non-Gaussian Information Research Articles

A complex-valued slow independent component analysis based incipient fault detection and diagnosis method with applications to wastewater treatment processes

Multivariate statistical process monitoring are the essential approaches to achieve better prognostics and health management (PHM) of process industries. However, incipient faults and complex behaviors (such as nonlinearity and dynamics) always render the traditional multivariate statistical process monitoring approaches inadequate. Thus, a complex-valued slow independent component analysis (CSICA) is proposed, which is able to extract optimized features from a complex-valued matrix containing both of raw data and their changing rates by resorting to a complex-valued independent component analysis operation and a batch of phase shifts. These features, named slow independent components (SICs), not only guarantee the statistical independence but also capture slowly-changing patterns, thus refining both dynamic and non-Gaussian information mostly related with incipient faults. The proposed algorithm together with novel statistics, Is2, If2 and SPE, as well as their control limits can sequentially detect incipient faults effectively. Then, together with the novel differential mapping reconstructed contribution plot (DM-RCP) and Granger causality analysis, the proposed method can accurately locate rooting causes of incipient faults. Finally, the proposed framework of process monitoring is validated through two data sets from a simulation platform and an oxidation-ditch-based wastewater treatment plant, respectively. The results demonstrate that the proposed method can achieve more accurate and efficient performances than conventional methods.

Skewing the CMB×LSS: a fast method for bispectrum analysis

Upcoming cosmic microwave background (CMB) lensing measurements and tomographic galaxy surveys are expected to provide us with high-precision data sets in the coming years, thus paving the way for fruitful cross-correlation analyses. In this paper we study the information content of the weighted skew-spectrum, a nearly-optimal estimator of the angular bispectrum amplitude, as a means to extract non-Gaussian information on both bias and cosmological parameters from the bispectra of galaxies cross-correlated with CMB lensing, while gaining significantly on speed.Our results show that for the combination of the Planck satellite and the Dark Energy Spectroscopic Instrument (DESI), the difference in the constraints on bias and cosmological parameters from the skew-spectrum and the bispectrum is at most 17%.We further compare and find agreement between our theoretical skew-spectra and those estimated from N-body simulations, for which it is important to include gravitational non-linearities beyond perturbation theory and the post-Born effect for CMB lensing. We define an algorithm to apply the skew-spectrum estimator to the data and, as a preliminary step, we use the skew-spectra to constrain bias parameters and the amplitude of shot noise from the simulations through a Markov chain Monte Carlo likelihood analysis, finding that it may be possible to reach percent-level estimates for the linear bias parameter b 1.

Towards an optimal estimation of cosmological parameters with the wavelet scattering transform

Optimal extraction of the non-Gaussian information encoded in the Large-Scale Structure (LSS) of the universe lies at the forefront of modern precision cosmology. We propose achieving this task through the use of the Wavelet Scattering Transform (WST), which subjects an input field to a layer of non-linear transformations that are sensitive to non-Gaussianity in spatial density distributions through a generated set of WST coefficients. In order to assess its applicability in the context of LSS surveys, we apply the WST on the 3D overdensity field obtained by the Quijote simulations, out of which we extract the Fisher information in 6 cosmological parameters. It is subsequently found to deliver a large improvement in the marginalized errors on all parameters, ranging between $1.2-4\times$ tighter than the corresponding ones obtained from the regular 3D cold dark matter + baryon power spectrum, as well as a $50 \%$ improvement over the neutrino mass constraint given by the marked power spectrum. Through this first application on 3D cosmological fields, we demonstrate the great promise held by this novel statistic and set the stage for its future application to actual galaxy observations.

Exploring the cosmic 21-cm signal from the epoch of reionization using the wavelet scattering transform

