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## Classifying real-world data with the DDα-procedure (2013)

Citations: | 1 - 1 self |

### Citations

12920 | The nature of statistical learning theory
- Vapnik
- 1995
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Citation Context ...12), where the polynomial degree is chosen by cross-validating. Other possible approaches are regression depth (Christmann & Rousseeuw, 2001; Christmann et al., 2002) or SVM (Christmann et al., 2002; =-=Vapnik, 1998-=-). It is clear that in general the obtained separating hypersurface is not the one minimizing EMR, if more than two features are needed. But in which of the applications are they really needed? Tables... |

3611 | Support vector networks
- Cortes, Vapnik
- 1995
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Citation Context ... classifiers: optimal Bayes (dashed lines), DDα (left, solid line), and SVM-simplified (right, solid line) classifiers. The SVM-s step consists in solving the following quadratic programming problem (=-=Cortes & Vapnik, 1995-=-): maximize λ W (λ) = λ′1− 1 2 λ′Dλ (7) subject to the constraints λ ≥ 0, (8) λ′y = 0. (9) Here we notate l = n1 + n2, λ = (λ1, ..., λl) ′, 1 = (1, . . . , 1)′ and 0 = (0, . . . , 0)′ ∈ Rl. y stands f... |

3388 | The elements of statistical learning, - Hastie, Tibshirani, et al. - 2009 |

1830 | Making large-scale SVM learning practical.,”
- Joachims
- 1999
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Citation Context ...tory. The max-Mahalanobis-depth classifier is calculated either with moment or MCD estimates, setting α = 0.75. As a basis for the SVM-simplified classifier Joachims’s C++ implementation of SVMlight (=-=Joachims, 1999-=-) is used with slight modifications and interfaced to the R-environment. We use an R-implementation for the traditional KNN with ties broken at random; similarly when treating the outsiders by the aff... |

1530 | The Use of Multiple Measurements in Taxonomic Problems - Fisher |

1446 |
Pattern Recognition and Neural Networks.
- Ripley
- 1997
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Citation Context ...taken from the R-packages ‘locfit’ and ‘rrcov’ respectively. The “pima” data set constitutes a training subsample of the “diabetes” (see below) and can be downloaded from www.stats.ox.ac.uk/pub/PRNN (=-=Ripley, 1996-=-). Datasets “baby”, “banknoten” (Flury & Riedwyl, 1988), “crab” (Ripley, 1996), “gemsen” , “groessen” (Galton, 1885), “tennis”, “tips” and “uscrime” (Hand et al., 1994) have been downloaded from the t... |

521 |
On the generalised distance in statistics,
- Mahalanobis
- 1936
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Citation Context ...the convex hull of the data. For a point z ∈ Rd and a random vector X in Rd (especially one having an empirical distribution on a set of d-variate observations {x1, . . . ,xn}) the Mahalanobis depth (=-=Mahalanobis, 1936-=-) of z w.r.t. X is defined as DMah(z|X) = (1 + (z− µX)′Σ−1X (z− µX))−1, (1) where µX measures the location (e.g. the mean) of X, and ΣX the scatter (e.g. the covariance matrix) of X. The affine invari... |

367 | Large margin DAGs for multiclass classification, - Platt, Cristianini, et al. - 1999 |

336 | A Fast algorithm for the Minimum Covariance Determinant Estimator. - Rousseeuw, Driessen - 1999 |

168 | Mathematics and the picturing of data - Tukey - 1975 |

166 | Multisurface method of pattern separation for medical diagnosis applied to breast cytology,” - Wolberg, Mangasarian - 1990 |

122 |
Regression towards mediocrity in hereditary stature
- Galton
- 1886
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Citation Context ...f the “diabetes” (see below) and can be downloaded from www.stats.ox.ac.uk/pub/PRNN (Ripley, 1996). Datasets “baby”, “banknoten” (Flury & Riedwyl, 1988), “crab” (Ripley, 1996), “gemsen” , “groessen” (=-=Galton, 1885-=-), “tennis”, “tips” and “uscrime” (Hand et al., 1994) have been downloaded from the teaching data base stat.ethz.ch/Teaching/Datasets. The rest of the data sets is taken from archive.ics.uci.edu/ml (F... |

120 |
UCI Machine Learning Repository [http://archive.ics.uci.edu/ml].
- Frank, Asuncion
- 2010
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Citation Context ...5), “tennis”, “tips” and “uscrime” (Hand et al., 1994) have been downloaded from the teaching data base stat.ethz.ch/Teaching/Datasets. The rest of the data sets is taken from archive.ics.uci.edu/ml (=-=Frank & Asuncion, 2010-=-); it in particular originates from Yeh et al. (2009) (“blood-transfusion”), Wolberg & Mangasarian (1990) (“breast-cancer-wisconsin”) and Turney (1993) (“vowel”). Multiclass problems were reasonably s... |

