Kernel Density Estimation (KDE) with adaptive bandwidth selection for environmental contours of extreme sea states.
Abstract not provided.
Eckert-Gallup, Aubrey Celia↗
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Kernel Density Estimation of Reaction Rates in Neutron Transport Simulations of Nuclear Reactors
Kernel density estimators (KDEs) are applied to estimate neutron scalar flux and reaction rate densities in Monte Carlo neutron transport simulations of heterogeneous nuclear reactors in continuous energy. The mean free path (MFP) KDE is introduced in order to handle the issues that arise from estimating the discontinuous reaction rate densities at material interfaces. Results show the MFP KDE is more accurate at estimating reaction rates compared with previous KDE formulations. An approximate MFP (aMFP) KDE is introduced to circumvent several practical issues presented by the MFP KDE. A volume-averaged KDE is derived and used to determine the bias introduced by the aMFP KDE. A KDE is formulated for cylindrical coordinates to better represent the geometry and capture the physics in two-dimensional reactor physics problems. The results here indicate that the cylindrical MFP KDE and cylindrical aMFP KDE are accurate tools for capturing reaction rates in heterogeneous reactor physics problems in continuous energy, with local biases of less than 1%.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
A Novel Non-Parametric Density Estimation Approach to Measuring Muon Ionization Cooling and Reverse Emittance Exchange in the MICE Experiment
The Muon Ionization Cooling Experiment (MICE) located at Rutherford Ap- pleton Laboratory in the U.K. has demonstrated muon ionization beam cooling for the rst time. A beam of muons in MICE is produced from high-energy proton beam collision with a xed target, generating pions which in turn decay into muons. Pion-decay muons, thus, are tertiary particles and, as a result, occupy a large vol- ume in position{momentum phase space. To t the muon beam into smaller and more cost-eective accelerating devices, muon beam phase{space volume needs to be reduced (beam cooling). Ionization beam cooling, which before MICE has never been demonstrated experimentally for muons, is the only technique fast enough to be used for muons within their short lifetime. Ionization cooling occurs when muons traverse an absorbing material and lose momentum through ionization energy loss. The cooling eect in MICE is measured using two scintillating-ber tracking detec- tors. These trackers, one upstream and one downstream of the absorber, reconstruct and measure the position and momentum coordinates of each muon. Given the pre- cision MICE needed to demonstrate beam cooling, it is necessary to develop analysis tools that can account for any eects that may lead to inaccurate measurement of cooling, such as non{linear eects in beam optics. Non{parametric density estima- tion techniques, such as kernel density estimation (KDE), provide a basis for creating analysis tools that are robust against these eects, directly calculating the muon beam phase-space density and volume for demonstrating beam cooling. This thesis focuses on the application of KDE to the measurement of beam cooling in MICE. The KDE technique is validated using known distributions and is applied to sim- ulated and experimental MICE data corresponding to the various magnet, optics, and absorber congurations. Using the KDE technique, muon beam cooling in the four{dimensional transverse phase space, as well as reverse emittance exchange using MICE data have been demonstrated.
43 PARTICLE ACCELERATORS↗
Spherical Kernel Density Estimators For Monte Carlo Radiation Transport Problems
Recently, there has been interest in creating alternatives to histograms (mesh tallies) for estimating spatially-distributed quantities such as scalar flux and reaction rate densities. Kernel Density Estimators (KDEs) are non-parametric estimators that obtain estimates of the underlying distribution at user-defined points without having to specify an underlying mesh, and the uncertainty at those points is independent of the number of tally points used in the simulation. This has led to the investigation of KDEs as an attractive potential alternative to histogram tallies in Monte Carlo radiation transport simulations for reactor physics problems and shielding applications. A KDE was recently created specifically for cylindrical geometries that is capable of producing smooth and accurate estimates of flux and reaction rate densities in reactor physics problems. This work extends the cylindrical KDE and cylindrical Mean Free Path (MFP) KDE to spherical geometries and verifies the spherical KDE on bare and reflected critical benchmark problems.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
Efficient screening of rare large pit anomalies on polished surfaces using a minimalist sampling scheme