ABSTRACT Detecting the cosmic 21-cm signal during the Epoch of Reionization and Cosmic Dawn will reveal insights into the properties of the first galaxies and advance cosmological parameter estimation. Until recently, the primary focus for astrophysical parameter inference from the 21-cm signal centred on the power spectrum (PS). However, the cosmic 21-cm signal is highly non-Gaussian rendering the PS suboptimal for characterizing the cosmic signal. In this work, we introduce a new technique to analyse the non-Gaussian information in images of the 21-cm signal called the Wavelet Scattering Transform (WST). This approach closely mirrors that of convolutional neural networks with the added advantage of not requiring tuning or training of a neural network. Instead, it compresses the 2D spatial information into a set of coefficients making it easier to interpret while also providing a robust statistical description of the non-Gaussian information contained in the cosmic 21-cm signal. First, we explore the application of the WST to mock 21-cm images to gain valuable physical insights by comparing to the known behaviour from the 21-cm PS. Then we quantitatively explore the WST applied to the 21-cm signal by extracting astrophysical parameter constraints using Fisher Matrices from a realistic 1000 h mock observation with the Square Kilometre Array. We find that: (i) the WST applied only to 2D images can outperform the 3D spherically averaged 21-cm PS, (ii) the excision of foreground contaminated modes can degrade the constraining power by a factor of ∼1.5–2 with the WST and (iii) higher cadences between the 21-cm images can further improve the constraining power.

Fault diagnosis of a planetary gearbox based on a local bi-spectrum and a convolutional neural network

The transmission paths of vibration signals in the planetary gearboxes are complex. The signals have the characteristics of strong background noise, instability and non-Gaussian. Bi-spectrums can suppress Gaussian colored noise and are suitable for vibration signal processing of planetary gearboxes. In the traditional fault diagnosis methods based on bi-spectrums, the amplitudes of fault characteristic frequency, or the other further quantitative calculations values, are generally used as the basis of fault diagnosis processes. It has been found that bi-spectrum images can directly characterize the faults of the planetary gearboxes. Convolutional neural networks (CNNs) have been used in mechanical fault diagnoses in recent years. One-dimensional original signals are converted into two-dimensional images as CNN input, which is an effective method for mechanical fault diagnoses. At the present time, there has not been any relevant research conducted using bi-spectral images as CNN input. In this study, a fault diagnosis method based on local bi-spectrum and CNN was proposed. A bi-spectral analysis of the vibration signals of the planetary gearbox was first carried out in order to reveal the fault information while retaining the non-Gaussian information. Then, according to the bi-spectrum symmetry, local images containing the main information were taken as the input of the CNN, which reduced the redundancy of the fault information. Then, in order to improve the diagnostic accuracy of the CNN, the key parameters of CNN architecture were optimized. Finally, a CNN diagnosis model was built to realize the classification diagnoses of different fault positions and different fault degrees of planetary gearboxes. This study’s comparison of the diagnosis results of the full bi-spectrum + CNN, original vibration signal + CNN, local bi-spectrum + (support vector machines), and local bi-spectrum + (stacked auto-encoder) showed that the proposed method in this study had achieved both accuracy and rapidity in the fault diagnoses of planetary gearboxes.

Dark energy survey year 3 results: Cosmology with peaks using an emulator approach

ABSTRACT We constrain the matter density Ωm and the amplitude of density fluctuations σ8 within the ΛCDM cosmological model with shear peak statistics and angular convergence power spectra using mass maps constructed from the first three years of data of the Dark Energy Survey (DES Y3). We use tomographic shear peak statistics, including cross-peaks: peak counts calculated on maps created by taking a harmonic space product of the convergence of two tomographic redshift bins. Our analysis follows a forward-modelling scheme to create a likelihood of these statistics using N-body simulations, using a Gaussian process emulator. We take into account the uncertainty from the remaining, largely unconstrained ΛCDM parameters (Ωb, ns, and h). We include the following lensing systematics: multiplicative shear bias, photometric redshift uncertainty, and galaxy intrinsic alignment. Stringent scale cuts are applied to avoid biases from unmodelled baryonic physics. We find that the additional non-Gaussian information leads to a tightening of the constraints on the structure growth parameter yielding $S_8~\equiv ~\sigma _8\sqrt{\Omega _{\mathrm{m}}/0.3}~=~0.797_{-0.013}^{+0.015}$ (68 per cent confidence limits), with a precision of 1.8 per cent, an improvement of 38 per cent compared to the angular power spectra only case. The results obtained with the angular power spectra and peak counts are found to be in agreement with each other and no significant difference in S8 is recorded. We find a mild tension of $1.5 \, \sigma$ between our study and the results from Planck 2018, with our analysis yielding a lower S8. Furthermore, we observe that the combination of angular power spectra and tomographic peak counts breaks the degeneracy between galaxy intrinsic alignment AIA and S8, improving cosmological constraints. We run a suite of tests concluding that our results are robust and consistent with the results from other studies using DES Y3 data.

