A novel scale-space approach for multinormality testing and the k-sample problem in the high dimension low sample size scenario

Abstract

Two classical multivariate statistical problems, testing of multivariate normality and the k-sample problem, are explored by a novel analysis on several resolutions simultaneously. The presented methods do not invert any estimated covariance matrix. Thereby, the methods work in the High Dimension Low Sample Size situation, i.e. when n ≤ p. The output, a significance map, is produced by doing a one-dimensional test for all possible resolution/position pairs. The significance map shows for which resolution/position pairs the null hypothesis is rejected. For the testing of multinormality, the Anderson-Darling test is utilized to detect potential departures from multinormality at different combinations of resolutions and positions. In the k-sample case, it is tested whether k data sets can be said to originate from the same unspecified discrete or continuous multivariate distribution. This is done by testing the k vectors corresponding to the same resolution/position pair of the k different data sets through the k-sample Anderson-Darling test. Successful demonstrations of the new methodology on artificial and real data sets are presented, and a feature selection scheme is demonstrated.

Early Detection of Change by Applying Scale-Space Methodology to Hyperspectral Images

“…In our test, the distorted set of signatures in Figure 1b was compared with 14 other acquisitions and the goal was to detect the small change we manually introduced. It turned out that such a small change is indeed detected by our new methodology but not by the method suggested in [7] nor by a standard approach such as principal components analysis (PCA). We will return in more detail to this example in Section 4.…”

“…At scale s = 3 and position d, a smoothing in terms of a weighted average of the observed values for spectral frequencies d − 1, d and d + 1 are used to test whether the acquisitions are different. The weights are calculated from an Epanechnikov kernel function (i.e., parabolic function) [10], the same as in [7]. For other scales, completely analogous smoothing over the frequency bands are made and used to perform the tests.…”

“…To simplify the description we illustrate the methodology by testing for change in the sample mean,X, over the pixels in the image. This training-procedure is the core difference between the method presented here and in [7], where there is a more direct comparison between curve families. Also, instead of the non-parametric Andersson-Darling test combined with either Bonferroni or False Discovery Rate correction for multiple hypothesis testing employed in [7], our novel method uses the t-test either with a Bonferroni correction or the inference approach suggested in [12].…”

“…The HSI system used for the mole example differs from the one described in Section 3 and a detailed description can be found in [14]. In our analysis, we compared the proposed novel technique to the method of Hindberg et al described in [7]. When applied in the present context, this method uses a two-sample test to decide if the test sample distribution differs from the distribution under the null hypothesis.…”

“…Recently, Hindberg et al proposed a scale-space method for testing whether k multivariate data sets of same dimension originate from the same distribution [7]. Thus, the proposed method solves the classical k-sample problem using scale-space analysis and the method has proven successful in many applications.…”