non-Gaussian Information Research Articles

Reionization Parameter Inference from 3D Minkowski Functionals of the 21 cm Signals

The Minkowski functionals (MFs), a set of topological summary statistics, have emerged as a powerful tool for extracting non-Gaussian information. We investigate the prospect of constraining the reionization parameters using the MFs of the 21 cm brightness temperature field from the epoch of reionization (EOR). Realistic effects, including thermal noise, synthesized beam, and foreground avoidance, are applied to the mock observations from radio interferometric array experiments such as the Hydrogen Epoch of Reionization Array (HERA) and the Square Kilometre Array (SKA). We demonstrate that the MFs of the 21 cm signal measured with SKA-Low can be used to distinguish different reionization models, whereas the MF measurement with a HERA-like array cannot be made accurately enough. We further forecast the accuracies with which the MF measurements can place constraints on reionization parameters, using the standard Markov Chain Monte Carlo analysis for parameter inference based on forward modeling. We find that for SKA-Low observation, MFs provide unbiased estimations of the reionization parameters with accuracies comparable to the power spectrum (PS) analysis. Furthermore, joint constraints using both MFs and PS can improve the constraint accuracies by up to 30% compared to those with the PS alone. Nevertheless, the constraint accuracies can be degraded if the EOR window is shrunk with strong foreground avoidance. Our analysis demonstrates the promise of MFs as a set of summary statistics that extract complementary information from the 21 cm EOR field to the two-point statistics, which suggests a strong motivation for incorporating the MFs into the data analysis of future 21 cm observations.

Unleashing cosmic shear information with the tomographic weak lensing PDF

In this work, we demonstrate the constraining power of the tomographic weak lensing convergence PDF for StageIV-like source galaxy redshift bins and shape noise. We focus on scales of 10 to 20 arcmin in the mildly nonlinear regime, where the convergence PDF and its changes with cosmological parameters can be predicted theoretically. We model the impact of reconstructing the convergence from the shear field using the well-known Kaiser-Squires formalism. We cross-validate the predicted and the measured convergence PDF derived from convergence maps reconstructed using simulated shear catalogues. Employing a Fisher forecast, we determine the constraining power for (Ωm,S8,w0). We find that adding a 5-bin tomography improves the κ−PDF constraints by a factor of {3.8,1.3,1.6} for (Ωm,S8,w0) respectively. Additionally, we perform a joint analysis with the shear two-point correlation functions, finding an enhancement of around a factor of 1.5 on all parameters with respect to the two-point statistics alone. These improved constraints come from disentangling from w0 by extracting non-Gaussian information, in particular, including the PDF skewness at different redshift bins. We also study the effect of varying the number of parameters to forecast, in particular we add h, finding that the convergence PDF maintains its constraining power while the precision from two-point correlations degrades by a factor of {1.7,1.4,1.8} for , respectively.

SUNBIRD: a simulation-based model for full-shape density-split clustering

ABSTRACT Combining galaxy clustering information from regions of different environmental densities can help break cosmological parameter degeneracies and access non-Gaussian information from the density field that is not readily captured by the standard two-point correlation function (2PCF) analyses. However, modelling these density-dependent statistics down to the non-linear regime has so far remained challenging. We present a simulation-based model that is able to capture the cosmological dependence of the full shape of the density-split clustering (DSC) statistics down to intra-halo scales. Our models are based on neural-network emulators that are trained on high-fidelity mock galaxy catalogues within an extended-ΛCDM framework, incorporating the effects of redshift-space, Alcock–Paczynski distortions, and models of the halo–galaxy connection. Our models reach sub-per cent level accuracy down to $1 \, h^{-1}\text{Mpc}$ and are robust against different choices of galaxy–halo connection modelling. When combined with the galaxy 2PCF, DSC can tighten the constraints on ωcdm, σ8, and ns by factors of 2.9, 1.9, and 2.1, respectively, compared to a 2PCF-only analysis. DSC additionally puts strong constraints on environment-based assembly bias parameters.

Forecast Cosmological Constraints with the 1D Wavelet Scattering Transform and the Lyman-α Forest.

