Kernel-smoothed Estimator Research Articles

FFT-Based Probability Density Imaging of Euler Solutions.

When using traditional Euler deconvolution optimization strategies, it is difficult to distinguish between anomalies and their corresponding Euler tails (those solutions are often distributed outside the anomaly source, forming "tail"-shaped spurious solutions, i.e., misplaced Euler solutions, which must be removed or marked) with only the structural index. The nonparametric estimation method based on the normalized B-spline probability density (BSS) is used to separate the Euler solution clusters and mark different anomaly sources according to the similarity and density characteristics of the Euler solutions. For display purposes, the BSS needs to map the samples onto the estimation grid at the points where density will be estimated in order to obtain the probability density distribution. However, if the size of the samples or the estimation grid is too large, this process can lead to high levels of memory consumption and excessive computation times. To address this issue, a fast linear binning approximation algorithm is introduced in the BSS to speed up the computation process and save time. Subsequently, the sample data are quickly projected onto the estimation grid to facilitate the discrete convolution between the grid and the density function using a fast Fourier transform. A method involving multivariate B-spline probability density estimation based on the FFT (BSSFFT), in conjunction with fast linear binning appropriation, is proposed in this paper. The results of two random normal distributions show the correctness of the BSS and BSSFFT algorithms, which is verified via a comparison with the true probability density function (pdf) and Gaussian kernel smoothing estimation algorithms. Then, the Euler solutions of the two synthetic models are analyzed using the BSS and BSSFFT algorithms. The results are consistent with their theoretical values, which verify their correctness regarding Euler solutions. Finally, the BSSFFT is applied to Bishop 5X data, and the numerical results show that the comprehensive analysis of the 3D probability density distributions using the BSSFFT algorithm, derived from the Euler solution subset of x0,y0,z0, can effectively separate and locate adjacent anomaly sources, demonstrating strong adaptability.

A Symmetric Kernel Smoothing Estimation of the Time-Varying Coefficient for Medical Costs

In longitudinal studies, subjects are repeatedly observed at a set of distinct time points until the terminal event time. The time-varying coefficient model extends the parametric method and captures the dynamic trajectories of time-dependent covariate effects, thus enabling it to describe the potential relationship between the longitudinal variable and the observed time points. In this study, we propose a novel approach to the estimation of medical costs using a symmetric kernel smoothing method in the time-varying coefficient joint model. A smooth function of medical costs is derived by weighting the values of longitudinal data at all distinct observed time points via the combination of the kernel method and the inverse probability weighting method. For the simulation study, we first set up the true functions of time-varying coefficients; we then generated random samples for covariates and censored survival times. Subsequently, the longitudinal data of response variables could be produced. Further, numerical simulation experiments were conducted by using the proposed method and applying R code to the generated data. The estimated results for the parameters and non-parametric functions were compared with different settings. The numerical results illustrate that as the sample size increases, the bias and model-based standard errors decrease, and the performance improves with larger sample sizes. The estimates of functions in the model almost coincide with the true functions, as shown in the figures of the simulation study. Furthermore, the consistency of the obtained estimator is demonstrated via theoretical analysis, and a numerical simulation is performed to illustrate the performance of the proposed estimators. The proposed model is applied to a real-world data set acquired from a multicenter automatic defibrillator implantation trial (MADIT).

Modelling spatial point patterns of Asian Giant Hornets occurrences

This study explores statistical methods for modelling the spatial patterns of Asian giant hornets using sighting records in the Republic of Korea from 2008 to 2013. The focus is on simulating the spatial distribution of hornets, with a key aspect being the statistical inference of their intensity. Gaussian kernel-smoothing estimation is utilized to model the hornets intensity based on sighting records, which also transforms the occurrences of honeybees, a primary prey of the hornets, into a covariate. Results show a significant dependency of hornets intensity on honeybees intensity, with a calculated probability that a hornet sighting location has higher honeybee intensity than a random location. By this finding, the parametric modelling of the hornets spatial intensity is applied with the covariate of honeybees in each year, with the basic inhomogeneous Poisson point process along with the log-linear model. The model is refined for each year using backward stepwise selection based on the Akaike Information Criteria. Model validation confirms the Poisson process assumption and shows promising results with raw residuals against honeybee intensity. The analysis demonstrates that the spatial pattern modelling method employed is both sensible and valid.

