Weighted Kernel Density Estimation Research Articles

Binary surrogates with stratified samples when weights are unknown

In clinical practice, surrogate variables are commonly used as an indirect measure when it is difficult or expensive to measure the primary outcome variable X, based on which the disease status is assessed. In this article, we consider the problem of constructing an optimal binary surrogate Y to substitute such the feature variable X. To retain samples that have rare values in X, the paired sample (X, Y) is usually selected based on stratified sampling, where the strata are constructed using the disjoint intervals with the support of X. For such a sampling design, the stratum proportions are usually unknown such that proportional allocation is infeasible and (X, Y)’s cannot be regarded as an i.i.d. sample between strata. We estimate the unknown cutoff determining higher/lower levels of X that optimally match the variable Y and provide the true positive rates (TPR) adjusted for the disproportionate stratum weights. Our approach is to estimate the underlying distribution of X, then conduct an ad-hoc estimation for the TPR and for the expected prediction errors under zero-one loss function. We develop parametric estimate of the distribution of X under exponential family assumption and a weighted-kernel density estimator when the distribution of X is unspecified. We illustrate our methods on various simulation studies and on a real example where binary surrogates were evaluated for a medical device. The simulation results indicate that our approach performs well.

Feature-Learning-Based Printed Circuit Board Inspection via Speeded-Up Robust Features and Random Forest

With the coming of the 4th industrial revolution era, manufacturers produce high-tech products. As the production process is refined, inspection technologies become more important. Specifically, the inspection of a printed circuit board (PCB), which is an indispensable part of electronic products, is an essential step to improve the quality of the process and yield. Image processing techniques are utilized for inspection, but there are limitations because the backgrounds of images are different and the kinds of defects increase. In order to overcome these limitations, methods based on machine learning have been used recently. These methods can inspect without a normal image by learning fault patterns. Therefore, this paper proposes a method can detect various types of defects using machine learning. The proposed method first extracts features through speeded-up robust features (SURF), then learns the fault pattern and calculates probabilities. After that, we generate a weighted kernel density estimation (WKDE) map weighted by the probabilities to consider the density of the features. Because the probability of the WKDE map can detect an area where the defects are concentrated, it improves the performance of the inspection. To verify the proposed method, we apply the method to PCB images and confirm the performance of the method.

Detailed Investigation into the Asphaltene Fraction of Hydrothermal Liquefaction Derived Bio-Crude and Hydrotreated Bio-Crudes

Hydrothermal liquefaction (HTL) is a wet thermo-chemical biomass conversion technology for the production of, e.g., liquid transportation fuels. The process yields a bio-crude with properties similar to petroleum crude but also with significant differences. Particularly the higher heteroatom content and high viscosity render bio-crudes unsuitable for a direct use in the current infrastructure as a drop-in fuel without an upgrading step. The presented work investigates the composition of an HTL bio-crude produced from lignocellulosic biomass at the pilot scale as well as the upgraded fuels obtained via catalytic hydrotreatment. Pentane solvent extraction is employed to fractionate the bio-crude and its hydrotreated counterpart into a light fuel extract and an asphaltene residue. The asphaltene fraction, which at room temperature is a solid material constituting 60 wt % of the bio-crude, is reduced upon hydrotreating to 34 wt%. The pentane extracts reveal a superior fuel quality in terms of higher heating value and composition. Detailed molecular and structural analysis via nuclear magnetic resonance (NMR), gas chromatography-mass spectrometry (GC-MS), and ultra high pressure liquid chromatography coupled to high-resolution mass spectrometry (UHPLC-HRMS) revealed an aromatic and heteroatomatic structure of the asphaltene fraction. UHPLC-HRMS analysis indicated an average molecular weight ranging from 150 until 300 Da, which was affected by hydrotreating. A novel analytical strategy for UHPLC-HRMS, including weighted kernel density estimation, was developed. The superior quality of solvent extracted fuels suggests this could be a simple step in improving the capability of HTL derived bio-crudes in a refinery context. Additionally, the difficulty in hydrotreating the asphaltenes suggests that removal of this fraction via solvent extraction prior to the upgrading could be a viable option for improving the efficiency of this post-HTL treatment step.

