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Improve the interpretability of convolutional neural networks with probability density function

INFORMATION SCIENCES 2025年 699卷

作者： Chen, Yueqi Pan, Tingting Yang, Jie Dalian Univ Technol Sch Math Sci Dalian 116024 Liaoning Peoples R China Key Lab Computat Math & Data Intelligence Liaoning Dalian 116024 Liaoning Peoples R China Dalian Polytech Univ Dept Basic Courses Teaching Dalian 116034 Peoples R China

Currently, Convolutional Neural Networks (CNNs) have demonstrated extensive success in numerous practical applications. Nevertheless, their limited interpretability remains a significant barrier to further advancement in certain crucial fields. Improving the interpretability of CNNs stands as an exceptionally compelling topic in the present time. This paper explores the interpretability of a basic CNN incorporating a convolution-pooled block and a fully connected layer from a statistical perspective. Assuming that the input variables adhere to a normal distribution and maintain independence from each other, the output variables subsequent to the convolution and pooling layers also conform to a normal distribution. Simultaneously, the probability density function (pdf) characterizing the final output variable belongs to an exponential family distribution. By introducing intermediate variables, the pdf of this output variable can be expressed as a linear combination of three distinct normal distributions. Furthermore, the likelihood of the predicted class label can be rewritten as a cumulative density function (cdf) of the standard normal distribution. The originality of this paper lies in its provision of a more innovative and intuitive perspective for dissecting the operational mechanism of CNNs, analyzing them layer by layer to improve their interpretability. Experimental results obtained from both an artificial dataset and the image datasets CIFAR-10 and ImageNet further validate the rationality of these conclusions.

关键词： Interpretability Convolutional neural networks probability density function

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Impact of Vertical Scaling on Normal probability density function Plots

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IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS 2025年第1期31卷 984-994页

作者： Fygenson, Racquel Padilla, Lace Northeastern Univ Boston MA 02115 USA

probability density function (PDF) curves are among the few charts on a Cartesian coordinate system that are commonly presented without y-axes. This design decision may be due to the lack of relevance of vertical scaling in normal PDFs. In fact, as long as two normal PDFs have the same means and standard deviations (SDs), they can be scaled to occupy different amounts of vertical space while still remaining statistically identical. Because unfixed PDF height increases as SD decreases, visualization designers may find themselves tempted to vertically shrink low-SD PDFs to avoid occlusion or save white space in their figures. Although irregular vertical scaling has been explored in bar and line charts, the visualization community has yet to investigate how this visual manipulation may affect reader comparisons of PDFs. In this paper, we present two preregistered experiments (n = 600, n = 401) that systematically demonstrate that vertical scaling can lead to misinterpretations of PDFs. We also test visual interventions to mitigate misinterpretation. In some contexts, we find including a y-axis can help reduce this effect. Overall, we find that keeping vertical scaling consistent, and therefore maintaining equal pixel areas under PDF curves, results in the highest likelihood of accurate comparisons. Our findings provide insights into the impact of vertical scaling on PDFs, and reveal the complicated nature of proportional area comparisons.

关键词： visualization probability density function uncertainty vertical scaling perception area chart

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Ensemble modeling of the two-dimensional stochastic confined groundwater flow through the evolution of the hydraulic head's probability density function

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JOURNAL OF HYDROLOGY 2025年 652卷

作者： Meza, Joaquin Kavvas, M. Levent Univ Tecn Federico Santa Maria Dept Obras Civiles Valparaiso Chile Univ Calif Davis Chile Dept Civil & Environm Engn Davis CA 95616 USA

Groundwater storage in aquifers has become a vital water source due to water scarcity in recent years. However, aquifer systems are full of uncertainties, which inevitably propagate throughout the modeling computations, mainly reducing the reliability of the model output. The fundamental science problem this study addresses is the development of a two-dimensional stochastic confined groundwater flow model, which determines the time-- space evolution of the ensemble mean and ensemble variance of the flow field over a model domain under uncertain parameters and uncertain sink/source conditions. This is achieved by linking the stochastic partial differential equation that governs the confined aquifer flow to a non-local Lagrangian-Eulerian extension to the Fokker-Planck equation (LEFPE). In the form of the LEFPE, the resulting deterministic governing equation describes the spatio-temporal evolution of the probability density function of the state variables in the confined groundwater flow process by one single numerical realization instead of requiring thousands of simulations in the Monte Carlo approach. As will be shown in the paper's text, the time-space evolving ensemble mean and ensemble variance of the flow process are then obtained from the pdf of the state variable (hydraulic head) of the process that is determined from the solution of the LEFPE of the process under specified initial and boundary conditions. As such, in the developed methodology, no assumption is made on the distribution of the time-space varying pdf of the flow process, which is obtained from the solution of the LEFPE of the process under specified initial and boundary conditions. Consequently, the groundwater flow process's mean and standard deviation behavior can be modeled under uncertainty in the transmissivity field and recharge and/or pumping conditions. In addition, an appropriate numerical method for LEFPE's solution is subsequently devised. Then, its solution is presented, discussed,