Data Forecasting of Air-Conditioning Load in Large Shopping Malls Based on Multiple Nonlinear Regression

Abstract This article applies multiple nonlinear regression methods to establish a forecasting model for the load characteristics of air conditioning in shopping malls at different times. Based on Python data, determine the functional relationship of refrigerant parameters concerning pressure and temperature. The article uses kernel smoothing estimation technology to calculate the room temperature probability density distribution of users participating in DLC to characterize the user’s comfort. The article’s research results show that the average error between the regression analysis results of refrigerant parameters and the reference value is within 1%. This model is suitable for medium and long-term load forecasting. It has high prediction accuracy for the sudden change trend with a turning point.

A law of the iterated logarithm for error density estimator in censored linear regression

We consider the strong consistency of the nonparametric estimation of error density in linear regression with right censored data. The estimator is defined to be the kernel-smoothed estimator of error density, which makes use of the Kaplan-Meier estimator of the error distribution. We establish a point-wise law of the iterated logarithm for kernel-type error density estimator in censored Linear Regression.

Minimax optimal conditional density estimation under total variation smoothness

This paper studies the minimax rate of nonparametric conditional density estimation under a weighted absolute value loss function in a multivariate setting. We first demonstrate that conditional density estimation is impossible if one only requires that $p_{X|Z}$ is smooth in $x$ for all values of $z$. This motivates us to consider a sub-class of absolutely continuous distributions, restricting the conditional density $p_{X|Z}(x|z)$ to not only be H\"older smooth in $x$, but also be total variation smooth in $z$. We propose a corresponding kernel-based estimator and prove that it achieves the minimax rate. We give some simple examples of densities satisfying our assumptions which imply that our results are not vacuous. Finally, we propose an estimator which achieves the minimax optimal rate adaptively, i.e., without the need to know the smoothness parameter values in advance. Crucially, both of our estimators (the adaptive and non-adaptive ones) impose no assumptions on the marginal density $p_Z$, and are not obtained as a ratio between two kernel smoothing estimators which may sound like a go to approach in this problem.

The hazard level set

The hazard rate function plays a fundamental role in survival analysis. Its statistical inference methods have been systemically and exclusively studied. When does the hazard rate reach a particular warning level? This is a basic question of interest to the investigator but largely left to be explored in practice. We define a level set of hazard rate to address this issue and propose a kernel smoothing estimator for such a level set. In terms of the Hausdorff distance, we establish the consistency, convergence rate and asymptotic distribution of the level set estimator. The validity of the proposed confidence set, based on the bootstrap method, for the level set of hazard rate function is theoretically justified. We conduct comprehensive simulation studies to assess the finite-sample performance of the proposed method, which is further illustrated with a breast cancer study.

The strong uniform consistency of kernel estimator of a smooth distribution function in censored linear regression

ABSTRACT This paper considers the kernel estimation of a smooth error distribution function in linear regression with right-censored data. Based on the Kaplan–Meier estimator of the residual distribution, we define the kernel-smoothed estimator of the error distribution function. Under the appropriate conditions, uniformly strong consistency of the proposed estimator is obtained.

A Non-Parametric Hawkes Process Model of Primary and Secondary Accidents on a UK Smart Motorway

Abstract A self-exciting spatiotemporal point process is fitted to incident data from the UK National Traffic Information Service to model the rates of primary and secondary accidents on the M25 motorway in a 12-month period during 2017–2018. This process uses a background component to represent primary accidents, and a self-exciting component to represent secondary accidents. The background consists of periodic daily and weekly components, a spatial component and a long-term trend. The self-exciting components are decaying, unidirectional functions of space and time. These components are determined via kernel smoothing and likelihood estimation. Temporally, the background is stable across seasons with a daily double peak structure reflecting commuting patterns. Spatially, there are two peaks in intensity, one of which becomes more pronounced during the study period. Self-excitation accounts for 6–7% of the data with associated time and length scales around 100 min and 1 km, respectively. In-sample and out-of-sample validation are performed to assess the model fit. When we restrict the data to incidents that resulted in large speed drops on the network, the results remain coherent.

