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Outline Of Regression Analysis
The following outline is provided as an overview of and topical guide to regression analysis: Regression analysis – use of statistical techniques for learning about the relationship between one or more dependent variables (''Y'') and one or more independent variables (''X''). Overview articles * Regression analysis * Linear regression Non-statistical articles related to regression * Least squares * Linear least squares (mathematics) * Non-linear least squares * Least absolute deviations * Curve fitting * Smoothing * Cross-sectional study Basic statistical ideas related to regression * Conditional expectation * Correlation * Correlation coefficient * Mean square error * Residual sum of squares * Explained sum of squares * Total sum of squares Visualization * Scatterplot Linear regression based on least squares * General linear model * Ordinary least squares * Generalized least squares * Simple linear regression * Trend estimation * Ridge regression * Polynomial regre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation
In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are '' linearly'' related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the demand curve. Correlations are useful because they can indicate a predictive relationship that can be exploited in practice. For example, an electrical utility may produce less power on a mild day based on the correlation between electricity demand and weather. In this example, there is a causal relationship, because extreme weather causes people to use more electricity for heating or cooling. However, in g ... [...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] |
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] |
Trend Estimation
Linear trend estimation is a statistical technique used to analyze data patterns. Data patterns, or trends, occur when the information gathered tends to increase or decrease over time or is influenced by changes in an external factor. Linear trend estimation essentially creates a straight line on a graph of data that models the general direction that the data is heading. Fitting a trend: Least-squares Given a set of data, there are a variety of functions that can be chosen to fit the data. The simplest function is a straight line with the dependent variable (typically the measured data) on the vertical axis and the independent variable (often time) on the horizontal axis. The least-squares fit is a common method to fit a straight line through the data. This method minimizes the sum of the squared errors in the data series y. Given a set of points in time t and data values y_t observed for those points in time, values of \hat a and \hat b are chosen to minimize the sum of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Linear Regression
In statistics, simple linear regression (SLR) is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the ''x'' and ''y'' coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variable. The adjective ''simple'' refers to the fact that the outcome variable is related to a single predictor. It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared '' residual'' (vertical distance between the point of the data set and the fitted line), and the goal is to make the sum of these squared deviations as small as possible. In this case, the slope of the fitted line is equal to the corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Least Squares
In statistics, generalized least squares (GLS) is a method used to estimate the unknown parameters in a Linear regression, linear regression model. It is used when there is a non-zero amount of correlation between the Residual (statistics), residuals in the regression model. GLS is employed to improve efficiency_(statistics), statistical efficiency and reduce the risk of drawing erroneous inferences, as compared to conventional least squares and weighted least squares methods. It was first described by Alexander Aitken in 1935. It requires knowledge of the covariance matrix for the residuals. If this is unknown, estimating the covariance matrix gives the method of feasible generalized least squares (FGLS). However, FGLS provides fewer guarantees of improvement. Method In standard linear regression models, one observes data \_ on ''n'' statistical units with ''k'' − 1 predictor values and one response value each. The response values are placed in a vector,\mathbf ... [...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] |
General Linear Model
The general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as : \mathbf = \mathbf\mathbf + \mathbf, where Y is a Matrix (mathematics), matrix with series of multivariate measurements (each column being a set of measurements on one of the dependent variables), X is a matrix of observations on independent variables that might be a design matrix (each column being a set of observations on one of the independent variables), B is a matrix containing parameters that are usually to be estimated and U is a matrix containing Errors and residuals in statistics, errors (noise). The errors are usually assumed to be uncorrelated across measurements, and follow a multivariate normal distribution. If the errors do not follow a multivariate normal distribution, generalized li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scatterplot
A scatter plot, also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram, is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of data. If the points are coded (color/shape/size), one additional variable can be displayed. The data are displayed as a collection of points, each having the value of one variable determining the position on the horizontal axis and the value of the other variable determining the position on the vertical axis. History According to Michael Friendly and Daniel Denis, the defining characteristic distinguishing scatter plots from line charts is the representation of specific observations of bivariate data where one variable is plotted on the horizontal axis and the other on the vertical axis. The two variables are often abstracted from a physical representation like the spread of bullets on a target or a geographic or celestial projection. Wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Sum Of Squares
In statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. For a set of observations, y_i, i\leq n, it is defined as the sum over all squared differences between the observations and their overall mean \bar.:Everitt, B.S. (2002) ''The Cambridge Dictionary of Statistics'', CUP, :\mathrm=\sum_^\left(y_-\bar\right)^2 For wide classes of linear models, the total sum of squares equals the explained sum of squares plus the residual sum of squares. For proof of this in the multivariate OLS case, see partitioning in the general OLS model. In analysis of variance (ANOVA) the total sum of squares is the sum of the so-called "within-samples" sum of squares and "between-samples" sum of squares, i.e., partitioning of the sum of squares. In multivariate analysis of variance (MANOVA) the following equation applies Especially chapters 11 and 12. :\mathbf = \mathbf + \mathbf, g where T is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Explained Sum Of Squares
In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression (SSR – not to be confused with the residual sum of squares (RSS) or sum of squares of errors), is a quantity used in describing how well a model, often a regression model, represents the data being modelled. In particular, the explained sum of squares measures how much variation there is in the modelled values and this is compared to the total sum of squares (TSS), which measures how much variation there is in the observed data, and to the residual sum of squares, which measures the variation in the error between the observed data and modelled values. Definition The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, , where ''y''''i'' is the ''i'' th observation of the response variable, ''x''''ji'' is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