Lawrence Livermore National Laboratory (LLNL) has made significant strides in generating clean energy through its inertial confinement fusion (ICF) experiments. These experiments rely on high-density carbon (HDC) coated shells to encapsulate the fusion fuel. The success of these experiments is heavily dependent on the surface quality of these shells, as even minor imperfections, such as deep pits, can negatively impact fusion yield. Ensuring the required smoothness involves an extensive surface-finishing process that spans approximately 20 stages, making it both time-intensive and resource-demanding. A critical challenge in this process is the need for high-resolution scans to detect rare deep pits, which can be costly and impractical if performed on every shell. This highlights the necessity of developing more efficient scanning methods to optimize time and cost without compromising accuracy. To address these challenges, we introduce a novel approach that employs the multivariate Dvoretzky–Kiefer–Wolfowitz (DKW) inequality to provide a probabilistic upper bound on the error in estimating pit distribution characteristics via a Kernel Density Estimator (KDE). This error bound enables efficient and reliable estimation of pit distribution characteristics at a specified statistical confidence level using a minimal number of surface scans. The integrated DKW-KDE approach was validated through surface-finishing experiments across two batches of HDC-coated shells, demonstrating consistent and robust performance across multiple stages of the surface-finishing experiments. The validation studies suggest that the integrated DKW-KDE approach achieves comparable accuracy in estimating the risk of deleterious large pits with six scans, thus conserving time and resources. Further evaluations show that performance remains consistent across batches and over multiple polishing stages. In conclusion, based on these findings, one can leverage the minimal-scan insights to strategically improve the bottleneck inspection process, thus enhancing the productivity and quality of shell polishing and similar challenging manufacturing processes.
Inertial confinement fusion↗
Quasar Identification Using Multivariate Probability Density Estimated from Nonparametric Conditional Probabilities
Nonparametric estimation for a probability density function that describes multivariate data has typically been addressed by kernel density estimation (KDE). A novel density estimator recently developed by Farmer and Jacobs offers an alternative high-throughput automated approach to univariate nonparametric density estimation based on maximum entropy and order statistics, improving accuracy over univariate KDE. This article presents an extension of the single variable case to multiple variables. The univariate estimator is used to recursively calculate a product array of one-dimensional conditional probabilities. In combination with interpolation methods, a complete joint probability density estimate is generated for multiple variables. Good accuracy and speed performance in synthetic data are demonstrated by a numerical study using known distributions over a range of sample sizes from 100 to 10 6 for two to six variables. Performance in terms of speed and accuracy is compared to KDE. The multivariate density estimate developed here tends to perform better as the number of samples and/or variables increases. As an example application, measurements are analyzed over five filters of photometric data from the Sloan Digital Sky Survey Data Release 17. The multivariate estimation is used to form the basis for a binary classifier that distinguishes quasars from galaxies and stars with up to 94% accuracy.
79 ASTRONOMY AND ASTROPHYSICS↗
Modeling error PDF optimization based wavelet neural network modeling of dynamic system and its application in blast furnace ironmaking
In general, the modeling errors of dynamic system model are a set of random variables. The traditional performance index of modeling such as means square error (MSE) and root means square error (RMSE) can not fully express the connotation of modeling errors with stochastic characteristics both in the dimension of time domain and space domain. Therefore, the probability density function (PDF) is introduced to completely describe the modeling errors in both time scales and space scales. Based on it, a novel wavelet neural network (WNN) modeling method is proposed by minimizing the two-dimensional (2D) PDF shaping of modeling errors. First, the modeling error PDF by the tradional WNN is estimated using data-driven kernel density estimation (KDE) technique. Then, the quadratic sum of 2D deviation between the modeling error PDF and the target PDF is utilized as performance index to optimize the WNN model parameters by gradient descent method. Since the WNN has strong nonlinear approximation and adaptive capability, and all the parameters are well optimized by the proposed method, the developed WNN model can make the modeling error PDF track the target PDF, eventually. Simulation example and application in a blast furnace ironmaking process show that the proposed method has a higher modeling precision and better generalization ability compared with the conventional WNN modeling based on MSE criteria. Furthermore, the proposed method has more desirable estimation for modeling error PDF that approximates to a Gaussian distribution whose shape is high and narrow.
