Kernel Estimation Research Articles

Blind image deblurring using both L0 and L1 regularization of Max-min prior

Image deblurring involves estimating the blur kernel from intermediate latent images to recover their corresponding sharp, deblurred versions. In recent years, sophisticated priors such as dark channel, extreme channel, local maximum gradient, and patch-wise minimal pixels have demonstrated promising effectiveness. In this study, we propose a Max-min (Mm) prior for effective blind image deblurring. This work is motivated by the observation that the 1−Mm map of blurred images is less sparse than that of the corresponding clean images. The difference between the highest and lowest intensities around dominant edges is greater than in smooth areas. We theoretically and empirically demonstrate that blurring greatly diminishes this inherent characteristic. This property enables us to establish a new energy function and propose a blur kernel estimation model using L0 and L1 regularization of the Mm prior. The proposed method employs a linear operator to compute the Mm map, combined with an efficient optimization scheme to handle various specific scenarios. Through visual and quantitative experiments, we show that the proposed algorithm outperforms state-of-the-art algorithms. In terms of deblurring quality, robustness, and computational efficiency, the developed model surpasses comparable methods. The implemented code is available at https://github.com/eqtedaei/mm-prior-deblur

TRANSFORMATION OPERATORS FOR THE PERTURBED HILL EQUATION WITH COMPLEX COEFFICIENTS

In this paper, we consider the perturbed Hill equation on the entire axis. Using a transformation operator that preserves asymptotics at infinity, triangular representations of special solutions of this equation are found. Estimates for the representation kernels are obtained.

Optimal smoothing parameter selection in single-index model derivative estimation

. Single-index model is one of the most popular semiparametric models in applied econometrics. Estimation of the derivative function is often of crucial importance, as studying “marginal effects” serves as a cornerstone of microeconomics. A prerequisite for the successful application of nonparametric/semiparametric kernel estimation methods is to select smoothing parameters properly to balance the estimation squared bias and variance. Henderson et al. (2015) propose a novel method for selecting the smoothing parameters optimally for derivative function estimation. However, their method suffers from the “curse of diemnsionality” problem in a multivariate nonparametric regression model. In this article, we extend the work of Henderson et al. (2015) to estimation of the derivative function of a single-index model. Specifically, we propose a data-driven method to select smoothing parameters optimally for single-index model derivative function estimation. We also derive the asymptotic distribution of the resulting local linear estimator of the derivative function. Both simulations and empirical applications show that the proposed method works well in practice.

Epidemiologia espacial aplicada à identificação de corredores da paisagem e dispersão da febre amarela no estado de Minas Gerais, sudeste do Brasil

This study aims to identify landscape corridors best able to support Yellow Fever (YF) virus spread and risk areas based on the disease transmission process. Spatial approaches tools were used, such as the land use and land cover to define buffers, kernel estimation to identify areas with greater intensity of ecological corridors, and the Moran Local Bivariate Index of human cases, epizootics of Non-Human Primates (NHP), and the number of corridor areas. Reported 975 Human cases of YF and 2,424 NHP epizootics, but only 576 were confirmed. The approach allowed the identification of the principal risk in corridors with a larger area, where groups of NHP can move more frequently, leading to crossings between different groups and providing populations with greater genetic diversity. In this situation, virus spread would be less likely to cause YF in many NHPs, acting as a regulator of virus circulation.

Bayesian taut splines for estimating the number of modes

The number of modes in a probability density function is representative of the complexity of a model and can also be viewed as the number of subpopulations. Despite its relevance, there has been limited research in this area. A novel approach to estimating the number of modes in the univariate setting is presented, focusing on prediction accuracy and inspired by some overlooked aspects of the problem: the need for structure in the solutions, the subjective and uncertain nature of modes, and the convenience of a holistic view that blends local and global density properties. The technique combines flexible kernel estimators and parsimonious compositional splines in the Bayesian inference paradigm, providing soft solutions and incorporating expert judgment. The procedure includes feature exploration, model selection, and mode testing, illustrated in a sports analytics case study showcasing multiple companion visualisation tools. A thorough simulation study also demonstrates that traditional modality-driven approaches paradoxically struggle to provide accurate results. In this context, the new method emerges as a top-tier alternative, offering innovative solutions for analysts.

