New interpretable statistics for large-scale structure analysis and generation

We introduce wavelet phase harmonics (WPH) statistics: interpretable low-dimensional statistics that describe 2D non-Gaussian fields. These statistics are built from WPH moments, which were recently introduced in the data science and machine learning community. We apply WPH statistics to projected 2D matter density fields from the Quijote $N$-body simulations of the large-scale structure of the Universe. By computing Fisher information matrices, we find that the WPH statistics place more stringent constraints on four of five cosmological parameters when compared to statistics based on the combination of the power spectrum and bispectrum. We also use the WPH statistics with a maximum entropy model to statistically generate new 2D density fields that accurately reproduce the probability density function, the mean and standard deviation of the power spectrum, the bispectrum, and Minkowski functionals of the input density fields. Although other methods are efficient for either parameter estimates or statistical syntheses of the large-scale structure, WPH statistics are the first statistics that achieve state-of-the-art results for both tasks as well as being interpretable.

Figure 1

Two-dimensional bump steerable wavelets. The real part of ${\psi }_{4,2}(\overrightarrow{x})$ (left) and the Fourier transform ${\widehat{\psi }}_{1,2}(\overrightarrow{k})$ (right). Axes are labeled with the units used when applying these wavelets to density maps of the large scale structure (LSS).

Figure 2

Left: typical projected 2D density map of the LSS from the Quijote simulations [23]. Center and right: real part of the same map convolved with wavelets ${\psi }_{1,+2}(\overrightarrow{x})$ and ${\psi }_{1,-2}(\overrightarrow{x})$, respectively. The dashed white circles highlight filaments captured by the first wavelet, the dotted red circles a filament captured by the second wavelet, and the plain black circles an intersection of filaments captured by both wavelets.

Figure 3

Illustration of wavelet phase harmonics (WPH) moments computation. A typical Quijote density field $\rho$ (far left) is convolved with two wavelets ${\psi }_{{\overrightarrow{\xi }}_1}$ and ${\psi }_{{\overrightarrow{\xi }}_2}$, with $(j_1,{\ell }_1)=(3,0)$ and $(j_2,{\ell }_2)=(4,0)$. The amplitude and phase of each convolution is shown in the central panel. From their phase, one sees that the $\rho *{\psi }_{{\overrightarrow{\xi }}_i}$ fields oscillate with different characteristic scales $2^{j_1}$ and $2^{j_2}$, respectively. Their covariance is therefore negligible. By applying the phase harmonic operator to $\rho *{\psi }_{\overrightarrow{{\xi }_2}}$, using harmonic exponent $p={\xi }_1/{\xi }_2=2^{j_2}/2^{j_1}$, one obtains a new field of the same amplitude but with a phase of characteristic scale $2^{j_1}$ (lower right). As the fields $\rho *{\psi }_{{\overrightarrow{\xi }}_1}$ and $[\rho *{\psi }_{{\overrightarrow{\xi }}_2}{]}^{{\xi }_1/{\xi }_2}$ have the same characteristic wavelength, their covariance may be non-negligible. This covariance is a WPH moment characterizing the relative phase alignment between the $\rho *{\psi }_{{\overrightarrow{\xi }}_1}$ and $\rho *{\psi }_{{\overrightarrow{\xi }}_2}$ fields. This type of WPH moment computation is illustrated in Fourier space in the left panel of Fig. 13.

Figure 4

Fisher matrix constraints for five cosmological parameters, based on WPH statistics (green), power spectrum (${\mathrm{P}}_k$, red), and joint power spectrum and bispectrum (${\mathrm{P}}_k+{\mathrm{B}}_k$, blue). These constraints were computed from $(1\mathrm{Gpc}/\mathrm{h}{)}^2$ maps of the projected matter density field (bottom) and its logarithm (top). Contours mark 95% confidence intervals. WPH statistics provide the best constraints for each parameter except ${\sigma }_8$, which is more tightly constrained by ${\mathrm{P}}_k+{\mathrm{B}}_k$ on the matter density field (bottom row).

Figure 5

Comparisons of the logarithm of the matter density field $\log (\rho )$ in Quijote simulation maps and in our statistically synthesized maps, showing how well the syntheses reproduce the statistical properties of $\log (\rho )$ in the Quijote maps. The error bars correspond to the realization-per-realization dispersion. (a) A map of $\log (\rho )$ from the Quijote simulations. (b) A map of $\log (\rho )$ synthesized based on WPH statistics of a sample of 30 Quijote maps (see Sec. 4a). (c)–(h) Statistics for $\log (\rho )$ estimated using 300 maps from the Quijote simulations (orange lines) and 300 syntheses (dashed blue lines). (c) Power spectrum, (d) standard deviation of the power spectrum, (e) pixel value PDF on a linear scale, (f) bispectrum in the flattened triangle configuration, $B(k/2,k/2,k)$, (g) bispectrum in the squeezed triangle configuration, $B(k,k,k_3)$, for $k_3\ll 1$, and (h) pixel value PDF on a logarithmic scale.

