ) × ln ⁡ ( 10 ) F W η  ( E )         ( 30  )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaWgaaWcbaGaemOvayLaeiilaWIaem4vaCfabeaakiabg2da9maalaaabaGaeeiiaaYaaWbaaSqabeaaieGacqGF0baDaaGccqGHhis0cqWGgbGrdaqhaaWcbaGaemOvayfabaGae8xTdugaaOGaeiikaGccbeGae0xrauKaeiykaKIaey41aqRae03qamKaey41aqRaey4bIeTaemOray0aa0baaSqaaiabdEfaxbqaaiab=D7aObaakiabcIcaOiab9veafjabcMcaPaqaaiGbcYgaSjabc6gaUjabcIcaOiabigdaXiabicdaWiabcMcaPiabdAeagnaaDaaaleaacqWGwbGvaeaacqWF1oqzaaGccqGGOaakcqqFfbqrcqGGPaqkcqGHxdaTcyGGSbaBcqGGUbGBcqGGOaakcqaIXaqmcqaIWaamcqGGPaqkcqWGgbGrdaqhaaWcbaGaem4vaCfabaGae83TdGgaaOGaeiikaGIae0xrauKaeiykaKcaaiaaxMaacaWLjaWaaeWaaeaacqaIZaWmcqaIWaamaiaawIcacaGLPaaaaaa@6CD8@
where
ε   ( resp .   η ) = { +  if pattern  V   ( resp .  W )  is over-represented    −  if pattern  V   ( resp .  W )  is unter-represented       .       ( 31  )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@AC70@
And after using results of sections "single pattern" and "under-represented pattern" we finally get
σ V , W   = ( Q V ε  Q W η   )  × (   t  ∇ G V  × C × ∇ G W   )        ( 32  )    MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFdpWCdaWgaaWcbaGaemOvayLaeiilaWIaem4vaCfabeaakiabg2da9maabmaabaGaemyuae1aa0baaSqaaiabdAfawbqaaiab=v7aLbaakiabdgfarnaaDaaaleaacqWGxbWvaeaacqWF3oaAaaaakiaawIcacaGLPaaacqGHxdaTdaqadaqaaiabbccaGmaaCaaaleqabaGaemiDaqhaaOGaey4bIencbeGae43raC0aaSbaaSqaaiabdAfawbqabaGccqGHxdaTcqGFdbWqcqGHxdaTcqGHhis0cqGFhbWrdaWgaaWcbaGaem4vaCfabeaaaOGaayjkaiaawMcaaiaaxMaacaWLjaWaaeWaaeaacqaIZaWmcqaIYaGmaiaawIcacaGLPaaaaaa@550C@
where QVε MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGrbqudaqhaaWcbaGaemOvayfabaacciGae8xTdugaaaaa@30E7@ (resp. W) and GV (resp. W) are the constant Q (Q+ and Q-) and the vector G for the pattern V (resp. W).