Nonparametric Smoothing Research Articles

Grid Point Approximation for Distributed Nonparametric Smoothing and Prediction

Kernel smoothing is a widely used nonparametric method in modern statistical analysis. The problem of efficiently conducting kernel smoothing for a massive dataset on a distributed system is a problem of great importance. In this work, we find that the popularly used one-shot type estimator is highly inefficient for prediction purposes. To this end, we propose a novel grid point approximation (GPA) method, which has the following advantages. First, the resulting GPA estimator is as statistically efficient as the global estimator under mild conditions. Second, it requires no communication and is extremely efficient in terms of computation for prediction. Third, it is applicable to the case where the data are not randomly distributed across different machines. To select a suitable bandwidth, two novel bandwidth selectors are further developed and theoretically supported. Extensive numerical studies are conducted to corroborate our theoretical findings. Two real data examples are also provided to demonstrate the usefulness of our GPA method.

An Improved Divide-and-Conquer Approach to Estimating Mean Functional, with Application to Average Treatment Effect Estimation

Mean estimation is an important issue in statistical inference and machine learning. We are concerned with estimating mean functional that is a function of several nonparametric functions when there is a large amount of observations. Directly estimating such mean functional through nonparametric smoothing has the complexity of at least a quadratic order of the sample size, which is computationally prohibitive for massive data. The divide-and-conquer approach are thus readily used to alleviate the computational complexity issue, which however imposes a stringent condition on the sample size in each local machine if a locally optimal bandwidth is used. To address this issue, we suggest to use a globally optimal bandwidth in each local machine, which alleviates the restriction on the local sample sizes substantially. We show that the divide-and-conquer approach with a globally optimal bandwidth achieves the estimation efficiency bound as if all observations were pooled together. In terms of computational efficiency, our proposal outperforms the pooled algorithm dramatically. We demonstrate these properties through average treatment effect estimation from both the asymptotic and the non-asymptotic perspectives.

Modelling Mixed-Frequency Time Series with Structural Change

AbstractPredictive ability of time series models is easily compromised in the presence of structural breaks, common among financial and economic variables amidst market shocks and policy regime shifts. We address this problem by estimating a semiparametric mixed-frequency model, that incorporate high frequency data either in the conditional mean or the conditional variance equation. The inclusion of high frequency data through non-parametric smoothing functions complements the low frequency data to capture possible non-linear relationships triggered by the structural change. Simulation studies indicate that in the presence of structural change, the varying frequency in the mean model provides improved in-sample fit and superior out-of-sample predictive ability relative to low frequency time series models. These hold across a broad range of simulation settings, such as varying time series lengths, nature of structural break points, and temporal dependencies. We illustrate the relative advantage of the method in predicting stock returns and foreign exchange rates in the case of the Philippines.

Towards optimal model evaluation: enhancing active testing with actively improved estimators

With rapid advancements in machine learning and statistical models, ensuring the reliability of these models through accurate evaluation has become imperative. Traditional evaluation methods often rely on fully labeled test data, a requirement that is becoming increasingly impractical due to the growing size of datasets. In this work, we address this issue by extending existing work on active testing (AT) methods which are designed to sequentially sample and label data for evaluating pre-trained models. We propose two novel estimators: the Actively Improved Levelled Unbiased Risk (AILUR) and the Actively Improved Inverse Probability Weighting (AIIPW) estimators which are derived from nonparametric smoothing estimation. In addition, a model recalibration process is designed for the AIIPW estimator to optimize the sampling probability within the AT framework. We evaluate the proposed estimators on four real-world datasets and demonstrate that they consistently outperform existing AT methods. Our study also shows that the proposed methods are robust to changes in subsample sizes, and effective at reducing labeling costs.

Graph-based Tracing of the Atmospheric Boundary Layer by a Ceilometer: A Case Study

The atmospheric boundary layer (ABL) is an important factor in the atmospheric processes. It serves as an intermediary between the earth's surface and the atmosphere in the exchange of mass, energy, and momentum. ABL height (ABLH) also restricts the dispersion volume of atmospheric pollutants. The ABLH can be retrieved from lidar and ceilometer (low power lidar) backscatter profiles by tracing the increased aerosol loading associated with ABL. Numerous lidar-based algorithms are currently implemented, but their ability to trace ABL evolution is often limited. The main issues are related to the selection of the correct layer when multiple aerosol layers are present. To overcome the problem, we employ a graph-based approach to post-process already identified ABL candidates (determined by the gradient in ceilometer backscatter) in two cases. A simple non-parametric smoothing method was also utilized for comparison. Our findings indicate that the graph-based algorithm is superior to the smoothing method in obtaining a consistent tracing of the diurnal evolution of ABL over Sofia, although some issues were also encountered during the evening ABL transition.

