5 Application to non-Gaussian signals

We apply our statistical discriminators to detect the non-Gaussian signature of different processes. We first study two sets of non-Gaussian maps, one constituted of a distribution of top-hat profiles and the other constituted of a distribution of Gaussian profiles, both having the same power spectrum as the Gaussian test maps used in the previous section. We also apply the statistical test to a combination of Gaussian and non-Gaussian signals with different mixing ratios.

5.1 The multi-scale gradient and its distribution

We compute the multi-scale gradient coefficients (${\mathcal{G}}_L$) using 100 statistical realisations of the non-Gaussian process (top-hat profiles). At the four decomposition scales, we plot the fitted histogram (right panels of Fig. 3). We note that the distribution of the multi-scale gradient coefficients also fits a Laplace distribution for small values of ${\mathcal{G}}_L$. However, there is a significant departure from this distribution for higher values. This is exhibited by the larger error bars and by the wings of the gradient distribution at large ${\mathcal{G}}_L$. Figure 3 (right panels) exhibits the non-Gaussian signatures mostly at the first three decomposition scales. At the fourth, the lack of coefficients enlarges the error bars but we still marginally distinguish the non-Gaussian signal.

  

Table 3: k₁ is the median excess of kurtosis, at four decomposition scales, computed with the 100 non-Gaussian maps (top-hat profile). k₂ is computed with the the 100 non-Gaussian maps (Gaussian profile). $\sigma_+$ and $\sigma_{-}$ give the confidence interval for one realisation

In our test case, the process is leptokurtic, that is the non-gaussianity is characterised by a positive excess of kurtosis. We quote, in the left panel of Table 3 the median excesses of kurtosis computed with the multi-scale gradient coefficients of the 100 maps. This is a more suitable quantity to characterise a non-Gaussian process, than the mean $\overline k$, as there is an important dispersion of the k values with a clear excess towards large values. The $\sigma_{\pm}$, which represents the rms excess of kurtosis with respect to the median for one realisation, takes naturally into account the non symmetric distribution of the multi-scale gradients. This results in a lower boundary ($\sigma_{-}$) for the confidence interval smaller that the upper boundary ($\sigma_+$). The latter is biased towards large values as we are studying a leptokurtic process. Therefore, the comparison between the values of k and $\sigma_{-}$ indicates the detectability of non-gaussianity. When $k-\sigma_{-}$ for one realisation differs from zero by a value of the order of, or larger than, $\sigma_{-}$ this suggests that the signal is non-Gaussian. For the top-hat profiles, there is an obvious excess of kurtosis at all scales.

In order to test the non-Gaussian signature arising form different processes with same power spectra, we analyse a set of 100 non-Gaussian maps made of the superposition of Gaussian profiles of different sizes and amplitudes. We compute the median value of the excess of kurtosis and the corresponding confidence intervals (Table 3, right panel). We note that k is different from zero at all scales, exhibiting the non-Gaussian nature of the studied process. However, it is smaller than in the case of the top-hat profiles. This decrease is due to the superposition of smoother profiles.

We now add one representative Gaussian map to 100 non-Gaussian maps (top-hat profiles). As the non-Gaussian signal is very strongly dependent on the studied map, it is necessary to span a large set of non-Gaussian statistical realisations in order to have a reliable statistical specification of non-gaussianity. The Gaussian and non-Gaussian signals were summed with different mixing ratios represented by the ratio of their rms amplitudes ($R_{\rm rms}=\sigma_{\rm gauss}/\sigma_{\rm non-gauss}$).

After wavelet decomposition, we compute the multi-scale gradient coefficients of the summed maps and derive the normalised median excess of kurtosis with respect to a Laplace distribution together with the confidence intervals. The results are quoted in Table 4 as a function of the mixing ratio $R_{\rm rms}$ and the wavelet decomposition scale.
For $R_{\rm rms}=1$, the excess of kurtosis is larger than that of the Gaussian test map and it is smaller than that of the purely non-Gaussian signal. The summation of the two processes has therefore, as expected, smoothed the gradients and diluted the non-Gaussian signal. For a non-Gaussian signal half that of the Gaussian signal, only the first three scales indicate an excess of kurtosis different from the Gaussian one. For the ratio $R_{\rm rms}=3$, only the first scale has an excess marginally different from the Gaussian signal. For larger ratios, the non-Gaussian signal is quite blurred.