Insight into soft chemometric computational learning for modelling oily-wastewater separation efficiency and permeate flux of polypyrrole-decorated ceramic-polymeric membranes

Reliable modeling of oily wastewater emphasizes the paramount importance of sustainable and health-conscious wastewater management practices, which directly aligns with the Sustainable Development Goals (SDG) while also meeting the guidelines of the World Health Organization (WHO). This research explores the efficiency of utilizing polypyrrole-coated ceramic-polymeric membranes to model oily wastewater separation efficiency (SE) and permeate flux (PF) based on established experimental procedures. In this area, computational simulation still needs to be explored. The study developed predictive regression models, including robust linear regression (RLR), stepwise linear regression (SWR) and linear regression (LR) for the ceramic-polymeric porous membrane, aiming to interpret its complex performance across diverse conditions and, thus, develop its utility in oily wastewater treatment applications. Subsequently, a novel, simple average ensemble paradigm was explored to reduce errors and improve prediction skills. Prior to the development of the model, stability and reliability analysis of the data was conducted based on Philip Perron tests with the Bartlett kernel estimation method. The accuracy of the SE exhibited a high consistency, averaging 99.92% with minimal variability (standard deviation of 0.026%), potentially simplifying its prediction compared to PF. The modes were validated and evaluated using metrics like MAE, RMSE, Speed, and MSE, in addition to 2D graphical and cumulative distribution function graphs. The LR model emerged as the best with the lowest RMSE =0.21951, indicating superior prediction accuracy, followed closely by RLR with an RMSE = 0.22359. SWLR, while having the highest RMSE = 0.34573, marked its dominance in prediction speed with 110 observations per second. Notably, the RLR model justified a reduction in error by approximately 35.29% compared to SWLR. Moreover, the training efficiency of the LR model exceeded, demanding a mere 2.9252 s, marking a reduction of about 32.54% compared to SWLR. The improved simple ensemble learning proved merit over the three models regarding error accuracy. This study emphasizes the essential role of soft-computing learning in optimizing the design and performance of ceramic-polymeric membranes.

Non-local degradation modeling for spatially adaptive single image super-resolution

Existing methods for single image super-resolution (SISR) model the blur kernel as spatially invariant across the entire image, and are susceptible to the adverse effects of textureless patches. To achieve improved results, adaptive estimation of the degradation kernel is necessary. We explore the synergy of joint global and local degradation modeling for spatially adaptive blind SISR. Our model, named spatially adaptive network for blind super-resolution (SASR), employs a simple encoder to estimate global degradation representations and a decoder to extract local degradation. These two representations are fused with a cross-attention mechanism and applied using spatially adaptive filtering to enhance the local image detail. Specifically, SASR contains two novel features: (1) a non-local degradation modeling with contrastive learning to learn global and local degradation representations, and (2) a non-local spatially adaptive filtering module (SAFM) that incorporates the global degradation and spatial-detail factors to preserve and enhance local details. We demonstrate that SASR can efficiently estimate degradation representations and handle multiple types of degradation. The local representations avoid the detrimental effect of estimating the entire super-resolved image with only one kernel through locally adaptive adjustments. Extensive experiments are performed to quantitatively and qualitatively demonstrate that SASR not only performs favorably for degradation estimation but also leads to state-of-the-art blind SISR performance when compared to alternative approaches.

A hybrid nonparametric multivariate density estimator with applications to risk management

. Multivariate density estimation is plagued by the curse of dimensionality in theory and practice. We propose a hybrid density estimator of a multivariate density f that combines the strengths of the kernel estimator and the exponential series estimator. This estimator refines a preliminary kernel estimate f ̂ 0 with a multiplicative correction that estimates the ratio r = f / f ̂ 0 with an exponential series estimator. Thanks to the consistency of the pilot estimate, the coefficients of the series expansion tend to approach zero asymptotically. Accordingly, we design a thresholding method for basis function selection. A major obstacle of multivariate exponential series estimator is the calculation of its normalization factor. We resolve this difficulty with Monte Carlo integration, using the pilot kernel estimate as the trial density for importance sampling. This approach greatly enhances the practicality of the hybrid estimator. Numerical simulations demonstrate the good finite sample performance of the hybrid estimator. We present one empirical application in financial risk management.

