In the field of science and engineering, the family of the Generalized Gaussian
Distributions (GGD) are applied to model and detect the Gaussian and non-Gaussian
signals (Miller and Thomas, 1972; Kassam,
1988). For example, in natural environments, the performance of communication
systems can significantly degrade due to the atmospheric noise caused by the
lightning activity in local and distant storm, or the ambient noise in the ocean.
In seismic research (Walden, 1985), the non-Gaussian
distribution has been used to model the amplitude distribution of the primary
reflection coefficients. The aim is to estimate the reflection coefficient sequences
as accurately as possible. In image processing, the fundamental work is fitting
the image intensity histogram data to gain underlying knowledge of data source.
Fan and Lin (2009) developed an expectation maximization
algorithm to estimate the parameter of the GGD mixture model that best fit the
observed image intensity histogram data. Other useful applications of the GGD
also include, video coding (Sharifi and Leon-Garcia, 1995),
handwritten character identification (Solihin and Leedham,
1999), speech recognition (Gazor and Zhang, 2003),
etc. Therefore, the choice between different families of distribution is important
and the estimation of the designed probability density function from observed
data is the major task.
In this study, the parameter estimator for the extended generalized Gaussian
distribution to include asymmetry is applied. The parameter estimation is determined
by the maximum likelihood scheme with iteration method. The key factor of the
iteration method depends on the choice of the initial and step values. Soderlind
(2006) presented and discussed the approaches to step size selection in
control theory and signal processing. Tlelo-Cuautle et
al. (2007) proposed a procedure to control the step-size, where the
step size is determined by the eigenvalues of its state matrixes. For saving
the computing cost and improving the accuracy, we proposed a table look-up method.
The generalized Gaussian distribution to include asymmetry proposed by Nandi
and Mampel (1995) is:
where, αn>0, αp>0, βn>0 and Γ is gamma function. By setting:
then the variance of Eq. 1 is chosen as one. By changing the parameters in Eq. 1, one can obtain symmetric (αn = αp) and asymmetric (αn ≠ αp) distributions. For symmetric cases, Fig. 1, (αn , αp) are (1,1) and (2,2) represent the Laplace distribution and Gaussian distribution, respectively. When αn tends to 0 the distribution tends to impulse and when αn tends to infinite the distribution tends to Uniform distribution. For asymmetric cases, (Fig. 2), the right skew distribution can be modeled by reducing the parameter αp and as αp decreases the tail length in the right side increases. Therefore, Eq. 1 provides a wide range of uni-modal distributions with symmetry and asymmetry.
||Symmetric model for varying the parameters of (αn
||Asymmetric model for varying the parameters of (αn
Given a series samples xi, where i = 1,2,3,...,N, the likelihood function of Eq. 1 is:
where, N1 (Nr) is the number of xi samples <0 (≥0). Taking the logarithm of Eq. 4, one gets:
If ∂ InL/∂αn, ∂InL/∂βn and ∂InL/∂αp are equate to zero, then the maximum likelihood estimates αn, βn and αp are obtained, i.e.,
and Ψ(•) is digamma function defined as:
To fit the observed data and approximate the solutions αn,
βn and αn, one need to solve (Eq.
6-8). In the following two iterated methods are developed.
Iterated method 1: Given a reasonable initial values αn0,
βn and αp, (Eq. 6-8)
can be solved iteratively for αn, βn and αp.
First step, compute Eq. 6 with (αn0±Δαn)
by fixed βn0 and αp0, then αn1
is selected by the value of (αn0±Δαn)
or (αn0-Δαn) so that the likelihood function
of Eq. 6 approaches zero. Second step, compute Eq.
7 with (βn0±Δβn) by fixed αn1
and αp0, then βn1 can be updated by the value
of (βn0±Δβn) or (βn0-Δβn)
so that Eq. 7 approaches zero. Third step, compute Eq.
8 with (αn0±Δαn) by fixed αn1
and βn1, then αp1 can be updated by the value
of (αn0+Δαn) or (αn0-Δαn)
so that Eq. 8 approaches zero. Thus, one iteration is completed
for updating αn1, βn1 and αn1.
