|
Shattered Set
The concept of shattered sets plays an important role in Vapnik–Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes as well as in statistical computational learning theory. Definition Suppose ''A'' is a set (mathematics), set and ''C'' is a class (set theory), class of sets. The class ''C'' shatters the set ''A'' if for each subset ''a'' of ''A'', there is some element ''c'' of ''C'' such that : a = c \cap A. Equivalently, ''C'' shatters ''A'' when their Growth function#Definitions, intersection is equal to ''As power set: ''P''(''A'') = . We employ the letter ''C'' to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set ''A'' is often assumed to be finite set, finite because, in empirical processes, we are interested in the shattering of finite sets of data points. Example We will show that the class of all disc (geometry), discs in the plane (geometry) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vapnik–Chervonenkis Theory
Vapnik–Chervonenkis theory (also known as VC theory) was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkis. The theory is a form of computational learning theory, which attempts to explain the learning process from a statistical point of view. Introduction VC theory covers at least four parts (as explained in ''The Nature of Statistical Learning Theory''): *Theory of consistency of learning processes **What are (necessary and sufficient) conditions for consistency of a learning process based on the empirical risk minimization principle? *Nonasymptotic theory of the rate of convergence of learning processes **How fast is the rate of convergence of the learning process? *Theory of controlling the generalization ability of learning processes **How can one control the rate of convergence (the generalization ability) of the learning process? *Theory of constructing learning machines **How can one construct algorithms that can control the generalization abilit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shattering01
Shatter or shattering may refer to: * The act of violently breaking into small pieces * Alan Shatter, Irish politician * Susan Louise Shatter (1943–2011), American landscape painter * Shattering (machine learning), a concept in mathematics, especially Vapnik–Chervonenkis theory * Shatter attack, in computing, a technique used against ''Microsoft Windows'' operating systems * Shattering (agriculture), an undesirable trait in crop plants that makes harvesting difficult * Shatter (cannabis), a concentrate made from cannabis * Brisance the shattering capability of a high explosive, determined mainly by its detonation pressure. Entertainment * ''Shatter'' (film), a 1974 film * ''Shatter'' (comics), the Marvel Comics fictional mutant character * ''Shatter'' (digital comic), the First Comics digital comic * ''Shatter'' (novel), a 2008 novel by Australian author Michael Robotham * ''Shatter'' (video game), a 2009 video game by Sidhe Interactive ;Music * ''Shatter'' (EP), a 201 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Probability
The ''Annals of Probability'' is a leading peer-reviewed probability journal published by the Institute of Mathematical Statistics, which is the main international society for researchers in the areas probability and statistics. The journal was started in 1973 as a continuation in part of the ''Annals of Mathematical Statistics'', which was split into the ''Annals of Statistics'' and this journal. In July 2021, the journal was ranked 7th in the field Probability & Statistics with Applications according to Google Scholar. It had an impact factor of 1.470 (as of 2010), according to the ''Journal Citation Reports''. The impact factor for 2018 is 2.085. Its CiteScore is 4.3, and SCImago Journal Rank is 3.184, both from 2020. Editors-in-Chief: Past and Present The following persons have been editor-in-chief of the journal: * Ronald Pyke (1972–1975) * Patrick Billingsley (1976–1978) * Richard M. Dudley (1979–1981) * Thomas M. Liggett (1985–1987) * Peter E. Ney (1988–1990) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The very first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 0.87. Notable publications * The 1972 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Family Of Sets
In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set family, or a set system. The term "collection" is used here because, in some contexts, a family of sets may be allowed to contain repeated copies of any given member, and in other contexts it may form a proper class rather than a set. A finite family of subsets of a finite set S is also called a ''hypergraph''. The subject of extremal set theory concerns the largest and smallest examples of families of sets satisfying certain restrictions. Examples The set of all subsets of a given set S is called the power set of S and is denoted by \wp(S). The power set \wp(S) of a given set S is a family of sets over S. A subset of S having k elements is called a k-subset of S. The k-subsets S^ of a set S form a family of sets. Let S = \. An ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sauer–Shelah Lemma
