Data Mining: The Textbook
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ψr(T, si, sj ): Probability that the rth position in sequence T corresponds to state si, the (r + 1)th position corresponds to sj .
γr(T, si): Probability that the rth position in sequence T corresponds to state si. The EM procedure starts with a random initialization of the model parameters and then iteratively estimates (α(·), β(·), ψ(·), γ(·)) from the model parameters, and vice versa. Specif-ically, the iteratively executed steps of the EM procedure are as follows: (E-step) Estimate (α(·), β(·), ψ(·), γ(·)) from currently estimated values of the model parameters (π(·), θ(·), p..). (M-step) Estimate model parameters (π(·), θ(·), p..) from currently estimated values of (α(·), β(·), ψ(·), γ(·)). It now remains to explain how each of the above estimations is performed. The values of α(·) and β(·) can be estimated using the forward and backward procedures, respectively. The forward procedure is already described in the evaluation section, and the backward procedure is analogous to the forward procedure, except that it works backward from the end of the sequence. The value of ψr(T, si, sj ) is equal to αr(T, si) · pij · θj (ar+1) · βr+1(T, sj ) because the sequence-generation procedure can be divided into three portions corresponding to that up to position r, the generation of the (r + 1)th symbol, and the portion after the
(r+1)th symbol. The estimated values of ψr(T, si, s j ) are normalized to a probability vector by ensuring that the sum over different pairs [i, j] is 1. The value of γ r(T, s i) is estimated by summing up the values of ψr(T, si, sj ) over fixed i and varying j. This completes the description of the E-step. The re-estimation formulas for the model parameters in the M-Step are relatively straightforward. Let I(ar, σ k) be a binary indicator function, which takes on the value of 1 when the two symbols are the same, and 0 otherwise. Then the estimations can be performed as follows:
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