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Isotonic Regression
In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing function, non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible. Applications Isotonic regression has applications in statistical inference. For example, one might use it to fit an isotonic curve to the means of some set of experimental results when an increase in those means according to some particular ordering is expected. A benefit of isotonic regression is that it is not constrained by any functional form, such as the linearity imposed by linear regression, as long as the function is monotonic increasing. Another application is nonmetric multidimensional scaling, where a low-dimensional embedding for data points is sought such that order of distances between points in the embedding matches order of dissimilarity between points. Isot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is Reflexive relation, reflexive, antisymmetric relation, antisymmetric, and Transitive relation, transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonparametric Regression
Nonparametric regression is a form of regression analysis where the predictor does not take a predetermined form but is completely constructed using information derived from the data. That is, no parametric equation is assumed for the relationship between predictors and dependent variable. A larger sample size is needed to build a nonparametric model having a level of uncertainty as a parametric model because the data must supply both the model structure and the parameter estimates. Definition Nonparametric regression assumes the following relationship, given the random variables X and Y: : \mathbb \mid X=x= m(x), where m(x) is some deterministic function. Linear regression is a restricted case of nonparametric regression where m(x) is assumed to be a linear function of the data. Sometimes a slightly stronger assumption of additive noise is used: : Y = m(X) + U, where the random variable U is the `noise term', with mean 0. Without the assumption that m belongs to a specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of logic gate, gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Active-set Method
In mathematical optimization, the active-set method is an algorithm used to identify the active constraints in a set of inequality constraints. The active constraints are then expressed as equality constraints, thereby transforming an inequality-constrained problem into a simpler equality-constrained subproblem. An optimization problem is defined using an objective function to minimize or maximize, and a set of constraints : g_1(x) \ge 0, \dots, g_k(x) \ge 0 that define the feasible region, that is, the set of all ''x'' to search for the optimal solution. Given a point x in the feasible region, a constraint : g_i(x) \ge 0 is called active at x_0 if g_i(x_0) = 0, and inactive at x_0 if g_i(x_0) > 0. Equality constraints are always active. The active set at x_0 is made up of those constraints g_i(x_0) that are active at the current point . The active set is particularly important in optimization theory, as it determines which constraints will influence the final result of opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pool Adjacent Violators Algorithm
Pool may refer to: Bodies of water * Swimming pool, usually an artificial structure containing a large body of water intended for swimming * Reflecting pool, a shallow pool designed to reflect a structure and its surroundings * Tide pool, a rocky pool on an ocean shore that remains filled with seawater when the tide goes out * Salt pannes and pools, a water-retaining depression located within salt and brackish marshes * Plunge pool, a small, deep body of water * Stream pool, a quiet slow-moving portion of a stream * Spent fuel pool, a storage facility for used fuel rods from a nuclear reactor * Vernal pools, seasonal pools of water that provide habitat for distinctive plants and animals. Sports and gambling * Pool (cards), the common pot for stakes or the stakes themselves in card games * Pool (dominoes), the stock or boneyard in dominoes * Pool (cue sports), a group of games played on a pool table * Pool (poker) or pot (poker), money wagered during a single hand of poker * Po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Algorithm
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called '' convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
WLOG
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate the assumption that what follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic. As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases. In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry. For example, if some property ''P''(''x'',''y'') of real numbers is known to be symmetric in ''x'' and ''y'', namely that ''P''(''x'',''y'') is equivalent to ''P''(''y'',''x''), then in proving that ''P''(''x'',''y'') holds for every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Ordered Set
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Acyclic Graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). Directed acyclic graphs are also called acyclic directed graphs or acyclic digraphs. Definitions A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph, each edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Programming
Quadratic programming (QP) is the process of solving certain mathematical optimization problems involving quadratic functions. Specifically, one seeks to optimize (minimize or maximize) a multivariate quadratic function subject to linear constraints on the variables. Quadratic programming is a type of nonlinear programming. "Programming" in this context refers to a formal procedure for solving mathematical problems. This usage dates to the 1940s and is not specifically tied to the more recent notion of "computer programming." To avoid confusion, some practitioners prefer the term "optimization" — e.g., "quadratic optimization." Problem formulation The quadratic programming problem with variables and constraints can be formulated as follows. Given: * a real-valued, -dimensional vector , * an -dimensional real symmetric matrix , * an -dimensional real matrix , and * an -dimensional real vector , the objective of quadratic programming is to find an -dimensional vector , that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