关键词： probability density function

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Modified Unscented Kalman Filter Considering Maximum Point of probability density function

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IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS 2025年第2期61卷 4926-4944页

作者： Liu, Hanyu Sun, Xiucong Bai, Shengzhou Beihang Univ Sch Astronaut Beijing 100191 Peoples R China

The unscented Kalman filter (UKF) has demonstrated its effectiveness for state estimation in highly nonlinear systems over the past two decades. However, the UKF's assumption of a single Gaussian probability density function (PDF) adversely affects its filtering accuracy. To address this limitation, this article proposes a variant of the UKF, referred to as MaxUKF, which enhances estimation accuracy by incorporating the maximum point of the PDF, denoting the point getting maximum value of the PDF. First, two mathematical theorems are presented, providing direct ways to accurately predict the maximum point of the PDF of the state after nonlinear transformation. Subsequently, a Gaussian sum PDF is constructed to accurately match the mean, covariance, and maximum point of the PDF after nonlinear transformation. Then, the MaxUKF algorithm is presented in a manner analogous to the UKF, following a two-step process: time-update step and measurement-update step, during which the non-Gaussian PDF of the state is represented by the constructed Gaussian sum PDF. Finally, the effectiveness of the proposed MaxUKF is verified through a series of numerical simulations. The results demonstrate that the MaxUKF algorithm offers higher estimation accuracy compared to the conventional UKF while requiring similar computational time.

关键词： probability density function Accuracy Kalman filters Random variables Gaussian distribution Aerospace and electronic systems Matched filters Nonlinear systems Mathematical models Kurtosis Gaussian sum maximum point non-Gaussian unscented Kalman filter (UKF)

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Identifying parametric expression of probability density function of slowly-varying processes: a data-driven method based on the Π theorem

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NONLINEAR DYNAMICS 2025年第10期113卷 10969-10987页

作者： Li, Xin Jin, Xiaoling Huang, Zhilong Zhejiang Univ Dept Engn Mech Key Lab Soft Machines & Smart Devices Zhejiang Pro Hangzhou 310027 Peoples R China

The response analysis of random vibration systems holds significant importance in comprehending intricate natural phenomena, bolstering the reliability of engineering design, and optimizing structural performance. Traditional analytical approaches frequently encounter hurdles, particularly in deriving parametric expressions for the probability density of system response. This paper aims to propose a novel data-driven methodology based on the Buckingham Pi theorem to identify the parametric expression of probability density of slowly-varying processes from state data. The proposed data-driven method is carried out by two successive steps. The first step involves identifying the parametric expressions of invariants of the corresponding conservative system within the framework of the Koopman operator theory in conjunction with the Pi theorem. This process yields the expressions for slowly-varying processes. In the second step, the parametric expression of probability density of the slowly-varying processes is identified from the data of slowly-varying processes. This identification is based on the principle of maximum entropy and the Pi theorem. The data of slowly-varying processes can be calculated from the random state data with the expression identified in the first step. Sparse optimizations are carried out in both steps. The applicability and effectiveness of the proposed method are illustrated through three typical systems, including the Duffing-van der Pol system, Coulomb friction, and a two-degree-of-freedom system. The extensionality of the parametric expression for the probability density of slowly-varying processes is demonstrated. The proposed data-driven method is capable of reducing system dimensions due to the identified slowly-varying processes with lower dimensions compared to the system state. Additionally, leveraging the Pi theorem results in a smaller number of involved dimensionless groups in the identification compared to the original parameters.

