Kernel Smoothing Research Articles

Estimation of neuronal firing rate using Bayesian Adaptive Kernel Smoother (BAKS).

Neurons use sequences of action potentials (spikes) to convey information across neuronal networks. In neurophysiology experiments, information about external stimuli or behavioral tasks has been frequently characterized in term of neuronal firing rate. The firing rate is conventionally estimated by averaging spiking responses across multiple similar experiments (or trials). However, there exist a number of applications in neuroscience research that require firing rate to be estimated on a single trial basis. Estimating firing rate from a single trial is a challenging problem and current state-of-the-art methods do not perform well. To address this issue, we develop a new method for estimating firing rate based on a kernel smoothing technique that considers the bandwidth as a random variable with prior distribution that is adaptively updated under an empirical Bayesian framework. By carefully selecting the prior distribution together with Gaussian kernel function, an analytical expression can be achieved for the kernel bandwidth. We refer to the proposed method as Bayesian Adaptive Kernel Smoother (BAKS). We evaluate the performance of BAKS using synthetic spike train data generated by biologically plausible models: inhomogeneous Gamma (IG) and inhomogeneous inverse Gaussian (IIG). We also apply BAKS to real spike train data from non-human primate (NHP) motor and visual cortex. We benchmark the proposed method against established and previously reported methods. These include: optimized kernel smoother (OKS), variable kernel smoother (VKS), local polynomial fit (Locfit), and Bayesian adaptive regression splines (BARS). Results using both synthetic and real data demonstrate that the proposed method achieves better performance compared to competing methods. This suggests that the proposed method could be useful for understanding the encoding mechanism of neurons in cognitive-related tasks. The proposed method could also potentially improve the performance of brain-machine interface (BMI) decoder that relies on estimated firing rate as the input.

Empirical Likelihood Inference for Functional Coefficient ARCH-M Model

Empirical likelihood inference for parametric and nonparametric parts in functional coefficient ARCH-M models is investigated in this paper. Firstly, the kernel smoothing technique is used to estimate coefficient function δ(x). In this way we obtain an estimated function with parameter β. Secondly, the empirical likelihood method is developed to estimate the parameter β. An estimated empirical log-likelohood ratio is proved to be asymptotically standard chi-squred, and the maximum empirical likelihood estimation (MELE) for β is shown to be asymptotically normal. Finally, based on the MELE of β, the empirical likelihood approach is again applied to reestimate the nonparametric part δ(x). The empirical log-likelohood ratio for δ(x) is proved to be also asymptotically standard chi-squred. Simulation study shows that the proposed method works better than the normal approximation method in terms of average areas of confidence regions for β, and the empirical likelihood confidence belt for δ(x) performs well.

Estimation of total electricity consumption curves by sampling in a finite population when some trajectories are partially unobserved

AbstractMillions of smart meters that are able to collect individual load curves, that is, electricity consumption time series, of residential and business customers at fine scale time grids are now deployed by electricity companies all around the world. It may be complex and costly to transmit and exploit such a large quantity of information, therefore it can be relevant to use survey sampling techniques to estimate mean load curves of specific groups of customers. Data collection, like every mass process, may undergo technical problems at every point of the metering and collection chain resulting in missing values. We consider imputation approaches (linear interpolation, kernel smoothing, nearest neighbours, principal analysis by conditional estimation) that take advantage of the specificities of the data, that is to say the strong relation between the consumption at different instants of time. The performances of these techniques are compared on a real example of Irish electricity load curves under various scenarios of missing data. A general variance approximation of total estimators is also given which encompasses nearest neighbours, kernel smoothers imputation and linear imputation methods. The Canadian Journal of Statistics 47: 65–89; 2019 © 2018 Statistical Society of Canada

Data-Adaptive Estimation of Time-Varying Spectral Densities

ABSTRACTThis article introduces a data-adaptive nonparametric approach for the estimation of time-varying spectral densities from nonstationary time series. Time-varying spectral densities are commonly estimated by local kernel smoothing. The performance of these nonparametric estimators, however, depends crucially on the smoothing bandwidths that need to be specified in both time and frequency direction. As an alternative and extension to traditional bandwidth selection methods, we propose an iterative algorithm for constructing localized smoothing kernels data-adaptively. The main idea, inspired by the concept of propagation-separation, is to determine for a point in the time-frequency plane the largest local vicinity over which smoothing is justified by the data. By shaping the smoothing kernels nonparametrically, our method not only avoids the problem of bandwidth selection in the strict sense but also becomes more flexible. It not only adapts to changing curvature in smoothly varying spectra but also adjusts for structural breaks in the time-varying spectrum. Supplementary materials, including the R package tvspecAdapt containing an implementation of the routine, are available online.

