4 Characterisation of Gaussian signals

4.1 The multi-scale gradient and its distribution

For the 100 Gaussian maps, we find that the histogram of the multi-scale gradient coefficients can be fitted by the positive wing of the Laplace probability distribution function:

$${\mathcal{H}}({\mathcal{G}}_L)=\frac{1}{\sqrt{2}\,\sigma}\, ... ...\left(-\frac{\sqrt{2}\,({\mathcal{G}_L}-\mu_1)}{\sigma}\right),$$ (9)

where $\mu_1$ is the mean of the distribution (theoretically equal to zero) and $\sigma^2$ is its second moment.

  

Figure 3: Fits of the multi-scale gradient coefficient histograms obtained with the 100 statistical realisations. In all plots, the error bar is for one realisation. The left panels are given for the Gaussian process whereas the right panels represent the non-Gaussian signal. The wings at large multi-scale gradient indicate the non-Gaussian signature

We plot in Fig. 3 (left panels) the distribution of the multi-scale 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 multi-scale gradient distribution through its nth-order moments ($\mu_n$). In particular, we compute the excess of kurtosis using the second and fourth moments of the distribution($\mu_2$ and $\mu_4$). For a Gaussian distribution, the normalised excess is zero. For a Laplace distribution, the fourth moment is given by $\mu_4=6\mu_2^2$. The normalised excess of kurtosis $k=\mu_4/\mu_2^2-6$ highlights the non-Gaussian signature of a signal through the departure of the multi-scale gradient from a Laplace distribution.

At each decomposition level, we compute the normalised excess of kurtosis of the multi-scale gradient coefficients for the 100 Gaussian maps, and we derive a representative value of the distribution that is the mean $\overline k$ which we quote in Table 1. The results show that $\overline k$ is very close to zero. The $\sigma$ values correspond to the root mean square values with respect to the mean $\overline k$. The $\sigma$ 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 $\sigma$ (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.

  

Table 1: The mean excess of kurtosis at four decomposition scales computed over the 100 Gaussian maps. The $\sigma_{\pm}$ values define the confidence intervals for one realisation

4.2 Partial derivatives

  

Table 2: The mean excess of kurtosis at four decomposition scales. $\overline k_1$ is computed using the wavelet coefficients associated with $\partial/\partial x$ and $\partial/\partial y$. $\overline k_2$ is computed using the coefficients of $\partial^2/\partial x\partial y$. The $\sigma_{\pm}$ values define the confidence intervals for one realisation

  

Figure 4: Excess of kurtosis computed over the wavelet coefficients of respectively $\partial/\partial x$ for the left panels, $\partial/\partial y$ for the centre panels and $\partial^2/\partial x\partial y$ for the right panels. The solid line is for the non-Gaussian process. The dashed line is for the Gaussian process with the same power spectrum

The wavelet coefficient distributions associated with the first and cross derivatives are Gaussian for the Gaussian maps. We thus compute the normalised excess of kurtosis with respect to a Gaussian distribution ($k=\mu_4/\mu_2^2-3$). These values are displayed in Fig. 4 (dashed line) for the Gaussian maps. In this figure and at each decomposition scale, the first set of 100 values stands for the excess of kurtosis of the wavelet coefficients associated with the horizontal gradient ($\partial/\partial y$). The second set of 100 values represents the same quantity computed for the vertical gradient ($\partial/\partial x$) and the last one represents the excess of kurtosis for the wavelet coefficients associated with cross derivative $\partial^2/\partial x\partial y$ (diagonal gradients). We note that for the Gaussian maps the excess is always centred around zero at all the decomposition scales. For the multi-scale gradient, the dispersion around the mean $\overline k$ increases with increasing decomposition scale. In Table 2, we quote the mean together with the confidence intervals at each scale. The results also show that the values are close to zero confirming the Gaussian nature of the signal.

As a result, we conclude that a Gaussian signal can be characterised by the distribution of the multi-scale gradient coefficients and of the coefficients associated with $\partial/\partial x$, $\partial/\partial y$ and $\partial^2/\partial x\partial y$. In the first case, the multi-scale 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 $\overline k=0$ 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 multi-scale 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 non-zero excess of kurtosis for the multi-scale gradient coefficients, as well as for the coefficients related to the first derivatives, which could be misinterpreted for a non-Gaussian signature. This non-zero 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 non-zero excess of kurtosis. We tested a wider filtering wavelet and found that the excess of kurtosis decreases. Nevertheless, a wider filtering wavelet smoothes the non-Gaussian signatures and reduces the efficiency of our discriminating tests. However, at the second decomposition scale, the wavelet coefficients associated with $\partial^2/\partial x\partial y$ 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 non-gaussianity, 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 non-zero excess of kurtosis at the corresponding scale.