Unknown Density Research Articles

Optimal Bandwidth Selection Methods with Application to Wind Speed Distribution

Accurate estimation of the unknown probability density functions of critical variables, such as wind speed—which plays a pivotal role in harnessing clean energy—is essential for various scientific and practical applications. This research conducts a comprehensive comparative analysis of seven distinct bandwidth calculation techniques across various normal distributions, using simulation as the evaluation method in the context of Kernel Density Estimation (KDE). This analysis includes the calculation of the optimal bandwidth and assessment of the performance of these methods with respect to Mean Squared Error (MSE), bias, and the optimal bandwidth value. The findings reveal that among the various bandwidth methods evaluated, the Bandwidth bandwidth-based Cross-Validation (BCV), especially for small sample sizes, consistently provides the closest result to the optimal bandwidth across most of the applied normal distributions. These results provide valuable insights into the selection of optimal bandwidths for accurate and reliable density estimation in the context of normal distributions. Another key aspect of this work is the extension of these methods to wind speed data in a specific region. Monthly wind speed kernel density estimates obtained using all seven bandwidth selection techniques show that Smoothed Cross-Validation (SCV) is suited for this type of real-world data.

Symmetries and Conservation Laws for a Class of Fourth-Order Reaction–Diffusion–Advection Equations

We have studied a class of (1+1)-dimensional equations that models phenomena with heterogeneous diffusion, advection, and reaction. We have analyzed these fourth-order partial differential equations within the framework of group methods. In this class, the diffusion coefficient is constant, while the coefficients of advection and the reaction term are assumed to depend on the unknown density u(t,x). We have identified the Lie symmetries extending the Principal Algebra along with all the conservation laws corresponding to the different forms of the coefficients, and have derived several brief applications.

Frictional boundary condition for lattice Boltzmann modelling of dense granular flows

Hydrodynamic approaches that treat granular materials as a continuum via the homogenization of discrete flow properties have become viable options for efficient predictions of bulk flow behaviours. However, simplified boundary conditions in computational fluid dynamics are often adopted, which have difficulty in describing the complex stick–slip phenomenon at the boundaries. This paper extends the lattice Boltzmann method for granular flow simulations by incorporating a novel frictional boundary condition. The wall slip velocity is first calculated based on the shear rate limited by the Coulomb friction, followed by the reconstruction of unknown density distribution functions through a modified bounce-back scheme. Validation is performed against a unique plane Couette flow configuration, and the analytical solutions for the flow velocity profile and the wall slip velocity, as functions of the friction coefficient, are reproduced by the numerical model. The transition between no-slip and partial-slip regimes is captured well, but the convergence rate drops from second order to first order when slip occurs. The rheological parameters and the basal friction coefficient are calibrated further against the discrete element simulation of a square granular column collapsing over a horizontal bottom plane. It is found that the calibrated continuum model can predict other granular column collapses with different initial aspect ratios and slope inclination angles, including the basal slip and the complex internal flow structures, without any further adjustments to the model parameters. This highlights the generalization ability of the numerical model, which has a wide range of application in granular flow predictions and controls.

Smooth deconvolution of low-field NMR signals

BackgroundLow resolution nuclear magnetic resonance (LR-NMR) is a common technique to identify the constituents of complex materials (such as food and biological samples). The output of LR-NMR experiments is a relaxation signal which can be modelled as a type of convolution of an unknown density of relaxation times with decaying exponential functions, plus random Gaussian noise. The challenge is to estimate that density, a severely ill-posed problem. A complication is that non-negativity constraints need to be imposed in order to obtain valid results. Significance and noveltyWe present a smooth deconvolution model for solution of the inverse estimation problem in LR-NMR relaxometry experiments. We model the logarithm of the relaxation time density as a smooth function using (adaptive) P-splines while matching the expected residual magnetisations with the observed ones. The roughness penalty removes the singularity of the deconvolution problem, and the estimated density is positive by design (since we model its logarithm). The model is non-linear, but it can be linearized easily. The penalty has to be tuned for each given sample. We describe an efficient EM-type algorithm to optimize the smoothing parameter(s). ResultsWe analyze a set of food samples (potato tubers). The relaxation spectra extracted using our method are similar to the ones described in the previous experiments but present sharper peaks. Using penalized signal regression we are able to accurately predict dry matter content of the samples using the estimated spectra as covariates.