Modelling Time-Varying Income Elasticities of Health Care Expenditure for the OECD

Income elasticity dynamics of health expenditure is considered for the OECD and the Eurozone over the period 1995-2014. This paper studies a novel non-linear cointegration model with fixed effects, controlling for cross-section dependence and unobserved heterogeneity. Most importantly, its coefficients can vary over time and its variables can be non-stationary. The resulting asymptotic theory is fundamentally different with a faster rate of convergence to similar kernel smoothing methodologies. A fully modified kernel regression method is also proposed to reduce the asymptotic bias. Results show a steep increase in the income elasticity for the OECD and a small increase for the Eurozone.

Star cluster detection and characterization using generalized Parzen density estimation

Star cluster studies hold the key to understanding star formation, stellar evolution, and origin of galaxies. The detection and characterization of clusters depend on the underlying background density and the cluster richness. We examine the ability of the Parzen Density Estimation (a.k.a. Parzen Windows) method, which is a generalization of the well-known Star Count method, to detect clusters and measure their properties. We apply it on a range of simulated and real star fields, considering square and circular windows, with and without Gaussian kernel smoothing. Our method successfully identifies clusters and we suggest an optimal standard deviation of the Gaussian Parzen window for obtaining the best estimates of these parameters. Finally, we demonstrate that the Parzen Windows with Gaussian kernels are able to detect small clusters in regions of relatively high background density where the Star Count method fails.

Modeling arterial travel time distribution by accounting for link correlations: a copula-based approach

The estimation of urban arterial travel time distribution (TTD) is critical to help implement Intelligent Transportation Systems (ITS) and provide travelers with timely and reliable route guidance. The state-of-practice procedure for arterial TTD estimation commonly assumes that the path travel time follows a certain distribution without considering link correlations. However, this approach appears inappropriate since travel times on successive links are essentially dependent along signalized arterials. In this study, a copula-based approach is proposed to model arterial TTD by accounting for spatial link correlations. First, TTDs on consecutive links along one arterial in Hangzhou, China are investigated. Link TTDs are estimated through the nonparametric kernel smoothing method. Link correlations are analyzed in both unfavorable and favorable coordination cases. Then, Gaussian copula models are introduced to model the dependent structure between link TTDs. The parameters of Gaussian copula are obtained by Maximum-Likelihood Estimation (MLE). Next, path TTDs covering consecutive links are estimated based on the estimated copula models. The results demonstrate the advantage of the proposed copula-based approach, compared with the convolution without capturing link correlations and the empirical distribution fitting methods in both unfavorable and favorable coordination cases.

Feature screening in ultrahigh-dimensional partially linear models with missing responses at random

This paper proposes a new feature screening procedure in ultrahigh-dimensional partially linear models with missing responses at random for longitudinal data based on the profile marginal kernel-assisted estimating equations imputation technique. The proposed feature screening procedure has three key merits. First, it is computationally efficient, and can be used to screen significant covariates in the presence of missing responses. Second, it does not require estimating respondent probability and is robust to the misspecification of respondent probability models. Third, the univariate kernel smoothing method is adopted to estimate nonparametric functions, and is employed to impute estimating equations with missing responses at random, which avoids the well-known “curse of dimensionality”. The ranking consistency property and the sure screening property are shown under some regularity conditions. Simulation studies are conducted to investigate the finite sample performance of the proposed screening procedure. An example is used to illustrate the proposed procedure.

