From: Implementing EM and Viterbi algorithms for Hidden Markov Model in linear memory
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 ≤ i ≤ N.
• Gaussian emission 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, j ≤ N.
b ^ j ( o ) → μ = ∑ t = 1 T o t γ t ( j ) ∑ t = 1 T γ t ( j ) , 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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@84B2@ , for 1 ≤ j ≤ N,
• Discrete emission b ^ j ( k ) = ∑ t = 1 T δ ( o t = v k ) γ t ( j ) ∑ t = 1 T γ t ( j ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaGafmOyaiMbaKaadaWgaaWcbaGaemOAaOgabeaakiabcIcaOiabdUgaRjabcMcaPiabg2da9KqbaoaalaaabaWaaabmaeaacqaH0oazcqGGOaakcqWGVbWBdaWgaaqaaiabdsha0bqabaGaeyypa0JaemODay3aaSbaaeaacqWGRbWAaeqaaiabcMcaPiabeo7aNnaaBaaabaGaemiDaqhabeaacqGGOaakcqWGQbGAcqGGPaqkaeaacqWG0baDcqGH9aqpcqaIXaqmaeaacqWGubavaiabggHiLdaabaWaaabmaeaacqaHZoWzdaWgaaqaaiabdsha0bqabaGaeiikaGIaemOAaOMaeiykaKcabaGaemiDaqNaeyypa0JaeGymaedabaGaemivaqfacqGHris5aaaaaaa@5759@ , for 1 ≤ j ≤ N. and 1 ≤ k ≤ K, where v1,..., v K is the set of possible dircrete observations.