关键词： Buckingham Pi theorem Data-driven approach probability density function Random vibration Maximum entropy principle

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Digital image stabilization based on time-varying filtering empirical mode decomposition and probability density function

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JOURNAL OF ELECTRONIC IMAGING 2025年第1期34卷

作者： Gao, Shiwei Wu, Yingnian Tan, Hao Wang, Hao Chen, Wenbai Beijing Informat Sci & Technol Univ Sch Automat Beijing Peoples R China High End Equipment Intelligent Percept & Control B Beijing Peoples R China Inst Intelligent IoT & Collaborat Control Beijing Peoples R China

The random movement of a non-stabilized camera platform during the shooting process can lead to jitter in the captured video sequences. We propose a digital image stabilization algorithm based on time-varying filter-based empirical mode decomposition (TVF-EMD) and probability density function (PDF) evaluation criterion. First, an accelerated robust feature algorithm obtains the global motion sequence. Then, TVF-EMD decomposes the global motion sequence to get the intrinsic mode functions (IMFs) and separates the intentional and jitter motion-dominated IMFs using PDFs. Finally, the deliberate motion sequence is reconstructed by summing the intentional motion-dominated IMFs. The experimental results demonstrate that, compared with existing methods, the video sequences reconstructed using the algorithm proposed in this paper exhibit a higher peak signal-to-noise ratio and structural similarity index, as well as a lower gradient magnitude similarity deviation, which proves the effectiveness of the technique.

关键词： digital image stabilization probability density function time-varying filter-based empirical mode decomposition intrinsic mode function

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Linear quadrature method of moments for solving the transported probability density function model for turbulent combustion

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COMBUSTION THEORY AND MODELLING 2025年第2期29卷 145-176页

作者： Wang, Yunchu Wang, Yize Yuan, Li Chinese Acad Sci ICMSEC Beijing Peoples R China Chinese Acad Sci Acad Math & Syst Sci LSEC Beijing Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing Peoples R China

Large eddy simulation (LES) coupled with the transported probability density function (PDF) model has become a promising strategy for modelling turbulent combustion. For solving the high-dimensional PDF transport equation numerically, deterministic quadrature-based moment methods have arisen as an alternative choice besides stochastic particle and stochastic field methods. In this work, we propose a new quadrature-based moment method called linear Quadrature Method Of Moments (LQMOM) for the univariate PDF case. This method assumes that the underlying PDF distribution is smooth and it uses a set of predefined quadrature nodes and weights in the sample space to establish a linear system of equations for the PDF values at the quadrature nodes from the moments. The LQMOM is more efficient than the original QMOM which needs to solve a nonlinear system of equations for the moment inversion. However, it may generate negative PDF values for non-smooth PDF distributions. To remedy this flaw, we construct the LQMOM-QMOM hybrid algorithm by using detectors to detect single- and two-peak distributions. Furthermore, we extend the LQMOM to the multivariate PDF case by utilizing the main idea of the conditional QMOM (CQMOM) to obtain the linear CQMOM (LCQMOM), and then reduce it to the simplified version (LCQMOM-S) to gain efficiency. The LQMOM-QMOM hybrid algorithm for univariate PDFs is tested in a typical example and the LCQMOM-S-CQMOM hybrid algorithm for multivariate PDFs is checked in compressible turbulent reactive 2D shear layer flow and 3D air/H2 jet. It is shown that the hybrid algorithms can reduce 20-35% simulation time and produce similar results compared with the respective QMOM and CQMOM.

关键词： turbulent combustion large eddy simulation probability density function quadrature-based moment method linear inversion hybrid algorithm

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Modeling chikungunya virus infection with Black-Karasinski process: stationary distribution, probability density function, and extinction

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ADVANCES IN CONTINUOUS AND DISCRETE MODELS 2025年第1期2025卷 1-29页

作者： Cao, Zhongwei Shi, Zhenfeng Song, Zhifei Zu, Li Xu, Hewei JiLin Univ Finance & Econ Off Res Affairs Changchun 130117 Peoples R China Southern Med Univ Sch Publ Hlth State Key Lab Organ Failure Res Dept BiostatGuangdong Prov Key Lab Trop Dis Res Guangzhou 510515 Peoples R China Jilin Univ Finance & Econ Personnel Off Changchun 130117 Peoples R China Hainan Normal Univ Coll Math & Stat Haikou 571158 Peoples R China Jilin Univ Finance & Econ Sch Accounting Changchun 130117 Peoples R China

