e semi-parametric approach

parametric approaches and the non-parametric approaches enjoy

antages when estimating a density for a data set. However they

some limitations. The inflexibility of the parametric approaches

pace costing problem of the non-parametric approaches lead to

deration of the semi-parametric approaches for density estimation.

-parametric approaches estimate a density function for a data set

ing that a data set is drawn from a mixture of the finite number

nents, which have the similar function as the kernels used in the

nsity estimation approach. Each component can be either a single

distribution or a single Gamma distribution or others [Olkin and

an, 1987; Duda, et al., 2000]. The number of components used

ure is pre-defined or can be optimised using a statistic metric.

he Gaussian mixture

hows the Gaussian mixture of K components, where ߤ௞, ߪ௞

ଶ, and

for the mean, the variance and the mixing coefficient (weight) of

mponent, respectively,

݂ሺݔ௜ሻൌ෍ݓ௞࣡൫ݔ௜|ߤ௞, ߪ௞

ଶ൯

௄

௞ୀଵ

(2.9)

og-likelihood function of such a model is defined as below, where

ଶ,

gࣦ∝െ෍log ෍ݓ௞ඥߚ௞exp ൬െ^ߚ௞

2 ^{ሺݔ௜െߤ௞ሻଶ൰}

௄

௞ୀଵ

ே

௜ୀଵ

(2.10)

maximum likelihood method is normally used to estimate the

rameters for a Gaussian mixture model. The following equations

to estimate ߤ௞, ߚ௞, and ݓ௞, respectively,