A general non-parametric technique is proposed for the analysis of a complex multimodal feature space and to delineate arbitrarily shaped clusters in it. The basic computational module of the technique is an old pattern recognition procedure: the mean shift. For discrete data, we prove the convergence of a recursive mean shift procedure to the nearest stationary point of the underlying density function and, thus, its utility in detecting the modes of the density. The relation of the mean shift procedure to the Nadaraya-Watson estimator from kernel regression and the robust M-estimators; of location is also established. Algorithms for two low-level vision tasks discontinuity-preserving smoothing and image segmentation - are described as applications. In these algorithms, the only user-set parameter is the resolution of the analysis, and either gray-level or color images are accepted as input. Extensive experimental results illustrate their excellent performance.

Mean shift: a robust approach toward feature space analysis

1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

This paper is concerned with the estimation of covariance matrices in the presence of heteroskedasticity and autocorrelation of unknown forms. Currently available estimators that are designed for this context depend upon the choice of a lag truncation parameter and a weighting scheme. Results in the literature provide a condition on the growth rate of the lag truncation parameter as T \rightarrow \infty that is sufficient for consistency. No results are available, however, regarding the choice of lag truncation parameter for a fixed sample size, regarding data-dependent automatic lag truncation parameters, or regarding the choice of weighting scheme. In consequence, available estimators are not entirely operational and the relative merits of the estimators are unknown. This paper addresses these problems. The asymptotic truncated mean squared errors of estimators in a given class are determined and compared. Asymptotically optimal kernel/weighting scheme and bandwidth/lag truncation parameters are obtained using an asymptotic truncated mean squared error criterion. Using these results, data-dependent automatic bandwidth/lag truncation parameters are introduced. The finite sample properties of the estimators are analyzed via Monte Carlo simulation.

/pdf/heteroskedasticity-and-autocorrelation-consistent-covariance-3tz1r06ty1.pdf

Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation

Purpose To improve on current standards for breast cancer prognosis and prediction of chemotherapy benefit by developing a risk model that incorporates the gene expression–based “intrinsic” subtypes luminal A, luminal B, HER2-enriched, and basal-like. Methods A 50-gene subtype predictor was developed using microarray and quantitative reverse transcriptase polymerase chain reaction data from 189 prototype samples. Test sets from 761 patients (no systemic therapy) were evaluated for prognosis, and 133 patients were evaluated for prediction of pathologic complete response (pCR) to a taxane and anthracycline regimen. Results The intrinsic subtypes as discrete entities showed prognostic significance (P = 2.26E-12) and remained significant in multivariable analyses that incorporated standard parameters (estrogen receptor status, histologic grade, tumor size, and node status). A prognostic model for node-negative breast cancer was built using intrinsic subtype and clinical information. The C-index estimate for t...

/pdf/supervised-risk-predictor-of-breast-cancer-based-on-32oqdfzz8t.pdf

Supervised Risk Predictor of Breast Cancer Based on Intrinsic Subtypes

B-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent B-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of B-splines, of their construction and of penalized likelihood is presented. We discuss properties of penalized B-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.

/pdf/flexible-smoothing-with-b-splines-and-penalties-aqyuxvm7bh.pdf

Flexible smoothing with B-splines and penalties

We present a new method for data-based selection of the bandwidth in kernel density estimation which has excellent properties It improves on a recent procedure of Park and Marron (which itself is a good method) in various ways First, the new method has superior theoretical performance; second, it also has a computational advantage; third, the new method has reliably good performance for smooth densities in simulations, performance that is second to none in the existing literature These methods are based on choosing the bandwidth to (approximately) minimize good quality estimates of the mean integrated squared error The key to the success of the current procedure is the reintroduction of a non- stochastic term which was previously omitted together with use of the bandwidth to reduce bias in estimation without inflating variance

A reliable data-based bandwidth selection method for kernel density estimation

Visual Analogue Scales (VAS) provide a simple technique for measuring subjective experience. They have been established as valid and reliable in a range of clinical and research applications, although there is also evidence of increased error and decreased sensitivity when used with some subject groups. Decisions concerned with the choice of scoring interval, experimental design, and statistical analysis for VAS have in some instances been based on convention, assumption and convenience, highlighting the need for more comprehensive assessment of individual scales if this versatile and sensitive measurement technique is to be used to full advantage.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0033291700009934

Clinical applications of visual analogue scales: a critical review.

There has been major progress in recent years in data-based bandwidth selection for kernel density estimation. Some “second generation” methods, including plug-in and smoothed bootstrap techniques, have been developed that are far superior to well-known “first generation” methods, such as rules of thumb, least squares cross-validation, and biased cross-validation. We recommend a “solve-the-equation” plug-in bandwidth selector as being most reliable in terms of overall performance. This article is intended to provide easy accessibility to the main ideas for nonexperts.

https://www.stat.washington.edu/courses/stat527/s13/readings/Jones_etal_JASA_1996.pdf

A Brief Survey of Bandwidth Selection for Density Estimation

Local least squares kernel regression provides an appealing solution to the nonparametric regression, or “scatterplot smoothing,” problem, as demonstrated by Fan, for example. The practical implementation of any scatterplot smoother is greatly enhanced by the availability of a reliable rule for automatic selection of the smoothing parameter. In this article we apply the ideas of plug-in bandwidth selection to develop strategies for choosing the smoothing parameter of local linear squares kernel estimators. Our results are applicable to odd-degree local polynomial fits and can be extended to other settings, such as derivative estimation and multiple nonparametric regression. An implementation in the important case of local linear fits with univariate predictors is shown to perform well in practice. A by-product of our work is the development of a class of nonparametric variance estimators, based on local least squares ideas, and plug-in rules for their implementation.

https://www.jstor.org/stable/pdf/2291516.pdf

An Effective Bandwidth Selector for Local Least Squares Regression

The Field of Statistics. Estimating Scale----Finite Sample Results. Estimating Scale----Asymptotic Results. Location--Dispersion Estimation. Testing for a Location Parameter. The Two--Sample Problem. Regression. Appendices. References. Author Index. Subject Index.

Simon J. Sheather

Papers

A reliable data-based bandwidth selection method for kernel density estimation

Clinical applications of visual analogue scales: a critical review.

A Brief Survey of Bandwidth Selection for Density Estimation

An Effective Bandwidth Selector for Local Least Squares Regression

Robust Estimation and Testing