This process continues until Eq. 6-8 simultaneously
approach zero with some error tolerances. The accuracy of the estimation of
this method depends on the choice of the step values Δαn,
Δβn and Δαp. Therefore, small step
value leads to accuracy of the estimation but requires more computing cost.
Iterated method 2: In this section the gradient method for iteration is developed. In the j-th iteration the Taylor series of Eq. 11 is applied to update the parameters αnj, βnj and αpj.
where, θ = (αn, βn, αp)T, grad In L = (∂In L/∂αn, ∂In L/∂βn, ∂In L/∂αp)T and:
is a Fisher information matrix. The inverse of the Fisher information matrix plays a key role for the iteration of Eq. 11. Given N samples of the observed data, the Fisher matrix can be obtained from Eq. 12. For each iteration, the expected value of the information matrix is calculated with updating parameters αnj, βnj and αpj. The estimation of the parameter is obtained as the value V(θ(j))-1 converge to zero with some error tolerances.
Cramer-Rao lower bound: It is important to understand the unbiasedness
of an estimator. The most common way to decide whether or not an estimators
distribution is unbiased is by looking at its expected value. The Cramer-Rao
Lower Bound (CRLB) for the standard deviation of every unbiased estimator is
defined as (Hogg and Tanis, 1989):
where, N is the samples of the every realizations and the E[•] is the
expected value. Note that Eq. 13-15 are
equivalence to the inverse of the square root of the diagonal elements in Eq.
RESULTS AND DISCUSSION
In the following simulation, the digamma function of Eq. 10
is computed using the algorithm in Char et al. (1991)
and for further details of the algorithms one can refer to Abramowitz
and Stegun (1972). The step value for method 1 is 0.005 and the error tolerance
for method 2 is 0.05. All simulation results are based on 50 Monte Carlo runs.
To obtain a good initial value, the tail length and skewness of the observed
data are measured and indicated by Q0 (tailness) and Q2
(skewness) (Lee, 2003):
the means of right αR and left αL intervals
and m(α) is α-trimmed mean. The Q0 for Gaussian distribution
is 2.58 and for Laplace distribution is 3.30, and Q2 is 1 for all
symmetric cases. Distributions with Q2>1 have relatively longer
right tail (right skewness), while those with 0<Q2<1 have relatively
longer left tail (left skewness). By choosing the parameter values (with αn
= αp for symmetry and αn ≠ αp
for asymmetry), the Q0-Q2 plot (Fig. 3)
for various tail length and skewness can be pre-calculated which corresponds
to the parameters under the Eq. 1. For example, the values
for Gaussian model the (Q0, Q2) is (2.58, 1) and the corresponding
(αn, βn, αp) is (2, 1.14, 2).
The results shown in Fig. 3 are various Q0-Q2
plots which cover the tailness up to 5.5 and right skewness up to 10. Using
the table look-up of Q0-Q2 plots and the corresponding
(αn, βn, αp), a reasonable initial
value of estimated parameter can be interpolated.
The samples applied in Table 1 are Gaussian distributions,
the results shown both iterated methods are reach the exact values of the parameters
with low bias. As the sample size increases the standard deviation of the estimation
decreases. It appears that the estimation is asymptotically unbiased and satisfies
the Cramer-Rao lower bound, where the standard deviations approach but greater
than the CRLB (Fig. 4). The results are shown in Table
2 and Fig. 5 are for the samples of Laplace distributions.
They are also in agreement with the theoretical values of the parameters.
||The Q0-Q2 plots
The parameter estimation of extended GGD model include asymmetric distribution is achieved in this study. This differs from the previous study of GGD mode, where more data can be fitted in the proposed model. Furthermore, the pre-calculated table of the Q0-Q2 plots save the computing cost and improve the accuracy of the estimation. The key factor of the iteration method depends on the choice of the initial and step values, in this study a table look-up method is provided for fast and accuracy estimation.
||Gaussian distribution with 50 realizations
||Standard deviation of the estimated parameters of Gaussian
||Standard deviation of the estimated parameters of Laplace
||Laplace distribution with 50 realizations
The maximum likelihood estimator is applied to fit three-parameter probability density function. The results showed that the proposed estimator is asymptotically unbiased for generalized Gaussian family distributions. They are useful for modeling and detecting of the observed data in real-time applications.
The author would like to acknowledge the valued comments made by Prof. A.K. Nandi.