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Spline Smoothing
Smoothing splines are function estimates, \hat f(x), obtained from a set of noisy observations y_i of the target f(x_i), in order to balance a measure of goodness of fit of \hat f(x_i) to y_i with a derivative based measure of the smoothness of \hat f(x). They provide a means for smoothing noisy x_i, y_i data. The most familiar example is the cubic smoothing spline, but there are many other possibilities, including for the case where x is a vector quantity. Cubic spline definition Let \ be a set of observations, modeled by the relation Y_i = f(x_i) + \epsilon_i where the \epsilon_i are independent, zero mean random variables (usually assumed to have constant variance). The cubic smoothing spline estimate \hat f of the function f is defined to be the minimizer (over the class of twice differentiable functions) of : \sum_^n \^2 + \lambda \int \hat f''(x)^2 \,dx. Remarks: * \lambda \ge 0 is a smoothing parameter, controlling the trade-off between fidelity to the data and roughnes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Goodness Of Fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measures can be used in statistical hypothesis testing, e.g. to test for normality of residuals, to test whether two samples are drawn from identical distributions (see Kolmogorov–Smirnov test), or whether outcome frequencies follow a specified distribution (see Pearson's chi-square test). In the analysis of variance, one of the components into which the variance is partitioned may be a lack-of-fit sum of squares. Fit of distributions In assessing whether a given distribution is suited to a data-set, the following tests and their underlying measures of fit can be used: * Bayesian information criterion *Kolmogorov–Smirnov test *Cramér–von Mises criterion *Anderson–Darling test * Shapiro–Wilk test *Chi-squared test *Akaike informat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thin Plate Splines
Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing. They were introduced to geometric design by Duchon. They are an important special case of a polyharmonic spline. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm. Physical analogy The name ''thin plate spline'' refers to a physical analogy involving the bending of a thin sheet of metal. Just as the metal has rigidity, the TPS fit resists bending also, implying a penalty involving the smoothness of the fitted surface. In the physical setting, the deflection is in the z direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the x or y coordinates within the plane. In 2D cases, given a set of K corresponding points, the TPS warp is described by 2(K+3) parameters which include 6 global affine motion parameters and 2K coefficients for corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Programming Language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language. The description of a programming language is usually split into the two components of syntax (form) and semantics (meaning), which are usually defined by a formal language. Some languages are defined by a specification document (for example, the C programming language is specified by an ISO Standard) while other languages (such as Perl) have a dominant implementation that is treated as a reference. Some languages have both, with the basic language defined by a standard and extensions taken from the dominant implementation being common. Programming language theory is the subfield of computer science that studies the design, implementation, analysis, characterization, and classification of programming languages. Definitions There are many considerations when defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl R
Carl may refer to: *Carl, Georgia, city in USA *Carl, West Virginia, an unincorporated community * Carl (name), includes info about the name, variations of the name, and a list of people with the name *Carl², a TV series * "Carl", an episode of television series ''Aqua Teen Hunger Force'' * An informal nickname for a student or alum of Carleton College CARL may refer to: *Canadian Association of Research Libraries *Colorado Alliance of Research Libraries See also * Carle (other) * Charles *Carle, a surname *Karl (other) *Karle (other) Karle may refer to: Places * Karle (Svitavy District), a municipality and village in the Czech Republic * Karli, India, a town in Maharashtra, India ** Karla Caves, a complex of Buddhist cave shrines * Karle, Belgaum, a settlement in Belgaum ... {{disambig ja:カール zh:卡尔 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting. When the problem has substantial uncertainties in the independent variable (the ''x'' variable), then simple regression and least-squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regressio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold Learning
Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. Applications of NLDR Consider a dataset represented as a matrix (or a database table), such that each row represents a set of attributes (or features or dimensions) that describe a particular instance of something. If the number of attributes is large, then the space of unique possible rows is exponentially large. Thus, the larger the dimensionality, the more difficult it becomes to sample the space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elastic Map
Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. GorbanA.Y. Zinovyevand A.A. Pitenko in 1996–1998. Energy of elastic map Let be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes _j in the same space. Each datapo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Adaptive Regression Splines
In statistics, multivariate adaptive regression splines (MARS) is a form of regression analysis introduced by Jerome H. Friedman in 1991. It is a non-parametric regression technique and can be seen as an extension of linear models that automatically models nonlinearities and interactions between variables. The term "MARS" is trademarked and licensed to Salford Systems. In order to avoid trademark infringements, many open-source implementations of MARS are called "Earth". The basics This section introduces MARS using a few examples. We start with a set of data: a matrix of input variables ''x'', and a vector of the observed responses ''y'', with a response for each row in ''x''. For example, the data could be: Here there is only one independent variable, so the ''x'' matrix is just a single column. Given these measurements, we would like to build a model which predicts the expected ''y'' for a given ''x''. A linear model for the above data is : \widehat = -37 + 5.1 x The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolating Spline
In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-ten polynomial to all of them. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low-degree polynomials for the spline. Spline interpolation also avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high-degree polynomials. Introduction Originally, '' spline'' was a term for elastic rulers that were bent to pass through a number of predefined points, or ''knots''. These were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Karl Longin Zeller
Karl Longin Zeller (December 28, 1924, Šiauliai, Lithuania – July 20, 2006, Tübingen) was a German mathematician and computer scientist who worked in numerical analysis and approximation theory.. He is the namesake of Zeller operators. Zeller was drafted into the Wehrmacht, and lost his right arm on the Soviet front of World War II. He earned his Ph.D. from the University of Tübingen in 1950, under the supervision of Konrad Knopp and Erich Kamke, and remained at Tübingen for most of his career as a professor and as director of the computer center. He left Tübingen in 1959 for a professorship in Stuttgart but returned to Tübingen in 1960 with a personal chair in "the mathematics of supercomputer facilities" (german: Mathematik der Hochleistungsrechenanlagen), making him one of the founders of computer science in Germany. He has over 200 academic descendants. In 1993, he was given an honorary doctorate by the University of Siegen The University of Siegen (german: Universi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thin Plate Spline
Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing. They were introduced to geometric design by Duchon. They are an important special case of a polyharmonic spline. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm. Physical analogy The name ''thin plate spline'' refers to a physical analogy involving the bending of a thin sheet of metal. Just as the metal has rigidity, the TPS fit resists bending also, implying a penalty involving the smoothness of the fitted surface. In the physical setting, the deflection is in the z direction, orthogonal to the plane. In order to apply this idea to the problem of coordinate transformation, one interprets the lifting of the plate as a displacement of the x or y coordinates within the plane. In 2D cases, given a set of K corresponding points, the TPS warp is described by 2(K+3) parameters which include 6 global affine motion parameters and 2K coefficients for corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
