|
Constrained Least Squares
In constrained least squares one solves a linear least squares problem with an additional constraint on the solution. This means, the unconstrained equation \mathbf \boldsymbol = \mathbf must be fit as closely as possible (in the least squares sense) while ensuring that some other property of \boldsymbol is maintained. There are often special-purpose algorithms for solving such problems efficiently. Some examples of constraints are given below: * Equality constrained least squares: the elements of \boldsymbol must exactly satisfy \mathbf \boldsymbol = \mathbf (see Ordinary least squares). * Stochastic (linearly) constrained least squares: the elements of \boldsymbol must satisfy \mathbf \boldsymbol = \mathbf + \mathbf , where \mathbf is a vector of random variables such that \operatorname(\mathbf ) = \mathbf and \operatorname(\mathbf \mathbf ^) = \tau^\mathbf. This effectively imposes a prior distribution for \boldsymbol and is therefore equivalent to Bayesian li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Least Squares (mathematics)
Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods. Basic formulation Consider the linear equation where A \in \mathbb^ and b \in \mathbb^m are given and x \in \mathbb^n is variable to be computed. When m > n, it is generally the case that () has no solution. For example, there is no value of x that satisfies \begin 1 & 0 \\ 0 & 1 \\ 1 & 1 \end x = \begin 1 \\ 1 \\ 0 \end, because the first two rows require that x = (1, 1), but then the third row is not satisfied. Thus, for m > n, the goal of solving () exactly is typically replaced by finding the value of x that minimizes some error. There are many ways t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constraint (mathematics)
In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set. Example The following is a simple optimization problem: :\min f(\mathbf x) = x_1^2+x_2^4 subject to :x_1 \ge 1 and :x_2 = 1, where \mathbf x denotes the vector (''x''1, ''x''2). In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint. These two constraints are hard constraints, meaning that it is required that they be satisfied; they define the feasible set of candidate solutions. Without the constraints, the solution would be (0,0), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constrained Generalized Inverse
In linear algebra, a constrained generalized inverse is obtained by solving a system of linear equations with an additional constraint that the solution is in a given subspace. One also says that the problem is described by a system of constrained linear equations. In many practical problems, the solution x of a linear system of equations : Ax=b\qquad (\textA\in\R^\text b\in\R^m) is acceptable only when it is in a certain linear subspace L of \R^n. In the following, the orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it we ... on L will be denoted by P_L. Constrained system of linear equations :Ax=b\qquad x\in L has a solution if and only if the unconstrained system of equations :(A P_L) x = b\qquad x\in\R^n is solvable. If the subspace L is a proper subspace of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Least Squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression In statistics, linear regression is a statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ... model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable. Some sources consider OLS to be linear regression. Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prior Distribution
A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election. The unknown quantity may be a parameter of the model or a latent variable rather than an observable variable. In Bayesian statistics, Bayes' rule prescribes how to update the prior with new information to obtain the posterior probability distribution, which is the conditional distribution of the uncertain quantity given new data. Historically, the choice of priors was often constrained to a conjugate family of a given likelihood function, so that it would result in a tractable posterior of the same family. The widespread availability of Markov chain Monte Carlo methods, however, has made this less of a concern. There are many ways to constru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bayesian Linear Regression
Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well as other parameters describing the distribution of the regressand) and ultimately allowing the out-of-sample prediction of the regressand (often labelled y) '' conditional on'' observed values of the regressors (usually X). The simplest and most widely used version of this model is the ''normal linear model'', in which y given X is distributed Gaussian. In this model, and under a particular choice of prior probabilities for the parameters—so-called conjugate priors—the posterior can be found analytically. With more arbitrarily chosen priors, the posteriors generally have to be approximated. Model setup Consider a standard linear regression problem, in which for i = 1, \ldots, n we specify the mean of the conditional distributio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tikhonov Regularization
Ridge regression (also known as Tikhonov regularization, named for Andrey Tikhonov) is a method of estimating the coefficients of multiple- regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. It is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff). The theory was first introduced by Hoerl and Kennard in 1970 in their ''Technometrics'' papers "Ridge regressions: biased estimation of nonorthogonal problems" and "Ridge regressions: applications in nonorthogonal problems". Ridge regression was developed as a possible solution to the imprecision of least s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Standard Deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. The standard deviation is commonly used in the determination of what constitutes an outlier and what does not. Standard deviation may be abbreviated SD or std dev, and is most commonly represented in mathematical texts and equations by the lowercase Greek alphabet, Greek letter Sigma, σ (sigma), for the population standard deviation, or the Latin script, Latin letter ''s'', for the sample standard deviation. The standard deviation of a random variable, Sample (statistics), sample, statistical population, data set, or probability distribution is the square root of its variance. (For a finite population, v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-negative Least Squares
In mathematical optimization, the problem of non-negative least squares (NNLS) is a type of constrained least squares problem where the coefficients are not allowed to become negative. That is, given a matrix and a (column) vector of response variables , the goal is to find :\operatorname\limits_\mathbf \, \mathbf - \mathbf\, _2^2 subject to . Here means that each component of the vector should be non-negative, and denotes the Euclidean norm. Non-negative least squares problems turn up as subproblems in matrix decomposition, e.g. in algorithms for PARAFAC and non-negative matrix/tensor factorization. The latter can be considered a generalization of NNLS. Another generalization of NNLS is bounded-variable least squares (BVLS), with simultaneous upper and lower bounds . Quadratic programming version The NNLS problem is equivalent to a quadratic programming problem :\operatorname\limits_\mathbf \left(\frac \mathbf^\mathsf \mathbf\mathbf + \mathbf^\mathsf \mathbf\right), wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Vector Space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Definition Given a vector space X over the real numbers \Reals and a preorder \,\leq\, on the set X, the pair (X, \leq) is called a preordered vector space and we say that the preorder \,\leq\, is compatible with the vector space structure of X and call \,\leq\, a vector preorder on X if for all x, y, z \in X and r \in \Reals with r \geq 0 the following two axioms are satisfied # x \leq y implies x + z \leq y + z, # y \leq x implies r y \leq r x. If \,\leq\, is a partial order compatible with the vector space structure of X then (X, \leq) is called an ordered vector space and \,\leq\, is called a vector partial order on X. The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping x \mapsto -x is an isomorphism to the dual order structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