A framework to select tuning parameters for nonparametric derivative estimation.

In this paper, we propose a general framework to select tuning parameters for the nonparametric derivative estimation. The new framework broadens the scope of the previously proposed generalized criterion by replacing the empirical derivative with any other linear nonparametric smoother. We provide the theoretical support of the proposed derivative estimation in a random design and justify it through simulation studies. The practical application of the proposed framework is demonstrated in the study of the age effect on hippocampal gray matter volume in healthy adults from the IXI dataset and the study of the effect of age and body mass index on blood pressure from the Pima Indiansdataset.

Nonparametric Smoothing Spline Approach in Examining Investor Interest Factors

The nonparametric approach is an appropriate approach for patterns of relationships between predictor variables and response variables that are not or have not been known in form. In other words, there is no complete information about the pattern of relationships between variables. Curve estimation is determined based on relationship patterns in existing data. The nonparametric approach has great flexibility for estimating regression curves. This study aims to form a model on investor interest factors in improving tourism investment decisions with a nonparametric approach. The nonparametric method used is the smoothing spline regression method. The smoothing spline method is used because the modeling results from the smoothing spline approach can follow the relationship model between variables contained in the data. Thus, this method really helps researchers to model relationships between variables that are not linear and whose linear form is unknown. The results of the analysis showed that the nonparametric smoothing spline regression analysis method could model data by 94.63%, indicates that data variance can be explained by 94.63% with models, while other variance outside the study explain the remaining 5.37%. That is, investment motivation is one of the most important factors to improve investment decisions.

A rank-based approach to evaluate a surrogate marker in a small sample setting.

In clinical studies of chronic diseases, the effectiveness of an intervention is often assessed using "high cost" outcomes that require long-term patient follow-up and/or are invasive to obtain. While much progress has been made in the development of statistical methods to identify surrogate markers, that is, measurements that could replace such costly outcomes, they are generally not applicable to studies with a small sample size. These methods either rely on nonparametric smoothing which requires a relatively large sample size or rely on strict model assumptions that are unlikely to hold in practice and empirically difficult to verify with a small sample size. In this paper, we develop a novel rank-based nonparametric approach to evaluate a surrogate marker in a small sample size setting. The method developed in this paper is motivated by a small study of children with nonalcoholic fatty liver disease (NAFLD), a diagnosis for a range of liver conditions in individuals without significant history of alcohol intake. Specifically, we examine whether change in alanine aminotransferase (ALT; measured in blood) is a surrogate marker for change in NAFLD activity score (obtained by biopsy) in a trial, which compared Vitamin E ($n=50$) versus placebo ($n=46$) among children with NAFLD.

The Factors Driving the Spatial Variation in the Selection of Spawning Grounds for Sepiella japonica in Offshore Zhejiang Province, China

Due to overfishing and marine pollution, the status of fishery resources and the ecological environment in the sea areas have considerably changed. The proliferation and release of fishery resources have become the main means of human intervention to maintain and conserve the resource populations. The annual output of Sepiella japonica (S. japonica), once one of the four major seafood species in the East China Sea, has now recovered to a level of 4000 t. However, do the traditional spawning grounds of S. japonica still exist? Have the spawning grounds shifted? These scientific questions are worthy of attention. Based on the survey data of fishery resources and environment at 120 stations in coastal Zhejiang waters from 2015 to 2022, the spatial and temporal distribution and resource center of gravity of S. japonica in springtime were analyzed. Random forest (RF) was used to explain the importance of six environmental variables, including water depth, sea surface temperature, sea surface salinity, chlorophyll a, pH, and dissolved oxygen. The generalized additive model (GAM) nonparametric smoothing function was used to analyze the relationship between environmental factors and the distribution of S. japonica inhabiting the offshore areas of Zhejiang province, and the effects of environmental factors on spawning habitat selection of S. japonica were revealed. This study found that there was a significant interannual variation in S. japonica resources, with an overall increasing trend in the resource. The spawning grounds were mainly distributed in the Jiushan Islands Marine Reserve, the Dachen Islands Marine Reserve, and their nearby sea areas. The resource peaked at latitude 28.3° N. Additionally, the most important variables affecting the distribution of S. japonica were depth of water, followed by sea surface salinity, pH, dissolved oxygen, sea surface temperature, and chlorophyll a. S. japonica mainly inhabited sea areas with a depth of 15~25 m and a sea surface salinity of 26~32. When the pH ranged from 7.6 to 8.3, dissolved oxygen ranged from 6 to 9 mg/L, sea surface temperature ranged from 14 to 17 °C, and chlorophyll a ranged from 2.5 to 5 µg/L, S. japonica was more likely to be present. This study provides insights into the spatial distribution of S. japonica in offshore Zhejiang province, offering a reference for the rational utilization and scientific protection of this resource.