Maximum likelihood estimation of the log-concave component in a semi-parametric mixture with a standard normal density

The two-component mixture model with known background density, unknown signal density, and unknown mixing proportion has been studied in many contexts. One such context is multiple testing, where the background and signal densities describe the distribution of the p-values under the null and alternative hypotheses, respectively. In this paper, we consider the log-concave MLE of the signal density using the estimator of Patra & Sen (2016) for the mixing probability. We show that it is consistent and converges at the global rate n−2/5. An EM-algorithm in combination with an active set algorithm implemented in the R-package logcondens was used to compute the log-concave MLE. When one is interested in estimation at a fixed point, a conjecture is made about the limit distribution of our estimator. The performance of our method is assessed through a simulation study.

Exact Boundary Correction Methods for Multivariate Kernel Density Estimation

This paper develops a method to obtain multivariate kernel functions for density estimation problems in which the density function is defined on compact support. If domain-specific knowledge requires certain conditions to be satisfied at the boundary of the support of an unknown density, the proposed method incorporates the information contained in the boundary conditions into the kernel density estimators. The proposed method provides an exact kernel function that satisfies the boundary conditions, even for small samples. Existing methods primarily deal with a one-sided boundary in a one-dimensional problem. We consider density in a two-sided interval and extend it to a multi-dimensional problem.

Testing symmetry of model errors for non linear multiplicative distortion measurement error models

To study the symmetry and asymmetry of the model error under multiplicative distortion measurement errors setting, we propose a correlation coefficient-based measure between the distribution function and the root of density function. The unknown distribution function and density function are estimated from four kinds of residuals: the conditional mean calibration-based residuals, the conditional absolute mean calibration-based residuals, the conditional variance calibration-based residuals, and the conditional absolute logarithmic calibration-based residuals. We study the asymptotic results of the estimators of correlation coefficient-based measure under four calibrations. Next, we consider statistical inference of the correlation coefficient-based measure by using the empirical likelihood method. The empirical likelihood statistics are shown to be an asymptotically standard chi-squared distribution. Simulation studies demonstrate the performance of the proposed estimators and test statistics. A real example is analyzed to illustrate its practical usage.

Isogeometric multi-resolution full waveform inversion based on the finite cell method

Full waveform inversion (FWI) is an iterative identification process that serves to minimize the misfit of model-based simulated and experimentally measured wave field data. Its goal is to identify a field of parameters for a given physical object. For many years, FWI is very successful in seismic imaging to deduce velocity models of the earth or of local geophysical exploration areas. FWI has also been successfully applied in various other fields, including non-destructive testing (NDT) and biomedical imaging. The inverse optimization process of FWI relies on forward and backward solutions of the (elastic or acoustic) wave equation, as well as on efficient computations of adequate optimization directions. Many approaches employ (low order) finite element or finite difference methods, using parameterized material fields whose resolution is chosen in relation to the elements or nodes of the discretized wave field. In our previous paper (Bürchner et al., 2023), we investigated the potential of using the finite cell method (FCM) as the wave field solver. The FCM offers the advantage that highly complex geometric models can be incorporated easily. Furthermore, we demonstrated that the identification of the model’s density outperforms that of the velocity — particularly in cases where unknown voids characterized by homogeneous Neumann boundary conditions need to be detected. The paper at hand extends this previous study in the following aspects: The isogeometric finite cell analysis (IGA-FCM) – a combination of isogeometric analysis (IGA) and FCM – is applied as the wave field solver, with the advantage that the polynomial degree and subsequently also the sampling frequency of the wave field can be increased quite easily. Since the inversion efficiency strongly depends on the accuracy of the forward and backward wave field solutions and of the gradients of the functional, consistent, and lumped mass matrix discretization are compared. The resolution of the grid describing the unknown material density – thus allowing to identify voids in a physical object – is then decoupled from the knot span grid. Finally, we propose an adaptive multi-resolution algorithm that locally refines the material grid using an image processing-based refinement indicator. The developed inversion framework allows fast and memory-efficient wave simulations and object identification. While we study the general behavior of the proposed approach using 2D benchmark problems, a final 3D problem shows that it can also be used to identify void regions in geometrically complex spatial structures.

