ON SOME TOPOLOGICAL PROPERTIES IN GRADUAL NORMED SPACES

In this paper, we have investigated some topological properties of sets in a given gradual normed space. We have stated gradual Hausdorff property and then,we have studied the relationship between gradual closed sets and gradual compact sets. Also, we have given a result about having the closure point for an innite set in a gradual normed space. In the end, we have provided some illustrative examples

Novel semi-metrics for multivariate change point analysis and anomaly detection

This paper proposes a new method for determining similarity and anomalies between time series, most practically effective in large collections of (likely related) time series, by measuring distances between structural breaks within such a collection. We introduce a class of \emph{semi-metric} distance measures, which we term \emph{MJ distances}. These semi-metrics provide an advantage over existing options such as the Hausdorff and Wasserstein metrics. We prove they have desirable properties, including better sensitivity to outliers, while experiments on simulated data demonstrate that they uncover similarity within collections of time series more effectively. Semi-metrics carry a potential disadvantage: without the triangle inequality, they may not satisfy a "transitivity property of closeness." We analyse this failure with proof and introduce an computational method to investigate, in which we demonstrate that our semi-metrics violate transitivity infrequently and mildly. Finally, we apply our methods to cryptocurrency and measles data, introducing a judicious application of eigenvalue analysis.Comment: Accepted manuscript. Minor edits since v2. Equal contribution from first two author