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.

Trustworthiness and metrics in visualizing similarity of gene expression.

Abstract: Background: Conventionally, the first step in analyzing the large and high-dimensional data sets measured by microarrays is visual exploration. Dendrograms of hierarchical clustering, selforganizing maps (SOMs), and multidimensional scaling have been used to visualize similarity relationships of data samples. We address two central properties of the methods: (i) Are the visualizations trustworthy, i.e., if two samples are visualized to be similar, are they really similar? (ii) The metric. The measure of similarity determines the result; we propose using a new learning metrics principle to derive a metric from interrelationships among data sets. Results: The trustworthiness of hierarchical clustering, multidimensional scaling, and the selforganizing map were compared in visualizing similarity relationships among gene expression profiles. The self-organizing map was the best except that hierarchical clustering was the most trustworthy for the most similar profiles. Trustworthiness can be further increased by treating separately those genes for which the visualization is least trustworthy. We then proceed to improve the metric. The distance measure between the expression profiles is adjusted to measure differences relevant to functional classes of the genes. The genes for which the new metric is the most different from the usual correlation metric are listed and visualized with one of the visualization methods, the self-organizing map, computed in the new metric. Conclusions: The conjecture from the methodological results is that the self-organizing map can be recommended to complement the usual hierarchical clustering for visualizing and exploring gene expression data. Discarding the least trustworthy samples and improving the metric still improves it.

Metric regularity, openness and Lipschitzian behavior of multifunctions

Abstract: On considere les proprietes de regularite metrique, d'ouverture et de comportement lipschitzien de fonctions multiformes

Three-feature based automatic lane detection algorithm (TFALDA) for autonomous driving

Abstract: Three-feature based automatic lane detection algorithm (TFALDA) is a new lane detection algorithm which is simple, robust, and efficient, thus suitable for real-time processing in cluttered road environments without a priori knowledge on them. Three features of a lane boundary - starting position, direction (or orientation), and its gray-level intensity features comprising a lane vector are obtained via simple image processing. Out of the many possible lane boundary candidates, the best one is then chosen as the one at a minimum distance from the previous lane vector according to a weighted distance metric in which each feature is assigned a different weight. An evolutionary algorithm then finds the optimal weights for combination of the three features that minimize the rate of detection error. The proposed algorithm was successfully applied to a series of actual road following experiments using the PRV (POSTECH research vehicle) II both on campus roads and nearby highways.

Coverage estimation for symbolic model checking

Abstract: Although model checking is an exhaustive formal verification method, a bug can still escape detection if the erroneous behavior does not violate any verified property. We propose a coverage metric to estimate the "completeness" of a set of properties verified by model checking. A symbolic algorithm is presented to compute this metric for a subset of the CTL property specification language. It has the same order of computational complexity as a model checking algorithm. Our coverage estimator has been applied in the course of some real-world model checking projects. We uncovered several coverage holes including one that eventually led to the discovery of a bug that escaped the initial model checking effort.