Modelling↗
Kernel Density Estimators for SAMPLER Analyses [Slides]
This presentation discusses Sampler, specifically in terms of uncertainty quantification and states that for many simulation tasks nominal input parameters have complicated influence on the uncertainty in the final result. Sampler provides a convenient means to perform stochastic uncertainty quantification, brute force sampling parameter space and tallying changes in outputs, and it can be coupled with any SCALE modules. Perturbed responses can be collected and analyzed with standard statistical measures; however, two problems have been identified. The first problem is that for Monte Carlo simulations the individual response uncertainties are ignored, and the second problem is that features of distributions are easily hidden in histograms. They found that the solution is to add Kernel Density Estimator (KDE) analysis to the Sampler output. In conclusion, Sampler is a unique tool for stochastic uncertainty propagation, and it can be coupled with any module in SCALE. Current functionality is limited to histograms, and it may not adequately account for sample uncertainties and/or not adequately display underlying distribution features. Work is in progress to implement KDE in the Sampler module and no additional flagging is needed.
73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
The effective number of parameters in kernel density estimation
We devise a new formula for measuring the effective degrees of freedom (EDoF) in kernel density estimation (KDE). Starting from the orthogonal polynomial sequence (OPS) expansion for the ratio of the empirical to the oracle density, we show how convolution with the kernel leads to a new OPS with respect to which one may express the resulting KDE. The expansion coefficients of the two OPS systems can then be related via a kernel sensitivity matrix, which leads to a natural oracle definition of EDoF through the trace operator. Asymptotic properties of the (empirical) plug-in EDoF are worked out through influence functions, and connections with other empirical EDoFs are established. Minimization of Kullback-Leibler divergence is investigated as an alternative to integrated squared error based bandwidth selection rules, yielding a new normal scale rule. The methodology, which arises from a proper oracle formulation and is not restricted to convolution kernels, suggests the possibility of a new bandwidth selection rule based on an information criterion such as AIC.
bandwidth selection↗
GalaxyFlow: upsampling hydrodynamical simulations for realistic mock stellar catalogues
ABSTRACT Cosmological N-body simulations of galaxies operate at the level of ‘star particles’ with a mass resolution on the scale of thousands of solar masses. Turning these simulations into stellar mock catalogues requires ‘upsampling’ the star particles into individual stars following the same phase-space density. In this paper, we introduce two new upsampling methods. First, we describe GalaxyFlow, a sophisticated upsampling method that utilizes normalizing flows to both estimate the stellar phase-space density and sample from it. Secondly, we improve on existing upsamplers based on adaptive kernel density estimation (KDE), using maximum likelihood estimation to fine-tune the bandwidth for such algorithms in a way that improves both the density estimation accuracy and upsampling results. We demonstrate our upsampling techniques on a neighbourhood of the Solar location in two simulated galaxies: Auriga 6 and h277. Both yield smooth stellar distributions that closely resemble the stellar densities seen in the Gaia DR3 catalogue. Furthermore, we introduce a novel multimodel classifier test to compare the accuracy of different upsampling methods quantitatively. This test confirms that GalaxyFlow more accurately estimates the density of the underlying star particles than methods based on KDE, at the cost of being more computationally intensive.
Lim, Sung Hak (ORCID:0000000330981092)↗
Detecting Anomalies in Time Series Using Kernel Density Approaches
This paper introduces a novel anomaly detection approach tailored for time series data with exclusive reliance on normal events during training. Our key innovation lies in the application of kernel-density estimation (KDE) to scrutinize reconstruction errors, providing an empirically derived probability distribution for normal events post-reconstruction. This non-parametric density estimation technique offers a nuanced understanding of anomaly detection, differentiating it from prevalent threshold-based mechanisms in existing methodologies. In post-training, events are encoded, decoded, and evaluated against the estimated density, providing a comprehensive notion of normality. In addition, we propose a data augmentation strategy involving variational autoencoder-generated events and a smoothing step for enhanced model robustness. The significance of our autoencoder-based approach is evident in its capacity to learn normal representation without prior anomaly knowledge. Through the KDE step on reconstruction errors, our method addresses the versatility of anomalies, departing from assumptions tied to larger reconstruction errors for anomalous events. Our proposed likelihood measure then distinguishes normal from anomalous events, providing a concise yet comprehensive anomaly detection solution. The extensive experimental results support the feasibility of our proposed method, yielding significantly improved classification performance by nearly 10% on the UCR benchmark data.
