Metric (mathematics) in the context of "Open (topology)"

In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space.

Several notions of a plane may be defined. The Euclidean plane follows Euclidean geometry, and in particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. The elliptic plane may be further defined by adding a metric to the real projective plane. One may also conceive of a hyperbolic plane, which obeys hyperbolic geometry and has a negative curvature.

Cluster analysis, or clustering, is a data analysis technique aimed at partitioning a set of objects into groups such that objects within the same group (called a cluster) exhibit greater similarity to one another (in some specific sense defined by the analyst) than to those in other groups (clusters). It is a main task of exploratory data analysis, and a common technique for statistical data analysis, used in many fields, including pattern recognition, image analysis, information retrieval, bioinformatics, data compression, computer graphics and machine learning.

Cluster analysis refers to a family of algorithms and tasks rather than one specific algorithm. It can be achieved by various algorithms that differ significantly in their understanding of what constitutes a cluster and how to efficiently find them. Popular notions of clusters include groups with small distances between cluster members, dense areas of the data space, intervals or particular statistical distributions. Clustering can therefore be formulated as a multi-objective optimization problem. The appropriate clustering algorithm and parameter settings (including parameters such as the distance function to use, a density threshold or the number of expected clusters) depend on the individual data set and intended use of the results. Cluster analysis as such is not an automatic task, but an iterative process of knowledge discovery or interactive multi-objective optimization that involves trial and failure. It is often necessary to modify data preprocessing and model parameters until the result achieves the desired properties.

In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object. A frequent notation for the symmetry group of an object X is G = Sym(X).

For an object in a metric space, its symmetries form a subgroup of the isometry group of the ambient space. This article mainly considers symmetry groups in Euclidean geometry, but the concept may also be studied for more general types of geometric structure.

In geometry, a motion is an isometry of a metric space. For instance, a plane equipped with the Euclidean distance metric is a metric space in which a mapping associating congruent figures is a motion.

Motions can be divided into direct (also known as proper or rigid) and indirect (or improper) motions.Direct motions include translations and rotations, which preserve the orientation of a chiral shape.Indirect motions include reflections, glide reflections, and Improper rotations, that invert the orientation of a chiral shape.Some geometers define motion in such a way that only direct motions are motions.

In mathematics, an open set is a generalization of an open interval in the real line.

In a metric space (a set with a distance defined between every two points), an open set is a set that, with every point P in it, contains all points of the metric space that are sufficiently near to P (that is, all points whose distance to P is less than some value depending on P).

When classification is performed by a computer, statistical methods are normally used to develop the algorithm.

Often, the individual observations are analyzed into a set of quantifiable properties, known variously as explanatory variables or features. These properties may variously be categorical (e.g. "A", "B", "AB" or "O", for blood type), ordinal (e.g. "large", "medium" or "small"), integer-valued (e.g. the number of occurrences of a particular word in an email) or real-valued (e.g. a measurement of blood pressure). Other classifiers work by comparing observations to previous observations by means of a similarity or distance function.

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite.

In the case of a directed graph the distance d(u,v) between two vertices u and v is defined as the length of a shortest directed path from u to v consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, d(u,v) does not necessarily coincide with d(v,u)—so it is just a quasi-metric, and it might be the case that one is defined while the other is not.

Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function (or metric) called the taxicab distance, Manhattan distance, or city block distance. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length.

The taxicab distance is also sometimes known as rectilinear distance or L distance (see L space). This geometry has been used in regression analysis since the 18th century, and is often referred to as LASSO. Its geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski.

In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such measures are in some sense the inverse of distance metrics: they take on large values for similar objects and either zero or a negative value for very dissimilar objects. Though, in more broad terms, a similarity function may also satisfy metric axioms.

Cosine similarity is a commonly used similarity measure for real-valued vectors, used in (among other fields) information retrieval to score the similarity of documents in the vector space model. In machine learning, common kernel functions such as the RBF kernel can be viewed as similarity functions.