Metric (mathematics)

About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.

Probability binning comparison: a metric for quantitating univariate distribution differences.

Abstract: Background While several algorithms for the comparison of univariate distributions arising from flow cytometric analyses have been developed and studied for many years, algorithms for comparing multivariate distributions remain elusive. Such algorithms could be useful for comparing differences between samples based on several independent measurements, rather than differences based on any single measurement. It is conceivable that distributions could be completely distinct in multivariate space, but unresolvable in any combination of univariate histograms. Multivariate comparisons could also be useful for providing feedback about instrument stability, when only subtle changes in measurements are occurring. Methods We apply a variant of Probability Binning, described in the accompanying article, to multidimensional data. In this approach, hyper-rectangles of n dimensions (where n is the number of measurements being compared) comprise the bins used for the chi-squared statistic. These hyper-dimensional bins are constructed such that the control sample has the same number of events in each bin; the bins are then applied to the test samples for chi-squared calculations. Results Using a Monte-Carlo simulation, we determined the distribution of chi-squared values obtained by comparing sets of events from the same distribution; this distribution of chi-squared values was identical as for the univariate algorithm. Hence, the same formulae can be used to construct a metric, analogous to a t-score, that estimates the probability with which distributions are distinct. As for univariate comparisons, this metric scales with the difference between two distributions, and can be used to rank samples according to similarity to a control. We apply the algorithm to multivariate immunophenotyping data, and demonstrate that it can be used to discriminate distinct samples and to rank samples according to a biologically-meaningful difference. Conclusion Probability binning, as shown here, provides a useful metric for determining the probability with which two or more multivariate distributions represent distinct sets of data. The metric can be used to identify the similarity or dissimilarity of samples. Finally, as demonstrated in the accompanying paper, the algorithm can be used to gate on events in one sample that are different from a control sample, even if those events cannot be distinguished on the basis of any combination of univariate or bivariate displays. Cytometry 45:47–55, 2001. Published 2001 Wiley-Liss, Inc.

A Dependence Metric for Possibly Nonlinear Processes

Abstract: . A transformed metric entropy measure of dependence is studied which satisfies many desirable properties, including being a proper measure of distance. It is capable of good performance in identifying dependence even in possibly nonlinear time series, and is applicable for both continuous and discrete variables. A nonparametric kernel density implementation is considered here for many stylized models including linear and nonlinear MA, AR, GARCH, integrated series and chaotic dynamics. A related permutation test of independence is proposed and compared with several alternatives.

Spatio-temporal data reduction with deterministic error bounds

Abstract: A common way of storing spatio-temporal information about mobile devices is in the form of a 3D (2D geography + time) trajectory. We argue that when cellular phones and Personal Digital Assistants become location-aware, the size of the spatio-temporal information generated may prohibit efficient processing. We propose to adopt a technique studied in computer graphics, namely line-simplification, as an approximation technique to solve this problem. Line simplification will reduce the size of the trajectories. Line simplification uses a distance function in producing the trajectory approximation. We postulate the desiderata for such a distance-function: it should be sound, namely the error of the answers to spatio-temporal queries must be bounded. We analyze several distance functions, and prove that some are sound in this sense for some types of queries, while others are not. A distance function that is sound for all common spatio-temporal query types is introduced and analyzed. Then we propose an aging mechanism which gradually shrinks the size of the trajectories as time progresses. We also propose to adopt existing linguistic constructs to manage the uncertainty introduced by the trajectory approximation. Finally, we analyze experimentally the effectiveness of line-simplification in reducing the size of a trajectories database.

Towards trajectory anonymization: a generalization-based approach

Abstract: Trajectory datasets are becoming more and more popular due to the massive usage of GPS and other location-based devices and services. In this paper, we address privacy issues regarding the identification of individuals in static trajectory datasets. We provide privacy protection by definig trajectory k-anonymity, meaning every released information refers to at least k users/trajectories. We propose a novel generalization-based approach that applies to trajectories and sequences in general. We also suggest the use of a simple random reconstruction of the original dataset from the anonymization, to overcome possible drawbacks of generalization approaches.We present a utility metric that maximizes the probability of a good representation and propose trajectory anonymization techniques to address time and space sensitive applications. The experimental results over synthetic trajectory datasets show the effectiveness of the proposed approach.