One-Point Statistics Matter in Extended Cosmologies

The late universe contains a wealth of information about fundamental physics and gravity, wrapped up in non-Gaussian fields. To make use of as much information as possible, it is necessary to go beyond two-point statistics. Rather than going to higher-order N-point correlation functions, we demonstrate that the probability distribution function (PDF) of spheres in the matter field (a one-point function) already contains a significant amount of this non-Gaussian information. The matter PDF dissects different density environments which are lumped together in two-point statistics, making it particularly useful for probing modifications of gravity or expansion history. Our approach in Cataneo et al. 2021 extends the success of Large Deviation Theory for predicting the matter PDF in ΛCDM in these “extended” cosmologies. A Fisher forecast demonstrates the information content in the matter PDF via constraints for a Euclid-like survey volume combining the 3D matter PDF with the 3D matter power spectrum. Adding the matter PDF halves the uncertainties on parameters in an evolving dark energy model, relative to the power spectrum alone. Additionally, the matter PDF contains enough non-linear information to substantially increase the detection significance of departures from General Relativity, with improvements up to six times the power spectrum alone. This analysis demonstrates that the matter PDF is a promising non-Gaussian statistic for extracting cosmological information, particularly for beyond ΛCDM models.

Detecting the non-Gaussianity of the 21-cm signal during reionization with the wavelet scattering transform

ABSTRACT Detecting the 21-cm hyperfine transition from neutral hydrogen in the intergalactic medium is our best probe for understanding the astrophysical processes driving the Epoch of Reionization (EoR). The primary means for a detection of this 21-cm signal is through a statistical measurement of the spatial fluctuations using the 21-cm power spectrum (PS). However, the 21-cm signal is non-Gaussian meaning the PS, which only measures the Gaussian fluctuations, is suboptimal for characterizing all of the available information. The upcoming Square Kilometre Array (SKA) will perform a deep, 1000 h observation over 100 deg2 specifically designed to recover direct images of the 21-cm signal. In this work, we use the Wavelet Scattering Transform (WST) to extract the non-Gaussian information directly from these 2D images of the 21-cm signal. The key advantage of the WST is its stability with respect to statistical noise for measuring non-Gaussian information, unlike the bispectrum whose statistical noise diverges. In this work, we specifically focus on introducing a novel method to isolate non-Gaussian information from an image and apply this methodology to individual mock 21-cm images to quantify the strength of the non-Gaussian information contained within a single image. For example, at 150 (177) MHz (z ∼ 8.5 and ∼7) for a fiducial reionization model we recover a signal to noise of ∼5 (8) for the non-Gaussian information assuming perfect foreground removal and ∼2 (3) assuming foreground wedge avoidance.

MCMC generation of cosmological fields far beyond Gaussianity

Structure formation in our Universe creates non-Gaussian random fields that will soon be observed over almost the entire sky by the Euclid satellite, the Vera-Rubin observatory, and the Square Kilometre Array. An unsolved problem is how to analyze best such non-Gaussian fields, e.g. to infer the physical laws that created them. This problem could be solved if a parametric non-Gaussian sampling distribution for such fields were known, as this distribution could serve as likelihood during inference. We therefore create a sampling distribution for non-Gaussian random fields. Our approach is capable of handling strong non-Gaussianity, while perturbative approaches such as the Edgeworth expansion cannot. To imitate cosmological structure formation, we enforce our fields to be (i) statistically isotropic, (ii) statistically homogeneous, and (iii) statistically independent at large distances. We generate such fields via a Monte Carlo Markov Chain technique and find that even strong non-Gaussianity is not necessarily visible to the human eye. We also find that sampled marginals for pixel pairs have an almost generic Gauss-like appearance, even if the joint distribution of all pixels is markedly non-Gaussian. This apparent Gaussianity is a consequence of the high dimensionality of random fields. We conclude that vast amounts of non-Gaussian information can be hidden in random fields that appear nearly Gaussian in simple tests, and that it would be short-sighted not to try and extract it.

Self-similarity of k-nearest neighbour distributions in scale-free simulations

ABSTRACT We use the k-nearest neighbour probability distribution function (kNN-PDF; Banerjee & Abel 2021a) to assess convergence in a scale-free N-body simulation. Compared to our previous two-point analysis, the kNN-PDF allows us to quantify our results in the language of haloes and numbers of particles, while also incorporating non-Gaussian information. We find good convergence for 32 particles and greater at densities typical of haloes, while 16 particles and fewer appear unconverged. Halving the softening length extends convergence to higher densities, but not to fewer particles. Our analysis is less sensitive to voids, but we analyse a limited range of underdensities and find evidence for convergence at 16 particles and greater even in sparse voids.