Limits and reproducibility of resting-state functional MRI definition of DLPFC targets for neuromodulation

BackgroundTranscranial magnetic stimulation (TMS) is a noninvasive neuromodulation technique with therapeutic applications for the treatment of major depressive disorder (MDD). The standard protocol uses high frequency stimulation over the left dorsolateral prefrontal cortex (DLPFC) identified in a heuristic manner leading to moderate clinical efficacy. A proposed strategy to increase the anatomical precision in targeting, based on resting-state functional MRI (rsfMRI), identifies the subregion within the DLPFC having the strongest anticorrelated functional connectivity with the subgenual cortex (SGC) for each individual subject. ObjectiveIn this work, we comprehensively test the reliability and reproducibility of this targeting method for different scan lengths on 100 subjects from the Human Connectome Project (HCP) where each subject had a four 15-min rsfMRI scan on 2 different days. MethodsWe quantified the inter-scan and inter-day distance between the rsfMRI-guided DLPFC targets for each subject controlling for a number of expected sources of noise using volumetric as well as surface analyses. ResultsOur results show that the average inter-day distance (with fMRI scans lasting 30 min on each day) is 25% less variable than the inter-scan distance, which uses 50% less data. Specifically, the inter-scan distance was more than 37 mm, while for the longer-scan, the inter-day distance had lower variability at 25 mm. Finally, we tested the same rsfMRI strategy using the nucleus accumbens (NAc) as a control region relevant to MDD but less susceptible to artifacts, using both volume and surface rsfMRI data. The results showed similar variability to the SGC-DLPFC functional connectivity. Moreover, our results suggest that a smoothing kernel with 12 mm full-width half maximum (FWHM) lead to more stable and reliable target estimates. ConclusionOur work provides a quantitative assessment of the topographic precision of this targeting method, describing an anatomical variability that may surpass the spatial resolution of some forms of focal TMS as it is commonly applied, and provides recommendations for improved accuracy.

Efficient Estimation of the Nonparametric Mean and Covariance Functions for Longitudinal and Sparse Functional Data

ABSTRACTWe consider the estimation of mean and covariance functions for longitudinal and sparse functional data by using the full quasi-likelihood coupling a modification of the local kernel smoothing method. The proposed estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient in terms of their linear functionals. Their superiority to the competitors is further illustrated numerically through simulation studies. The method is applied to analyze AIDS study and atmospheric study. Supplementary materials for this article are available online.

Three‐dimensional image registration using distributed parallel computing

Three-dimensional (3D) images have become increasingly popular in practice. They are commonly used in medical imaging applications. In such applications, it is often critical to compare two 3D images, or monitor a sequence of 3D images. To make the image comparison or image monitoring valid, the related 3D images should be geometrically aligned first, which is called image registration (IR). However, IR for 3D images would take much computing time, especially when a flexible method is considered, which does not impose any parametric form on the underlying geometric transformation. Here, the authors explore a fast-computing environment for 3D IR based on the distributed parallel computing. The selected 3D IR method is based on the Taylor's expansion and 3D local kernel smoothing. It is flexible, but involves much computation. The authors demonstrate that this fast-computing environment can effectively handle the computing problem while keeping the good properties of the 3D IR method. The method discussed here is therefore useful for applications involving big data.

Quantifying and Visualizing Intraregional Connectivity in Resting-State Functional Magnetic Resonance Imaging with Correlation Densities.

The use of correlation densities is introduced to quantify and provide visual interpretation for intraregional functional connectivity in the brain. For each brain region, pairwise correlations are computed between a seed voxel and other gray matter voxels within the region, and the distribution of the ensemble of these correlation values is represented as a probability density, the correlation density. The correlation density can be estimated by kernel smoothing. It provides an intuitive and comprehensive representation of subject-specific functional connectivity strength at the local level for each region. To address the challenge of interpreting and utilizing this rich connectivity information when multiple regions are considered, methods from functional data analysis are implemented, including a recently developed method of dimensionality reduction specifically tailored to the analysis of probability distributions. To illustrate the utility of these methods in neuroimaging, experiments were carried out to identify the associations between local functional connectivity and a battery of neurocognitive scores. These experiments demonstrate that correlation densities facilitate the discovery and interpretation of specific region-score associations.