A novel variational robust filter with Gaussian mixture model for unknown non-Gaussian noises

In numerous practical applications, the presence of unpredictable abnormal behaviors often leads to random measurement outliers, causing the random measurement noises are no longer Gaussian but unknown non-Gaussian distributions, and it is intractable to obtain accurate a priori information about the noises. To address this challenge and enable nonlinear state estimation under such measurement noises, we propose a novel robust filter based on Gaussian mixture model (GMM). Firstly, a GMM is introduced to model the unknown non-Gaussian measurement likelihood probability density function (ML-PDF). Without relying on any prior noise statistics, the GMM can accurately model the ML-PDF utilizing a set of model parameters adaptively learned by the variational Bayesian (VB) method. Then, considering the high computational burden associated with the existing variational iterative process, we develop an improved VB method, which independently calculates the posterior distributions of state and unknown model parameters, thereby improving the computational efficiency. Finally, a new robust filter is derived by integrating the GMM with the improved VB approach, The effectiveness and superiority of the filter are demonstrated through both numerical simulations and real-world data analysis.

Intensity Tomography Method for Secure and High-Performance Quantum Key Distribution

Decoy-state method has greatly filled in loopholes from multi-photon pulses in the field of quantum key distribution (QKD) and it has been employed in almost all practical QKD systems in recent years. However, security and performance of the practical decoy-state QKDs are still limited by inaccurate modulations such as random intensity fluctuation and correlated intensity fluctuation. In this work, an intensity tomography method is proposed as a countermeasure against these fluctuations. It has an ability to tightly bound the unknown density matrices of photon-number state and accurately calibrate the fluctuations by measuring the decoy states with a local single-photon detector. We verified the tomography method by experiments and simulations. The results indicated that our method can efficiently mitigate the flaws from these fluctuations to improve the protocol performance and practical security. Noting that these fluctuations are inevitable in practical QKDs, our method would pave the avenue for QKD's practical applications.

A modified Christoffel function and its asymptotic properties

We introduce a certain variant (or regularization) Λ̃nμ of the standard Christoffel function Λnμ associated with a measure μ on a compact set Ω⊂Rd. Its reciprocal is now a sum-of-squares polynomial in the variables (x,ɛ), ɛ>0. It shares the same dichotomy property of the standard Christoffel function, that is, the growth with n of its inverse is at most polynomial inside and exponential outside the support of the measure. Its distinguishing and crucial feature states that for fixed ɛ>0, and under weak assumptions, limn→∞ɛ−dΛ̃nμ(ξ,ɛ)=f(ζɛ) where f (assumed to be continuous) is the unknown density of μ w.r.t. Lebesgue measure on Ω, and ζɛ∈B∞(ξ,ɛ) (and so f(ζɛ)≈f(ξ) when ɛ>0 is small). This is in contrast with the standard Christoffel function where if limn→∞ndΛnμ(ξ) exists, it is of the form f(ξ)/ωE(ξ) where ωE is the density of the equilibrium measure of Ω, usually unknown. At last but not least, the additional computational burden (when compared to computing Λnμ) is just integrating symbolically the monomial basis (xα)α∈Nnd on the box {x:‖x−ξ‖∞<ɛ/2}, so that 1/Λ̃nμ is obtained as an explicit polynomial of (ξ,ɛ).

