Fast DD-classification of functional data

Mosler, Karl; Mozharovskyi, Pavlo

doi:10.1007/s00362-015-0738-3

Cited by 26 publications

(25 citation statements)

References 42 publications

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“…Methods of functional data analysis are becoming increasingly popular, e.g. in the cluster analysis (Jacques and Preda 2013;James and Sugar 2003;Peng and Müller 2008), classification (Chamroukhi et al 2013;Delaigle and Hall 2012;Mosler and Mozharovskyi 2015;Rossi and Villa 2006) and regression (Ferraty et al 2012;Goia and Vieu 2014;Kudraszow and Vieu 2013;Peng et al 2015;Rachdi and Vieu 2006;Wang et al 2015). Unfortunately, multivariate data methods cannot be directly used for functional data, because of the problem of dimensionality and difficulty in putting functional data into order.…”

Section: Introductionmentioning

confidence: 99%

Selected statistical methods of data analysis for multivariate functional data

et al. 2016

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Data in the form of a continuous vector function on a given interval are referred to as multivariate functional data. These data are treated as realizations of multivariate random processes. The paper is devoted to three statistical dimension reduction techniques for multivariate data. For the first one, principal components analysis, the authors present a review of a recent paper (Jacques and Preda in, Comput Stat Data Anal, 71:92-106, 2014). For two others one, canonical variables and discriminant coordinates, the authors extend existing works for univariate functional data to multivariate. These methods for multivariate functional data are presented, illustrated and discussed in the context of analyzing real data sets. Each of these techniques is applied on real data set.

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Section: Introductionmentioning

confidence: 99%

Selected statistical methods of data analysis for multivariate functional data

et al. 2016

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“…However, such weights are difficult to obtain. Rather, as in Mosler and Mozharovskyi (2014), I suggest that the whole interval be divided into a number of subintervals of equal length, and the functions be averaged over the subintervals. If the number of subintervals is m, we obtain a problem of outlier detection in R m , which may be solved by employing a properly chosen multivariate data depth.…”

Section: Local Versus Global Outlyingness: Considering Subintervalsmentioning

confidence: 99%

“…Therefore, in comparing functions not only their levels may be taken into account but also their increasing behavior. For example, in the classic Berkeley data the growth curves of girls and boys are most easily distinguished by looking at the average slope of the curves above the age of 10; see Mosler and Mozharovskyi (2014). Most existing procedures for functional data analysis largely disregard this aspect by considering levels of functions only and aggregating this information symmetrically in the time parameter.…”

Section: Features Of Functional Datamentioning

confidence: 99%

“…As it is piece-wise linear, the projection depth attains its maximum at the edges of direction cones of constant linearity, a randomly chosen direction yields the exact depth value with probability zero. Consequently, it has to be evaluated in a huge number of directions (which moreover should increase exponentially with dimension k), each of which involves the calculation of the median and MAD of a univariate distribution; see Mosler and Mozharovskyi (2014).…”

Section: Depth Statistics For Functional Datamentioning

confidence: 99%

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Discussion of “Multivariate functional outlier detection” by Mia Hubert, Peter Rousseeuw and Pieter Segaert

Mosler

2015

Stat Methods Appl

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subsequently HRS) are to be praised for having developed a visual and well applicable methodology of treating outliers in functional data. Their principal achievements include a systematics of functional outliers, novel visualizations and a measure of outlyingness (bagdistance), which nicely combines Euclidean distance with location depth. Most important, the authors offer a consequent multivariate view on the functions, either assuming genuinely multidimensional functions or considering multiple aspects of such functions. Several real-data examples illuminate the capacity and the possible output of the new procedures. Also, an R-package of the methodology is announced, which will be greatly welcomed.In the following remarks, I shall first try to put the new methodology into a broader context of outlier search. Then some specifics of functional data are discussed, which call for particular treatments. Third a refined approach is sketched that considers integrals over subintervals. Finally, I will address the use of different functional depth notions and the computational problems arising with them.

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“…The idea behind is that, while being conservative, the bound can still provide insightful ordering of the LS-pairs, especially in the case when the empirical risk and the bound have the same order of magnitude. Given a set of considerable pairs S = {(l i , s i )|i = 1, ..., N ls }, for each its element calculate the Vapnik-Chervonenkis bound (see Mosler and Mozharovskyi, 2015, for this particular derivation)…”

Section: An Extension To Functional Datamentioning

confidence: 99%

Depth and Depth-Based Classification with R Package ddalpha

Pokotylo¹,

Mozharovskyi²,

Dyckerhoff³

2019

J. Stat. Soft.

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Following the seminal idea of Tukey (1975), data depth is a function that measures how close an arbitrary point of the space is located to an implicitly defined center of a data cloud. Having undergone theoretical and computational developments, it is now employed in numerous applications with classification being the most popular one. The R-package ddalpha is a software directed to fuse experience of the applicant with recent achievements in the area of data depth and depth-based classification.ddalpha provides an implementation for exact and approximate computation of most reasonable and widely applied notions of data depth. These can be further used in the depth-based multivariate and functional classifiers implemented in the package, where the DDα-procedure is in the main focus. The package is expandable with user-defined custom depth methods and separators. The implemented functions for depth visualization and the built-in benchmark procedures may also serve to provide insights into the geometry of the data and the quality of pattern recognition.Being intrinsically nonparametric, a depth function captures the geometrical features of given data in an affine-invariant way. By that, it appears to be useful for description of data's location, scatter, and shape, allowing for multivariate inference, detection of outliers, ordering of multivariate distributions, and in particular classification, that recently became an important and rapidly developing application of the depth machinery. While the parameter-free nature of data depth ensures attractive theoretical properties of classifiers, its ability to reflect data topology provides promising predicting results on finite samples. Classification in the depth spaceConsider the following setting for supervised classification: Given a training sample consisting of q classes X 1 , ..., X q , each containing n i , i = 1, ..., q, observations in R d . For a new observation x 0 , a class should be determined, to which it most probably belongs. Depth-based learning started with plug-in type classifiers. Ghosh and Chaudhuri (2005b) construct a depth-based classifier, which, in its naïve form, assigns the observation x 0 to the class in which it has maximal depth. They suggest an extension of the classifier, that is consistent w.r.t. Bayes risk for classes stemming from elliptically symmetric distributions. Further Dutta and Ghosh (2011, 2012) suggest a robust classifier and a classifier for L p -symmetric distributions, see also Cui et al. (2008), Mosler and Hoberg (2006), and additionally Jörnsten (2004) for unsupervised classification.A novel way to perform depth-based classification has been suggested by Li et al. (2012): first map a pair of training classes into a two-dimensional depth space, which is called the DD-plot, and then perform classification by selecting a polynomial that minimizes empirical risk. Finding such an optimal polynomial numerically is a very challenging and -when done appropriatelycomputationally involved task, with a solution that in practice ca...

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Fast DD-classification of functional data

Cited by 26 publications

References 42 publications

Selected statistical methods of data analysis for multivariate functional data

Selected statistical methods of data analysis for multivariate functional data

Discussion of “Multivariate functional outlier detection” by Mia Hubert, Peter Rousseeuw and Pieter Segaert

Depth and Depth-Based Classification with R Package ddalpha

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