A practical implementation of weighted kernel density estimation for handling shape constraints

The weighted kernel density estimator is an attractive option for shape‐restricted density estimation, because it is simple, familiar, and potentially applicable to many different shape constraints. Despite this, no reliable software implementation has appeared since the method's original proposal in 2002. We found that serious numerical and practical difficulties arise when attempting to implement the method. We overcame these difficulties and in the process discovered that the weighted method and our own recently proposed method—controlling the shape of a kernel density using an adjustment curve—can be unified in a single computational framework. This article describes our findings and introduces the R package scdensity, which can be used to easily obtain density estimates that are unimodal, bimodal, symmetric, and more. © 2018 The Authors. Stat Published by John Wiley & Sons Ltd

Human gait pattern changes detection system: A multimodal vision-based and novelty detection learning approach

This paper proposes a novel gait rehabilitation analysis system, based on an innovative multimodal vision-based sensor setup, focused on detecting gait pattern changes over time. The proposed setup is based on inexpensive technologies, without compromising performance, and was designed to be deployed on walkers, which are a typical assistive aid used in gait rehabilitation. In the medical field the evaluation of a patient's rehabilitation progress is typically performed by a medical professional through subjective techniques based on the professional's visual perception and experience. In this context, we are proposing an automatic system to detect the progress of patients undergoing rehabilitation therapy. Our approach is able to perform novelty detection for gait pattern classification based on One-Class Support Vector Machines (OC-SVM). Using point-cloud and RGB-D data, we detect the lower limbs (waist, legs and feet) by using Weighted Kernel-Density Estimation and Weighted Least-Squares to segment the legs into several parts (thighs and shins), and by fitting 3D ellipsoids to model them. Feet are detected using k-means clustering and a Circular Hough Transform. A temporal analysis of the feet's depth is also performed to detect heel strike events. Spatiotemporal and kinematic features are extracted from the lower limbs’ model and fed to a classifier based on the fusion of several OC-SVMs. Experiments with volunteers using the robotic walker platform ISR-AIWALKER, where the proposed system was deployed, revealed a lower limbs tracking accuracy of 3° and that our novelty detection approach successfully identified novel gait patterns, evidencing an overall 97.89% sensitivity.

Finding their way in the world: Using acoustic telemetry to evaluate relative movement patterns of hatchery-reared fish in the period following release

Patterns in space use and activity were compared for wild and hatchery-reared Mulloway (Argyrosomus japonicus) using acoustic telemetry. Acoustic tags were implanted in four wild (42±2cm; mean±SE), and four hatchery-reared (42±1cm; mean±SE) Mulloway, and movements were simultaneously monitored for up to 288h in an 11km section of river. Across all fish, space utilisation contours developed from weighted kernel density estimates ranged between 2–51ha (90% or total area) and 1–28ha (50% or core area), and contained up to 99% optimal habitat area. Hatchery-reared fish used significantly larger total and core areas, and activity rates of hatchery-reared fish were consistently higher than wild fish. A period of settlement or acclimation appeared to occur during the first 5days following release for hatchery-reared fish, and their movement tended to contract back to within the habitat patch into which they were released from the 6–12th day following release. The movement ranges of wild fish were largely invariant between the two periods. The relevance of this research to understanding broader ecological processes is discussed. Acoustic telemetry presents a useful approach for studying post-release dynamics of hatchery-reared fish and their wild counterparts.

Pricing Kernel Modeling

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

Asset Structure and Solvency of Insurance Companies in China with Path Identification Model

Based on weighted kernel density estimation and the nonparametric path identification model, this paper explores the impact of asset structure on solvency and risk exposure by insurance companies. To test the differences in solvency and risk exposure of admitted assets under stock and time deposit investment paths, we compared distributions under different investment paths to benchmark distribution. Our results suggest that both stock and time deposit investment impact positively on solvency, meaning that, all other things being equal, solvency increases when investment assets are expanded but risk increases simultaneously. The impact of stock investment is also greater than that of time deposit investment. The extent of these differences increased gradually over the year under the stock investment path, but reduced under the time deposit investment. Under the stock investment path, the value of the high quantiles of the distribution are likely to shift from that of low quantiles. Expected value and risk exposure under time deposit investment are not necessarily reduced, so regulators should both emphasize solvency indicators in daily supervision and also take into account changes in the investment structure, improve how admissible assets are calculated in case the insurance company increases the solvency margin by expanding high-risk investment.

Pricing kernel modeling

We propose a new method to estimate the empirical pricing kernel based on option data. We estimate the pricing kernel nonparametrically by using the ratio of the risk-neutral density estimator and the subjective density estimator. The risk-neutral density is approximated by a weighted kernel density estimator with varying unknown weights for different observations, and the subjective density is approximated by a kernel density estimator with equal weights. We represent the European call option price function by the second order integration of the risk-neutral density, so that the unknown weights are obtained through one-step penalized least squares estimation with the Kullback-Leibler divergence as the penalty function. Asymptotic results of the resulting estimators are established. The performance of the proposed method is illustrated empirically by simulation and real data application studies.

