6 Detection strategy of non-gaussianity

 We have characterised the Gaussian signal through the excess of kurtosis of the multi-scale gradient and partial (first and cross) derivative coefficients using processes with different power spectra. When the power spectrum of the process exhibits a sharp cut off in one of the wavelet decomposition windows, we find that the excess of kurtosis, associated with the coefficients of the first derivatives, and consequently of the multi-scale gradient, is non-zero at the filtering level of the cut off. On the contrary, the excess of kurtosis computed with the cross derivative coefficients is zero.

Accordingly, we propose a detection strategy to test for non-gaussianity. We compare a set of maps of the "real'' observed sky to a set of Gaussian realisations having the power spectrum of the "real sky''. Our proposed method overcomes the problems arising from eventual cut off in the power spectrum of the studied process, and the consequent possible misinterpretations on the statistical signature. It also constitutes the most general approach to exhibit the statistical nature (Gaussian or not) of a signal and quantify its detectability through our statistical tests. Our detection strategy of the non-gaussianity is based on the following steps:

$\bullet$ using observed maps of the "real sky'' we compute the angular power spectrum of the signal, regardless of its statistical nature.

$\bullet$ We simulate Gaussian synthetic realisations of a process having the power spectrum of the "real'' process. On the obtained Gaussian test maps filtered with the wavelet function, we compute the excess of kurtosis for the multi-scale gradient and derivative coefficients. This analysis allows us to characterise completely the Gaussian maps, naturally taking into account eventual sample variance effects due to cut off at any scale.

$\bullet$ For the set of observed maps of the "real sky'', we compute the excesses of kurtosis associated with the multi-scale gradient, and the derivative coefficients.

$\bullet$ Assuming that the realisations (maps) are independent, each value of the excess of kurtosis has a probability of 1/N, where N is the number of maps. Using the computed excesses of kurtosis of both the Gaussian and non-Gaussian realisations, we deduce the probability distribution function (PDF) of the excess of kurtosis, for the multi-scale gradient coefficients and for the coefficients related to the derivatives.

$\bullet$ The last step consists in quantifying the detectability of the non-Gaussian signature. That is to compare the PDF of the Gaussian process to the PDF of the "real sky''. In practice this can be done by computing at each decomposition scale the probability that the median excess of kurtosis of the non-Gaussian maps belongs to the PDF of the synthetic Gaussian counterparts. It is the probability that a random variable is greater or equal to the real median k. We take k which is the asymptotic value given by the central limit theorem. Another way of comparing the two PDFs is to use the Kolmogorov-Smirnov (K-S) test (Press et al. 1992) which gives the probability for two distributions to be identical. This test for non-gaussianity is more global than the previous test because it is sensitive to the shift in the PDFs, especially the median value, and to the spread of the distributions. This property makes it more sensitive to non-gaussianity especially in the case where we only have a small number of observed maps.

We apply our detection strategy of non-gaussianity to the non-Gaussian test maps constituted of the top-hat and Gaussian profiles. For illustrative purposes, we give the results of the multi-scale gradient coefficient only. At the first three decomposition scales and for both sets of maps, we find that the probability for the signal to be non-Gaussian is 100% using the probability of the measured k to belong to the Gaussian PDF. At the fourth scale, the probability is 99.99% and 99.95% for respectively the top-hat and Gaussian profile distribution. The K-S test gives a 100% probability of detecting non-gaussianity. The detectability of the non-Gaussian signature, for the sum of the Gaussian and non-Gaussian (top-hat profile) maps with mixing ratio $R_{\rm rms}=1$, is 100% at the first decomposition scale. It is 99.96% and 93.4% at the second and third scale, and 76.43% at the fourth scale. For $R_{\rm rms}=2$, the first scale is still perfectly non-Gaussian, and only the second scale is detected with a probability of 72.4%. The K-S test gives more or less the same results for both mixing ratios. The results are illustrated in Fig. 6 (for the multi-scale gradient coefficients) and in Fig. 7 (for the cross derivative coefficients). In these plots, the solid line represents the PDF of the excess of kurtosis for the non-Gaussian measured signal. The dashed line represents the PDF of the synthetic Gaussian maps with same power spectrum. In both figures the left panels are for the non-Gaussian signal alone, whereas the right panels are for the sum with a mixing ratio of one.

  

Figure 6: Probability distribution functions of the excess of kurtosis, as percentages, computed with the multi-scale gradient coefficients. In the two panels, the dashed line represents the PDF of the Gaussian test maps. In the left panels, the solid line represents the non-Gaussian signal alone (top-hat profiles). In the right panels, the solid line represents the sum of the Gaussian and non-Gaussian processes with $R_{\rm rms}=1$

  

Figure 7: Probability distribution functions of the excess of kurtosis, as percentages, computed with the coefficients associated with the cross derivative. In the two panels, the dashed line represents the PDF of the Gaussian test maps. In the left panels, the solid line represents the non-Gaussian signal alone (top-hat profiles). In the right panels, the solid line represents the sum of the Gaussian and non-Gaussian processes with $R_{\rm rms}=1$