next up previous
次へ: Another Interpretation as Clustering 上へ: A Maximum Likelihood Formulation 戻る: Model of Harmonic Structures


Model Parameter Estimation using EM Algorithm

Since the observed spectral density function , where denotes log-frequency, is considered to be generated from the model of multiple harmonic structures, the log-likelihood difference in accordance with an update of the model parameter to is

    (2)



Although Dempster formulated EM algorithm [8] in order to maximize the mean log-likelihood considering as a probabilistic density function, it can also be formulated in a same way even if is replaced with spectral density function. By taking expectation of both sides with respect to which represents the probability of the -labeled Gaussian distribution from which is generated, -function will be derived in the right-hand side. Given -function as

    (3)



thus it yields

     



    (4)



By obtaining which maximizes the function, the log-likelihood of the model of multiple harmonic structures with respect to every will be monotonously increased. A posteriori probability in equation (3) is given as

(5)
  (6)
(7)

where is a Gaussian distribution. Log-likelihood of the -labeled Gaussian, , is given as
    (8)

By the iterative procedure of the two steps as follows, the model parameter locally converges to ML estimates.


【Iterative procedure of EM algorithm】
Initial-step

Initialize the model parameter .
Expectaion-step

Calculate with equation (3).
Maximization-step

Maximize to obtain the next estimate

    (9)



Replace with and repeat from the Expectation-step.


next up previous
次へ: Another Interpretation as Clustering 上へ: A Maximum Likelihood Formulation 戻る: Model of Harmonic Structures
平成16年3月25日