A PRECIPITAÇÃO E O CRITÉRIO DE ESCOLHA DE ANOS-PADRÃO

A proposta deste estudo é desenvolver com base no elemento precipitação, as etapas para a eleição de anos-padrão ao estado do Ceará, considerando a aplicação do método de Tavares (1976). Como critério de escolha, fez-se uso também da espacialização geoestatística dos dados por meio do estimador Kernel Smoothing. A base de dados utilizada é composta de 184 postos pluviométricos da Fundação Cearense de Meteorologia e Recursos Hídricos (FUNCEME), e compreende uma escala temporal de 20 anos, entre 1991-2010. Os dados foram organizados conforme os critérios adotados por Tavares (1976), sendo estes validados por meio das ferramentas de validação cruzada e validação/previsão. Feito a validação, os dados foram quantificados em histogramas e espacializados a partir do estimador Kernel Smoothing. Pela quantificação dos histogramas e análises dos mapas, foram selecionados como anos-padrão para o estado do Ceará: 2005, como ano seco, 2004, como ano habitual e 2009, como ano chuvoso.

Is Switching from Oral Antidiabetic Therapy to Insulin Associated with an Increased Fracture Risk?

Observational studies showed that exposure to exogenous insulin increases fracture risk. However, it remains unclear whether the observed association is a function of the severity of underlying type 2 diabetes mellitus, complications, therapies, comorbidities, or all these factors combined. That being so, and because of the relative infrequency of these events, it is important to study this further in a large-database setting. QUESTION/PURPOSES: (1) Is switching from oral antidiabetic agents to insulin associated with an increased fracture risk? (2) How soon after switching does the increased risk appear, and for how long does this increased risk persist? Data from healthcare utilization databases of the Italian region of Lombardy were used. These healthcare utilization databases report accurate, complete, and interconnectable information of inpatient and outpatient diagnoses, therapies, and services provided to the almost 10 million residents in the region. The 216,624 patients on treatment with oral antidiabetic therapy from 2005 to 2009 were followed until 2010 to identify those who modified their antidiabetic therapy (step 1 cohort). Among the 63% (136,307 patients) who experienced a therapy modification, 21% (28,420 patients) switched to insulin (active exposure), and the remaining 79% (107,887 patients) changed to another oral medication (referent exposure). A 1:1 high-dimension propensity score matching design was adopted for balancing patients on active and referent exposure. Matching failed for 3% of patients (926 patients), so the cohort of interest was formed by 27,494 insulin-referent couples. The latter were followed until 2012 to identify those who experienced hospital admission for fracture (outcome). A Cox proportional hazard model was fitted to estimate the hazard ratio (HR) for the outcome risk associated with active-exposure (first research question). Between-exposure comparison of daily fracture hazard rates from switching until the 24 successive months was explored through the Kernel-smoothed estimator (second research question). Compared with patients on referent exposure, those who switched to insulin had an increased risk of experiencing any fracture (HR = 1.5 [95% CI 1.3 to 1.6]; p < 0.001). The same risk was observed for hip and vertebral fractures, with HRs of 1.6 (95% CI 1.4 to 1.8; p < 0.001) and 1.8 (95% 1.5 to 2.3; p < 0.001), respectively. Differences in the daily pattern of outcome rates mainly appeared the first 2 months after switching, when the hazard rate of patients on active exposure (9 cases for every 100,000 person-days) was higher than that of patients on referent exposure (4 cases for every 100,000 person-days). These differences persisted during the remaining follow-up, though with reduced intensity. We found quantitative evidence that switching from oral antidiabetic therapy to insulin is associated with an increased fracture risk, mainly in the period immediately after the start of insulin therapy. The observed association may result from higher hypoglycemia risk among patients on insulin, which leads to a greater number of falls and resulting fractures. However, although our study was based on a large sample size and highly accurate data, its observational design and the lack of clinical data suggest that future research will need to replicate or refute our findings and address the issue of causality, if any. Until then, though, prescribers and patients should be aware of this risk. Careful control of insulin dosage should be maintained and measures taken to reduce fall risk in these patients. Level III, therapeutic study.