Memory effects in disease modelling through kernel estimates with oscillatory time history

We design a linear chain trick algorithm for dynamical systems for which we have oscillatory time histories in the distributed time delay. We make use of this algorithmic framework to analyse memory effects in disease evolution in a population. The modelling is based on a susceptible-infected-recovered SIR—model and on a susceptible-exposed-infected-recovered SEIR—model through a kernel that dampens the activity based on the recent history of infectious individuals. This corresponds to adaptive behavior in the population or through governmental non-pharmaceutical interventions. We use the linear chain trick to show that such a model may be written in a Markovian way, and we analyze the stability of the system. We find that the adaptive behavior gives rise to either a stable equilibrium point or a stable limit cycle for a close to constant number of susceptibles, i.e. locally in time. We also show that the attack rate for this model is lower than it would be without the dampening, although the adaptive behavior disappears as time goes to infinity and the number of infected goes to zero.

The Law of the Iterated Logarithm for Lp-Norms of Kernel Estimators of Cumulative Distribution Functions

In this paper, we consider the strong convergence of Lp-norms (p≥1) of a kernel estimator of a cumulative distribution function (CDF). Under some mild conditions, the law of the iterated logarithm (LIL) for the Lp-norms of empirical processes is extended to the kernel estimator of the CDF.

Consistency of the frequency polygon estimators of density function and density mode under m-END samples

. The frequency polygon estimation, which is based on the histogram technique, has a similar convergence rate as those of modern non negative kernel estimators and the advantage of computational simplicity. This article will study the uniformly complete consistency and the rate of uniformly complete consistency of frequency polygon estimation for m-END samples under some general conditions, which improve and extend the corresponding ones in the literature. As an application, the uniformly complete consistency and the corresponding rate of density mode estimator are also obtained. Some numerical simulations are also carried out to support the theoretical results.

A new nonparametric copula framework for the joint analysis of river water temperature and low flow characteristics for aquatic habitat risk assessment

This study proposed a nonparametric copula hazard framework in the joint risk of river water temperature (RWT) and low flow (LF) events for aquatic ecosystems, specifically ectotherm fish. This nonparametric copula density can adapt to any mutual dependence structure, providing greater flexibility. This can reduce the risk of misspecification if the underlying assumption is violated compared to conventional parametric or semiparametric copula settings. The analysis uses nonparametric copulas densities like Beta kernel copula estimator (BKCE), Bernstein copula estimator (BCE), and Transformation kernel estimator (TKE), conjoined with Gaussian Kernel density estimations (GKDEs) and parametric marginals for joint annual maximum RWT (AMRWT) and LF. The study compares different models for analyzing five Swiss river basins: parametric copulas with best-fitted GKDE and parametric margins, versus nonparametric copula. Six bandwidth selectors are estimated to fit GKDE. BKCE with GKDE margins outperformed most stations, while TKE and BCE density with GKDE margins work best for only one station. However, at one station, BKCE with best-fitted parametric margins is outperformed. All stations except Station 2473 are characterized by temperature and low flows that may be conducive to stress of a number of aquatic species, causing high AMRWT (exceeding 19 °C) and minimum LF or specific discharge, SD) quantiles at low AND-joint return periods (RP). AND (i.e. low flow AND high temperature) hazard events are less likely to occur together than OR hazard events, while univariate RPs happen more often than OR-joint RPs. In addition, the joint RP of AMRWT, given LF at different percentiles, significantly affected AMRWT for various LF conditions. Higher AMRWT with low flow conditions results in lower joint RP (except station 2473), and this stress level would be reduced when conditioned with high LF events at the same AMRWT. Station 2473 has high LF even at low percentiles, and with low AMRWT makes it very less stressful than other station under different LF conditions. Summer river flow maintenance can improve aquatic environments with high RWT. Analyzing joint statistics is crucial to understanding mutual risk in freshwater ecosystems.

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model

We present on-diagonal heat kernel estimates and quantitative homogenization statements for the one-dimensional Bouchaud trap model. The heat kernel estimates are obtained using standard techniques, with key inputs coming from a careful analysis of the volume growth of the invariant measure of the process under study. As for the quantitative homogenization results, these include both quenched and annealed Berry–Esseen-type theorems, as well as a quantitative quenched local limit theorem. Whilst the model we study here is a particularly simple example of a random walk in a random environment, we believe the roadmap we provide for establishing the latter result in particular will be useful for deriving quantitative local limit theorems in other, more challenging, settings.

