Kernel ellipsoidal trimming

Abstract

Ellipsoid estimation is an issue of primary importance in many practical areas such as control, system identification, visual/audio tracking, experimental design, data mining, robust statistics and novelty/outlier detection. This paper presents a new method of kernel information matrix ellipsoid estimation (KIMEE) that finds an ellipsoid in a kernel defined feature space based on a centered information matrix. Although the method is very general and can be applied to many of the aforementioned problems, the main focus in this paper is the problem of novelty or outlier detection associated with fault detection. A simple iterative algorithm based on Titterington's minimum volume ellipsoid method is proposed for practical implementation. The KIMEE method demonstrates very good performance on a set of real-life and simulated datasets compared with support vector machine methods.

Molecular evaluation of vitamin D responsiveness of healthy young adults

“…All possible correlations were computed by testing the line that fits best the maximum number of participants following the concepts of Dolia et al [28]. We simplified the latter method by substituting ellipsoidal kernels by lines.…”

The Minimum Volume Covering Ellipsoid Estimation in Kernel-Defined Feature Spaces

“…Then we can write i α i x i x i + γI = X A 2 X + γI. Note that the matrices (AX) (AX) = X A 2 X and (AX)(AX) = AXX A = AKA have the same nonzero eigenvalues λ i , equal to the squares of the singular values of AX [2]. With d the dimensionality of the space and the number of data points x i , it is now easy to show that:…”

“…We should be able to compute the Mahalanobis distance for a test point exclusively using kernel evaluations and the vector α. Recall the eigenvalue decompositions of i α i x i x i = X A 2 X = UΛU and AXX A = AKA = VΛV [2]. We then have that…”

“…The minimum volume covering ellipsoid (MVCE) [2,3,10,12], the ellipsoid smallest in volume that covers all of a given set of points, has many applications in areas ranging from systems and control to robust statistics. In this paper we focus on its application to novelty detection (also known as support estimation or domain description): then, all data points from a training set {x i } i=1 are specified to be sampled from an unknown distribution D, and the support of D is be estimated as the inside region of the MVCE.…”

Distributed Anomaly Detection Using Minimum Volume Elliptical Principal Component Analysis

“…Pauwels and Ambekar [29] reformulate the cost function for the one-class SVM (OCSVM) so that the centre of the sphere is a weighted median of the support vectors, rather than the weighted mean of the support vectors. Dolia et al [30] use kernel ellipsoidal trimming where the outliers are removed from the training set and the algorithm rerun. Both OCSVM and kernel ellipsoidal trimming use the boundary for anomaly detection.…”