Kernel estimation for panel data with heterogeneous dynamics

Summary This paper proposes nonparametric kernel-smoothing estimation for panel data to examine the degree of heterogeneity across cross-sectional units. We first estimate the sample mean, autocovariances, and autocorrelations for each unit and then apply kernel smoothing to compute their density functions. The dependence of the kernel estimator on bandwidth makes asymptotic bias of very high order affect the required condition on the relative magnitudes of the cross-sectional sample size (N) and the time-series length (T). In particular, it makes the condition on N and T stronger and more complicated than those typically observed in the long-panel literature without kernel smoothing. We also consider a split-panel jackknife method to correct bias and construction of confidence intervals. An empirical application illustrates our procedure.

Spatiotemporal distribution of a non-haematophagous bat community and rabies virus circulation: a proposal for urban rabies surveillance in Brazil.

In Brazil, rabies surveillance is based on monitoring domestic and wild animals, although the most prevalent lineage of the rabies virus (RABV) currently diagnosed in Brazil is associated with bats, particularly non-haematophagous bats. Disease control is based on the mass vaccination of dogs and cats. We used data collected by the passive surveillance system of the city of Campinas from 2011 to 2015, to describe the temporal and geographic distributions of the bat specimens and RABV and discuss the current rabies surveillance with the advent of the declaration of canine and feline rabies-free areas in Brazil. We described the species, locations and health statuses of the collected bat specimens. Moreover, all samples were submitted for RABV diagnosis. Then, we performed a time series decomposition for each bat family. Additionally, we determined the spatiotemporal relative risk for RABV infection using the ratio of the kernel-smoothed estimates of spatiotemporal densities of RABV-positive and RABV-negative bats. From the 2537 bat specimens, the most numerous family was Molossidae (72%), followed by Vespertilionidae (14%) and Phyllostomidae (13%). The bat families behaved differently in terms of seasonal and spatial patterns. The distribution of bats varied geographically in the urban environment, with Molossidae and Phyllostomidae being observed downtown and Vespertilionidae being observed in peripheral zones. Concurrently, a significant relative risk of RABV infection was observed downtown for Vespertilionidae and in peripheral zones for Molossidae. No RABV-positive sample clusters were observed. As a result of the official declaration of RABV-free areas in southern Brazil, mass dog and cat vaccinations are expected to halt in the near future. This stoppage would make most dog and cat populations susceptible to other RABV lineages, such as those maintained by non-haematophagous bats. In this scenario, all information available on bats and RABV distribution in urban areas is essential. Currently, few studies have been conducted. Some local health authorities, such as that in Campinas, are spontaneously basing their surveillance efforts on bat rabies, which is the alternative in reality scenario of increased susceptibility to bat-associated RABV that is developing in Brazil.

Glivenko–Cantelli Theorem for the kernel error distribution estimator in the first-order autoregressive model

This paper considers the uniform strong consistency of the kernel estimator of the error cumulative distribution function (CDF) in the first-order autoregressive model. The classical Glivenko–Cantelli Theorem is extended to the residual based kernel smooth CDF estimator in the autoregressive model.

Jackknife empirical likelihood-based inferences for Lorenz curve with kernel smoothing

ABSTRACTThe Lorenz curve describes the wealth proportion for an income-ordered population. In this paper, we introduce a kernel smoothing estimator for the Lorenz curve and propose a smoothed jackknife empirical likelihood method for constructing confidence intervals of Lorenz ordinates. Extensive simulation studies are conducted to evaluate finite sample performances of the proposed methods. A real dataset of Georgia professor’s income is used to illustrate the proposed methods.