Enhanced Trajectory Based Similarity Prediction with Uncertainty Quantification


 
 
 Today, data driven prognostics acquires historic data to generate degradation path and estimate the Remaining Useful Life (RUL) of a system. A successful methodology, Trajectory Similarity Based Prediction (TSBP) that details the process of predicting the system RUL and evaluating the performance metrics of the estimate was proposed in 2008. Two essential components of TSBP identified for potential improvement include 1) a distance or similarity measure that is capable of determining which degradation model the testing data is most similar to and 2) computation of uncertainty in the remaining useful life prediction, instead of a point estimate. In this paper, the Trajectory Based Similarity Prediction approach is evaluated to include Similarity Linear Regression (SLR) based on Pearson Correlation and Dynamic Time Warping (DTW) for determining the degradation models that are most similar to the testing data. A computational approach for uncertainty quantification is implemented using the principle of weighted kernel density estimation in order to quantify the uncertainty in the remaining useful life prediction. The revised approach is measured against the same dataset and performance metrics evaluation method used in the original TBSP approach. The result is documented and discussed in the paper. Future research is expected to augment TSBP methodology with higher accuracy and stronger anticipation of uncertainty quantification.
 
 

A note on nonparametric density deconvolution by weighted kernel estimators

Abstract Recently Hazelton and Turlach (2009) proposed a weighted kernel density estimatorfor the deconvolution problem. In the case of Gaussian kernels and measurement er-ror, they argued that the weighted kernel density estimator is a competitive estimatorover the classical deconvolution kernel estimator. In this paper we consider weightedkernel density estimators when sample observations are contaminated by double expo-nentially distributed errors. The performance of the weighted kernel density estimatorsis compared over the classical deconvolution kernel estimator and the kernel density es-timator based on the support vector regression method by means of a simulation study.The weighted density estimator with the Gaussian kernel shows numerical instabilityin practical implementation of optimization function. However the weighted densityestimates with the double exponential kernel has very similar patterns to the classicalkernel density estimates in the simulations, but the shape is less satisfactory than theclassical kernel density estimator with the Gaussian kernel.Keywords: Deconvolution, kernel density estimator, support vector regression, weightedkernel density estimator.

THE RADIUS DISTRIBUTION OF PLANETS AROUND COOL STARS

We calculate an empirical, non-parametric estimate of the shape of the period-marginalized radius distribution of planets with periods less than 150 days using the small yet well-characterized sample of cool ($T_{\rm eff} <4000 $K) dwarf stars in the Kepler catalog. In particular, we present and validate a new procedure, based on weighted kernel density estimation, to reconstruct the shape of the planet radius function down to radii smaller than the completeness limit of the survey at the longest periods. Under the assumption that the period distribution of planets does not change dramatically with planet radius, we show that the occurrence of planets around these stars continues to increase to below 1 $R_\oplus$, and that there is no strong evidence for a turnover in the planet radius function. In fact, we demonstrate using many iterations of simulated data that a spurious turnover may be inferred from data even when the true distribution continues to rise toward smaller radii. Finally, the sharp rise in the radius distribution below $\sim$3 $R_\oplus$ implies that a large number of planets await discovery around cool dwarfs as the sensitivities of ground-based transit surveys increase.

Density estimation and nonparametric inferences using maximum likelihood weighted kernels

We show that maximum likelihood weighted kernel density estimation offers a unified approach to density estimation and nonparametric inferences. For density estimation, the approach is a generalisation of the standard kernel density estimator that allows the weights attached to each kernel to be chosen by maximum likelihood, instead of being set to n −1 from the outset (see also Jones, M.C., and Henderson, D.A. (2005), ‘Maximum Likelihood Kernel Density Estimation’, Technical Report 01/05, Department of Statistics, The Open University, UK). For nonparametric inferences, the approach offers a natural, smoothed analogue to empirical likelihood (Owen, A.B. (2001), Empirical Likelihood, Boca Raton, FL: Chapman and Hall/CRC) for inferences on functionals of the underlying distribution, such as its mean or median. Numerical results demonstrate that the proposed method is comparable to the standard kernel density estimator (of the same bandwidth) for density estimation, but can offer noticeable small-sample improvements over empirical likelihood for inferences when the underlying distribution is continuous.

Trading Industry Clusters amid the Legacy of Industrial Land-Use Planning in Southern California

This paper tests for excessive employment agglomeration among twenty southern California manufacturing industry clusters using two methods claiming to control for the natural and planned geography constraining industrial location generally. We find broad consistency across the alternative tests; although fewer industry clusters test statistically significant for agglomeration or dispersion using the weighted kernel density estimator than when we use the more established weighted case-control approach. Similarly, the observed values for the former statistic test significant over a subset of the spatial scales over which the case-control method tests significant.

