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Set Estimation
In statistics, a random vector ''x'' is classically represented by a probability density function. In a set-membership approach or set estimation, ''x'' is represented by a set ''X'' to which ''x'' is assumed to belong. This means that the support of the probability distribution function of ''x'' is included inside ''X''. On the one hand, representing random vectors by sets makes it possible to provide fewer assumptions on the random variables (such as independence) and dealing with 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 approach which considers that measurements are represented by a set ''Y'' (most of the time a box of R''m'', where ''m'' is the number of measurements) of the measurement space. If ''p'' is the parameter vector and ''f'' is the model function, then t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "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.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Vector
In probability, and statistics, a multivariate random variable or random vector is a list 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. More formally, a multivariate random variable is a column vector \mathbf = (X_1,\dots,X_n)^\mathsf (or its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a ''relative likelihood'' that the value of the random variable would be close to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. In a more precise sense, the PDF is used to specify the probability of the random variable falling ''within a particular range of values'', as opposed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used very widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Estimation Theory
Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An ''estimator'' attempts to approximate the unknown parameters using the measurements. In estimation theory, two approaches are generally considered: * The probabilistic approach (described in this article) assumes that the measured data is random with probability distribution dependent on the parameters of interest * The set-membership approach assumes that the measured data vector belongs to a set which depends on the parameter vector. Examples For example, it is desired to estimate the proportion of a population of voters who will vote for a particular candidate. That proportion is the parameter sought; the estimate is based on a small random sample of voters. Alternatively, it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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''t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 polyhedron. A linear programming algorithm finds a point in the polytope where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Arithmetic
Interval arithmetic (also known as interval mathematics, interval analysis, or interval computation) is a mathematical technique used to put bounds on rounding errors and measurement errors in mathematical computation. Numerical methods using interval arithmetic can guarantee reliable and mathematically correct results. Instead of representing a value as a single number, interval arithmetic represents each value as a range of possibilities. For example, instead of saying the height of someone is approximately 2 meters, one could using interval arithmetic, say that the height of the person is definitely between 1.97 meters and 2.03 meters. Mathematically, using interval arithmetic, instead of working with an uncertain real-valued variable x, one 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, yields an inexact value; f ins ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feasible Region
In mathematical optimization, a feasible region, feasible set, search space, 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 or more ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Wiki Estim Data
A wiki ( ) is an online hypertext publication collaboratively edited and managed by its own audience, using a web browser. A typical wiki contains multiple pages for the subjects or scope of the project, and could be either open to the public or limited to use within an organization for maintaining its internal knowledge base. Wikis are enabled by wiki software, otherwise known as wiki engines. A wiki engine, being a form of a content management system, differs from other web-based systems such as blog software, in that the content is created without any defined owner or leader, and wikis have little inherent structure, allowing structure to emerge according to the needs of the users. Wiki engines usually allow content to be written using a simplified markup language and sometimes edited with the help of a rich-text editor. There are dozens of different wiki engines in use, both standalone and part of other software, such as bug tracking systems. Some wiki engines are open ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiki Estim Param
A wiki ( ) is an online hypertext publication collaboratively edited and managed by its own audience, using a web browser. A typical wiki contains multiple pages for the subjects or scope of the project, and could be either open to the public or limited to use within an organization for maintaining its internal knowledge base. Wikis are enabled by wiki software, otherwise known as wiki engines. A wiki engine, being a form of a content management system, differs from other web-based systems such as blog software, in that the content is created without any defined owner or leader, and wikis have little inherent structure, allowing structure to emerge according to the needs of the users. Wiki engines usually allow content to be written using a simplified markup language and sometimes edited with the help of a rich-text editor. There are dozens of different wiki engines in use, both standalone and part of other software, such as bug tracking systems. Some wiki engines are open ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