Pseudo-True SDFs in Conditional Asset Pricing Models*

Abstract This article is motivated by the need to bridge some gap between modern asset pricing theory and recent developments in econometric methodology. While asset pricing theory enhances the use of conditional pricing models, econometric inference of conditional models can be challenging due to misspecification or weak identification. To tackle the case of misspecification, we utilize the conditional Hansen and Jagannathan (1997) (HJ) distance as studied by Gagliardini and Ronchetti (2016), but we set the focus on interpretation and estimation of the pseudo-true value defined as the argument of the minimum of this distance. While efficient Generalized Method of Moments (GMM) has no meaning for estimation of a pseudo-true value, the HJ-distance not only delivers a meaningful loss function, but also features an additional advantage for the interpretation and estimation of managed portfolios whose exact pricing characterizes the pseudo-true pricing kernel (stochastic discount factor (SDF)). For conditionally affine pricing kernels, we can display some managed portfolios which are well-defined independently of the pseudo-true value of the parameters, although their exact pricing is achieved by the pseudo-true SDF. For the general case of nonlinear SDFs, we propose a smooth minimum distance (SMD) estimator (Lavergne and Patilea, 2013) that avoids a focus on specific directions as in the case of managed portfolios. Albeit based on kernel smoothing, the SMD approach avoids instabilities and the resulting need of trimming strategies displayed by classical local GMM estimators when the density function of the conditioning variables may take arbitrarily small values. In addition, the fact that SMD may allow fixed bandwidth asymptotics is helpful regarding the curse of dimensionality. In contrast with the true unknown value for a well-specified model, the estimated pseudo-true value, albeit defined in a time-invariant (unconditional) way, may actually depend on the choice of the state variables that define fundamental factors and their scaling weights. Therefore, we may not want to be overly parsimonious about the set of explanatory variables. Finally, following Antoine and Lavergne (2014), we show how SMD can be further robustified to deal with weaker identification contexts. Since SMD can be seen as a local extension of the method of jackknife GMM (Newey and Windmeijer, 2009), we characterize the Gaussian asymptotic distribution of the estimator of the pseudo-true value using classical U-statistic theorems.

Bayesian nonstationary Gaussian process models via treed process convolutions

The Gaussian process is a common model in a wide variety of applications, such as environmental modeling, computer experiments, and geology. Two major challenges often arise: First, assuming that the process of interest is stationary over the entire domain often proves to be untenable. Second, the traditional Gaussian process model formulation is computationally inefficient for large datasets. In this paper, we propose a new Gaussian process model to tackle these problems based on the convolution of a smoothing kernel with a partitioned latent process. Nonstationarity can be modeled by allowing a separate latent process for each partition, which approximates a regional clustering structure. Partitioning follows a binary tree generating process similar to that of Classification and Regression Trees. A Bayesian approach is used to estimate the partitioning structure and model parameters simultaneously. Our motivating dataset consists of 11918 precipitation anomalies. Results show that our model has promising prediction performance and is computationally efficient for large datasets.

Estimation of Regression Function in Multi-Response Nonparametric Regression Model Using Smoothing Spline and Kernel Estimators

The functions which describe relationship of more than one response variables observed at several values of the predictor variables in which there are correlations among the responses can be estimated by using a multi-response nonparametric regression model approach. In this study, we discuss about how we estimate the regression function of the multi-response nonparametric regression model by using both smoothing spline and kernel estimators. The principal objective is determining the smoothing spline and kernel estimators to estimate the regression function of the multi-response nonparametric regression model. The obtained results show that the regression functions obtained by using smoothing spline and kernel estimators are mathematically just distinguished by their smoother matrices. In addition, they are linear in observation and bias estimators.

Bootstrap testing for first-order stationarity on irregular windows in spatial point patterns

Kernel smoothing is commonly used in spatial point patterns to construct intensity plots. Kernels allow for visually and subjectively inferring on first-order stationarity. Formal objective tests exist for testing first-order stationarity that assume independence of spatial regions. We propose to extend inference for first-order stationary by using bootstrapping in existing hypothesis tests to deal with the violation of independence. More specifically we compare Poisson intensities from bootstrapped spatial quadrat samples, providing a test for first-order stationarity without violating the assumption of independence of the tests. Five hypothesis testing methods are investigated. The choice of grid mesh size and window shape used in these tests is discussed and guidance is provided through testing the power of the tests. The application considers the household locations in rural villages in Northern Tanzania as an unmarked point pattern. A clear effect of the village sizes on the relation between grid mesh size and confidence intervals of bootstrap sampling is shown. We conclude that bootstrapping provides a novel contribution to inference of first-order stationarity for spatial point patterns.