A powerful nonparametric test of the effect of dementia duration on mortality

A continuous-time multi-state history is semi-Markovian, if an intensity to migrate from one state into a second, depends on the duration in the first state. Such duration can be formalised as marker, entering the intensity process of the transition counts. We derive the intensity process, prove its predictability and the martingale property of the residual to the integrated intensity. In particular, we verify the usual conditions for the respective filtration. As a consequence, according to Nielsen and Linton (1995), a kernel estimator of the transition intensity, including the duration dependence, converges point-wise at a slow rate, compared to the fast rate of the Markovian kernel estimator, i.e. when ignoring duration dependence. By using the inequality of the two rates, we follow Gozalo (1993) and show that the (properly penalised) maximal difference of the two kernel estimators on a random grid of points is asymptotically χ 1 2 -distributed. As compared to the χ 2 goodness-of-fit test on a fixed grid, with less power, the estimator of the variance needs to be shown to converge uniformly. The data example is a sample of 130,000 German women, observed over a period of 10 years, and we model the intensity to die with dementia, potentially dependent on the disease duration. As usual, the models under the null and the alternative hypothesis need to be enlarged to allow for independent right-censoring. We find a significant effect of dementia duration, nearly independent of the bandwidth. A Monte Carlo simulation confirms level and high power of the test with the penalised maximal difference, under the conditions of our application. It also reminds that hyperparameter selection is sensitive to the application.

Fundamental properties of Cauchy–Szegő projection on quaternionic Siegel upper half space and applications

Abstract We investigate the Cauchy–Szegő projection for quaternionic Siegel upper half space to obtain the pointwise (higher order) regularity estimates for Cauchy–Szegő kernel and prove that the Cauchy–Szegő kernel is nonzero everywhere, which further yields a non-degenerated pointwise lower bound. As applications, we prove the uniform boundedness of Cauchy–Szegő projection on every atom on the quaternionic Heisenberg group, which is used to give an atomic decomposition of regular Hardy space H p {H^{p}} on quaternionic Siegel upper half space for 2 3 < p ≤ 1 {\frac{2}{3}<p\leq 1} . Moreover, we establish the characterisation of singular values of the commutator of Cauchy–Szegő projection based on the kernel estimates. The quaternionic structure (lack of commutativity) is encoded in the symmetry groups of regular functions and the associated partial differential equations.

Weber's Law-based Regularization for Blind Image Deblurring

Blind image deblurring aims to recover an output latent image and a blur kernel from a given blurred image. Kernel estimation is a significant step in blind image deblurring and requires a regularization technique to minimize the cost function and the edges of objects to generate a sharp image in a better way. This study proposes a new image regularization technique called Weber's Law Regularization (WLR) based on the Weber law phenomenon. The Weber ratio was used to preserve the edges of small salient objects and to minimize the cost function to obtain a sharp image while minimizing the ringing effect. To validate the WLR, experiments were conducted on benchmark synthetic and real word images and compared with existing state-of-the-art methods. The experimental results showed that WLR can effectively and efficiently deblur images even in the absence of prior knowledge.

Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.

Nonparametric predictive model for sparse and irregular longitudinal data.

We propose a kernel-based estimator to predict the mean response trajectory for sparse and irregularly measured longitudinal data. The kernel estimator is constructed by imposing weights based on the subject-wise similarity on L2 metric space between predictor trajectories, where we assume that an analogous fashion in predictor trajectories over time would result in a similar trend in the response trajectory among subjects. In order to deal with the curse of dimensionality caused by the multiple predictors, we propose an appealing multiplicative model with multivariate Gaussian kernels. This model is capable of achieving dimension reduction as well as selecting functional covariates with predictive significance. The asymptotic properties of the proposed nonparametric estimator are investigated under mild regularity conditions. We illustrate the robustness and flexibility of our proposed method via extensive simulation studies and an application to the Framingham Heart Study.

Sobolev embeddings for kinetic Fokker-Planck equations

We introduce intrinsic Sobolev-Slobodeckij spaces for a class of ultra-parabolic Kolmogorov type operators satisfying the weak Hörmander condition. We prove continuous embeddings into Lorentz and intrinsic Hölder spaces. We also prove approximation and interpolation inequalities by means of an intrinsic Taylor expansion, extending analogous results for Hölder spaces. The embedding at first order is proved by adapting a method by Luc Tartar which only exploits scaling properties of the intrinsic quasi-norm, while for higher orders we use uniform kernel estimates.