Kernel Estimation for Panel Data with Heterogeneous Dynamics

This paper proposes nonparametric kernel-smoothing estimation for panel data to examine the degree of heterogeneity across cross-sectional units. We first estimate the sample mean, autocovariances, and autocorrelations for each unit and then apply kernel smoothing to compute their density functions. The dependence of the kernel estimator on bandwidth makes asymptotic bias of very high order affect the required condition on the relative magnitudes of the cross-sectional sample size (N) and the time-series length (T). In particular, it makes the condition on N and T stronger and more complicated than those typically observed in the long-panel literature without kernel smoothing. We also consider a split-panel jackknife method to correct bias and construction of confidence intervals. An empirical application and Monte Carlo simulations illustrate our procedure in finite samples.

Semiparametric outlier detection in nonstationary times series: Case study for atmospheric pollution in Brno, Czech Republic

Large environmental datasets usually include outliers which can have significant effects on further analysis and modelling. There exist various outlier detection methods that depend on the distribution of the analysed variable. However quite often the distribution of environmental variables can not be estimated. This paper presents an approach for identification of outliers in environmental time series which does not impose restrictions on the distribution of observed variables. The suggested algorithm combines kernel smoothing and extreme value estimation techniques for stochastic processes within considerations of nonstationary expected value of the process. The nonstationarity in variance is evaded by change point analysis which precedes the proposed algorithm. Possible outliers are identified as observations with rare occurrence and, in correspondence to extreme value methodology, the confidence limits for high values of observed variables are constructed. The proposed methodology can be especially convenient for cases where validation of the data has to be carried out manually, since it significantly reduces the number of implausible observations. For a case study, the technique is applied for outlier detection in time series of hourly PM10 concentrations in Brno, Czech Republic. The methodology is derived on solid theoretical results and seems to perform well for the series of PM10. However its flexibility makes it generally applicable not only to series of atmospheric pollutants. On the other hand, the choice of return level turns out to be crucial in sensitivity to the outliers. This issue should be left to the practitioners to decide with respect to specific application conditions.

Asymptotic Properties of Hazard Rate Estimator in Censored Linear Regression

We consider nonparametric inference for error hazard rates in linear regression with right censored data. The estimator for hazard rate function is defined based on the kernel-smoothed estimator of error density, which makes use of the Kaplan-Meier estimator of the error distribution. We obtain the limit distribution for the maximum deviation of the hazard rate estimator.

A new method of kernel-smoothing estimation of the ROC curve

The receiver operating characteristic (ROC) curve is a popular graphical tool for describing the accuracy of a diagnostic test. Based on the idea of estimating the ROC curve as a distribution function, we propose a new kernel smoothing estimator of the ROC curve which is invariant under nondecreasing data transformations. We prove that the estimator has better asymptotic mean squared error properties than some other estimators involving kernel smoothing and we present an easy method of bandwidth selection. By simulation studies, we show that for the limited sample sizes, our proposed estimator is competitive with some other nonparametric estimators of the ROC curve. We also give an example of applying the estimator to a real data set.

Application of a hazard-based visual predictive check to evaluate parametric hazard models.

Parametric models used in time to event analyses are evaluated typically by survival-based visual predictive checks (VPC). Kaplan-Meier survival curves for the observed data are compared with those estimated using model-simulated data. Because the derivative of the log of the survival curve is related to the hazard--the typical quantity modeled in parametric analysis--isolation, interpretation and correction of deficiencies in the hazard model determined by inspection of survival-based VPC's is indirect and thus more difficult. The purpose of this study is to assess the performance of nonparametric hazard estimators of hazard functions to evaluate their viability as VPC diagnostics. Histogram-based and kernel-smoothing estimators were evaluated in terms of bias of estimating the hazard for Weibull and bathtub-shape hazard scenarios. After the evaluation of bias, these nonparametric estimators were assessed as a method for VPC evaluation of the hazard model. The results showed that nonparametric hazard estimators performed reasonably at the sample sizes studied with greater bias near the boundaries (time equal to 0 and last observation) as expected. Flexible bandwidth and boundary correction methods reduced these biases. All the nonparametric estimators indicated a misfit of the Weibull model when the true hazard was a bathtub shape. Overall, hazard-based VPC plots enabled more direct interpretation of the VPC results compared to survival-based VPC plots.