Incorporating stochastic processes into biological models is crucial for capturing the inherent variability and uncertainties within biological systems. This paper explores the benefits of introducing Black-Karasinski process into chikungunya virus infection modeling. By utilizing the Black-Karasinski process researchers can capture the inherent variability in biological processes and account for uncertainties. This paper highlights the advantages of Black-Karasinski processes in biological modeling. We investigate the dynamical behavior of a stochastic model for chikungunya virus infection incorporating a Black-Karasinski process. Firstly, we establish sufficient conditions for the existence of a stationary distribution in the model. By solving the corresponding Fokker-Planck equation we obtain the local probability density function near the quasi-endemic equilibrium, which provides insights into the statistical characteristics of the stochastic system. Additionally, we present sufficient conditions for the extinction of infected host cells and chikungunya virus particles. Finally, we supplement the analytical results with numerical simulations to investigate the impact of random noise.

关键词： Chikungunya virus infection model Black-Karasinski process Stationary distribution probability density function Extinction

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Microfluidic droplet generation: an experimental study of size distribution using probability density function analysis

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JOURNAL OF MICROMECHANICS AND MICROENGINEERING 2025年第1期35卷 015001-015001页

作者： alvarez-Martinez, J. U. Medina-Cazares, O. Gonzalez-Vega, A. Segura-Gomez, G. Gutierrez-Juarez, G. Castro-Beltran, R. Univ Guanajuato Div Ingn & Ciencias Campus LeonLoma del Bosque 103 Leon 37150 Guanajuato Mexico

Controlling the volume and geometrical ratio of microdroplets offers numerous advantages, particularly in fields like pharmaceutics, photonics, soft electronics, and robotics. For microdroplets generated via microfluidics, precise control of their length-to-width ( Lw) ratio enhances their overall performance and contributes to a comprehensive understanding of the results. Advanced experimental tools, including droplet generation systems, visualization techniques, digital processing algorithms, and analysis, are crucial, especially for in situ control of microdroplet size during generation. This study investigates the variation in microdroplet Lw ratio and their generation frequencies using two novel approaches: the probability density function (derived both from the Gaussian assumption and directly from experimental data distribution) and the fast Fourier transform. The configuration utilizing one syringe pump (1mp) and syringes (made entirely of plastic) demonstrated the best monodispersity in microdroplet generation with volume differences ranged from 40 +/- 2.2 nl to 65 +/- 6.7 nl, corresponding to coefficient of variation values of 5.5 % and 11 %, respectively. The analysis of similar to 7 million Lw data points revealed different types of oscillation frequencies. Specifically, a frequency of around 1 mHz originated from the syringe pump, while frequencies between similar to 0.5 and 3 Hz were attributed to the syringe materials and the interaction of a combination of syringe pumps. These results contribute to the phenomenological understanding and classification of droplet generation within the T-junction microfluidic system. Furthermore, they present an enhanced method for visualizing the geometrical variations of the generated microdroplets.

关键词： microfluidics microdroplets probability density function fast Fourier Transform

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On one-dimensional surrogacy of probability density function of enstrophy in isotropic turbulence

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Physics of Fluids 2025年第2期37卷

作者： Gotoh, Toshiyuki Yeung, P.K. Department of Physical Science and Engineering Nagoya Institute of Technology Nagoya Japan Research and Education Center for Natural Sciences Keio University Yokohama Japan Schools of Aerospace Engineering and Mechanical Engineering Georgia Institute of Technology AtlantaGA United States

One-dimensional surrogates of fully three-dimensional (3D) quantities such as fluctuations of dissipation rate and enstrophy in turbulent flows have played an important role in many experimental and computational studies of intermittency. This paper addresses the connections between the probability density functions (PDFs) of the square of the magnitude of a fluctuating vector and that of a single Cartesian coordinate component under the condition of isotropy. A pair of rigorous relations between the PDFs of 3D enstrophy and its one-dimensional (1D) surrogate is derived mathematically. These relations are general, and independent of the functional form of the PDFs, degree of non-Gaussianity or the Reynolds number. The 1D surrogate is substantially more intermittent than its 3D counterpart. The functional forms of the respective PDFs are also examined using numerical simulation data, especially for regimes of very small and very large amplitudes. In the latter case, stretched exponential fits with different algebraic pre-factors but the same exponents at a given Reynolds number are found to agree very well with the data available. © 2025 Author(s).

关键词： probability density function

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