Dynamic Non-Gaussian hybrid serial modeling for industrial process monitoring

Process monitoring has been widely used for fault detection and performance supervision in modern industrial processes. Nevertheless, hybrid characteristics including Gaussianity, non-Gaussianity and dynamic usually coexist in process variables, which brings new challenge to obtain satisfactory monitoring performance. Aiming at the hybrid characteristics problem, this paper proposes a dynamic non-Gaussian hybrid serial modeling method for industrial process monitoring. First, a multivariate non-Gaussianity evaluation method is utilized to divide industrial process variables into the Gaussian variable subspace and the non-Gaussian variable subspace. Afterwards considering the hybrid characteristics including Gaussianity, non-Gaussianity and dynamic at information level, a dynamic principal component analysis (DPCA)-dynamic independent component analysis (DICA)-based hybrid serial modeling method is presented for analyzing simultaneously the dynamic Gaussian and non-Gaussian information in each variable subspace. Subsequently, the final monitoring results are obtained using Bayesian inference and the DPCA-DICA-based hybrid serial similarity factor is proposed for fault identification. Unlike the existing methods, the proposed method analyzes simultaneously the Gaussianity, non-Gaussianity and dynamic at different levels of variable and information for improving performance. The case studies including a numerical system, the Tennessee Eastman process and a practical industrial process demonstrate its feasibility and effectiveness.

Cosmic shear cosmology beyond two-point statistics: a combined peak count and correlation function analysis of DES-Y1

ABSTRACT We constrain cosmological parameters from a joint cosmic shear analysis of peak-counts and the two-point shear correlation functions, as measured from the Dark Energy Survey (DES-Y1). We find the structure growth parameter $S_8\equiv \sigma _8\sqrt{\Omega _{\rm m}/0.3} = 0.766^{+0.033}_{-0.038}$ which, at 4.8 per cent precision, provides one of the tightest constraints on S8 from the DES-Y1 weak lensing data. In our simulation-based method we determine the expected DES-Y1 peak-count signal for a range of cosmologies sampled in four w cold dark matter parameters (Ωm, σ8, h, w0). We also determine the joint covariance matrix with over 1000 realizations at our fiducial cosmology. With mock DES-Y1 data we calibrate the impact of photometric redshift and shear calibration uncertainty on the peak-count, marginalizing over these uncertainties in our cosmological analysis. Using dedicated training samples we show that our measurements are unaffected by mass resolution limits in the simulation, and that our constraints are robust against uncertainty in the effect of baryon feedback. Accurate modelling for the impact of intrinsic alignments on the tomographic peak-count remains a challenge, currently limiting our exploitation of cross-correlated peak counts between high and low redshift bins. We demonstrate that once calibrated, a fully tomographic joint peak-count and correlation functions analysis has the potential to reach a 3 per cent precision on S8 for DES-Y1. Our methodology can be adopted to model any statistic that is sensitive to the non-Gaussian information encoded in the shear field. In order to accelerate the development of these beyond-two-point cosmic shear studies, our simulations are made available to the community upon request.

Fault detection based on ICA-GLR for non-Gaussian industrial processes

ABSTRACT As the growth of Industry 4.0, online fault detection plays a crucial role in ensuring the manufacturing quality. Generally, the fault detection methods can be classified into model-based and data-driven methods. There are advantages/disadvantages between two methods. In this study, we integrated both methods in order to develop an efficient fault detection method for non-Gaussian industrial processes. The data-driven method, independent component analysis (ICA) is used to extract non-Gaussian information and dimensionality reduction. Meanwhile, the model-based method, generalized likelihood ratio (GLR) test is adopted as the charting statistic. The proposed ICA-GLR method has advantages of 1) detecting a wide range of process changes, 2) estimating the change points and 3) needless prior parameters to be specified by practitioner. The efficiency of the proposed ICA-GLR fault detection method will be verified via implementing one simulated non-Gaussian process and two real manufacturing processes: Tennessee Eastman process and semiconductor manufacturing process. Results demonstrate that the proposed ICA-GLR method has superior fault detectability when compared to traditional methods, such as principal component analysis and ICA.

