The Mode Tree: A Tool for Visualization of Nonparametric Density Features

Abstract

Recognition and extraction of features in a nonparametric density estimate are highly dependent on correct calibration. The data-driven choice of bandwidth h in kernel density estimation is a difficult one that is compounded by the fact that the globally optimal h is not generally optimal for all values of x. In recognition of this fact a new type of graphical tool, the mode tree, is proposed. The basic mode tree plot relates the locations of modes in density estimates with the bandwidths of those estimates. Additional information can be included on the plot indicating factors such as the size of modes, how modes split, and the locations of antimodes and bumps. The use of a mode tree in adaptive multimodality investigations is proposed, and an example is given to show the value in using a normal kernel, as opposed to the biweight or other kernels, in such investigations. Examples of such investigations are provided for Ahrens's chondrite data and van Winkle's Hidalgo stamp data. Finally, the bivariate mode tree is introduced, together with an example using Scott's lipid data.     

Algorithms for manipulation of level sets of nonparametric density estimates

Summary  We present algorithms for finding the level set tree of a multivariate density estimate. That is, we find the separated components of level sets of the estimate for a series of levels, gather information on the separated components, such as volume and barycenter, and present the information together with the tree structure of the separated components. The algorithm proceeds by first building a binary tree which partitions the support of the density estimate, followed by bottom-up travels of this tree during which we join those parts of the level sets which touch each other. As a byproduct we present an algorithm for evaluating a kernel estimate on a large multidimensional grid. Since we find the barycenters of the separated components of the level sets also for high levels, our method finds the locations of local extremes of the estimate. Writing of this article was financed by Deutsche Forschungsgemeinschaft under project MA1026/8-1.     

The Accuracy and the Computational Complexity of a Multivariate Binned Kernel Density Estimator

The computational cost of multivariate kernel density estimation can be reduced by prebinning the data. The data are discretized to a grid and a weighted kernel estimator is computed. We report results on the accuracy of such a binned kernel estimator and discuss the computational complexity of the estimator as measured by its average number of nonzero terms.     

Mode Trees for Multivariate Data

We consider visualizing scales of multivariate density estimates with the help of mode trees. Mode trees visualize the locations of the modes of density estimates, when the smoothing parameter of the estimates ranges over an interval. We define multiframe mode graphs which generalize classical mode trees to the multivariate setting. We give examples of the application of multiframe mode graphs with kernel estimates and with multivariate adaptive histograms.     

A quantile-copula approach to conditional density estimation

A new kernel-type estimator of the conditional density is proposed. It is based on an efficient quantile transformation of the data. The proposed estimator, which is based on the copula representation, turns out to have a remarkable product form. Its large-sample properties are considered and comparisons in terms of bias and variance are made with competitors based on nonparametric regression. A comparative simulation study is also provided.     

Laws of the Iterated Logarithm for the Local U-Statistic Process

Laws of the iterated logarithm are established for the local U-statistic process. This entails the development of probability inequalities and moment bounds for U-processes that should be of separate interest. The local U-statistic process is based upon an estimator of the density of a function of several i.i.d. variables proposed by Frees (J. Am. Stat. Assoc. 89, 517–525, 1994). As a consequence, our results are directly applicable to the derivation of exact rates of uniform in bandwidth consistency in the sup and in the L _p norms for these estimators. Research of E. Giné partially supported by NSA Grant H98230-04-1-0075. Research of D.M. Mason partially supported by NSA Grant MDA904-02-1-0034 and NSF Grant DMS-0503908.     

Estimation of heavy-tailed probability density function with application to Web data

Summary  Common non-parametric estimators of a probability density function (PDF) show bad performance for heavy-tailed PDFs. Using a parametric approximation of the true cumulative distribution function (CDF), the transformation-retransformation of the data is explored here as a useful tool for the reliable PDF prediction. The PDF estimators are compared by their capacity to solve a classification problem. Simulation results and an application to Web data analysis are presented, too.     

核估计在医疗费用分布研究中的应用

医疗费用的分布是健康保险的精算基础之一.本文研究了某团体一年的住院医疗费用记录,运用非参数模型中的核密度估计方法,对每次住院费用对数的分布密度进行了估计,并用它进一步估计了每次住院费用的均值和方差,取得了较好的效果.在此基础上,本文对拟合结果及其对健康保险的启示作了一些讨论.     

InfoVis Is So Much More: A Comment on Gelman and Unwin and an Invitation to Consider the Opportunities

We propose new tools for visualizing large amounts of functional data in the form of smooth curves. The proposed tools include functional versions of the bagplot and boxplot, which make use of the first two robust principal component scores, Tukey’s data depth and highest density regions.

By-products of our graphical displays are outlier detection methods for functional data. We compare these new outlier detection methods with existing methods for detecting outliers in functional data, and show that our methods are better able to identify outliers.

An R-package containing computer code and datasets is available in the online supplements.     

Continuous Contour Monte Carlo for Marginal Density Estimation With an Application to a Spatial Statistical Model

The problem of marginal density estimation for a multivariate density function f(x) can be generally stated as a problem of density function estimation for a random vector λ(x) of dimension lower than that of x. In this article, we propose a technique, the so-called continuous Contour Monte Carlo (CCMC) algorithm, for solving this problem. CCMC can be viewed as a continuous version of the contour Monte Carlo (CMC) algorithm recently proposed in the literature. CCMC abandons the use of sample space partitioning and incorporates the techniques of kernel density estimation into its simulations. CCMC is more general than other marginal density estimation algorithms. First, it works for any density functions, even for those having a rugged or unbalanced energy landscape. Second, it works for any transformation λ(x) regardless of the availability of the analytical form of the inverse transformation. In this article, CCMC is applied to estimate the unknown normalizing constant function for a spatial autologistic model, and the estimate is then used in a Bayesian analysis for the spatial autologistic model in place of the true normalizing constant function. Numerical results on the U.S. cancer mortality data indicate that the Bayesian method can produce much more accurate estimates than the MPLE and MCMLE methods for the parameters of the spatial autologistic model.     

