Information Theory and Principal Component Analysis - Biomedical Signal Analysis: Contemporary Methods and Applications - page 81

Laplacian Density
Definition 3.13:

n is said to be Lapla-
cian if its density function p X exists and is of the form

A random vector X :Ω → R

2 exp

n

p X ( x )= λ | 1 )= λ

2 exp (

− λ | x − λ | x i |

i=1

for a fixed λ> 0.

| 1 := i=1 | Here denotes the 1 -norm of x .

More generally, we can take the p -norm on

| x x i |

n to generate γ-
distributions or generalized Laplacians or generalized Gaussians [152].
They have the density

R

γ = C ( γ )exp γ

n

p X ( x )= C ( γ )exp − λ | x | − λ | x i |

i=1

for fixed γ> 0. For the case γ = 2 we get an independent Gaussian
distribution, for γ = 1 a Laplacian, and for smaller γ we get distributions
with even higher kurtosis.
In figure 3.3 the density of a two-dimensional Laplacian is plotted.

Higher-Order Moments and Kurtosis

The covariance is the main second-order statistical measure used to
compare two or more random variables. It basically consists of the second
moment α 2 ( X ):= E ( X 2 ) of a random variable and combinations. In
so-called higher-order statistics ,too,higher moments α j ( X ):= E ( X j )
or central moments μ j ( X ):= E (( X E ( X )) j ) are used to analyze a −

random variable X :Ω .

By definition, we have α 1 ( X )= E ( X )and μ 2 ( X )=var( X ). The
third central moment μ 3 ( X )= E (( X

→ R

E ( X )) 3 ), is called skewness of
X . It measures asymmetry of its density; obviously it vanishes if X is
distributed symmetrically around its mean.
Consider now the fourth moment α 4 ( X )= E ( X 4 ) and the central
moment μ 4 ( X )= E (( X

−

E ( X )) 4 ). They are often used in order to
determine how much a random variable is Gaussian. Instead of using
the moments themselves, a combination called kurtosis is used.

−