Signal detection for FTN system under unknown noise model with imperfect CSI

The problem of signal detection for faster-than-Nyquist (FTN) signaling system with imperfect channel state information (CSI) is studied. Also, the non-Gaussian noise in real applications is considered. A general model, Gaussian mixture noise (GMN) model, is applied to describe the unknown probability density function (PDF) of the non-Gaussian noise. After that, a framework over variational inference is constructed to simplify the problem through finding an auxiliary PDF with factorization form. Based on the factor graph over the signal, joint response and the parameters of GMN, an iterative solution is obtained. Simulations demonstrate the effectiveness of the proposed approach.

Zoea Crab Larva Counter (CLARCO) Based On Image Processing With Adaptive Gaussian Filter Algorithm And Blob Detection Technique

The size of crab larvae is very small, so there is no accurate and easy-to-use crab larvae counting tool at an affordable price. The low larval survival rate is due to the unknown larval stocking density, which causes cannibalism, a feed-to-larvae ratio that is out of proportion to the number of larvae needed to maintain water quality, a water supply that is out of proportion to stocking density, and is economically unfavorable in terms of cultivation, feed management, maintaining water quality, and the buying and selling process. Accurately estimating the amount of crab larvae is anticipated to boost their survival rate and make them more economically successful. This research is a continuation of previous research with the addition of different methods using adaptive gaussian filter algorithms and blob detection techniques to count crab larvae in the zoea-2, zoea-3, and zoea-4 phases where an increase in accuracy was obtained with an average of 97.67%.

On an Inverse Problem for the Plate Equation with Passive Measurement

This paper focuses on an inverse problem associated with the plate equation which is derived from models in fluid mechanics and elasticity. We establish the unique identifying results in simultaneously determining both the unknown density and the internal sources from the passive boundary measurement. The proof mainly relies on the asymptotic analysis and harmonic analysis on integral transforms.

A dislocation-based model for shear cracks in arbitrary orientations under contact loading

Mechanical transmission components are generally under the risk of contact fatigue, resulting in cracks initiated from the subsurface. When subject to compressive stress, these cracks might be forced to be closed, thus their faces are in partial slip contact. Under such conditions, the interactions of stick and slip zones cannot be directly determined, which is one of the greatest challenges of contact fatigue problems. In this study, the distributed dislocation technique is adopted to investigate the stick–slip friction along the alignment of shear cracks in arbitrary orientations in a half-plane under contact loading. The unknown dislocation density describing the crack face displacement is iteratively calculated based on the stress conditions and then introduced to update the surface gap between contacting bodies. The complex expressions of the influence coefficients used to determine the stress and displacement induced by the dislocations are derived and the FORTRAN codes are provided in the Supplementary Materials. A step-by-step loading method is proposed to progressively capture the slip history of crack faces. The model is also capable of simulating rolling contact problems. The results are comparable to those predicted using the finite element method with more robust convergence properties. The effects of friction coefficient, crack alignment, and locations of crack faces are discussed in detail. The model paves the foundation for further studies on competing mechanisms of crack propagation and crack face wear, and provides insights into the frictional fracture of materials under contact loading.

Augmented Complex Minimization of Error Entropy With Fiducial Points for Power System Frequency Estimation

Frequency estimation plays an important role in ensuring the stability of the power system. The augmented complex minimum error entropy (ACMEE) algorithm has achieved better performance in frequency estimation under impulsive noise environments. However, it has the problem of unknown error probability density function (PDF) in frequency tracking, the improved algorithms called the complex minimization of error entropy with fiducial points (CMEEF) and the augmented CMEEF (ACMEEF) are proposed in this paper. The ACMEEF algorithm based on widely linear modeling uses second-order statistical information to process the non-circular signal through the complex voltage signal. At the same time, it incorporates a location-sensitive cost into minimum error entropy (MEE) to automatically locate the peak of the error PDF at the origin, which can accurately estimate the grid frequency under the unbalanced system and interference of noise. The stability of the proposed algorithm and the computational complexity are analyzed. Finally, the enhanced frequency estimation performance of the proposed ACMEEF algorithm is verified through simulation results of different synthetic signals and real-world data.