We make forecasts for the constraining power of the 1D wavelet scattering transform when used with a Lyman-α forest cosmology survey. Using mock simulations and a Fisher matrix, we show that there is considerable cosmological information in the scattering transform coefficients not captured by the flux power spectrum. We estimate mock covariance matrices assuming uncorrelated Gaussian pixel noise for each quasar at a level drawn from a simple log-normal model. The extra information comes from a smaller estimated covariance in the first-order wavelet power and from second-order wavelet coefficients that probe non-Gaussian information in the forest. Forecast constraints on cosmological parameters from the wavelet scattering transform are more than an order of magnitude tighter than for the power spectrum, shrinking a 4D parameter space by a factor of 10^{6}. Should these improvements be realized with the Dark Energy Spectroscopic Instrument, inflationary running would be constrained to test common inflationary models predicting α_{s}=-6×10^{-4} and neutrino mass constraints would be improved enough for a 5-σ detection of the minimal neutrino mass.

Cosmological constraints from the redshift-space galaxy skew spectra

Extracting the non-Gaussian information of the cosmic large-scale structure (LSS) is vital in unlocking the full potential of the rich datasets from the upcoming stage-IV galaxy surveys. Galaxy skew spectra serve as efficient beyond-two-point statistics, encapsulating essential bispectrum information with computational efficiency akin to power spectrum analysis. This paper presents the first cosmological constraints from analyzing the full set of redshift-space galaxy skew spectra of the data from the SDSS-III BOSS, accessing cosmological information down to nonlinear scales. Employing the forward modeling framework and simulation-based inference via normalizing flows, we analyze the CMASS-SGC subsample, which constitute approximately 10% of the full BOSS data. Analyzing the scales up to kmax=0.5 h−1 Mpc, we find that the skew spectra improve the constraints on Ωm,Ωb,h, and ns by 34%, 35%, 18%, 10%, respectively, compared to constraints from previous power spectrum multipoles analysis, yielding Ωm=0.288−0.034+0.024, Ωb=0.043−0.007+0.005, h=0.759−0.050+0.104, ns=0.918−0.090+0.041 (at 68% confidence limit). On the other hand, the constraints on σ8 are weaker than from the power spectrum. Including the big bang nucleosynthesis (BBN) prior on baryon density reduces the uncertainty on the Hubble parameter further, achieving h=0.750−0.032+0.034, which is a 38% improvement over the constraint from the power spectrum with the same prior. Compared to the bispectrum (monopole) analysis, skew spectra offer comparable constraints on larger scales (kmax<0.3 h−1 Mpc) for most parameters except for σ8. Published by the American Physical Society 2024

Analysis of BOSS galaxy data with weighted skew-spectra

We present the first application of the weighted skew-spectra to analyze non-Gaussian information in galaxy survey data. Using the tree-level galaxy skew-spectra together with the one-loop power spectrum multipoles, we analyze the Sloan Digital Sky Survey (SDSS)-III Baryon Oscillation Spectroscopic Survey (BOSS) galaxy clustering data, and target our search towards the equilateral bispectrum shape of primordial non-Gaussianity. We use the Effective Field Theory model for the galaxy power spectrum and bispectrum, and account for systematic effects, such as the survey geometry. From our likelihood analysis, we find f NL equil = -34+296 -334 at 68% CL, consistent with previous works, while systematic errors from our treatment of the survey geometry lead to an unreliable estimation of f NL ortho. We further constrain the bias and counterterm parameters, while keeping the cosmology fixed to Planck 2018 values. As a check, we also validate our analysis pipeline using the Nseries simulation suite.

Cosmology from weak lensing peaks and minima with Subaru Hyper Suprime-Cam Survey first-year data

ABSTRACT We present cosmological constraints derived from peak counts, minimum counts, and the angular power spectrum of the Subaru Hyper Suprime-Cam first-year (HSC Y1) weak lensing shear catalogue. Weak lensing peak and minimum counts contain non-Gaussian information and hence are complementary to the conventional two-point statistics in constraining cosmology. In this work, we forward-model the three summary statistics and their dependence on cosmology, using a suite of N-body simulations tailored to the HSC Y1 data. We investigate systematic and astrophysical effects including intrinsic alignments, baryon feedback, multiplicative bias, and photometric redshift uncertainties. We mitigate the impact of these systematics by applying cuts on angular scales, smoothing scales, signal-to-noise ratio bins, and tomographic redshift bins. By combining peaks, minima, and the power spectrum, assuming a flat-ΛCDM model, we obtain $S_{8} \equiv \sigma _8\sqrt{\Omega _m/0.3}= 0.810^{+0.022}_{-0.026}$, a 35 per cent tighter constraint than that obtained from the angular power spectrum alone. Our results are in agreement with other studies using HSC weak lensing shear data, as well as with Planck 2018 cosmology and recent CMB lensing constraints from the Atacama Cosmology Telescope and the South Pole Telescope.