97 |
Multivariate statistics. A practical approach
- Flury, Riedwyl
- 1988
(Show Context)
Citation Context ...’ respectively. The “pima” data set constitutes a training subsample of the “diabetes” (see below) and can be downloaded from www.stats.ox.ac.uk/pub/PRNN (Ripley, 1996). Datasets “baby”, “banknoten” (=-=Flury & Riedwyl, 1988-=-), “crab” (Ripley, 1996), “gemsen” , “groessen” (Galton, 1885), “tennis”, “tips” and “uscrime” (Hand et al., 1994) have been downloaded from the teaching data base stat.ethz.ch/Teaching/Datasets. The ... |

67 |
A Handbook of Small Data Sets
- Hand, J
- 1994
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Citation Context ...ded from www.stats.ox.ac.uk/pub/PRNN (Ripley, 1996). Datasets “baby”, “banknoten” (Flury & Riedwyl, 1988), “crab” (Ripley, 1996), “gemsen” , “groessen” (Galton, 1885), “tennis”, “tips” and “uscrime” (=-=Hand et al., 1994-=-) have been downloaded from the teaching data base stat.ethz.ch/Teaching/Datasets. The rest of the data sets is taken from archive.ics.uci.edu/ml (Frank & Asuncion, 2010); it in particular originates ... |

48 |
Computing location depth and regression depth in higher dimensions
- Rousseeuw, Struyf
- 1998
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Citation Context ... Using the random Tukey depth as an efficient approximation of the Tukey depth and selecting the random directions has many applications. Available algorithms for exactly calculating the Tukey depth (=-=Rousseeuw & Struyf, 1998-=-; Liu & Zuo, 2014a) are computationally expensive, but can serve as a benchmark. Finally, the SVM-simplified method is introduced and appears as a simple and efficient way to avoid the computational b... |

46 |
Multivariate Dispersion, Central Regions and Depth: the Lift Zonoid Approach
- Mosler
- 2002
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Citation Context ...ively q one-against-all separators can be used. We restrict the present study to the case q = 2, see Lange et al. (2014a) for q > 2. In Lange et al. (2014a) the zonoid depth (Koshevoy & Mosler, 1997; =-=Mosler, 2002-=-) is applied, which is efficiently computed also in higher dimensions. Here we employ four alternative depths: the Mahalanobis depth, the spatial depth, the projection depth and the Tukey depth. The f... |

35 | Regression with frailty in survival analysis, Biometrics - McGilchrist, Aisbett - 1991 |

31 | Fast and robust discriminant analysis - Hubert, Driessen - 2003 |

27 |
Zonoid trimming for multivariate distributions
- Koshevoy, Mosler
- 1998
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Citation Context ...ty rule applied; alternatively q one-against-all separators can be used. We restrict the present study to the case q = 2, see Lange et al. (2014a) for q > 2. In Lange et al. (2014a) the zonoid depth (=-=Koshevoy & Mosler, 1997-=-; Mosler, 2002) is applied, which is efficiently computed also in higher dimensions. Here we employ four alternative depths: the Mahalanobis depth, the spatial depth, the projection depth and the Tuke... |

27 |
An Attempt to Define the Nature of Chemical Diabetes Using a Multidimensional Analysis
- REAVEN, MILLER
- 1979
(Show Context)
Citation Context ...ma-retinol”), Timothy & Raftery (1993) (“socmob”) and Kalbfleisch & Prentice (1980) (“veteran-lung-cancer”); these data sets have been downloaded from lib.stat.cmu.edu/datasets. Data sets “chemdiab” (=-=Reaven & Miller, 1979-=-) and “hemophilia” (Habemma et al., 1974) have been taken from the R-packages ‘locfit’ and ‘rrcov’ respectively. The “pima” data set constitutes a training subsample of the “diabetes” (see below) and ... |

23 | Robust classification with contextsensitive features - Turney - 1993 |

21 |
The random Tukey depth
- Cuesta-Albertos, Nieto-Reyes
- 2008
(Show Context)
Citation Context ...hape of the data much better than the previous three do and it is more robust against outliers than these (and than the zonoid depth as well). For computational reasons we use the random Tukey depth (=-=Cuesta-Albertos & Nieto-Reyes, 2008-=-), which approximates the Tukey (= location) depth by minimizing univariate Tukey depths over a finite number of directions. When using the random Tukey depth (or another depth that vanishes outside t... |

18 | A depth function and a scale curve based on spatial quantiles
- Serfling
- 2002
(Show Context)
Citation Context ...= (1 + (z− µX)′Σ−1X (z− µX))−1, (1) where µX measures the location (e.g. the mean) of X, and ΣX the scatter (e.g. the covariance matrix) of X. The affine invariant spatial depth (Vardi & Zhang, 2000; =-=Serfling, 2002-=-) of z regarding X is defined as DSpt(z|X) = 1− ‖EX [ v(Σ −1/2 X (z−X)) ] ‖ . (2) Here v(y) = ‖y‖−1y for y 6= 0 and v(0) = 0, and ΣX is the covariance matrix of X. As the Mahalanobis and spatial depth... |

