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Table 2 Maximization step in HMM learning.

From: Clustering ionic flow blockade toggles with a Mixture of HMMs

Initial probability estimate

Transition probability estimate

Emission parameters estimate

π ^ i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafqiWdaNbaKaadaWgaaWcbaGaemyAaKgabeaaaaa@2F29@ = γ1(i), for 1 ≤ iN.

a ^ i , j = t = 1 T 1 ξ t ( i , j ) t = 1 T 1 γ t ( i ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmyyaeMbaKaadaWgaaWcbaGaemyAaKMaeiilaWIaemOAaOgabeaakiabg2da9KqbaoaalaaabaWaaabmaeaacqaH+oaEdaWgaaqaaiabdsha0bqabaGaeiikaGIaemyAaKMaeiilaWIaemOAaOMaeiykaKcabaGaemiDaqNaeyypa0JaeGymaedabaGaemivaqLaeyOeI0IaeGymaedacqGHris5aaqaamaaqadabaGaeq4SdC2aaSbaaeaacqWG0baDaeqaaiabcIcaOiabdMgaPjabcMcaPaqaaiabdsha0jabg2da9iabigdaXaqaaiabdsfaujabgkHiTiabigdaXaGaeyyeIuoaaaaaaa@5245@ , for 1 ≤ i, jN.

Gaussian emission b ^ j ( o ) μ = t = 1 T o t γ t ( j ) t = 1 T γ t ( j ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOyaiMbaKaadaWgaaWcbaGaemOAaOgabeaakiabcIcaOiabd+gaVjabcMcaPiabgkziUkabeY7aTjabg2da9KqbaoaalaaabaWaaabmaeaacqWGVbWBdaWgaaqaaiabdsha0bqabaGaeq4SdC2aaSbaaeaacqWG0baDaeqaaiabcIcaOiabdQgaQjabcMcaPaqaaiabdsha0jabg2da9iabigdaXaqaaiabdsfaubGaeyyeIuoaaeaadaaeWaqaaiabeo7aNnaaBaaabaGaemiDaqhabeaacqGGOaakcqWGQbGAcqGGPaqkaeaacqWG0baDcqGH9aqpcqaIXaqmaeaacqWGubavaiabggHiLdaaaaaa@53B2@ , b ^ j ( o ) σ 2 = t = 1 T ( o t μ ^ j ) 2 γ t ( j ) t = 1 T γ t ( j ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@5BDF@ , for 1 ≤ jN.