Figure 6

Left: comparison of the bispectrum of $\log (\rho )$ in the equilateral configuration computed from 300 Quijote simulations (orange) and 300 syntheses (dashed blue). The error bars correspond to the realization-per-realization dispersion. Right: plot of the absolute value of the same quantities in logarithmic scale. Orange dashed vertical lines show the change of sign of the bispectrum of the Quijote simulation and blue dotted lines the change of sign of the bispectrum of the syntheses. Please notice the change of the vertical order of the lines due to the absolute value.

Figure 7

Further comparisons of $\log (\rho )$ in 300 Quijote maps and our 300 syntheses, showing how well the syntheses reproduce the statistical properties of the Quijote maps. (a) Correlation matrix of the power spectrum of $\log (\rho )$ in Quijote maps (left) and syntheses (right). Notice that this matrix is mainly diagonal because we consider $\log (\rho )$. The correlation matrix of the power spectrum of $\rho$ is highly nondiagonal, as also seen on Fig (3) of [23]. (b) First, second, and third Minkowski functionals of Quijote maps (thick orange lines) and syntheses (dashed blue lines). The error bars correspond to the realization-per-realization dispersion. The error bars have been inflated by a factor of 5 in these three plots in order to be visible. (c) Histograms of the distribution of the power spectrum, at the frequencies $k=0.1$, 0.17, and 0.32 h/Mpc for Quijote maps (orange) and syntheses (blue).

Figure 8

Left: improvement in the marginalized errors on cosmological parameters obtained with Fisher analysis using models I to V of WPH statistics, applied to (a) the projected matter density field $\rho$ and (b) its logarithm. In both cases, the results are normalized by those obtained with power spectrum statistics of the projected matter density field. Right: syntheses of $\log (\rho )$ for models I to IV [see Fig. 5 for the corresponding model V synthesis]. Both the Fisher analysis and the syntheses improve as the WPH statistics expand from model I to model V.

Figure 9

Subset of WPH moments, estimated on the logarithm of the matter density fields from Quijote simulations at fiducial cosmology (solid orange line, 15000 maps), and from model I (top) or model V (bottom) syntheses (blue dashed line, 300 maps). The model I syntheses are a good approximation of a Gaussian field (see Sec. 5a). The model V corresponds to the WPH statistics used in Sec. 4, which accordingly reproduce those of the Quijote simulation. The error bars, which characterize the variability of WPH moments from one map to another, have been multiplied by a factor of 3 for readability reasons. These moments are a subset of those used to estimate cosmological Fisher information in Sec. 3. For $\mathcal{S}$-type moments, all $j_1=j_2$ values are given. $\mathcal{C}$-type moments are drawn for $j_1=1$ and $j_1=2$ only and for $j_1<j_2\le 7$. All plotted coefficients satisfy $\delta \ell =0$ and $n=0$, see Appendix pp2-s2 for details.

Figure 10

Correlation matrix of a subset of WPH moments, estimated on the logarithm of the matter density field from Quijote simulations at fiducial cosmology (15000 maps). The WPH moments are those described in Fig. 9, with the same order, and the reader is referred to this figure for more details.

Figure 11

Probability density function of statistical syntheses of $\log (\rho )$ as performed with models I to V, along with the PDF of the initial Quijote field. Each line is estimated from 300 maps, as in Fig. 5.

Figure 12

Improvement of the power spectrum of model I syntheses of $\log (\rho )$ as the number ${\mathrm{\Delta }}_n$ of spatial translations in each direction is increased from 0 to 2. The imprint of the wavelet spectral bands are clearly visible at ${\mathrm{\Delta }}_n=0$ (i.e., no spatial translation) and are greatly reduced at ${\mathrm{\Delta }}_n=2$. The error bars correspond to the realization-per-realization dispersion.

Figure 13

Coupling terms depicted in Fourier space for (from left to right) the ${\mathcal{C}}^{\text{phase}}$, ${\mathcal{C}}^{(0,0)}$, and ${\mathcal{C}}^{(0,1)}$ types of coupling. In each case the frequency support of the filtered fields (filled circles) are modified by the phase harmonic operator to share common frequencies (dotted circles). Note that this figure is for explanatory purposes only, since in practice the spectral support of a filtered field and its modulus overlap.

Figure 14

Improvements of the marginalized errors on cosmological parameters obtained with Fisher analysis for model V of WPH statistics with increasing ${\mathrm{\Delta }}_j$ values, for the LSS matter density field (left) and its logarithm (right). Errors are normalized by those obtained with ${\mathrm{\Delta }}_j=0$. When limited to ${\mathrm{\Delta }}_j=0$, model V characterizes only local couplings and corresponds to model III.

Figure 15

Squeezed-triangle bispectrum of $\log (\rho )$ of the Quijote simulation (orange) and its syntheses from model V of WPH statistics with $\mathrm{\Delta }j=2$ (green) and $\mathrm{\Delta }j=5$ (blue). The $\mathrm{\Delta }j=2$ synthesis does not characterize the couplings between distant scales of the LSS that are needed to reproduce this bispectrum. The error bars correspond to the realization-per-realization dispersion.

Figure 16

Left: histogram of the Quijote density field $\rho$ and of the syntheses of $\rho$. The syntheses are not strictly positive, contrary to the initial maps. Middle: example of a syntheses of $\rho$ shown in logarithmic scales. The white pixels correspond to negative value of the field. Right: example of a Quijote simulation $\rho$.