Download Characterizing Non-Gaussianites or Why a dog is more than

Characterizing NonGaussianities
or
How to tell a Dog from an
Elephant
Jesús Pando
DePaul University
Contents
• Gaussian Signals
• Distinguishing among Gaussian signals
• The non-Gaussian domain and the failure of
spectral methods
• Wavelets
• Distinguishing dogs from elephants
• Conclusion
Gaussian Signals

Normal distributions are common,
because the central limit theorem states
that the sum of independent random
variables with finite variance will
result in normal distributions.
The probability density is centered at the mean and
68% of the distribution lies within the square root of
the second central moment, or the standard deviation.
However, distributions caused by different physical
processes with different time scales can have similar
means and variances.
Two Gaussian signals with mean = 0 and variance = 1.5
Spectral Methods

We need the variance as a function of scale in
order to distinguish these distributions of the same
mean and variance.
The Power Spectrum reveals
differences
Spectral Methods


Spectral techniques are effective in
untangling Gaussian signals.
The power spectrum, or variance as a
function of scale, breaks down
contributions of different physical
processes to a signal.
Non-Gaussian Signals

Non-Gaussian distributions
 Non-Linear dynamics and chaos
 Multi-component systems


Non-Gaussian signals are much harder to
characterize and detect.
One easy way to distinguish between
Gaussian and non-Gaussian signals is by
the use of cumulants.
Moments and Cumulants

The nth moment of distribution having probability density
f and mean µ is:



 xn f(x) dx
 (x - µ)n f(x) dx is the nth central moment.
Cumulants are defined via moments. For the the 1st and
2nd central moment (mean and variance), cumulants are
equal to moments. However, higher order cumulants are
given by the recursion formula:
•For instance, the first 3 cumulants defined in
terms of the central moments are:
2 = 2
3 = 3
4 = 4 - 3 22
• For a normal distribution
•1 = 1 (mean)
•2 = 2 (variance)
•n = 0 for n > 2
•Cumulants are a way to detect non-Gaussian
signals.
•However, we are faced with same problem as
before; that is, it may be possible to have two very
different signals with the nth cumulant equal.
Wavelet Transform
 The wavelet transform is an integral
transform whose basis functions are well
localized in time and frequency (or space
and scale).
 Wavelets have become increasingly
important because of the ability to localize
a signal efficiently in both time and
frequency.
Unlike the Fourier transform, there is no unique wavelet
basis.
Instead, the wavelets are defined by a function, , that is
rescaled and translated:
Wavelet Properties:
• Most useful wavelets have compact support.
• Wavelets can be classified as continuous or
discrete.
• The discrete wavelet transform (DWT)
produces two sets of coefficients; the scaling
coefficients which give a local average, and the
wavelet coefficients which give the fluctuation
from the local average.
• The most useful DWT are also orthogonal.
Especially, the wavelet coefficients are
orthogonal in both space and scale (time and
frequency).
• WFC’s, j,l , measure changes from local mean. A large
WFC indicates a large local fluctuation.
• WFC’s look suspiciously like a central moment.
• As with the Fourier spectrum, we can define the Wavelet
Variance Spectrum: Pj = <  l,j2 > where the average is done
over position l at scale j.
• With wavelets, higher order cumulant spectra are readily
defined.
• We define the third and fourth order cumulants as:
Sj = Mj3/(Mj2)3/2
Kj = Mj4 /(Mj2)2 - 3
where
Mn = < (j,l - j,l ) n >
• Thus, the wavelet gives a simple way to
characterize some non-Gaussian distributions.
Gaussian distribution with power spectrum, P(k) = k/(1 + a k4)
where a is constant.
Non-Gaussian Simulations
 Clumps
or valleys with a signal/noise = 2.0
and random width between 1-5 bins are
embedded in a Gaussian background.
 Distributions with16, 32, and 48 clumps (or
valleys) are generated.
 100 realizations of each is done and 95%
confidence levels computed.
Cumulant, 3
Cumulant, 4
Scale-Scale Correlations
• The DWT cumulant spectra give a way to
characterize different non-Gaussian
signals.
• DWT measure can also give clues to the
dynamics behind the non-Gaussian
distributions.
• In scale dependent processes, one such
measure is the scale-scale correlation.


The Gaussian Block model results in a
final distribution that is Gaussian since it is
formed at each level by Gaussian random
variables.
The Branching block model results in a
final distribution that is not Gaussian since
each level(scale) has a memory of how it
got there.


The usual statistical measures fail to
distinguish these distributions.
We introduce the scale-scale DWT correlation:
For a Gaussian distribution, Cjp,p = 1 for p  2.
This statistical measure can therefore detect
some types of dynamics (hierarchical).
Conclusion




One point measures detect non-Gaussianities,
but provide limited information about the signal.
Traditional Fourier spectral methods are not
ideal for higher order cumulants.
Wavelets allow one to construct cumulant
spectra.
Wavelet versatility allows for the construction of
customized measures and sometimes help us to
say more than just a dog is not an elephant.