Characterizing Non- Gaussianities 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. 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 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. 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. 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 n th moment of distribution having probability density f and mean µ is:  x n f(x) dx  x n f(x) dx  (x - µ) n f(x) dx is the n th central moment.  (x - µ) n f(x) dx is the n th 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:For instance, the first 3 cumulants defined in terms of the central moments are:  2 =  2  3 =  3  4 =   2 2  4 =   2 2 For a normal distribution For a normal distribution  1 =  1 (mean)  1 =  1 (mean)  2 =  2 (variance)  2 =  2 (variance)  n = 0 for n > 2  n = 0 for n > 2 Cumulants are a way to detect non-Gaussian signals.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 n th cumulant equal.However, we are faced with same problem as before; that is, it may be possible to have two very different signals with the n th 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. 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 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). 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. WFC’s look suspiciously like a central moment. As with the Fourier spectrum, we can define the Wavelet Variance Spectrum: P j = where the average is done over position l at scale j. As with the Fourier spectrum, we can define the Wavelet Variance Spectrum: P j = where the average is done over position l at scale j. With wavelets, higher order cumulant spectra are readily defined. With wavelets, higher order cumulant spectra are readily defined.

We define the third and fourth order cumulants as: S j = M j 3 /(M j 2 ) 3/2 K j = M j 4 /(M j 2 ) where M n = Thus, the wavelet gives a simple way to characterize some non-Gaussian distributions.

Gaussian distribution with power spectrum whereis constant Gaussian distribution with power spectrum, P(k) = k/(1 + a k 4 ) 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 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 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. The usual statistical measures fail to distinguish these distributions. We introduce the scale-scale DWT correlation: We introduce the scale-scale DWT correlation: For a Gaussian distribution, C j p,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. One point measures detect non-Gaussianities, but provide limited information about the signal. Traditional Fourier spectral methods are not ideal for higher order cumulants. Traditional Fourier spectral methods are not ideal for higher order cumulants. Wavelets allow one to construct cumulant spectra. 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. Wavelet versatility allows for the construction of customized measures and sometimes help us to say more than just a dog is not an elephant.