Noise reduction for weak lensing mass mapping: an application of generative adversarial networks to Subaru Hyper Suprime-Cam first-year data

ABSTRACT We propose a deep-learning approach based on generative adversarial networks (GANs) to reduce noise in weak lensing mass maps under realistic conditions. We apply image-to-image translation using conditional GANs to the mass map obtained from the first-year data of Subaru Hyper Suprime-Cam (HSC) Survey. We train the conditional GANs by using 25 000 mock HSC catalogues that directly incorporate a variety of observational effects. We study the non-Gaussian information in denoised maps using one-point probability distribution functions (PDFs) and also perform matching analysis for positive peaks and massive clusters. An ensemble learning technique with our GANs is successfully applied to reproduce the PDFs of the lensing convergence. About $60{{\ \rm per\ cent}}$ of the peaks in the denoised maps with height greater than 5σ have counterparts of massive clusters within a separation of 6 arcmin. We show that PDFs in the denoised maps are not compromised by details of multiplicative biases and photometric redshift distributions, nor by shape measurement errors, and that the PDFs show stronger cosmological dependence compared to the noisy counterpart. We apply our denoising method to a part of the first-year HSC data to show that the observed mass distribution is statistically consistent with the prediction from the standard ΛCDM model.

Impact of baryons in cosmic shear analyses with tomographic aperture mass statistics

NonGaussian cosmic shear statistics based on weak-lensing aperture mass (Map) maps can outperform the classical shear two-point correlation function (γ-2PCF) in terms of cosmological constraining power. However, reaching the full potential of these new estimators requires accurate modeling of the physics of baryons as the extra nonGaussian information mostly resides at small scales. We present one such modeling based on the Magneticum hydrodynamical simulation for the KiDS-450 and DES-Y1 surveys and a Euclid-like survey. We compute the bias due to baryons on the lensing PDF and the distribution of peaks and voids in Map maps and propagate it to the cosmological forecasts on the structure growth parameter S8, the matter density parameter Ωm, and the dark energy equation of state w0 using the SLICS and cosmo-SLICS sets of dark-matter-only simulations. We report a negative bias of a few percent on S8 and Ωm and also measure a positive bias of the same level on w0 when including a tomographic decomposition. These biases reach ∼5% when combining Map statistics with the γ-2PCF as these estimators show similar dependency on the AGN feedback. We verify that these biases constitute a less than 1σ shift on the probed cosmological parameters for current cosmic shear surveys. However, baryons need to be accounted for at the percentage level for future Stage IV surveys and we propose to include the uncertainty on the AGN feedback amplitude by marginalizing over this parameter using multiple simulations such as those presented in this paper. Finally, we explore the possibility of mitigating the impact of baryons by filtering the Map map but find that this process would require suppressing the small-scale information to a point where the constraints would no longer be competitive.

Over-complete deep recurrent neutral network based on wastewater treatment process soft sensor application

The wastewater treatment process (WWTP) is a complex biochemical reaction process in which sensor data has strong nonlinear, non-Gaussian and time correlation characteristics. The traditional methods ignore to consider the aforementioned three characteristics simultaneously, which may have insufficient feature extraction of WWTP. In this work, an Over-Complete Deep Recurrent Neural Network (ODRNN) method is proposed to solve the above issues. The ODRNN combines the over-complete independent component analysis (OICA) and binary particle swarm optimization (BPSO) to efficiently extract the non-Gaussian information, and then the extracted information is fed into DRNN to obtain the time correlation characteristics. In this way, the method can not only capture the non-linear and non-Gaussian information but also extract temporal correlation of WWTP data. Simulation results on BSM1 showed that the ODRNN based soft sensor method has higher accuracy and robustness than other state-of-the-art methods.

Cosmological forecast for non-Gaussian statistics in large-scale weak lensing surveys