The Cluster Analysis by Using Nonparametric Cubic B-Spline Modeling for Longitudinal Data

Longitudinal data is becoming increasingly common, especially in the medical and economic fields, and various methods have been analyzed and developed to analyze this type of data.
 In this research, the focus was on compiling and analyzing this data, as cluster analysis plays an important role in identifying and grouping co-expressed subfiles over time and employing them on the nonparametric smoothing cubic B-spline model, which is characterized by providing continuous first and second derivatives, resulting in a smoother curve with fewer abrupt changes in slope. It is also more flexible and can pick up on more complex patterns and fluctuations in the data.
 The longitudinal balanced data profile was compiled into subgroups by penalizing the pairwise distances between the coefficients of the cubic B-spline model using one of the common penalize functions, the Minimax Concave Penalty function (MCP). This method, in turn, works to determine the number of clusters through one of the model selection criteria, Bayesian information criteria (BIC), and we used optimization methods to solve their equations. Therefore, we applied the alternative direction method of the ADMM multiplier algorithm to reach approximate solutions to find the estimators of the nonparametric model using R statistical software.Longitudinally balanced data were generated in the simulation study, as the number of subjects was 60 and the number of repeats (time) was 10 for each subject. The simulation was iterated 100 times, and it showed that employing the MCP partial methods on the cubic model can group profiles into clusters, which is the aim of this paper.
 
 Paper type: Research paper.

Convex Robust Recovery of Corrupted Tensors via Tensor Singular Value Decomposition and Local Low-Rank Approximation

This paper discusses the recovery of tensor data corrupted by random noise. Our approach assumes that the potential structure of data is a linear combination of several low-rank tensor subspaces. The goal is to recover exactly these local low-rank tensors and remove random noise as much as possible. Non-parametric kernel smoothing technique is employed to establish an effective mathematical notion of local models. After that, each local model can be robustly separated into a low-rank tensor and a sparse tensor. The low-rank tensor can be recovered by minimizing a weighted combination of the norm and the tensor nuclear norm (TNN) obtained as the tightest convex relaxation of tensor multi-linear rank defined in Tensor Singular Value Decomposition (TSVD). Numerical simulation experiments verify that our proposed approach can effectively denoise the tensor data such as color images with random noise and has superior performance compared to the existing methods.

TWO-DIMENSIONAL MODELING OF CAR RELIABILITY DURING WARRANTY PERIOD

The paper focuses on presenting the concept of two-dimensional modeling of passenger car reliability during the warranty period. The main objective of this paper is to detect the regularity in the intensity of the number of first failure reports during the warranty period. The two-dimensional distribution of the time and mileage of failure-free exploitation is estimated. The period from the date of purchase to the first warranty repair is analysed. The concept presented incorporates the existing state of knowledge on two-dimensional warranties, expanding it through the use of a nonparametric approach and probability smoothing with the use of P-splines. The estimation involved censored data, i.e., data on vehicles that were not submitted for warranty repair within the warranty limits of time and mileage. The originality of this paper entails the combination of a nonparametric approach with probability smoothing. The statistical analyses presented in the paper were carried out on a population of 1005 vehicles of two car brands sold and serviced in 2011-2021 at the Authorized Service Station (Dealership). There were sales, repair, and warranty claim databases.

1287. Assessing the Utility of Vertebral Body / Disc Space Fluid Cell Differential in the Diagnosis of Native Vertebral Osteomyelitis