Evaluating temporally weighted kernel density methods for predicting the next event location in a series

One aspect of tactical crime or terrorism analysis is predicting the location of the next event in a series. The objective of this article is to present a methodology to identify the optimal parameters and to test the performance of temporally weighted kernel density estimation models for predicting the next event in a criminal or terrorist event series. By placing event series in a space–time point pattern framework, the next event prediction models are shown to be based on estimating a conditional spatial density function. We use temporal weights that indicate how much influence past events have toward predicting future event locations, which can also incorporate uncertainty in the event timing. Results of applying this methodology to crime series in Baltimore County, MD, indicate that performance can vary greatly by crime type and little by series length and is fairly robust to choice of bandwidth.

Weighted Kernel Density Estimation of the Prepulse Inhibition Test

Problem statement: The goal of this study was to devise a more reliable and sensitive method for analysis of experimental data of the Prepulse Inhibition (PPI), the reduction in startle reaction towards a startle-eliciting “pulse” stimulus when it is shortly preceded by a sub-threshold “prepulse” stimulus. Approach: Different from the conventional simple averaging-based method, we proposed a probabilistic approach to modeling the PPI data. With this probabilistic description, we reconstructed complete response signals from the PPI data and devised a nonparametric weighted Kernel Density Estimation (KDE) method to tackle two important issues in PPI data related density estimation: instability and limited number of samples. We designed two sets of animal experiments using different medicines and compared the KDE based method with the conventional simple-averaging based method. Results: Our results showed that the KDE method performed better than the conventional method and offered some advantages over the conventional method. Conclusion: The new method provided a more reliable and sensitive approach to the post-session analysis of PPI data.

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.

Empirical evidence of income dynamics across EU regions

AbstractThis paper analyses the distribution of purchasing power standardized per capita income across EU‐12 regions between 1977 and 1996. Dispersion of incomes between regions is measured taking into account their population sizes. The cross‐sectional distributions are initially described by weighted kernel density estimates, revealing a multimodal structure of the distributions, less evident over the period. This evidence is supported by a bootstrap test. To detect homogeneous groups of regions, the empirical distributions are approximated by a finite mixture of normal densities. The components of the mixture represent clusters of poor/rich regions, while the mixing proportions the allocation over the poor and the rich components. The number of components is assessed by a bootstrap LR test, and the goodness of fit by a kernel density‐based test. Income mobility is modelled by the stochastic kernel, the continuous counterpart of the transition probability matrix. The main implication is a very slow process of catching up of the poorest regions with the richer ones and a process of shifting away of a small group of very rich regions. This evidence is reflected in the shape of the ergodic distribution, which is well fitted by a two‐component mixture model. Copyright © 2006 John Wiley &amp; Sons, Ltd.

State-space likelihoods for nonlinear fisheries time-series

State-space models are commonly used to incorporate process and observation errors in analysis of fisheries time series. A gap in analysis methods has been the lack of classical likelihood methods for nonlinear state-space models. We evaluate a method that uses weighted kernel density estimates of Bayesian posterior samples to estimate likelihoods (Monte Carlo Kernel Likelihoods, MCKL). Classical likelihoods require integration over the state-space, and we compare MCKL to the widely used errors-in-variables (EV) method, which estimates states jointly with parameters by maximizing a nonintegrated likelihood. For a simulated, linear, autoregressive model and a Schaefer model fit to cape hake (Merluccius capensis × M. paradoxus) data, classical likelihoods outperform EV likelihoods, which give asymptotically biased parameter estimates and inaccurate confidence regions. Our results on the importance of integrated state-space likelihoods also support the value of Bayesian analysis with Monte Carlo posterior integration. Both approaches provide valuable insights and can be used complementarily. Previously, Bayesian analysis was the only option for incorporating process and observation errors with complex nonlinear models. The MCKL method provides a classical approach for such models, so that choice of analysis approach need not depend on model complexity.

A new model validation tool using kernel regression and density estimation

In physiological system modelling for control or decision support, model validation is a critical element. A nonparametric approach for assessing the validity of deterministic dynamic models against empirical data is developed, based on kernel regression and kernel density estimation, yielding visual graphical assessment tools as well as numerical metrics of compatibility between the model and the data. Nonparametric regression has been suggested for assessing a parametric statistical model by constructing a confidence band for the proposed model and then checking whether the nonparametric regression curve lies within the band. However, for deterministic models, there is no confidence band that can be constructed. A reversal of roles is therefore suggested--construct a probability band for the nonparametric regression curve and check whether the proposed model lies within the band. This approach extends the utility of nonparametric regression for model assessment to deterministic models. Weighted kernel density estimation is incorporated to derive a density profile for the regression curve, creating a local graphical validation tool. In addition, the density profile is used to define and compute two numerical measures--average normalized density (AND) and relative average normalized density (RAND), representing global statistical validity measures. These tools are demonstrated using a biomedical system model for agitation-sedation and sedation management control.