Frehner, Robin↗
Notos - a galaxy tool to analyze CpN observed expected ratios for inferring DNA methylation types
Background: DNA methylation patterns store epigenetic information in the vast majority of eukaryotic species. The relatively high costs and technical challenges associated with the detection of DNA methylation however have created a bias in the number of methylation studies towards model organisms. Consequently, it remains challenging to infer kingdom-wide general rules about the functions and evolutionary conservation of DNA methylation. Methylated cytosine is often found in specific CpN dinucleotides, and the frequency distributions of, for instance, CpG observed/expected (CpG o/e) ratios have been used to infer DNA methylation types based on higher mutability of methylated CpG. Results: Predominantly model-based approaches essentially founded on mixtures of Gaussian distributions are currently used to investigate questions related to the number and position of modes of CpG o/e ratios. These approaches require the selection of an appropriate criterion for determining the best model and will fail if empirical distributions are complex or even merely moderately skewed. We use a kernel density estimation (KDE) based technique for robust and precise characterization of complex CpN o/e distributions without a priori assumptions about the underlying distributions. Conclusions: We show that KDE delivers robust descriptions of CpN o/e distributions. For straightforward processing, we have developed a Galaxy tool, called Notos and available at the ToolShed, that calculates these ratios of input FASTA files and fits a density to their empirical distribution. Based on the estimated density the number and shape of modes of the distribution is determined, providing a rational for the prediction of the number and the types of different methylation classes. Notos is written in R and Perl.
59 BASIC BIOLOGICAL SCIENCES↗
Learning to Optimize Variational Quantum Circuits to Solve Combinatorial Problems
Quantum computing is a computational paradigm with the potential to outperform classical methods for a variety of problems. Proposed recently, the Quantum Approximate Optimization Algorithm (QAOA) is considered as one of the leading candidates for demonstrating quantum advantage in the near term. QAOA is a variational hybrid quantum-classical algorithm for approximately solving combinatorial optimization problems. The quality of the solution obtained by QAOA for a given problem instance depends on the performance of the classical optimizer used to optimize the variational parameters. In this paper, we formulate the problem of finding optimal QAOA parameters as a learning task in which the knowledge gained from solving training instances can be leveraged to find high-quality solutions for unseen test instances. To this end, we develop two machine-learning-based approaches. Our first approach adopts a reinforcement learning (RL) framework to learn a policy network to optimize QAOA circuits. Our second approach adopts a kernel density estimation (KDE) technique to learn a generative model of optimal QAOA parameters. In both approaches, the training procedure is performed on small-sized problem instances that can be simulated on a classical computer; yet the learned RL policy and the generative model can be used to efficiently solve larger problems. Furthermore, extensive simulations using the IBM Qiskit Aer quantum circuit simulator demonstrate that our proposed RL- and KDE-based approaches reduce the optimality gap by factors up to 30.15 when compared with other commonly used off-the-shelf optimizers.
97 MATHEMATICS AND COMPUTING↗
Stereokinetic effect and its relation to the kinetic depth effect
The stereokinetic effect (SKE) has been defined and studied by nested circular patterns rotating on a turntable. Circles must appear not to rotate as they revolve, which in turn results in their appearing to translate relative to one another. A powerful illusion of object depth results even though the individual circles do not undergo an appropriate foreshortening consistent with their apparent changes in slant. It is suggested and tested that the SKE is based on the changing positions between the nested contours despite the absence of any change within each contour, whereas the kinetic depth effect (KDE) entails both kinds of change. It follows that a turntable method of presentation is not necessary, and between-contour transformations can be simulated by computer animation. Displays consisting of simple translations were shown to evoke robust depth impressions as were patterns consisting of contours of varying shapes. Comparisons of the depth, compellingness, and rigidity of matched SKE and KDE displays are reported. The SKE is taken to be paradigmatic for how the visual system perceives depth when observing small object rotations that occur in everyday situations.