13 | DD-classifier: nonparametric classification procedure based on DD-plot - Li, Cuesta-Albertos, et al. - 2012 |

13 | Determinants of plasma levels of beta-carotene and retinol - Nierenberg, Stukel, et al. - 1989 |

12 |
Comparison between various regression depth methods and the support vector machine to approximate the minimum number of misclassifications
- Christmann, Fischer, et al.
- 2002
(Show Context)
Citation Context ...parating rule is similar to that proposed in Li et al. (2012), where the polynomial degree is chosen by cross-validating. Other possible approaches are regression depth (Christmann & Rousseeuw, 2001; =-=Christmann et al., 2002-=-) or SVM (Christmann et al., 2002; Vapnik, 1998). It is clear that in general the obtained separating hypersurface is not the one minimizing EMR, if more than two features are needed. But in which of ... |

8 | Equality of Opportunity - Greany, Kelleghan - 1984 |

8 | Knowledge discovery on RFM model using Bernoulli sequence, Expert Systems with Applications - Yeh, Yang, et al. - 2009 |

7 | Data analysis and classification with the zonoid depth - Mosler, Hobert |

7 | Nonparametric consistent depth-based classifiers. ECARES working paper 2012-014 - Paindaveine, Bever - 2012 |

6 | Data depths satisfying the projection property - Dyckerhoff - 2004 |

6 |
Den Broek. Stepwise Discriminant Analysis Program Using Density Estimation
- Habemma, Hermans, et al.
- 1974
(Show Context)
Citation Context ...socmob”) and Kalbfleisch & Prentice (1980) (“veteran-lung-cancer”); these data sets have been downloaded from lib.stat.cmu.edu/datasets. Data sets “chemdiab” (Reaven & Miller, 1979) and “hemophilia” (=-=Habemma et al., 1974-=-) have been taken from the R-packages ‘locfit’ and ‘rrcov’ respectively. The “pima” data set constitutes a training subsample of the “diabetes” (see below) and can be downloaded from www.stats.ox.ac.u... |

6 | Depth statistics - Mosler - 2013 |

5 |
Computing projection depth and its associated estimators
- Liu, Zuo
- 2014
(Show Context)
Citation Context ...bertos & NietoReyes, 2008), which is the minimum univariate Tukey depth over a set of unidimensional projections in randomly selected directions. As the exact calculation of DPrj is rather elaborate (=-=Liu & Zuo, 2014-=-b), we approximate it in the same way. 2.2 α-separation For each binary separation the α-procedure constructs a decision hyperplane in the extended property space E1 = [0, 1] r, r = ( p+q q ) − 1. The... |

4 |
Measuring overlap in binary regression
- Christmann, Rousseeuw
- 2001
(Show Context)
Citation Context ...By its nature the resulting separating rule is similar to that proposed in Li et al. (2012), where the polynomial degree is chosen by cross-validating. Other possible approaches are regression depth (=-=Christmann & Rousseeuw, 2001-=-; Christmann et al., 2002) or SVM (Christmann et al., 2002; Vapnik, 1998). It is clear that in general the obtained separating hypersurface is not the one minimizing EMR, if more than two features are... |

4 |
Computing halfspace depth and regression depth
- Liu, Zuo
- 2014
(Show Context)
Citation Context ...bertos & NietoReyes, 2008), which is the minimum univariate Tukey depth over a set of unidimensional projections in randomly selected directions. As the exact calculation of DPrj is rather elaborate (=-=Liu & Zuo, 2014-=-b), we approximate it in the same way. 2.2 α-separation For each binary separation the α-procedure constructs a decision hyperplane in the extended property space E1 = [0, 1] r, r = ( p+q q ) − 1. The... |

3 | The Statistical Analysis of Survival Time Data - Kalbfleisch, Prentice - 1980 |

3 |
Fast nonparametric classification based on data depth
- Lange, Mosler, et al.
- 2014
(Show Context)
Citation Context ...and its general fitness can be only established by successful application to a large variety of such data. In the sequel this is done for a newly developed nonparametric classifier, the DDαprocedure (=-=Lange et al., 2014-=-a). It is applied to fifty binary classification problems regarding real-world benchmark data. The DDα-procedure first transforms the data from their original property space into a depth space, which ... |

3 | The duality principle in learning for pattern recognition (in Russian). Kibernetika i Vytschislit’elnaya Technika 121 - Vasil’ev, Lange - 1998 |

2 | DDα-classification of asymmetric and fattailed data - Lange, Mosler, et al. |

2 | Two approaches for solving tasks of pattern recognition and reconstruction of functional dependencies - Lange, Mozharovskyi, et al. - 2011 |

2 | The reduction principle in pattern recognition learning (PRL) problem - Vasil’ev - 2000 |

2 | The reduction principle in problems of revealing regularities I - Vasil’ev - 2003 |

1 | Janning (eds.) Data Analysis - Schmidt-Thieme - 2014 |

1 | Analyzing the results of a cloudseeding experiment - Miller, Shaw, et al. - 1979 |