Collective Cognition on Global Density in Dynamic Swarm.

Swarm density plays a key role in the performance of a robot swarm, which can be averagely measured by swarm size and the area of a workspace. In some scenarios, the swarm workspace may not be fully or partially observable, or the swarm size may decrease over time due to out-of-battery or faulty individuals during operation. This can result in the average swarm density over the whole workspace being unable to be measured or changed in real-time. The swarm performance may not be optimal due to unknown swarm density. If the swarm density is too low, inter-robot communication will rarely be established, and robot swarm cooperation will not be effective. Meanwhile, a densely-packed swarm compels robots to permanently solve collision avoidance issues rather than performing the main task. To address this issue, in this work, the distributed algorithm for collective cognition on the average global density is proposed. The main idea of the proposed algorithm is to help the swarm make a collective decision on whether the current global density is larger, smaller or approximately equal to the desired density. During the estimation process, the swarm size adjustment is acceptable for the proposed method in order to reach the desired swarm density.

A High-Order-Accurate 3D Surface Integral Equation Solver for Uniaxial Anisotropic Media

This paper introduces a high-order accurate surface integral equation method for solving 3D electromagnetic scattering for dielectric objects with uniaxially anisotropic permittivity tensors. The N-Müller formulation is leveraged resulting in a second-kind integral formulation, and a finite-difference-based approach is used to deal with the strongly singular terms resulting from the dyadic Green’s functions for uniaxially anisotropic media while maintaining the high-order accuracy of the discretization strategy. The integral operators are discretized via a Nyström-collocation approach, which represents the unknown surface densities in terms of Chebyshev polynomials on curvilinear quadrilateral surface patches. The convergence is investigated for various geometries, including a sphere, cube, a complicated NURBS geometry imported from a 3D CAD modeler software, and a nanophotonic silicon waveguide, and results are compared against a commercial finite element solver. To the best of our knowledge, this is the first demonstration of high-order accuracy for objects with uniaxially anisotropic materials using surface integral equations.

THE USE OF MODEL DISTRIBUTIONS IN THE ANALYSIS OF QUEUING SYSTEMS WITH CORRELATED ARRIVALS

The article considers the task of determining the waiting time for a request in a queue in a general-type single-channel queueing system when time intervals between incoming requests are correlated. For solving the task, it is proposed to carry out decorrelation of specified time intervals using the discrete cosine transform. Generally, to analyze arbitrary distributions included in the expression for a decorrelated sequence, one can suggest an approach based on the approximation of unknown densities by a model distribution. The calculation of cumulants and moments of random variables is very simple compared to the analytical methods. Based on the found final set of cumulants, a model probability density can always be reproduced with a certain error. Generally, to calculate the average waiting time of a request in a queue, it is necessary to determine the correlation properties and one-dimensional probability density of time intervals between incoming requests, synthesize a two-dimensional probability density of a sequence of given intervals that has a measured correlation function, and, further, calculate joint moments and cumulants for correlated values, on which their model density is built.

Superpolynomial quantum-classical separation for density modeling

Density modelling is the task of learning an unknown probability density function from samples, and is one of the central problems of unsupervised machine learning. In this work, we show that there exists a density modelling problem for which fault-tolerant quantum computers can offer a super-polynomial advantage over classical learning algorithms, given standard cryptographic assumptions. Along the way, we provide a variety of additional results and insights, of potential interest for proving future distribution learning separations between quantum and classical learning algorithms. Specifically, we (a) provide an overview of the relationships between hardness results in supervised learning and distribution learning, and (b) show that any weak pseudo-random function can be used to construct a classically hard density modelling problem. The latter result opens up the possibility of proving quantum-classical separations for density modelling based on weaker assumptions than those necessary for pseudo-random functions.