Profitability in the car industry: New measures for estimating targets and target directions

In this paper we study the profitability of car manufacturers in relation to industry-wide profitability targets such as industry averages. Specifically we are interested in whether firms adjust their profitability in the direction of these targets, whether it is possible to detect any such change, and, if so, what the precise nature is of these changes.     

An integral transform method for estimating the central mean and central subspaces

The central mean and central subspaces of generalized multiple index model are the main inference targets of sufficient dimension reduction in regression. In this article, we propose an integral transform (ITM) method for estimating these two subspaces. Applying the ITM method, estimates are derived, separately, for two scenarios: (i) No distributional assumptions are imposed on the predictors, and (ii) the predictors are assumed to follow an elliptically contoured distribution. These estimates are shown to be asymptotically normal with the usual root-n convergence rate. The ITM method is different from other existing methods in that it avoids estimation of the unknown link function between the response and the predictors and it does not rely on distributional assumptions of the predictors under scenario (i) mentioned above.     

Simple consistent cluster methods based on redescending M-estimators with an application to edge identification in images

We use the local maxima of a redescending M-estimator to identify cluster, a method proposed already by Morgenthaler (in: H.D. Lawrence, S. Arthur (Eds.), Robust Regression, Dekker, New York, 1990, pp. 105–128) for finding regression clusters. We work out the method not only for classical regression but also for orthogonal regression and multivariate location and show that all three approaches are special cases of a general approach which includes also other cluster problems. For the general case we show consistency for an asymptotic objective function which generalizes the density in the multivariate case. The approach of orthogonal regression is applied to the identification of edges in noisy images.     

C-D生产函数在林产品投入产出中的应用研究

本文运用C-D生产函数，并结合闽西北毛竹的密度效应模型和产量模型，对其进行弹性分析和边际产量分析。提出用C-D生产函数建立毛竹密度效应模型的优越性，为达到合理、高效的营林工作提供科学依据。     

The Asymptotic Variance of the Continuous-Time Kernel Estimator with Applications to Bandwidth Selection

We derive simple expressions for the asymptotic variance of the kernel-density estimator of a stationary continuous-time process in one and d dimensions and relate convergence rates to sample path smoothness. Important applications include methods for selecting optimal smoothing parameters and construction of confidence bands for testing hypotheses about the density. In a simulation study the results are applied to bandwidth selection for discrete-time processes that can be modelled as continuous-time processes sampled at a high rate.     

恒星对增长螺旋引力扰动的非线性响应及其对增长模式的致稳作用*

本文计算了在增长的螺旋扰动引力场中恒星响应的非线性效应.结果表明,这一非线性效应会导致Q值的增加,从而降低增长率.对于振幅很小,增长率也较小的螺旋形模式,Q值增加较慢;对于振幅较大或增民率较大的模式,Q值也增加较快,从而有效地抑正振幅的增长.这一调节机制是使得由线性理论所得出的增长模式最终达到准稳状态的原因之一。     

Confidence regions for the intensity function of a cyclic Poisson process

A classical approach to constructing simultaneous confidence intervals (i.e., confidence bands or regions) for a function is via establishing a limiting process of the appropriately normalized difference between the function and its empirical estimator. In the present paper we depart from this approach and construct confidence bands for the intensity function of a cyclic Poisson process via extreme value type asymptotic results for the appropriately normalized supremum of the difference between the intensity function and its empirical estimator.      

Using the bootstrap to estimate mean squared error and select smoothing parameter in nonparametric problems

We describe a bootstrap method for estimating mean squared error and smoothing parameter in nonparametric problems. The method involves using a resample of smaller size than the original sample. There are many applications, which are illustrated using the special cases of nonparametric density estimation, nonparametric regression, and tail parameter estimation.     

The Effect of Inhomogeneity of Elastic Properties on the Stress State in Composite Cylindrical Panels

The effect of inhomogeneity of elastic properties in the circumferential direction on the distribution of stress and displacement fields in orthotropic cylindrical panels is studied. The mechanical properties of the panels and the load acting on them are constant in the axial direction, which makes it possible to neglect the influence of the curvilinear ends. From the initial relations of a three-dimensional problem of the elasticity theory of inhomogeneous anisotropic bodies, a resolving system of partial differential equations is obtained, whose solution is presented in the form of truncated Fourier series, so that the conditions of free support of the rectilinear ends are satisfied. This allows us to separate the variables and to get a system of ordinary high-order differential equations, which is integrated by a stable numerical method. The problem on the stress-strain state of an orthotropic composite panel with a varying relative volume content of reinforcing elements in the circumferential direction is solved. The effect of the change in the reinforcement density on the stresses and displacements of the panel is studied.     

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Besides the claims data in the past, certain assumptions aboutthe distribution of claimsare required to derive the credibilitypremium in the classical theory. In the paper, the credibility premium can be calculatedvia the maximum entropy method if we know nothing about the distribution of claims. Furthermore, two corollaries are obtained under certain assumptions, that is, new claims have more weight than the old ones and the classical credibilityformula is a special case of the credibility premium derived in the present paper.Finally, the simulation study is presented to illustrate that the credibility premiumin the present paper is better than other models if the mean square error is taken asthe evaluation criterion.