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Backfitting Algorithm
In statistics, the backfitting algorithm is a simple iterative procedure used to fit a generalized additive model. It was introduced in 1985 by Leo Breiman and Jerome Friedman along with generalized additive models. In most cases, the backfitting algorithm is equivalent to the Gauss–Seidel method, an algorithm used for solving a certain linear system of equations. Algorithm Additive models are a class of non-parametric regression models of the form: : Y_i = \alpha + \sum_^p f_j(X_) + \epsilon_i where each X_1, X_2, \ldots, X_p is a variable in our p-dimensional predictor X, and Y is our outcome variable. \epsilon represents our inherent error, which is assumed to have mean zero. The f_j represent unspecified smooth functions of a single X_j. Given the flexibility in the f_j, we typically do not have a unique solution: \alpha is left unidentifiable as one can add any constants to any of the f_j and subtract this value from \alpha. It is common to rectify this by constr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Generalized Additive Model
In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear response variable depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. GAMs were originally developed by Trevor Hastie and Robert Tibshirani to blend properties of generalized linear models with additive models. They can be interpreted as the discriminative generalization of the naive Bayes generative model. The model relates a univariate response variable, ''Y'', to some predictor variables, ''x''''i''. An exponential family distribution is specified for Y (for example normal, binomial or Poisson distributions) along with a link function ''g'' (for example the identity or log functions) relating the expected value of ''Y'' to the predictor variables via a structure such as : g(\operatorname(Y))=\beta_0 + f_1(x_1) + f_2(x_2)+ \cdots + f_m(x_m).\,\! The functions ''f''''i'' may be functions with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leo Breiman
Leo Breiman (January 27, 1928 – July 5, 2005) was an American statistician at the University of California, Berkeley and a member of the United States National Academy of Sciences. Breiman's work helped to bridge the gap between statistics and computer science, particularly in the field of machine learning Machine learning (ML) is a field of study in artificial intelligence concerned with the development and study of Computational statistics, statistical algorithms that can learn from data and generalise to unseen data, and thus perform Task ( .... His most important contributions were his work on classification and regression trees and ensembles of trees fit to bootstrap samples. Bootstrap aggregation was given the name ''bagging'' by Breiman. Another of Breiman's ensemble approaches is the random forest. See also * Shannon–McMillan–Breiman theorem Further reading * Leo Breimaobituary from the University of California, Berkeley * Richard A. OlshenA Con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jerome H
Jerome (; ; ; – 30 September 420), also known as Jerome of Stridon, was an early Christian presbyter, priest, Confessor of the Faith, confessor, theologian, translator, and historian; he is commonly known as Saint Jerome. He is best known for his translation of the Bible into Latin (the translation that became known as the Vulgate) and his commentaries on the whole Bible. Jerome attempted to create a translation of the Old Testament based on a Hebrew version, rather than the Septuagint, as Vetus Latina, prior Latin Bible translations had done. His list of writings is extensive. In addition to his biblical works, he wrote polemical and historical essays, always from a theologian's perspective. Jerome was known for his teachings on Christian moral life, especially those in cosmopolitan centers such as Rome. He often focused on women's lives and identified how a woman devoted to Jesus should live her life. This focus stemmed from his close patron relationships with several pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear System Of Equations
In mathematics, a system of linear equations (or linear system) is a collection of two or more linear equations involving the same variables. For example, : \begin 3x+2y-z=1\\ 2x-2y+4z=-2\\ -x+\fracy-z=0 \end is a system of three equations in the three variables . A '' solution'' to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied. In the example above, a solution is given by the ordered triple (x,y,z)=(1,-2,-2), since it makes all three equations valid. Linear systems are a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and play a prominent role in engineering, physics, chemistry, computer science, and economics. A system of non-linear equations can often be approximated by a linear system (see linearization), a helpful technique when making a mathematical model or computer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonparametric Regression
