Efficient estimation of the mode of continuous multivariate data

Abstract

Mode estimation is an important task, because it has applications to data from a wide variety of sources. Many mode estimates have been proposed with most based on nonparametric density estimates. However, mode estimates obtained by such methods, although they perform excellently with large sample sizes, perform non-satisfactorily with practical (i.e., small to moderate) sample sizes. Recently, Bickel (2003) proposed an efficient method to estimate the mode of continuous univariate data, and showed that its performance is excellent with small to moderate sample sizes. In this paper, we extend Bickel’s method to continuous multivariate data by using the multivariate Box–Cox transform. The excellent performance of the proposed method at practical sample sizes is demonstrated by simulation examples and two real examples from the fields of climatology and image recognition.

Introduction

Mode estimation is an important task, because it has applications to data from a wide variety of sources. Its applications, for example, include 1-D (one-dimensional, similarly, 2-D, etc.) measurements of cellular DNA content (Marchette and Wegman, 1997), 1-D time estimates in molecular clock studies (Hedges and Shah, 2003), 2-D predictions of the greatest air-pollution (Sager, 1979), seismic tremor locations (Ooi, 2002), 3-D color histograms (Comaniciu, 2003), 5-D color image segmentation (Comaniciu and Meer, 2002), 13-D flow cytometry measurements (Jing et al., 2012), 16-D image recognition (Griffin and Lillholm, 2003), 66-D gene expression time-courses (Liu and Muller, 2003), and 365-D daily rainfall rates (Hall and Heckman, 2002). However, it is difficult to estimate the mode(s) precisely when the underlying distribution is continuous but otherwise unknown.

Many mode estimates have been proposed for univariate data, with most based on nonparametric kernel density estimates, i.e., the population modes are estimated by those of the density estimates (see, e.g., Parzen, 1962, Eddy, 1980, Silverman, 1981, Vieu, 1996 and Minnotte et al., 1998, among many others). Although the NKDM (nonparametric kernel density mode) performs excellently with large sample sizes, its performance at small to moderate ones is not satisfactory. Recently, Bickel (2003) proposed a parametric mode, using the univariate Box–Cox transform, which performs very well at small to moderate sample sizes. For a review of methods for univariate data, the reader is referred to Bickel and Frühwirth (2006) and the references cited therein.

Although many practical applications use multivariate data, much less work has been done in the area of mode estimation for this. This may be due to the fact that multivariate cases are more complex than univariate ones. The selection of the bandwidth, i.e., scale parameter, is crucial to the performance of the NKDM. Gasser et al. (1998), Abraham et al. (2003) and Klemelä (2005) proposed multivariate NKDMs with a fixed bandwidth, and Jing et al. (2012) proposed a similar one using polynomial histograms instead of kernels. Griffin and Lillholm, 2003, Griffin and Lillholm, submitted for publication proposed multivariate NKDMs using the PSST (pessimistic scale space tracking) and MSMS (multiscale mean shift) methods. Both methods are variants of the bandwidth reduction method, where the bandwidth varies from large to small and the path of the mode varies continuously, with the tracking process continuing until some stopping rule is satisfied. For an approach which is different from the preceding nonparametric kernel or polynomial histogram approaches, the reader is referred to Sager (1979).

In this paper, we propose a parametric mode estimate for continuous multivariate data using the joint Box–Cox transform. Our estimate is the multivariate extension of the one by Bickel (2003). Section 2 gives the details and theoretical properties of the proposed estimate. Section 3 contains simulation studies. Sections 3.1 Simulation study for i.i.d. data from a unimodal density, 3.2 Sensitivity to multimodalities deal with simulated examples for i.i.d. data from unimodal- and multimodal-densities, respectively. These clearly show that for unimodal densities our estimate is superior to other estimates included in this study when the sample size $n$ is small to moderate. Further, our estimate is more reliable and less sensitive to multimodalities than other estimates in estimating the major mode when $n$ is relatively small and/or the amount of data clustering around the minor mode(s) is substantially less than that around the major one. Sections 3.3 Simulation study for dependent data, 3.4 Simulation study for functional data explore the possibility of extending our mode estimate to the case of dependent data and i.i.d. functional data, respectively, and very encouraging results are obtained when $n$ is relatively small. In Section 3.5, we illustrate the performance of our estimate by two real examples from the fields of climatology and image recognition. Finally, the Appendix contains the proofs.