Proffitt, Dennis R.↗
Acoustic Optimization for Anti-Phase Asymmetric Rotor
This investigation seeks to optimize the implementation of anti-phase alternating trailing edge (TE) patterns for rotor noise suppression. The design objective is to maximize reduction of noise perceived by the community while maintaining the aerodynamic thrust. Computations using a three-dimensional Unsteady-Reynolds-Averaged-Navier-Stokes (URANS) with k-w Shear Stress Transport (SST) turbulence model and Ffowcs-Williams and Hawkings (FW-H) formula are used to obtain aerodynamic thrust and far-field noise level. A parametric acoustic study of 13 configurations of KDE rotor with variable alternating trailing edge period, alternating trailing edge length, and trailing edge deflection angle is conducted. The best design candidate for the KDE rotor has a four-period TE waveform which results in a reduction in far-field noise level of 2.1 dB in the hover condition and a reduction of 1.1 dB in the forward flight condition at 9.7 m/s. A further parametric acoustic study is conducted for a different rotor manufactured by APC. Six APC rotor design candidates are simulated. The best design candidate 4H for the APC rotor results in a reduction in far-field noise level of 4.0 dB in the hover condition and a reduction of 1.3 dB in the forward flight condition at 9.7 m/s. A series of acoustic experiments in the Penn State University (PSU) anechoic chamber have been conducted. In the forward flight condition at 9.7 m/s, the APC anti-phase 4H rotor offers clear evidence of noise suppression capability across a wide range of the azimuthal angle. In the broadband frequency range of 2000-4000 Hz, the APC anti-phase 4H rotor produces as much as 6 dB noise reduction. The experimental results appear to confirm the noise suppression capability of the proposed anti-phase rotor design concepts.
Xiong, Juntao↗
Dynamic Performance Enhancement for Nonlinear Stochastic Systems Using RBF Driven Nonlinear Compensation with Extended Kalman Filter
In this paper, a novel hybrid control method is proposed to enhance the control performance of the PI based control system for a class of nonlinear and non-Gaussian stochastic dynamic processes with unmeasurable states. Firstly, to enhance the tracking performance of the PI controller where the PI parameters are fixed in many actual control systems, the compensative signal is formed using the extended Kalman filter(EKF) based state estimator and driven by the radial basis function (RBF) neural network based compensator. In addition, the weights of RBF is trained to minimize the entropy criterion of tracking error as the process is subjected to non-Gaussian disturbances. Meanwhile, since the precise statistical property of noises is hard to obtain for many industrial processes, the kernel density estimation (KDE) technique is employed in this paper to estimate the entropy. The convergence of RBF network is discussed and the stability of the resulted closed-loop hybrid control system is analyzed in mean square sense. Finally, a numerical example and a practical system testing are given to illustrate the effectiveness of the proposed control method.
stochastic systems, PI Controller, Entropy, Extend↗
An extended polynomial chaos expansion for PDF characterization and variation with aleatory and epistemic uncertainties
This paper presents an extended polynomial chaos formalism for epistemic uncertainties and a new framework for evaluating sensitivities and variations of output probability density functions (PDF) to uncertainty in probabilistic models of input variables. An ”extended” polynomial chaos expansion (PCE) approach is developed that accounts for both aleatory and epistemic uncertainties, modeled as random variables, thus allowing a unified treatment of both types of uncertainty. We explore in particular epistemic uncertainty associated with the choice of prior probabilistic models for input parameters. A PCE-based Kernel Density (KDE) construction provides a composite map from the PCE coefficients and germ to the PDF of quantities of interest (QoI). Here, the sensitivities of these PDF with respect to the input parameters are then evaluated. Input parameters of the probabilistic models are considered. By sampling over the epistemic random variable, a family of PDFs is generated and the failure probability is itself estimated as a random variable with its own PCE. Integrating epistemic uncertainties within the PCE framework results in a computationally efficient paradigm for propagation and sensitivity evaluation. Two typical illustrative examples are used to demonstrate the proposed approach.
Aleatory uncertainty↗
A functional global sensitivity measure and efficient reliability sensitivity analysis with respect to statistical parameters
Sensitivity analysis and reliability assessment are two important aspects of structural and system safety. Epistemic uncertainty with respect to probabilistic model of input parameters due to lack of knowledge is present in many scarce-data applications and complicates the characterization of uncertainty in model response. In this article, we present two importance measures to evaluate the impact of distribution parameters on the probability distribution function (PDF) of the output and the failure probability. The epistemic uncertainty associated with the distribution parameters is modeled as random variables. Additionally, a modified extended polynomial chaos expansion (MEPCE) approach is introduced in which aleatory and epistemic random variables are modeled and propagated simultaneously while allowing the separate assessment for any single epistemic variable. A MEPCE-based kernel density estimation (KDE) construction provides a composite map from each epistemic variable to the response PDF. The functional global sensitivity index of the PDF with respect to the distribution parameters is thus derived, as a function of output, which is both more informative and more efficient than standard scalar sensitivity measures. Reliability sensitivity indices can be readily evaluated by integrating the global sensitivity index function over the failure zone. Three illustrative examples are used to demonstrate the proposed methodology.
42 ENGINEERING↗