|
Set Estimation
In statistics, a random vector is classically represented by a probability density function. In a set-membership approach or set estimation, is represented by a set to which is assumed to belong. This means that the Support (mathematics), support of the probability distribution function of is included inside . On the one hand, representing random vectors by sets makes it possible to provide fewer assumptions on the random variables (such as Independence (probability theory), independence) and dealing with nonlinearity, nonlinearities is easier. On the other hand, a probability distribution function provides a more accurate information than a set enclosing its support. Set-membership estimation Set membership estimation (or ''set estimation'' for short) is an Estimation theory, estimation approach which considers that measurements are represented by a set (most of the time a box of where is the number of measurements) of the measurement space. If is the parameter vector a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear Programming
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomial function, linear relationships. Linear programming is a special case of mathematical programming (also known as mathematical optimization). More formally, linear programming is a technique for the mathematical optimization, optimization of a linear objective function, subject to linear equality and linear inequality Constraint (mathematics), constraints. Its feasible region is a convex polytope, which is a set defined as the intersection (mathematics), intersection of finitely many Half-space (geometry), half spaces, each of which is defined by a linear inequality. Its objective function is a real number, real-valued affine function, affine (linear) function defined on this polytope. A linear programming algorithm finds a point in the po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Relaxed Intersection
The ''relaxed intersection'' of ''m'' sets corresponds to the classical intersection between sets except that it is allowed to relax few sets in order to avoid an empty intersection. This notion can be used to solve Constraint satisfaction problem, constraints satisfaction problems that are inconsistent by Maximum satisfiability problem, relaxing a small number of constraints. When a set estimation, bounded-error approach is considered for parameter estimation, the relaxed intersection makes it possible to be robust with respect to some outliers. Definition The ''q''-relaxed intersection of the ''m'' subsets X_,\dots ,X_ of R^, denoted by X^=\bigcap^X_ is the set of all x \in R^ which belong to all X_ 's, except q at most. This definition is illustrated by Figure 1. Define \lambda (x) =\text \left\. We have X^=\lambda ^([m-q,m]) . Characterizing the q-relaxed intersection is a thus a set inversion problem. Example Consider 8 intervals: X_=[1,4], X_=\ [2,4], ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Wiki Estim Param
A wiki ( ) is a form of hypertext publication on the internet which is collaboratively edited and managed by its audience directly through a web browser. A typical wiki contains multiple pages that can either be edited by the public or limited to use within an organization for maintaining its internal knowledge base. Its name derives from the first user-editable website called "WikiWikiWeb," with "wiki" being a Hawaiian word meaning "quick." Wikis are powered by wiki software, also known as wiki engines. Being a form of content management system, these differ from other web-based systems such as blog software or static site generators in that the content is created without any defined owner or leader. Wikis have little inherent structure, allowing one to emerge according to the needs of the users. Wiki engines usually allow content to be written using a lightweight markup language and sometimes edited with the help of a rich-text editor. There are dozens of different wik ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Subpaving
In mathematics, a subpaving is a set of nonoverlapping boxes of R⁺. A subset ''X'' of Rⁿ can be approximated by two subpavings ''X⁻'' and ''X⁺'' such that ''X⁻'' ⊂ ''X'' ⊂ ''X⁺''. In R¹ the boxes are line segments, in R² rectangles and in Rⁿ hyperrectangles. A R² subpaving can be also a " non-regular tiling by rectangles", when it has no holes. Boxes present the advantage of being very easily manipulated by computers, as they form the heart of interval analysis. Many interval algorithms naturally provide solutions that are regular subpavings. In computation, a well-known application of subpaving in R² is the Quadtree data structure. In image tracing context and other applications is important to see ''X⁻'' as topological interior, as illustrated. Example The three figures on the right below show an approximation of the set ''X'' = with different accuracies. The set ''X⁻'' corresponds to red boxes and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Feasible Region
In mathematical optimization and computer science, a feasible region, feasible set, or solution space is the set of all possible points (sets of values of the choice variables) of an optimization problem that satisfy the problem's constraints, potentially including inequalities, equalities, and integer constraints. This is the initial set of candidate solutions to the problem, before the set of candidates has been narrowed down. For example, consider the problem of minimizing the function x^2+y^4 with respect to the variables x and y, subject to 1 \le x \le 10 and 5 \le y \le 12. \, Here the feasible set is the set of pairs (''x'', ''y'') in which the value of ''x'' is at least 1 and at most 10 and the value of ''y'' is at least 5 and at most 12. The feasible set of the problem is separate from the objective function, which states the criterion to be optimized and which in the above example is x^2+y^4. In many problems, the feasible set reflects a constraint that one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Interval Arithmetic
Interval arithmetic (also known as interval mathematics; interval analysis or interval computation) is a mathematical technique used to mitigate rounding and measurement errors in mathematical computation by computing function bounds. Numerical methods involving interval arithmetic can guarantee relatively reliable and mathematically correct results. Instead of representing a value as a single number, interval arithmetic or interval mathematics represents each value as a range of possibilities. Mathematically, instead of working with an uncertain real-valued variable x, interval arithmetic works with an interval ,b/math> that defines the range of values that x can have. In other words, any value of the variable x lies in the closed interval between a and b. A function f, when applied to x, produces an interval ,d/math> which includes all the possible values for f(x) for all x \in ,b/math>. Interval arithmetic is suitable for a variety of purposes; the most common use is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Set Inversion
In mathematics, set inversion is the problem of characterizing the preimage ''X'' of a set ''Y'' by a function ''f'', i.e., ''X'' = ''f''−1(''Y'' ) = . It can also be viewed as the problem of describing the solution set of the quantified constraint "''Y''(''f''(''x''))", where ''Y''(''y'') is a constraint, e.g. an inequality, describing the set ''Y''. In most applications, ''f'' is a function from R''n'' to R''p'' and the set ''Y'' is a box of R''p'' (i.e. a Cartesian product of ''p'' intervals of R). When ''f'' is nonlinear the set inversion problem can be solved using interval analysis combined with a branch-and-bound algorithm. The main idea consists in building a paving of R''p'' made with non-overlapping boxes. For each box 'x'' we perform the following tests: # if ''f''( 'x'' ⊂ ''Y'' we conclude that 'x''⊂ ''X''; # if ''f''( 'x'' ∩ ''Y'' = ∅ we conclude that 'x''∩ ''X'' = ∅; # Otherwise, the box 'x''the box is bisected except if its width is small ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Random Vector
In probability, and statistics, a multivariate random variable or random vector is a list or vector of mathematical variables each of whose value is unknown, either because the value has not yet occurred or because there is imperfect knowledge of its value. The individual variables in a random vector are grouped together because they are all part of a single mathematical system — often they represent different properties of an individual statistical unit. For example, while a given person has a specific age, height and weight, the representation of these features of ''an unspecified person'' from within a group would be a random vector. Normally each element of a random vector is a real number. Random vectors are often used as the underlying implementation of various types of aggregate random variables, e.g. a random matrix, random tree, random sequence, stochastic process, etc. Formally, a multivariate random variable is a column vector \mathbf = (X_1,\dots,X_n)^\maths ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
