Numerous extensions of linear regression have been developed, which allow some or all of the assumptions underlying the basic model to be relaxed. The very simplest case of a single scalar predictor variable x and a single scalar response variable y is known as simple linear regression. Nearly all real-world regression models involve multiple predictors, and basic descriptions of linear regression are often phrased in terms of the multiple regression model.

Note, however, that in these cases the response variable y is still a scalar. Another term, multivariate linear regression , refers to cases where y is a vector, i. The general linear model considers the situation when the response variable is not a scalar for each observation but a vector, y i. These are not the same as multivariable linear models also called "multiple linear models". Various models have been created that allow for heteroscedasticity , i. For example, weighted least squares is a method for estimating linear regression models when the response variables may have different error variances, possibly with correlated errors.

See also Weighted linear least squares , and Generalized least squares. Heteroscedasticity-consistent standard errors is an improved method for use with uncorrelated but potentially heteroscedastic errors. Generalized linear models GLMs are a framework for modeling response variables that are bounded or discrete.

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This is used, for example:. Hierarchical linear models or multilevel regression organizes the data into a hierarchy of regressions, for example where A is regressed on B , and B is regressed on C. It is often used where the variables of interest have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms are nested in schools, and schools are nested in some administrative grouping, such as a school district.

The response variable might be a measure of student achievement such as a test score, and different covariates would be collected at the classroom, school, and school district levels. Errors-in-variables models or "measurement error models" extend the traditional linear regression model to allow the predictor variables X to be observed with error.

Generally, the form of bias is an attenuation, meaning that the effects are biased toward zero. A large number of procedures have been developed for parameter estimation and inference in linear regression. These methods differ in computational simplicity of algorithms, presence of a closed-form solution, robustness with respect to heavy-tailed distributions, and theoretical assumptions needed to validate desirable statistical properties such as consistency and asymptotic efficiency.

Linear regression is widely used in biological, behavioral and social sciences to describe possible relationships between variables. It ranks as one of the most important tools used in these disciplines. A trend line represents a trend, the long-term movement in time series data after other components have been accounted for. It tells whether a particular data set say GDP, oil prices or stock prices have increased or decreased over the period of time.

A trend line could simply be drawn by eye through a set of data points, but more properly their position and slope is calculated using statistical techniques like linear regression. Trend lines typically are straight lines, although some variations use higher degree polynomials depending on the degree of curvature desired in the line. Trend lines are sometimes used in business analytics to show changes in data over time.

This has the advantage of being simple. Trend lines are often used to argue that a particular action or event such as training, or an advertising campaign caused observed changes at a point in time. This is a simple technique, and does not require a control group, experimental design, or a sophisticated analysis technique. However, it suffers from a lack of scientific validity in cases where other potential changes can affect the data.

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## Regression analysis - GIS Wiki | The GIS Encyclopedia

Early evidence relating tobacco smoking to mortality and morbidity came from observational studies employing regression analysis. In order to reduce spurious correlations when analyzing observational data, researchers usually include several variables in their regression models in addition to the variable of primary interest.

For example, in a regression model in which cigarette smoking is the independent variable of primary interest and the dependent variable is lifespan measured in years, researchers might include education and income as additional independent variables, to ensure that any observed effect of smoking on lifespan is not due to those other socio-economic factors. However, it is never possible to include all possible confounding variables in an empirical analysis. For example, a hypothetical gene might increase mortality and also cause people to smoke more. For this reason, randomized controlled trials are often able to generate more compelling evidence of causal relationships than can be obtained using regression analyses of observational data.

When controlled experiments are not feasible, variants of regression analysis such as instrumental variables regression may be used to attempt to estimate causal relationships from observational data. The capital asset pricing model uses linear regression as well as the concept of beta for analyzing and quantifying the systematic risk of an investment. This comes directly from the beta coefficient of the linear regression model that relates the return on the investment to the return on all risky assets. Linear regression is the predominant empirical tool in economics.

## Linear regression

For example, it is used to predict consumption spending , [20] fixed investment spending, inventory investment , purchases of a country's exports , [21] spending on imports , [21] the demand to hold liquid assets , [22] labor demand , [23] and labor supply. Linear regression finds application in a wide range of environmental science applications. In Canada, the Environmental Effects Monitoring Program uses statistical analyses on fish and benthic surveys to measure the effects of pulp mill or metal mine effluent on the aquatic ecosystem.

Linear regression plays an important role in the field of artificial intelligence such as machine learning. The linear regression algorithm is one of the fundamental supervised machine-learning algorithms due to its relative simplicity and well-known properties. Least squares linear regression, as a means of finding a good rough linear fit to a set of points was performed by Legendre and Gauss for the prediction of planetary movement.

Quetelet was responsible for making the procedure well-known and for using it extensively in the social sciences. From Wikipedia, the free encyclopedia. Main article: Linear least squares. Main article: Trend estimation. Main article: Econometrics.

This section needs expansion. You can help by adding to it. January Statistics portal. Analysis of variance Blinder—Oaxaca decomposition Censored regression model Cross-sectional regression Curve fitting Empirical Bayes methods Errors and residuals Lack-of-fit sum of squares Line fitting Linear classifier Linear equation Logistic regression M-estimator Multivariate adaptive regression splines Nonlinear regression Nonparametric regression Normal equations Projection pursuit regression Segmented linear regression Stepwise regression Structural break Support vector machine Truncated regression model.

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Cambridge University Press. A simple regression equation has on the right hand side an intercept and an explanatory variable with a slope coefficient. Seal The earliest form of the linear regression was the least squares method, which was published by Legendre in , and by Gauss in Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the sun.

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### Notes for ST/ST Linear Models 1

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Cohen, J. The Variation of Animals and Plants under Domestication. Darwin uses the term "reversion". Draper, N. Applied Regression Analysis 3rd ed. John Wiley. Francis Galton. Facsimile at: [1] Robert S. Pindyck and Daniel L. Rubinfeld , 4h ed. Econometric Models and Economic Forecasts , ch. Free download. Book file PDF easily for everyone and every device. This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats. See a Problem? Stock photo. Brand new: lowest price The lowest-priced brand-new, unused, unopened, undamaged item in its original packaging where packaging is applicable.

Printed in and for South Asia. Actual Image of this Book Title. It has been and still is readily readable and understandable. Carrying out a successful application of regression analysis, however, requires a balance of theoretical results, empirical rules, and subjective judgment.