Abstract Background Current approaches to diagnosing suspected native vertebral osteomyelitis (NVO) rely on imaging and microbiological workup, including blood and CT-guided biopsy cultures. These existing diagnostic methods often result in ambiguity, necessitating additional tests to distinguish NVO from its mimics. Our study aims to validate pilot findings demonstrating that an elevated neutrophil differential (PMN) in vertebral bone biopsy/ disc space fluid aspirate is associated with NVO. This may allow stratification of patients with suspected NVO into those who can be monitored closely versus those who may benefit from further invasive testing or empiric antibiotic therapy (Fig 1). Figure 1: Proposed workup of patients with suspected native vertebral osteomyelitis guided by disc space/ vertebral body neutrophil (PMN) differential. Methods This analysis examines a subset of patients from a prospective cohort study, enrolling adults referred to Mayo Clinic's neuroradiology department for spine biopsy. After collecting routine clinical specimens, the needle is rinsed with saline into an EDTA tube for automated cell count and differential analysis. To evaluate the predictive capacity of neutrophil (PMN) differential for NVO, logistic regression models were employed, and receiver operating characteristic (ROC) curve analyses were conducted for various PMN percentage cutoff points. Results In this preliminary analysis, 39 patients were included, comprising 7 patients with NVO and 32 patients with alternative diagnoses. Baseline characteristics are presented in Table 1. All NVO patients received antibiotics within two weeks of the spine biopsy. The median biopsy sample PMN percentage for NVO patients was 83.0 (77.5-88.0), compared to 65.0 (56.5-69.5) for those without NVO (p=0.005) (Fig 2A). The estimated relationship between the probability of NVO and each measure across the full range of percentages is illustrated in Fig 2B. The optimal ROC curve point corresponds to a PMN differential of 72.5% (sensitivity=0.86, specificity=0.81) (Fig 3).Table 1:Baseline Characteristics of Patients Included in the Preliminary AnalysisFigure 2:A) Distribution of spine biopsy neutrophil percent by native vertebral osteomyelitis (NVO) status, B) Loess nonparametric smoother showing the relationship between the probability of NVO and biopsy sample neutrophil percentage.Figure 3:ROC curve analysis of neutrophil percent and NVO Conclusion This preliminary analysis indicates that an elevated PMN differential in spine biopsy samples may serve as a valuable diagnostic tool for NVO, with results seemingly unaffected by recent antibiotic intake. The PMN differential cutoff of 72.5% exhibited notable sensitivity and specificity. Investigations are ongoing to further validate these findings Disclosures Brett Freedman, MD, Clear Choice Therapeutics: Stocks/Bonds|Medtronic: Grant/Research Support|Neuroinnovations: Stocks/Bonds Aaron J. Tande, M.D., Musculoskeletal Infection Society: Board Member|Wolters Kluwer Health - Lippincott Williams & Wilkins: Publishing royalties, financial or material support

Bayesian and Classical Semi-parametric Estimation of the Balanced Longitudinal Data Model

The primary objective of this study is to employ semi-parametric regression techniques in the balanced longitudinal data model. Where the parametric regression models are plagued by the problem of strict constraints, while non-parametric regression models, despite their flexibility, suffer from the problem of the curse of dimensionality. Consequently, semi-parametric regression presents a suitable solution to address the problems in parametric and non-parametric regression methods. The advantage of this model is that it contains all the positive properties included in the previous two models such as containing strict restrictions in its parametric component, complete flexibility in its non-parametric component, and clarity of the interaction between its parametric and non-parametric components. According to the above, two methods were used to estimate a semi-parametric balanced longitudinal data model. The first is the Bayesian estimating method; the second is the Speckman method, which estimated the unknown nonparametric smoothing function by employing the kernel smoothing Nadaraya & Watson method. The Aim was to make a comparison between the Bayesian estimation method and the classical estimation method. Based on simulation experiments conducted on three different sample sizes (50, 100, and 200), it was concluded that the Bayes method is best at the variance levels (1,5). In contrast, the Profile least square method is best at the variance level (10).

Test of conditional independence in factor models via Hilbert–Schmidt independence criterion

This work is concerned with testing conditional independence under a factor model setting. We propose a novel multivariate test for non-Gaussian data based on the Hilbert–Schmidt independence criterion (HSIC). Theoretically, we investigate the convergence of our test statistic under both the null and the alternative hypotheses, and devise a bootstrap scheme to approximate its null distribution, showing that its consistency is justified. Methodologically, we generalize the HSIC-based independence test approach to a situation where data follow a factor model structure. Our test requires no nonparametric smoothing estimation of functional forms including conditional probability density functions, conditional cumulative distribution functions and conditional characteristic functions under the null or alternative, is computationally efficient and is dimension-free in the sense that the dimension of the conditioning variable is allowed to be large but finite. Further extension to nonlinear, non-Gaussian structure equation models is also described in detail and asymptotic properties are rigorously justified. Numerical studies demonstrate the effectiveness of our proposed test relative to that of several existing tests.

