Topic
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.
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25 Feb 2019
TL;DR: In psychology, ordinal variables, although extremely common in psychology, are almost exclusively analyzed with statistical models that falsely assume them to be metric as discussed by the authors, which can lead to distorted effect.
Abstract: Ordinal variables, although extremely common in psychology, are almost exclusively analyzed with statistical models that falsely assume them to be metric. This practice can lead to distorted effect...
287 citations
TL;DR: A [phi]-entropy functional is defined on the probability space and its Hessian along a direction of the tangent space of the parameter space is taken as the metric, and the distance between two probability distributions is computed as the geodesic distance induced by the metric.
287 citations
TL;DR: A simulation study using data models and analysis of real microarray data shows that for small samples the root mean square differences of the estimated and true metrics are considerable, and even for large samples, there is only weak correlation between the true and estimated metrics.
Abstract: Motivation: The receiver operator characteristic (ROC) curves are commonly used in biomedical applications to judge the performance of a discriminant across varying decision thresholds. The estimated ROC curve depends on the true positive rate (TPR) and false positive rate (FPR), with the key metric being the area under the curve (AUC). With small samples these rates need to be estimated from the training data, so a natural question arises: How well do the estimates of the AUC, TPR and FPR compare with the true metrics?
Results: Through a simulation study using data models and analysis of real microarray data, we show that (i) for small samples the root mean square differences of the estimated and true metrics are considerable; (ii) even for large samples, there is only weak correlation between the true and estimated metrics; and (iii) generally, there is weak regression of the true metric on the estimated metric. For classification rules, we consider linear discriminant analysis, linear support vector machine (SVM) and radial basis function SVM. For error estimation, we consider resubstitution, three kinds of cross-validation and bootstrap. Using resampling, we show the unreliability of some published ROC results.
Availability: Companion web site at http://compbio.tgen.org/paper_supp/ROC/roc.html
Contact: edward@mail.ece.tamu.edu
286 citations
Book•
30 Aug 2008
TL;DR: The Fefferman Metric and Yamabe Problem were used in this article to define pseudoharmonic maps on CR Manifolds. But they did not consider pseudoeinsteinian manifolds and pseudo-Hermitian immersions.
Abstract: CR Manifolds.- The Fefferman Metric.- The CR Yamabe Problem.- Pseudoharmonic Maps.- Pseudo-Einsteinian Manifolds.- Pseudo-Hermitian Immersions.- Quasiconformal Mappings.- Yang-Mills Fields on CR Manifolds.- Spectral Geometry.
286 citations
TL;DR: The multivantage point tree structure (mvp-tree) that uses more than one vantage point to partiton the space into spherical cuts at each level is introduced and generalize the idea of using multiple vantage points and the results show that, after all, it may be best to use a large number of vantage points in an internal node in order to end up with a single directory node and keep as many of the precomputed distances as possible to provide more efficient filtering during search operations.
Abstract: One of the common queries in many database applications is finding approximate matches to a given query item from a collection of data items. For example, given an image database, one may want to retrieve all images that are similar to a given query image. Distance-based index structures are proposed for applications where the distance computations between objects of the data domain are expensive (such as high-dimensional data) and the distance function is metric. In this paper we consider using distance-based index structures for similarity queries on large metric spaces. We elaborate on the approach that uses reference points (vantage points) to partition the data space into spherical shell-like regions in a hierarchical manner. We introduce the multivantage point tree structure (mvp-tree) that uses more than one vantage point to partiton the space into spherical cuts at each level. In answering similarity-based queries, the mvp-tree also utilizes the precomputed (at construction time) distances between the data points and the vantage points.We summarize the experiments comparing mvp-trees to vp-trees that have a similar partitioning strategy, but use only one vantage point at each level and do not make use of the precomputed distances. Empirical studies show that the mvp-tree outperforms the vp-tree by 20% to 80% for varying query ranges and different distance distributions. Next, we generalize the idea of using multiple vantage points and discuss the results of experiments we have made to see how varying the number of vantage points in a node affects affects performance and how much is gained in performance by making use of precomputed distances. The results show that, after all, it may be best to use a large number of vantage points in an internal node in order to end up with a single directory node and keep as many of the precomputed distances as possible to provide more efficient filtering during search operations. Finally, we provide some experimental results that compare mvp-trees with M-trees, which is a dynamic distance-based index structure for metric domains.
285 citations