رده‌بندی ‌چند-رده‌ای مبتنی بر تابع ژرفا برای داده‌های چندمتغیره

این مقاله به بیان یک رویکرد ناپارامتری بر اساس تابع ژرفا برای رده‌بندی داده‌های چند‌متغیره به چندین رده می‌پردازد. پیاده‌سازی این روش برخلاف اغلب روش‌های ناپارامتری دارای پیچیدگی محاسباتی نیست و در صورت برقراری فرض تقارن بیضوی مشاهدات، با قاعده  بهینه بیزی معادل است. ارزیابی عملکرد این رده‌بندی‌‌ساز بر اساس توابع ژرفای مختلف، بر اساس مطالعات شبیه‌سازی و تحلیل داده‌های واقعی انجام می‌شود.

multi-class depth-based classification for multivariate data

in a supervised classification problem with j classes, we have nj labeled observations from the j-th class, j = 1, ..., j. we use this training sample consisting of n = pj j=1 nj observations to construct a decision rule for classifying an unlabeled observation x to one of these j classes. popular parametric classifiers like linear discriminant analysis and quadratic discriminantanalysis are motivated by parametric model assumptions. so, theymay lead to poor classification when these assumptions fail to hold. this paperpresents a nonparametric multi-class depth-based classification approachfor multivariate data. this approach is easy to implement rather than mostexisting nonparametric methods that have computational complexity. if theassumption of the elliptical symmetry holds, this method is equivalent tothe bayes optimal rule. the performance of the depth-based classifiers isevaluated using a monte carlo study as well as a real data example and iscompared to that of several competitors.material and methodsa depth function is aimed at ordering points in a space according to theircentrality with respect to a distribution f. the larger the depth of a pointx, the more central x is with respect to f. it is a useful tool in the multidimensionalcase where ordering the points from the inner to the outer part ofa distribution or sample is not trivial task. formally, a depth function is abounded function that satisfies certain properties including affine invariance,maximised somewhere in the center of the distribution, monotonicity relativeto deepest point, and vanishing at infinity. therefore, the depth functions of fer a chance to exploit nonparametric methods in multivariate data analysis.accordingly, several notions of multivariate depth have been proposed in theliterature, and the field of applications of data depth is vast and still growing.this paper presents an overview of the depth function concept, and a nonparametricmulti-class depth-based classification approach for multivariatedata has been investigated. moreover, the methodology has been verifiedusing a simulation study and a real data set example. calculations havebeen performed in r software, and r codes are available by request to theauthor.results and discussionthe depth-based classification approach for multivariate data is equivalentto the bayes optimal rule under elliptic symmetry. indeed, under ellipticsymmetry, the density function of a class can be expressed as a function ofaffine invariant depth functions, and hence the depth contours coincide withthe density contours. using both empirical and theoretical arguments, weconsider the properties of the depth-based classification methods, and showthat these classifiers can be particularly effective. we have used some simulateddata sets to evaluate the performance of the depth-based classifiers.moreover, to illustrate the method developed in this paper, we have appliedthe classifiers on a well-known data set.conclusionin this article, the depth function and related classifiers have been investigated.we have analyzed several data sets simulated from elliptic as well asnon-elliptic multivariate distributions. moreover, the swiss banknote dataset has been analyzed for further evaluation of depth-based classifiers. theresults showed that the maximum depth classifier based on the mahalanobisand half-space depth functions has a good performance.