Cosmic shear data contains a large amount of cosmological information encapsulated in the non-Gaussian features of the weak lensing mass maps. Weak lensing studies mostly rely on two-point statistics to constrain cosmology from cosmic shear data, that do not capture all of this information. Additional non-Gaussian information can be extracted using non-Gaussian statistics. We compare the constraining power in the Ωm–σ8 plane of three map-based non-Gaussian statistics with the angular power spectrum, namely; peak counts, minimum counts and Minkowski functionals. We further analyze the impact of tomography and systematic effects originating from galaxy intrinsic alignments, multiplicative shear bias and photometric redshift systematics. We forecast the performance of the statistics for a stage-3-like weak lensing survey, spanning an area of 5000 deg2 and restrict ourselves to scales ⩾ 10 arcmin to avoid baryonic effects. The study follows a forward modelling scheme to predict the statistics at different cosmologies based on N-Body simulations. We find, that in our setup, the considered non-Gaussian statistics provide tighter constraints than the angular power spectrum. The peak counts show the greatest potential, increasing the figure-of-Merit (FoM) in the Ωm–σ8 plane by a factor of about 6, while the minimum counts and the Minkowski functionals yield an increase by a factor of about 3 and 2, respectively. A combined analysis using all non-Gaussian statistics in addition to the power spectrum increases the FoM by a factor of 9 and reduces the error on S8 by ≈ 30%. We find that the importance of tomography is diminished when combining non-Gaussian statistics with the angular power spectrum. The non-Gaussian statistics indeed profit less from tomography and the minimum counts and Minkowski functionals add some robustness against galaxy intrinsic alignment in a non-tomographic setting. We further find that a combination of the angular power spectrum and the non-Gaussian statistics allows us to apply conservative scale cuts in the analysis, thus helping to minimize the impact of baryonic and relativistic effects, while conserving the cosmological constraining power. We make the code that was used to conduct this analysis publicly available to simplify performing such analyses in the future.

Process monitoring of batch process based on overcomplete broad learning network

Process monitoring is a common strategy for monitoring the industrial production operation state. It can detect abnormal conditions in industrial processes and provide certain guidance for production. Many classical data-driven process monitoring approaches neglect the non-Gaussian and nonlinearity of the data. For solving the above problems, this paper designs an overcomplete broad learning system (OBLS) with incremental learning ability. The method combines multiple fault data into one data matrix. Then the overcomplete approach is employed to capture the non-Gaussian information from the original data to obtain a mixed matrix. Next, the weights of the OBLS network are trained according to the extracted feature matrix containing non-Gaussian information and its corresponding fault label. Meanwhile, the nonlinearity of the data is addressed by the OBLS network. Finally, the incremental learning capabilities of the OBLS network enable it to be updated quickly when new fault samples are added to the training set without entire retraining process. The experimental results in numerical examples, penicillin fermentation simulation platform and real-world industrial process demonstrate the superiority and feasibility of the OBLS model.

Interpreting deep learning models for weak lensing

Deep Neural Networks (DNNs) are powerful algorithms that have been proven capable of extracting non-Gaussian information from weak lensing (WL) data sets. Understanding which features in the data determine the output of these nested, non-linear algorithms is an important but challenging task. We analyze a DNN that has been found in previous work to accurately recover cosmological parameters in simulated maps of the WL convergence ($\kappa$). We derive constraints on the cosmological parameter pair $(\Omega_m,\sigma_8)$ from a combination of three commonly used WL statistics (power spectrum, lensing peaks, and Minkowski functionals), using ray-traced simulated $\kappa$ maps. We show that the network can improve the inferred parameter constraints relative to this combination by $20\%$ even in the presence of realistic levels of shape noise. We apply a series of well established saliency methods to interpret the DNN and find that the most relevant pixels are those with extreme $\kappa$ values. For noiseless maps, regions with negative $\kappa$ account for $86-69\%$ of the attribution of the DNN output, defined as the square of the saliency in input space. In the presence of shape nose, the attribution concentrates in high convergence regions, with $36-68\%$ of the attribution in regions with $\kappa > 3 \sigma_{\kappa}$.

A new approach to observational cosmology using the scattering transform

ABSTRACT Parameter estimation with non-Gaussian stochastic fields is a common challenge in astrophysics and cosmology. In this paper, we advocate performing this task using the scattering transform, a statistical tool sharing ideas with convolutional neural networks (CNNs) but requiring neither training nor tuning. It generates a compact set of coefficients, which can be used as robust summary statistics for non-Gaussian information. It is especially suited for fields presenting localized structures and hierarchical clustering, such as the cosmological density field. To demonstrate its power, we apply this estimator to a cosmological parameter inference problem in the context of weak lensing. On simulated convergence maps with realistic noise, the scattering transform outperforms classic estimators and is on a par with the state-of-the-art CNN. It retains advantages of traditional statistical descriptors, has provable stability properties, allows to check for systematics, and importantly, the scattering coefficients are interpretable. It is a powerful and attractive estimator for observational cosmology and the study of physical fields in general.