Nonparametric regression is a form of regression analysis where the predictor does not take a predetermined form but is completely constructed using information derived from the data. That is, no parametric equation is assumed for the relationship between predictors and dependent variable. A larger sample size is needed to build a nonparametric model having a level of uncertainty as a parametric model because the data must supply both the model structure and the parameter estimates. Definition Nonparametric regression assumes the following relationship, given the random variables X and Y: : \mathbb \mid X=x= m(x), where m(x) is some deterministic function. Linear regression is a restricted case of nonparametric regression where m(x) is assumed to be a linear function of the data. Sometimes a slightly stronger assumption of additive noise is used: : Y = m(X) + U, where the random variable U is the `noise term', with mean 0. Without the assumption that m belongs to a specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothing Spline
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. The cubic smoothing spline estimate \hat f of the function f is defined to be the unique minimizer, in the Sobolev space W^2_2 on a compact interval, of : \sum_^n \^2 + \lambda \int \hat^(x)^2 \,dx. Remarks: * \lambda \ge 0 is a smoothing parameter, controlling the trade-off between fidelity to the data and roughness of the function estimate. This is oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Regression
In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable ''x'' and the dependent variable ''y'' is modeled as a polynomial in ''x''. Polynomial regression fits a nonlinear relationship between the value of ''x'' and the corresponding conditional mean of ''y'', denoted E(''y'' , ''x''). Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(''y'' , ''x'') is linear in the unknown parameters that are estimated from the data. Thus, polynomial regression is a special case of linear regression. The explanatory (independent) variables resulting from the polynomial expansion of the "baseline" variables are known as higher-degree terms. Such variables are also used in classification settings. History Polynomial regression models are usually fit using the method of least squares. The leas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel Smoothing
A kernel smoother is a statistics, statistical technique to estimate a real valued function (mathematics), function f: \mathbb^p \to \mathbb as the weighted average of neighboring observed data. The weight is defined by the ''Kernel (statistics), kernel'', such that closer points are given higher weights. The estimated function is smooth, and the level of smoothness is set by a single parameter. Kernel smoothing is a type of weighted moving average. Definitions Let K_(X_0 ,X) be a kernel defined by :K_(X_0 ,X) = D\left( \frac \right) where: * X,X_0 \in \mathbb^p * \left\, \cdot \right\, is the Euclidean norm * h_\lambda (X_0) is a parameter (kernel radius) * ''D''(''t'') is typically a positive real valued function, whose value is decreasing (or not increasing) for the increasing distance between the ''X'' and ''X''0. Popular Kernel (statistics), kernels used for smoothing include parabolic (Epanechnikov), tricube, and Gaussian function, Gaussian kernels. Let Y(X):\mathbb^p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trevor Hastie
Trevor John Hastie (born 27 June 1953) is an American statistician and computer scientist. He is currently serving as the John A. Overdeck Professor of Mathematical Sciences and Professor of Statistics at Stanford University. Hastie is known for his contributions to applied statistics, especially in the field of machine learning, data mining, and bioinformatics. He has authored several popular books in statistical learning, including ''The Elements of Statistical Learning: Data Mining, Inference, and Prediction''. Hastie has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge. He also contributed to the development of S. Education and career Hastie was born on 27 June 1953 in South Africa. He received his B.S. in statistics from the Rhodes University in 1976 and master's degree from University of Cape Town in 1979. Hastie joined the doctoral program at Stanford University in 1980 and received his Ph.D. in 1984 under the supervision of Werner St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Tibshirani
Robert Tibshirani (born July 10, 1956) is a professor in the Departments of Statistics and Biomedical Data Science at Stanford University. He was a professor at the University of Toronto from 1985 to 1998. In his work, he develops statistical tools for the analysis of complex datasets, most recently in genomics and proteomics. His most well-known contributions are the Lasso method, which proposed the use of L1 penalization in regression and related problems, and Significance Analysis of Microarrays. Education and early life Tibshirani was born on 10 July 1956 in Niagara Falls, Ontario, Canada. He received his B. Math. in statistics and computer science from the University of Waterloo in 1979 and a Master's degree in Statistics from the University of Toronto in 1980. Tibshirani joined the doctoral program at Stanford University in 1981 and received his Ph.D. in 1984 under the supervision of Bradley Efron. His dissertation was entitled "Local likelihood estimation". Honors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
