(9) 
We plot in Fig. 3 (left panels) the distribution of the multiscale gradient coefficients in the four decomposition scales for the Gaussian signal. For reasons of legibility, we have just plotted the fit obtained with the 100 maps. The error bars represent a confidence interval (for one realisation) and account for the statistical dispersion of the realisations.
We can analyse the multiscale gradient distribution through its nthorder moments (). In particular, we compute the excess of kurtosis using the second and fourth moments of the distribution( and ). For a Gaussian distribution, the normalised excess is zero. For a Laplace distribution, the fourth moment is given by . The normalised excess of kurtosis highlights the nonGaussian signature of a signal through the departure of the multiscale gradient from a Laplace distribution.
At each decomposition level, we compute the normalised excess of kurtosis of
the multiscale gradient coefficients for the 100 Gaussian maps, and we derive
a representative value of the distribution that is the mean
which we quote in Table
1. The results show that
is very close to zero. The values correspond to the root mean square
values with respect to the mean . The
values define a confidence interval, or a probability distribution
of the excess of kurtosis.
For the Gaussian signal the upper and lower boundaries of this interval are
equal suggesting that the k values are Gaussian distributed. The increasing
(Fig. 3, left panels)
with the decomposition scale is due to the larger dispersion. This feature is
also the consequence of the smaller number of
wavelet coefficients at higher decomposition scales.


As a result, we conclude that a Gaussian signal can be characterised
by the distribution of the multiscale gradient coefficients and of the
coefficients associated with ,
and . In the first
case, the multiscale gradient coefficient distribution is fitted by a
Laplace distribution and
the excess of kurtosis is zero. In the second case, the excesses of kurtosis
are Gaussian distributed with for the first and the cross
derivatives.
We check these two characteristics on other Gaussian processes with different power spectra. For a white noise spectrum, we find identical results as in the study case: zero excess of kurtosis for the multiscale gradient and the coefficients of the partial derivatives. Since our statistical tests are based on the statistics of the wavelet coefficients at each decomposition scale, we expect that a sharp cut off in the power spectrum of the Gaussian signal will induce a sample variance problem. We check this behaviour using a Gaussian process exhibiting a very sharp cut off with a shape close to a Heaviside function, at the second decomposition scale. This cut off is similar to the cut off expected in the CMB power spectrum in a standard cold dark matter model. At all the decomposition scales except the second, we find the expected zero excess of kurtosis for the wavelet coefficients of Gaussian signals. At the second decomposition scale, we find a nonzero excess of kurtosis for the multiscale gradient coefficients, as well as for the coefficients related to the first derivatives, which could be misinterpreted for a nonGaussian signature. This nonzero excess has nothing to do with an intrinsic property of the studied signal, as the latter is Gaussian at all scales. It comes from the very sharp decrease in power combined with the narrow filter associated with the wavelet basis. In fact, the contribution of the Gaussian process, at this scale, is sparse. Therefore, it induces a sample variance effect which in turn results in a nonzero excess of kurtosis. We tested a wider filtering wavelet and found that the excess of kurtosis decreases. Nevertheless, a wider filtering wavelet smoothes the nonGaussian signatures and reduces the efficiency of our discriminating tests. However, at the second decomposition scale, the wavelet coefficients associated with exhibit no excess of kurtosis for a process with sharp cut off. This indicates that the excess of kurtosis computed using the cross derivative coefficients is more reliable in characterising a Gaussian process, and consequently nongaussianity, regardless of the power spectrum.
We also analysed the sum of Gaussian signals. When there is no cut off in the power, the excess of kurtosis for the sum of Gaussian signals is zero for both discriminators. By contrast, if one of the Gaussian signals presents a cut off at any decomposition scale, we again find a nonzero excess of kurtosis at the corresponding scale.
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