In combinatorial mathematics and extremal set theory, the Sauer–Shelah lemma states that every family of sets with small VC dimension consists of a small number of sets. It is named after Norbert Sauer and Saharon Shelah, who published it independently of each other in 1972. The same result was also published slightly earlier and again independently, by Vladimir Vapnik and Alexey Chervonenkis, after whom the VC dimension is named. In his paper containing the lemma, Shelah gives credit also to Micha Perles, and for this reason the lemma has also been called the Perles–Sauer–Shelah lemma.. Buzaglo et al. call this lemma "one of the most fundamental results on VC-dimension", and it has applications in many areas. Sauer's motivation was in the combinatorics of set systems, while Shelah's was in model theory and that of Vapnik and Chervonenkis was in statistics. It has also been applied in discrete geometry. and graph theory.. Definitions and statement If \textstyle \mathcal=\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
VC Dimension
VC may refer to: Military decorations * Victoria Cross, a military decoration awarded by the United Kingdom and also by certain Commonwealth nations ** Victoria Cross for Australia ** Victoria Cross (Canada) ** Victoria Cross for New Zealand * Victorious Cross, Idi Amin's self-bestowed military decoration Organisations * Ocean Airlines (IATA airline designator 2003-2008), Italian cargo airline * Voyageur Airways (IATA airline designator since 1968), Canadian charter airline * Visual Communications, an Asian-Pacific-American media arts organization in Los Angeles, US * Viet Cong (also Victor Charlie or Vietnamese Communists), a political and military organization from the Vietnam War (1959–1975) Education * Vanier College, Canada * Vassar College, US * Velez College, Philippines * Virginia College, US Places * Saint Vincent and the Grenadines (ISO country code), a state in the Caribbean * Sri Lanka (ICAO airport prefix code) * Watsonian vice-counties, subdivisions of Great Brita ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized to infinite sets, which allows one to distinguish between different types of infinity, and to perform arithmetic on them. There are two approaches to cardinality: one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is also called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted , A, , with a vertical bar on each side; this is the same notation as absolute value, and the meaning depends on context. The cardinality of a set A may alternatively be denoted by n(A), , \operatorname(A), or \#A. History A crude sense of cardinality, an awareness that groups of things or events compare with other grou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sample Space
In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible ordered outcomes, or sample points, are listed as elements in the set. It is common to refer to a sample space by the labels ''S'', Ω, or ''U'' (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite. A subset of the sample space is an event, denoted by E. If the outcome of an experiment is included in E, then event E has occurred. For example, if the experiment is tossing a single coin, the sample space is the set \, where the outcome H means that the coin is heads and the outcome T means that the coin is tails. The possible events are E=\, E = \, and E = \. For tossing two coins, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve. The intersection of all the convex sets that contain a given subset of Euclidean space is called the convex hull of . It is the smallest convex set containing . A convex function is a real-valued function defined on an interval with the property that its epigraph (the set of points on or above the graph of the function) is a convex set. Convex minimization is a subfield of optimization that studies the problem of minimizing convex functions over convex se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Empirical Process
In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state. For a process in a discrete state space a population continuous time Markov chain or Markov population model is a process which counts the number of objects in a given state (without rescaling). In mean field theory, limit theorems (as the number of objects becomes large) are considered and generalise the central limit theorem for empirical measures. Applications of the theory of empirical processes arise in non-parametric statistics. Definition For ''X''1, ''X''2, ... ''X''''n'' independent and identically-distributed random variables in R with common cumulative distribution function ''F''(''x''), the empirical distribution function is defined by :F_n(x)=\frac\sum_^n I_(X_i), where I''C'' is the indicator function of the set ''C''. For every (fixed) ''x'', ''F''''n''(''x'') is a sequence of random variables which converge to ''F''(''x'') almost ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