Exploring Seismic Hazard in the Groningen Gas Field Using Adaptive Kernel Smoothing

The discovery of gas in Groningen in 1959 has been a massive boon to the Dutch economy. From the 1990s onwards, though, gas production has led to induced seismicity. In this paper, we carry out a comprehensive exploratory analysis of the spatio-temporal earthquake catalogue. We develop a non-parametric adaptive kernel smoothing technique to estimate the spatio-temporal hazard map and to interpolate monthly well-based gas production statistics. Second- and higher-order inhomogeneous summary statistics are used to show that the state-of-the-art rate-and-state models for the prediction of seismic hazard fail to capture inter-event interaction in the earthquake catalogue. Based on these findings, we suggest a modified rate-and-state model that also takes into account changes in gas production volumes and uncertainty in the pore pressure field.

Neural Networks for Partially Linear Quantile Regression

Deep learning has enjoyed tremendous success in a variety of applications but its application to quantile regression remains scarce. A major advantage of the deep learning approach is its flexibility to model complex data in a more parsimonious way than nonparametric smoothing methods. However, while deep learning brought breakthroughs in prediction, it is not well suited for statistical inference due to its black box nature. In this article, we leverage the advantages of deep learning and apply it to quantile regression where the goal is to produce interpretable results and perform statistical inference. We achieve this by adopting a semiparametric approach based on the partially linear quantile regression model, where covariates of primary interest for statistical inference are modeled linearly and all other covariates are modeled nonparametrically by means of a deep neural network. In addition to the new methodology, we provide theoretical justification for the proposed model by establishing the root-n consistency and asymptotically normality of the parametric coefficient estimator and the minimax optimal convergence rate of the neural nonparametric function estimator. Across several simulated and real data examples, the proposed model empirically produces superior estimates and more accurate predictions than various alternative approaches.

Adaptive Functional Thresholding for Sparse Covariance Function Estimation in High Dimensions

Covariance function estimation is a fundamental task in multivariate functional data analysis and arises in many applications. In this article, we consider estimating sparse covariance functions for high-dimensional functional data, where the number of random functions p is comparable to, or even larger than the sample size n. Aided by the Hilbert–Schmidt norm of functions, we introduce a new class of functional thresholding operators that combine functional versions of thresholding and shrinkage, and propose the adaptive functional thresholding estimator by incorporating the variance effects of individual entries of the sample covariance function into functional thresholding. To handle the practical scenario where curves are partially observed with errors, we also develop a nonparametric smoothing approach to obtain the smoothed adaptive functional thresholding estimator and its binned implementation to accelerate the computation. We investigate the theoretical properties of our proposals when p grows exponentially with n under both fully and partially observed functional scenarios. Finally, we demonstrate that the proposed adaptive functional thresholding estimators significantly outperform the competitors through extensive simulations and the functional connectivity analysis of two neuroimaging datasets. Supplementary materials for this article are available online.

Practical guide to the typical analysis of prognostic factors and biomarkers without the use of P-values

ObjectivesWe have seen an increasing number of studies evaluating biomarkers and prognostic factors. Biomedical researchers like to draw conclusions based on P-values. However, P-values are often not needed for this type of study. In this article, we show how most biomedical research problems in this area could be organized into three main analyses, each avoiding the use of P-values. MethodsThe three main analyses follow the framework of prediction modeling when the outcome of interest is binary or time-to-event. The analyses make use of figures such as boxplots, nonparametric smoothing line, and nomogram, and also incorporate prediction performance measures such as the Area under the receiver operating characteristic curve, and index of predictive accuracy. ResultsOur proposed framework is easy to follow. It is also in line with most of the research in the field of biomarkers and prognostic factors evaluation, such as reclassification table, net reclassification index, Akaike information criterion/Bayesian information criterion, receiver operating characteristic curve, and decision curve analysis. ConclusionOverall, we provide a step-by-step guideline that biomedical researchers could easily follow to conduct statistical analysis without using P-values, especially with the goal of evaluating biomarkers and prognostic factors.

Nonparametric testing for the specification of spatial trend functions

Correct specification of a spatial trend constitutes an important topic in statistics because it facilitates spatial kriging and inference. This paper proposes a global integrated squared error (GISE) statistics between the nonparametric smoothing surface and the parametric hypothesized model to test for the goodness-of-fit of spatial trends. By virtue of the m-dependence approximation of a stationary random field, it is shown that under certain regularity conditions, the proposed GISE statistics has an asymptotic normal distribution. Further, a grid-based block bootstrap (GBBB) procedure is also proposed to deal with the complicated asymptotic variance involved in the limit distribution. Numerical studies are also presented to illustrate the performance of the proposed method.