5.2 Partial derivatives

For the 100 non-Gaussian maps (top-hat profile) with the same power spectrum as the Gaussian test maps, we compute the normalised excess of kurtosis, with respect to a Gaussian, of the wavelet coefficients associated with $\partial/\partial x$, $\partial/\partial y$ and $\partial^2/\partial x\partial y$. As for the multi-scale gradient, we derive the median excess of kurtosis and the upper and lower boundaries of the confidence intervals. The results, given in Table 5, show non-zero excesses of kurtosis for the first and cross derivatives at all decomposition scales.

  

Table 4: The median excess of kurtosis k, at four decomposition scales, computed over the 100 non-Gaussian maps added to the Gaussian map as a function of the mixing ratio $R_{\rm rms}$. The $\sigma_+$ and $\sigma_{-}$ values represent respectively the upper and lower boundaries of the confidence interval for one realisation. They are the rms values with respect to the median excess of kurtosis

  

Table 5: The median excess of kurtosis, at four decomposition scales. k₁ is computed using the wavelet coefficients associated with $\partial/\partial x$ and $\partial/\partial y$. k₂ is given for $\partial^2/\partial x\partial y$. The 100 non-Gaussian maps (top-hat profile) have the same power spectrum as the Gaussian test maps. The $\sigma_+$ and $\sigma_{-}$ values are the rms values for one realisation with respect to the median excess of kurtosis

In Fig. 4, the solid line represents the values of the excess of kurtosis of each non-Gaussian realisation. The dashed line represents the same quantity for the Gaussian test maps. We first note the overall shift of the values towards non-zero positive values (leptokurtic signal), with some very large values compared to the median. A second characteristic worth noting is the difference in amplitudes between, on the one hand, the excess of kurtosis of the coefficients associated with $\partial^2/\partial x\partial y$ and, on the other hand, those associated with $\partial/\partial x$ and $\partial/\partial y$. The former are indeed smaller. As the excess of kurtosis of the first derivative coefficients is of the same order, we compute one median k over $\partial/\partial x$ and $\partial/\partial y$coefficients, and compare it to the excess of kurtosis of the cross derivative. At the first two decomposition scales there is an important and noticeable difference between the two sets of values $\partial/\partial x$ and $\partial/\partial y$,and $\partial^2/\partial x\partial y$. At the third and fourth decomposition scales, the difference decreases but is still present.

  

Table 6: The median excess of kurtosis, at four decomposition scales, computed over the 100 non-Gaussian maps (Gaussian profile). k₁ is the median excess computed with the coefficients associated with the vertical and horizontal gradients, and k₂ is given for the cross derivative. The $\sigma$ values are the boundaries of the confidence interval for one statistical realisation

For the non-Gaussian process made of the superposition of Gaussian profiles, we compute the median excess of kurtosis associated with the wavelet coefficients of the first and cross derivatives (Table 6). As for the multi-scale gradient coefficients, we find that the excess of kurtosis is smaller for this type of non-Gaussian maps but it is still significantly different from zero at all scales except the fourth.

  

Table 7: The median excess of kurtosis, at four decomposition scales, for 100 non-Gaussian maps (top-hat profile) added to one Gaussian map (with same power spectrum) as a function of the mixing ratio $R_{\rm rms}$. k₁ is computed with the coefficients associated with $\partial/\partial x$ and $\partial/\partial y$, and k₂ is given for $\partial^2/\partial x\partial y$. The $\sigma$ values are the boundaries of the confidence interval for one statistical realisation

.D..-1

  

Figure 5: Excess of kurtosis computed with the wavelet coefficients of $\partial/\partial x$ and $\partial/\partial y$ for respectively the left and centre panels. The right panels are associated with the coefficients of $\partial^2/\partial x\partial y$. The dashed line represents the Gaussian process alone and the solid line represents the sum of the Gaussian and non-Gaussian processes. The Gaussian and non-Gaussian processes have same power spectra and have been summed with a mixing ratio $R_{\rm rms}=1$

We analyse the sum of a representative Gaussian map and the set of 100 non-Gaussian maps (top-hat profile). The sum of the two processes has been performed again with different mixing ratios. The results we obtain are given in Table 7. An accompanying figure (Fig. 5) illustrates the corresponding results for a mixing ratio $R_{\rm rms}=1$ (solid line). In this figure, the dashed line represents the Gaussian process alone. For $R_{\rm rms}=1$, we find that non-gaussianity is detected at all decomposition scales for both first and cross derivative coefficients. For a mixing ratio $R_{\rm rms}=2$, we observe a significant excess of kurtosis only at the first decomposition scale. For $R_{\rm rms}\ge 3$, the excess becomes marginal for both $\partial/\partial x$ and $\partial/\partial y$ coefficients at the first decomposition scale, all other scales showing no departure from gaussianity. The same tendency is noted for the coefficients associated with $\partial^2/\partial x\partial y$.