Mechanical seal friction condition monitoring based on bispectral characteristics

PurposeThe purpose of this paper is to investigate the monitoring of the friction condition of mechanical seals.Design/methodology/approachAcoustic emission signals from the friction of the seal end face were obtained, and their bispectral characteristics were extracted. The variation of non-Gaussian information with the degree of friction was investigated, and by combining bispectral characteristics with information entropy, a bispectral entropy index was established to represent the friction level of the seal end face.FindingsIn the start-up stage, the characteristic frequency amplitude of the micro-convex body contact is obvious, the friction of the end face is abnormal, the complexity of the system increases in a short time and the bispectral entropy rises continuously in a short time. In the stable operation stage, the characteristic frequency amplitude of the micro-convex body contact varies with the intensity of the seal face friction, the seal face friction is stable and the bispectral entropy fluctuates up and down for a period of time.Originality/valueThe bispectral analysis method is applied to the seal friction monitoring, the seal frequency domain characteristics are extracted, the micro-convex body contact characteristic frequency is defined and the bispectral entropy characteristic index is proposed, which provides a certain theoretical basis for the mechanical seal friction monitoring.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-07-2023-0242/

Beyond 3×2-point cosmology: the integrated shear and galaxy 3-point correlation functions

We present the integrated 3-point correlation functions (3PCF) involving both the cosmic shear and the galaxy density fields. These are a set of higher-order statistics that describe the modulation of local 2-point correlation functions (2PCF) by large-scale features in the fields, and which are easy to measure from galaxy imaging surveys. Based on previous works on the shear-only integrated 3PCF, we develop the theoretical framework for modelling 5 new statistics involving the galaxy field and its cross-correlations with cosmic shear. Using realistic galaxy and cosmic shear mocks from simulations, we determine the regime of validity of our models based on leading-order standard perturbation theory with an MCMC analysis that recovers unbiased constraints of the amplitude of fluctuations parameter A s and the linear and quadratic galaxy bias parameters b 1 and b 2. Using Fisher matrix forecasts for a DES-Y3-like survey, relative to baseline analyses with conventional 3×2PCFs, we find that the addition of the shear-only integrated 3PCF can improve cosmological parameter constraints by 20–40%. The subsequent addition of the new statistics introduced in this paper can lead to further improvements of 10–20%, even when utilizing only conservatively large scales where the tree-level models are valid. Our results motivate future work on the galaxy and shear integrated 3PCFs, which offer a practical way to extend standard analyses based on 3×2PCFs to systematically probe the non-Gaussian information content of cosmic density fields.

Cosmology inference at the field level from biased tracers in redshift-space

Cosmology inference of galaxy clustering at the field level with the EFT likelihood in principle allows for extracting all non-Gaussian information from quasi-linear scales, while robustly marginalizing over any astrophysical uncertainties. A pipeline in this spirit is implemented in the LEFTfield code, which we extend in this work to describe the clustering of galaxies in redshift space. Our main additions are: the computation of the velocity field in the LPT gravity model, the fully nonlinear displacement of the evolved, biased density field to redshift space, and a systematic expansion of velocity bias. We test the resulting analysis pipeline by applying it to synthetic data sets with a known ground truth at increasing complexity: mock data generated from the perturbative forward model itself, sub-sampled matter particles, and dark matter halos in N-body simulations. By fixing the initial-time density contrast to the ground truth, while varying the growth rate f, bias coefficients and noise amplitudes, we perform a stringent set of checks. These show that indeed a systematic higher-order expansion of the velocity bias is required to infer a growth rate consistent with the ground truth within errors. Applied to dark matter halos, our analysis yields unbiased constraints on f at the level of a few percent for a variety of halo masses at redshifts z = 0, 0.5, 1 and for a broad range of cutoff scales 0.08 h/Mpc≤ Λ ≤ 0.20 h/Mpc. Importantly, deviations between true and inferred growth rate exhibit the scaling with halo mass, redshift and cutoff that one expects based on the EFT of Large Scale Structure. Further, we obtain a robust detection of velocity bias through its effect on the redshift-space density field and are able to disentangle it from higher-derivative bias contributions.