Spatial Prediction of Nitrate Concentration Using GIS and ANFIS Modelling in Groundwater.

Groundwater contamination by nitrate becomes a major problem in agricultural areas. Nitrate pollution in the groundwater of Coimbatore district of India has been investigated. Seventy-one observation wells within the study area were selected. The samples were collected in three stages with 6months interval during the years 2011 and 2012. The input for creating an Adaptive Neuro-Fuzzy Inference Systems (ANFIS) model is the average of calcium, electrical conductivity, sodium, chlorides, hardness, potassium and the concentration of nitrate was the output from the model. Results of the best ANFIS model were assessed by deterministic, geostatistical and kernel smoothing methods using GIS. Out of the five models developed, ANFIS model V predicts nitrate concentration with 90% accuracy and less root mean square error (0.0934). The results from model V are given as input in GIS to predict spatial nitrate concentration in groundwater. Kriging method is more accurate in estimating nitrate concentration over the study area.

Constructing disease onset signatures using multi-dimensional network-structured biomarkers

Potential disease-modifying therapies for neurodegenerative disorders need to be introduced prior to the symptomatic stage in order to be effective. However, current diagnosis of neurological disorders mostly rely on measurements of clinical symptoms and thus only identify symptomatic subjects in their late disease course. Thus, it is of interest to select and integrate biomarkers that may reflect early disease-related pathological changes for earlier diagnosis and recruiting pre-sypmtomatic subjects in a prevention clinical trial. Two sources of biological information are relevant to the construction of biomarker signatures for time to disease onset that is subject to right censoring. First, biomarkers' effects on disease onset may vary with a subject's baseline disease stage indicated by a particular marker. Second, biomarkers may be connected through networks, and their effects on disease may be informed by this network structure. To leverage these information, we propose a varying-coefficient hazards model to induce double smoothness over the dimension of the disease stage and over the space of network-structured biomarkers. The distinctive feature of the model is a non-parametric effect that captures non-linear change according to the disease stage and similarity among the effects of linked biomarkers. For estimation and feature selection, we use kernel smoothing of a regularized local partial likelihood and derive an efficient algorithm. Numeric simulations demonstrate significant improvements over existing methods in performance and computational efficiency. Finally, the methods are applied to our motivating study, a recently completed study of Huntington's disease (HD), where structural brain imaging measures are used to inform age-at-onset of HD and assist clinical trial design. The analysis offers new insights on the structural network signatures for premanifest HD subjects.

Discrete Heat Kernel Smoothing in Irregular Image Domains.

We present the discrete version of heat kernel smoothing on graph data structure. The method is used to smooth data in an irregularly shaped domains in 3D images. New statistical properties of heat kernel smoothing are derived. As an application, we show how to filter out noisy data in the lung blood vessel trees obtained from computed tomography. The method can be further used in representing the complex vessel trees parametrically as a linear combination of basis functions and extracting the skeleton representation of the trees.

Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams

Nonlocal strain gradient integral model of elasticity, extension of the fully nonlocal integral law, is widely adopted to assess size effects in nano-beams. The bending moment is sum of convolutions of elastic curvature and of its derivative with a smoothing kernel. For nanomechanical problems on unbounded domains, such as in wave propagation, the nonlocal strain gradient integral relation is equivalent to a differential law with constitutive conditions of vanishing at infinity. For bounded nano-beams, the constitutive boundary conditions (CBC) must be added to close the constitutive model. The formulation of these CBC is an original contribution of the paper. Equivalence between nonlocal strain gradient integral model of elasticity and the differential problem with CBC is proven. It is shown that the CBC do not conflict with equilibrium and provide a viable approach to study size-dependent phenomena in nano-beams of applicative interest. Theoretical outcomes are illustrated by examining the static scheme of a nano-actuator modelled by a nano-cantilever inflected by an end-point load. The relevance of a proper formulation of boundary conditions is elucidated by comparing the numerical results with previous attempts in literature.