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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] |
Andrey Nikolayevich Tikhonov
Andrey Nikolayevich Tikhonov (; 17 October 1906 – 7 October 1993) was a leading Soviet Russian mathematician and geophysicist known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems. He was also one of the inventors of the magnetotellurics method in geophysics. Other transliterations of his surname include "Tychonoff", "Tychonov", "Tihonov", "Tichonov". Biography Born in Gzhatsk, he studied at the Moscow State University where he received a Ph.D. in 1927 under the direction of Pavel Sergeevich Alexandrov. In 1933 he was appointed as a professor at Moscow State University. He became a corresponding member of the USSR Academy of Sciences on 29 January 1939 and a full member of the USSR Academy of Sciences on 1 July 1966. Research work Tikhonov worked in a number of different fields in mathematics. He made important contributions to topology, functional analysis, mathematical physics, and certain classes of ill- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Main Diagonal
In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A is the list of entries a_ where i = j. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones: \begin \color & 0 & 0\\ 0 & \color & 0\\ 0 & 0 & \color\end \qquad \begin \color & 0 & 0 & 0 \\ 0 & \color & 0 & 0 \\ 0 & 0 & \color & 0 \end \qquad \begin \color & 0 & 0 \\ 0 & \color & 0 \\ 0 & 0 & \color \\ 0 & 0 & 0 \end \qquad \begin \color & 0 & 0 & 0 \\ 0 & \color & 0 & 0 \\ 0 & 0 & \color & 0 \\ 0 & 0 & 0 & \color \end Square matrices For a square matrix, the ''diagonal'' (or ''main diagonal'' or ''principal diagonal'') is the diagonal line of entries running from the top-left corner to the bottom-right corner. For a matrix A with row index specified by i and column index specified by j, these would be entries A_ with i = j. For example, the iden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur E
Arthur is a masculine given name of uncertain etymology. Its popularity derives from it being the name of the legendary hero King Arthur. A common spelling variant used in many Slavic, Romance, and Germanic languages is Artur. In Spanish and Italian it is Arturo. Etymology The earliest attestation of the name Arthur is in the early 9th century Welsh-Latin text '' Historia Brittonum'', where it refers to a circa 5th century Romano-British general who fought against the invading Saxons, and who later gave rise to the famous King Arthur of medieval legend and literature. A possible earlier mention of the same man is to be found in the epic Welsh poem '' Y Gododdin'' by Aneirin, which some scholars assign to the late 6th century, though this is still a matter of debate and the poem only survives in a late 13th century manuscript entitled the Book of Aneirin. A 9th-century Breton landowner named Arthur witnessed several charters collected in the '' Cartulary of Redon''. The Iris ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doklady Akademii Nauk SSSR
The ''Proceedings of the USSR Academy of Sciences'' (, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), ) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. The journal is indexed in Russian Science Citation Index. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dating from 1726. ''Doklady Akademii Nauk SSSR-Comptes Rendus de l'Académie des Sciences de l'URSS, Seriya ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-binding Constraint
In mathematics, a constraint is a condition of an optimization (mathematics), optimization problem that the solution must satisfy. There are several types of constraints—primarily Equality (mathematics), equality constraints, Inequality (mathematics), inequality constraints, and Integer programming, 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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ridge Regression
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function (mathematics), function subject to constraint (mathematics), equation constraints (i.e., subject to the condition that one or more equations have to be satisfied exactly by the chosen values of the variable (mathematics), variables). It is named after the mathematician Joseph-Louis Lagrange. Summary and rationale The basic idea is to convert a constrained problem into a form such that the derivative test of an unconstrained problem can still be applied. The relationship between the gradient of the function and gradients of the constraints rather naturally leads to a reformulation of the original problem, known as the Lagrangian function or Lagrangian. In the general case, the Lagrangian is defined as \mathcal(x, \lambda) \equiv f(x) + \langle \lambda, g(x)\rangle for functions f, g; the notation \langle \cdot, \cdot \rangle denotes an inner prod ... [...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] |
Least Squares
The method of least squares is a mathematical optimization technique that aims to determine the best fit function by minimizing the sum of the squares of the differences between the observed values and the predicted values of the model. The method is widely used in areas such as regression analysis, curve fitting and data modeling. The least squares method can be categorized into linear and nonlinear forms, depending on the relationship between the model parameters and the observed data. The method was first proposed by Adrien-Marie Legendre in 1805 and further developed by Carl Friedrich Gauss. History Founding The method of least squares grew out of the fields of astronomy and geodesy, as scientists and mathematicians sought to provide solutions to the challenges of navigating the Earth's oceans during the Age of Discovery. The accurate description of the behavior of celestial bodies was the key to enabling ships to sail in open seas, where sailors could no longer rely on la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications In Statistics – Theory And Methods
''Communications in Statistics'' is a peer-reviewed scientific journal that publishes papers related to statistics. It is published by Taylor & Francis in three series, ''Theory and Methods'', ''Simulation and Computation'', and ''Case Studies, Data Analysis and Applications''. ''Communications in Statistics – Theory and Methods'' This series started publishing in 1970 and publishes papers related to statistical theory and methods. It publishes 20 issues each year. Based on Web of Science, the five most cited papers in the journal are: * Kulldorff M. A spatial scan statistic, 1997, 982 cites. * Holland PW, Welsch RE. Robust regression using iteratively reweighted least-squares, 1977, 526 cites. * Sugiura N. Further analysts of the data by Akaike's information criterion and the finite corrections, 1978, 490 cites. * Hosmer DW, Lemeshow S. Goodness of fit tests for the multiple logistic regression model, 1980, 401 cites. * Iman RL, Conover WJ. Small sample sensitivity analysis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Design Matrix
In statistics and in particular in regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables of a set of objects. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object. The design matrix is used in certain statistical models, e.g., the general linear model. It can contain indicator variables (ones and zeros) that indicate group membership in an ANOVA, or it can contain values of continuous variables. The design matrix contains data on the independent variables (also called explanatory variables), in a statistical model that is intended to explain observed data on a response variable (often called a dependent variable). The theory relating to such models uses the design matrix as input to some linear algebra : see for example linear regression. A notable feature of the concept of a design matrix i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