Testing charge quantization with axion string-induced cosmic birefringence

We demonstrate that the Peccei-Quinn-electromagnetic anomaly coefficient 𝒜 can be directly measured from axion string-induced cosmic birefringence by applying scattering transform to the anisotropic polarization rotation of the cosmic microwave background. This breaks the degeneracy between 𝒜 and the effective number of string loops in traditional inference analyses that are solely based on the spatial power spectrum of polarization rotation. Carrying out likelihood-based parameter inference on mock rotation realizations generated according to phenomenological string network models, we show that scattering transform is able to extract enough non-Gaussian information to clearly distinguish a number of discrete 𝒜 values, for instance 𝒜=1/9, 1/3, 2/3, in the ideal case of noise-free rotation reconstruction, and, to a lesser but interesting degree at reconstruction noise levels comparable to that expected for the proposed CMB-HD concept. In the event of a statistical detection of cosmic birefringence by Stage III or IV CMB experiments, our technique can be applied to test the stringy nature of the birefringence pattern and extract fundamental information about the smallest unit of charge in theories beyond the Standard Model.

Heart Sound Classification using the Nonlinear Dynamic Feature Approach along with Conventional Classifiers

Heart sounds show chaotic and complex behavior when murmurs are present, containing nonlinear and non-Gaussian information. This paper studies ways to extract features from nonlinear dynamic models. The features frequently used to describe the underlying dynamics of the heart are derived from nonlinear dynamical modeling of heart sound signals. This study incorporates nonlinear dynamic features alongside conventional classifiers in the analysis of phonocardiograms (PCGs), achieving a significant improvement in the classification performance with 0.90 sensitivity and 0.92 specificity.

Step-wise segment partition based stationary subspace analysis and Gaussian mixture model for nonstationary process performance assessment

This article focuses on solving the problem of performance assessment for nonstationary processes without significant cointegrated correlations, which is a basic assumption of traditional methods. The time axis is replaced by indicator axis of working condition. Thus, the irregular non-stationarity along time axis could turns to be typical piecewise stationarity along indicator axis of working condition. The proposed method provides a general tool to deal with non-stationarities and relax the cointegrated restriction on variable correlations. Since a nonstationary process has specific working patterns under different conditions, the process can thus be divided into multiple segments along the axis of working condition. To characterize the properties of different performances within a segment meticulously, random sampling-based stationary subspace analysis (SSA) is conducted to separate the Gaussian and non-Gaussian subspaces. Since the operating performance is mainly distinguied by the non-Gaussian information, the non-Gaussian subspace obtained by SSA is further characterized by Gaussian mixture model (GMM). For online application, the segment model is selected by the indicator of working condition, and then the performance is assessed using Bayesian inference to provide a probabilistic result. The efficacy of the proposed method is validated by a coal mill from a real thermal power plant.

Constraining νΛCDM with density-split clustering

ABSTRACTThe dependence of galaxy clustering on local density provides an effective method for extracting non-Gaussian information from galaxy surveys. The two-point correlation function (2PCF) provides a complete statistical description of a Gaussian density field. However, the late-time density field becomes non-Gaussian due to non-linear gravitational evolution and higher-order summary statistics are required to capture all of its cosmological information. Using a Fisher formalism based on halo catalogues from the Quijote simulations, we explore the possibility of retrieving this information using the density-split clustering (DS) method, which combines clustering statistics from regions of different environmental density. We show that DS provides more precise constraints on the parameters of the νΛCDM model compared to the 2PCF, and we provide suggestions for where the extra information may come from. DS improves the constraints on the sum of neutrino masses by a factor of 7 and by factors of 4, 3, 3, 6, and 5 for Ωm, Ωb, h, ns, and σ8, respectively. We compare DS statistics when the local density environment is estimated from the real or redshift-space positions of haloes. The inclusion of DS autocorrelation functions, in addition to the cross-correlation functions between DS environments and haloes, recovers most of the information that is lost when using the redshift-space halo positions to estimate the environment. We discuss the possibility of constructing simulation-based methods to model DS clustering statistics in different scenarios.

