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.