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Convex Layers
In computational geometry, the convex layers of a set of points in the Euclidean plane are a sequence of nested convex polygons having the points as their vertices. The outermost one is the convex hull of the points and the rest are formed in the same way recursively. The innermost layer may be degenerate, consisting only of one or two points. The problem of constructing convex layers has also been called onion peeling or onion decomposition. Although constructing the convex layers by repeatedly finding convex hulls would be slower, it is possible to partition any set of n points into its convex layers in time O(n\log n). An early application of the convex layers was in robust statistics, as a way of identifying outliers and measuring the central tendency of a set of sample points. In this context, the number of convex layers surrounding a given point is called its convex hull peeling depth, and the convex layers themselves are the depth contours for this notion of data depth. Co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Layers Halfspace
Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope, a polytope with a convex set of points ** Convex metric space, a generalization of the convexity notion in abstract metric spaces * Convex function, when the line segment between any two points on the graph of the function lies above or on the graph * Convex conjugate, of a function * Convexity (algebraic geometry), a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces Economics and finance * Convexity (finance), second derivatives in financial modeling generally * Convexity in economics * Bond convexity, a measure of the sensitivity of the duration of a bond to changes in interest rates * Convex preferences, an individual's ordering of various outcomes Other uses * Convex Com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Geometry
Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. While modern computational geometry is a recent development, it is one of the oldest fields of computing with a history stretching back to antiquity. Analysis of algorithms, Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points. For such sets, the difference between O(''n''2) and O(''n'' log ''n'') may be the difference between days and seconds of computation. The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (Computer-aided design, CAD/Compu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Plane
In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of parallel lines, and also metrical notions of distance, circles, and angle measurement. The set \mathbb^2 of pairs of real numbers (the real coordinate plane) augmented by appropriate structure often serves as the canonical example. History Books I through IV and VI of Euclid's Elements dealt with two-dimensional geometry, developing such notions as similarity of shapes, the Pythagorean theorem (Proposition 47), equality of angles and areas, parallelism, the sum of the angles in a triangle, and the three cases in which triangles are "equal" (have the same area), among many other topics. Later, the plane was described in a so-called '' Cartesian coordinate system'', a coordinate system that specifies each point uniquely in a plane by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points. A strictly convex polygon is a convex polygon such that no line contains two of its edges. In a convex polygon, all interior angles are less than or equal to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees. Properties The following properties of a simple polygon are all equivalent to convexity: *Every internal angle is strictly less than 180 degrees. *Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the boundary. *The polygon is entirely contained in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references ("crock recursion") can occur. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduces all successive cases toward the base case. For example, the following is a recursive definition of a person's ''ancestor''. One's ances ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robust Statistics
Robust statistics are statistics with good performance for data drawn from a wide range of probability distributions, especially for distributions that are not normal. Robust statistical methods have been developed for many common problems, such as estimating location, scale, and regression parameters. One motivation is to produce statistical methods that are not unduly affected by outliers. Another motivation is to provide methods with good performance when there are small departures from a parametric distribution. For example, robust methods work well for mixtures of two normal distributions with different standard deviations; under this model, non-robust methods like a t-test work poorly. Introduction Robust statistics seek to provide methods that emulate popular statistical methods, but which are not unduly affected by outliers or other small departures from Statistical assumption, model assumptions. In statistics, classical estimation methods rely heavily on assumpti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are sometimes excluded from the data set. An outlier can be an indication of exciting possibility, but can also cause serious problems in statistical analyses. Outliers can occur by chance in any distribution, but they can indicate novel behaviour or structures in the data-set, measurement error, or that the population has a heavy-tailed distribution. In the case of measurement error, one wishes to discard them or use statistics that are robust to outliers, while in the case of heavy-tailed distributions, they indicate that the distribution has high skewness and that one should be very cautious in using tools or intuitions that assume a normal distribution. A frequent cause of outliers is a mixture of two distributions, which may be two dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Tendency
In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in the Social Sciences, p.2 Colloquially, measures of central tendency are often called ''averages.'' The term ''central tendency'' dates from the late 1920s. The most common measures of central tendency are the arithmetic mean, the median, and the mode. A middle tendency can be calculated for either a finite set of values or for a theoretical distribution, such as the normal distribution. Occasionally authors use central tendency to denote "the tendency of quantitative data to cluster around some central value."Upton, G.; Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP (entry for "central tendency")Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP for International Statistical Institute. (entry for "ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Statistics
The ''Annals of Statistics'' is a peer-reviewed statistics journal published by the Institute of Mathematical Statistics. It was started in 1973 as a continuation in part of the '' Annals of Mathematical Statistics (1930)'', which was split into the ''Annals of Statistics'' and the ''Annals of Probability''. The journal CiteScore is 5.8, and its SCImago Journal Rank is 5.877, both from 2020. Articles older than 3 years are available on JSTOR, and all articles since 2004 are freely available on the arXiv. Editorial board The following persons have been editors of the journal: * Ingram Olkin (1972–1973) * I. Richard Savage (1974–1976) * Rupert Miller (1977–1979) * David V. Hinkley (1980–1982) * Michael D. Perlman (1983–1985) * Willem van Zwet (1986–1988) * Arthur Cohen (1988–1991) * Michael Woodroofe (1992–1994) * Larry Brown and John Rice (1995–1997) * Hans-Rudolf Künsch and James O. Berger (1998–2000) * John Marden and Jon A. Wellner (2001–2003) * M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Range Reporting
In computational geometry and database theory, a range reporting query asks for a list of the points that match the query. The query is often specified by a geometric shape, containing all the points that should match, and is called a range. Range reporting is a special case of range searching, in which queries may return other kinds of aggregate information about points in a range. Range reporting queries are often handled by building a data structure from a collection of points that can answer queries efficiently. Because the worst case output size for a range reporting query, measured as a function of the data set size , can be itself, much of the research on range reporting data structures has investigated output-sensitive algorithms, where the query time is analyzed in terms of both and the number of reported points (often denoted ). For example, for one-dimensional (numeric) data with query ranges that are intervals, range reporting queries can be handled by storing the data ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