Cosmological information in skew spectra of biased tracers in redshift space

Extracting the non-Gaussian information encoded in the higher-order clustering statistics of the large-scale structure is key to fully realizing the potential of upcoming galaxy surveys. We investigate the information content of the redshift-space weighted skew spectra of biased tracers as efficient estimators for 3-point clustering statistics. The skew spectra are constructed by correlating the observed galaxy field with an appropriately-weighted square of it. We perform numerical Fisher forecasts using two synthetic datasets: the halo catalogs from the Quijote N-body simulations and the galaxy catalogs from the Molino suite. The latter serves to understand the effect of marginalization over a more complex matter-tracer biasing relation. Compared to the power spectrum multipoles, we show that the skew spectra substantially improve the constraints on six parameters of the νΛCDM model, {Ω m , Ω b , h, ns , σ8 , Mν }. Imposing a small-scale cutoff of kmax = 0.25 Mpc-1 h, the improvements in parameter constraints from skew spectra alone range from 23% to 62% for the Quijote halos and from 32% to 71% for the Molino galaxies. Compared to the previous analysis of the bispectrum monopole on the same data and using the same range of scales, the skew spectra of Quijote halos provide competitive constraints. At the same time, the skew spectra outperform the bispectrum monopole for all cosmological parameters for the Molino catalogs. This may result from additional anisotropic information, particularly enhanced in the Molino sample, that is captured by the skew spectra but not by the bispectrum monopole. Our stability analysis of the numerical derivatives shows comparable convergence rates for the power spectrum and the skew spectra, indicating potential underestimation of parameter uncertainties by at most 30%.

Enhancing cosmic shear with the multiscale lensing probability density function

ABSTRACT We quantify the cosmological constraining power of the ‘lensing probability density function (PDF)’ – the one-point probability density of weak lensing convergence maps – by modelling this statistic numerically with an emulator trained on w cold dark matter cosmic shear simulations. After validating our methods on Gaussian and lognormal fields, we show that ‘multiscale’ PDFs – measured from maps with multiple levels of smoothing – offer considerable gains over two-point statistics, owing to their ability to extract non-Gaussian information: For a mock Stage-III survey, lensing PDFs yield 33 per cent tighter constraints on the clustering parameter $S_8=\sigma _8\sqrt{\Omega _{\rm m}/0.3}$ than the two-point shear correlation functions. For Stage-IV surveys, we achieve >90 per cent tighter constraints on S8, but also on the Hubble and dark energy equation-of-state parameters. Interestingly, we find improvements when combining these two probes only in our Stage-III set-up; in the Stage-IV scenario the lensing PDFs contain all information from the standard two-point statistics and more. This suggests that while these two probes are currently complementary, the lower noise levels of upcoming surveys will unleash the constraining power of the PDF.

KiDS-1000 cosmology: Constraints from density split statistics

Context. Weak lensing and clustering statistics beyond two-point functions can capture non-Gaussian information about the matter density field, thereby improving the constraints on cosmological parameters relative to the mainstream methods based on correlation functions and power spectra. Aims. This paper presents a cosmological analysis of the fourth data release of the Kilo Degree Survey based on the density split statistics, which measures the mean shear profiles around regions classified according to foreground densities. The latter is constructed from a bright galaxy sample, which we further split into red and blue samples, allowing us to probe their respective connection to the underlying dark matter density. Methods. We used the state-of-the-art model of the density splitting statistics and validated its robustness against mock data infused with known systematic effects such as intrinsic galaxy alignment and baryonic feedback. Results. After marginalising over the photometric redshift uncertainty and the residual shear calibration bias, we measured for the full KiDS-bright sample a structure growth parameter of $ S_8\equiv \sigma_8 \sqrt{\Omega_{\mathrm{m}}/0.3}=0.73^{+0.03}_{-0.02} $ that is competitive and consistent with two-point cosmic shear results, a matter density of Ωm = 0.27 ± 0.02, and a constant galaxy bias of b = 1.37 ± 0.10.

A tomographic spherical mass map emulator of the KiDS-1000 survey using conditional generative adversarial networks

Large sets of matter density simulations are becoming increasingly important in large-scale structure cosmology. Matter power spectra emulators, such as the Euclid Emulator and CosmicEmu, are trained on simulations to correct the non-linear part of the power spectrum. Map-based analyses retrieve additional non-Gaussian information from the density field, whether through human-designed statistics such as peak counts, or via machine learning methods such as convolutional neural networks. The simulations required for these methods are very resource-intensive, both in terms of computing time and storage. This creates a computational bottleneck for future cosmological analyses, as well as an entry barrier for testing new, innovative ideas in the area of cosmological information retrieval. Map-level density field emulators, based on deep generative models, have recently been proposed to address these challenges. In this work, we present a novel mass map emulator of the KiDS-1000 survey footprint, which generates noise-free spherical maps in a fraction of a second. It takes a set of cosmological parameters (Ω M , σ 8) as input and produces a consistent set of 5 maps, corresponding to the KiDS-1000 tomographic redshift bins. To construct the emulator, we use a conditional generative adversarial network architecture and the spherical convolutional neural network DeepSphere, and train it on N-body-simulated mass maps. We compare its performance using an array of quantitative comparison metrics: angular power spectra Cℓ , pixel/peaks distributions, Cℓ correlation matrices, and Structural Similarity Index. Overall, the average agreement on these summary statistics is <10% for the cosmologies at the centre of the simulation grid, and degrades slightly on grid edges. However, the quality of the generated maps is worse at high negative κ values or large scale, which can significantly affect summaries sensitive to such observables. Finally, we perform a mock cosmological parameter estimation using the emulator and the original simulation set. We find good agreement in these constraints, for both likelihood and likelihood-free approaches. The emulator is available at tfhub.dev/cosmo-group-ethz/models/kids-cgan.

Dark Energy Survey Year 3 results: Cosmology with moments of weak lensing mass maps

We present a cosmological analysis using the second and third moments of the weak lensing mass (convergence) maps from the first three years of data (Y3) data of the Dark Energy Survey. The survey spans an effective area of 4139 square degrees and uses the images of over 100 million galaxies to reconstruct the convergence field. The second moment of the convergence as a function of smoothing scale contains information similar to standard shear 2-point statistics. The third moment, or the skewness, contains additional non-Gaussian information. The data is analyzed in the context of the ΛCDM model, varying five cosmological parameters and 19 nuisance parameters modeling astrophysical and measurement systematics. Our modeling of the observables is completely analytical, and has been tested with simulations in our previous methodology study. We obtain a 1.7% measurement of the amplitude of fluctuations parameter S8≡σ8(Ωm/0.3)0.5=0.784±0.013. The measurements are shown to be internally consistent across redshift bins, angular scales, and between second and third moments. In particular, the measured third moment is consistent with the expectation of gravitational clustering under the ΛCDM model. The addition of the third moment improves the constraints on S8 and Ωm by ∼15% and ∼25% compared to an analysis that only uses second moments. We compare our results with Planck constraints from the cosmic microwave background, finding a 2.2–2.8σ tension in the full parameter space, depending on the combination of moments considered. The third moment, independently, is in 2.8σ tension with Planck, and thus provides a cross-check on the analyses of 2-point correlations.13 MoreReceived 13 November 2021Accepted 30 August 2022DOI:https://doi.org/10.1103/PhysRevD.106.083509© 2022 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCosmological parametersCosmologyGravitation, Cosmology & Astrophysics

Deep Learning of Latent Variable Models for Industrial Process Monitoring

Data-driven process monitoring based on latent variable models are widely employed in industry. This article proposes a novel monitoring framework for latent variable models using hierarchical feature extraction, Bayesian inference, and weighting strategy. We first establish a deep structure to implement hierarchical latent variables extraction, the extracted features are used to construct diverse monitoring statistics. Then, we utilize Bayesian inference and proper weighting strategy to fuse various useful information. In line with the different characteristics of principal component analysis (PCA) and independent component analysis (ICA), we construct a deep PCA-ICA model for process monitoring according to the proposed framework. The deep PCA-ICA model performs hierarchical feature extraction, which can simultaneously extract deep Gaussian information and deep non-Gaussian information. The features extracted by different layers are then transformed to posterior probabilities through Bayesian inference. After that, different posterior probabilities are combined through appropriate weighting strategy to build new probabilistic statistic, which can give more synthetic monitoring results. Moreover, the Bayesian inference and weighting strategy are further used to integrate the advantages of different models by transforming various probabilistic statistics into an overall monitoring index, which can comprehensively indicate the process status. The Tennessee Eastman process is used to validate the superiority of the proposed model over the existing methods. Besides, the extracted features are further analyzed to show the effectiveness and benefits of the deep hierarchical feature extraction structure.