1 the reasoning involved when you assume the conclusion is true and reason backward to the evidence [syn: reasoning backward]
1 go back to a statistical means
2 go back to a previous state; "We reverted to the old rules" [syn: revert, return, retrovert, turn back]
3 get worse; fall back to a previous or worse condition [syn: retrograde, retrogress] [ant: progress]
4 go back to bad behavior; "Those who recidivate are often minor criminals" [syn: relapse, lapse, recidivate, retrogress, fall back]
- Rhymes: -ɛs
- The act of passing back; passage back; return; retrogression.
- What is the progress or regress of man?
- The power or liberty of passing back.
- To move backwards to an earlier stage; to devolve.
Regression analysis is a technique used for the modeling and analysis of numerical data consisting of values of a dependent variable (response variable) and of one or more independent variables (explanatory variables). The dependent variable in the regression equation is modeled as a function of the independent variables, corresponding parameters ("constants"), and an error term. The error term is treated as a random variable. It represents unexplained variation in the dependent variable. The parameters are estimated so as to give a "best fit" of the data. Most commonly the best fit is evaluated by using the least squares method, but other criteria have also been used.
Data modeling can be used without there being any knowledge about the underlying processes that have generated the data; in this case the model is an empirical model. Moreover, in modelling, knowledge of the probability distribution of the errors is not required. Regression analysis requires assumptions to be made regarding probability distribution of the errors. Statistical tests are made on the basis of these assumptions. In regression analysis the term "model" embraces both the function used to model the data and the assumptions concerning probability distributions.
Regression can be used for prediction (including forecasting of time-series data), inference, hypothesis testing, and modeling of causal relationships. These uses of regression rely heavily on the underlying assumptions being satisfied. Regression analysis has been criticized as being misused for these purposes in many cases where the appropriate assumptions cannot be verified to hold. One factor contributing to the misuse of regression is that it can take considerably more skill to critique a model than to fit a model.
History of regression analysisThe earliest form of regression was the method of least squares, which was published by Legendre in 1805, and by Gauss in 1809. The term “least squares” is from Legendre’s term, moindres carrés. However, Gauss claimed that he had known the method since 1795.
Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun. Euler had worked on the same problem (1748) without success. Gauss published a further development of the theory of least squares in 1821, including a version of the Gauss–Markov theorem.
The term "regression" was coined in the nineteenth century to describe a biological phenomenon, namely that the progeny of exceptional individuals tend on average to be less exceptional than their parents and more like their more distant ancestors. Francis Galton, a cousin of Charles Darwin, studied this phenomenon and applied the slightly misleading term "regression towards mediocrity" to it. For Galton, regression had only this biological meaning, but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context. Nowadays the term "regression" is often synonymous with "least squares curve fitting".
- The sample must be representative of the population for the inference prediction.
- The dependent variable is subject to error. This error is assumed to be a random variable, with a mean of zero. Systematic error may be present but its treatment is outside the scope of regression analysis.
- The independent variable is error-free. If this is not so, modeling should be done using errors-in-variables model techniques.
- The predictors must be linearly independent, i.e. it must not be possible to express any predictor as a linear combination of the others. See Multicollinearity.
- The errors are uncorrelated, that is, the variance-covariance matrix of the errors is diagonal and each non-zero element is the variance of the error.
- The variance of the error is constant (homoscedasticity). If not, weights should be used.
- The errors follow a normal distribution. If not, the generalized linear model should be used.
Linear regressionIn linear regression, the model specification is that the dependent variable, y_i is a linear combination of the parameters (but need not be linear in the independent variables). For example, in simple linear regression for modeling N data points there is one independent variable: x_i , and two parameters, \beta_0 and \beta_1:
- straight line: y_i=\beta_0 +\beta_1 x_i +\epsilon_i,\quad i=1,\dots,N\!
In multiple linear regression, there are several independent variables or functions of independent variables. For example, adding a term in xi2 to the preceding regression gives:
- parabola: y_i=\beta_0 +\beta_1 x_i +\beta_2 x_i^2+\epsilon_i,\ i=1,N\!
In both cases, \epsilon_i is an error term and the subscript i indexes a particular observation. Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model: y_i = \widehat_0 + \widehat_1 X_i + e_i The term e_i is the residual, e_i = y_i - \widehat_i . One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSE:
In the case of simple regression, the formulas for the least squares estimates are
- \widehat=\frac and \hat=\bar-\widehat\bar
- \hat\sigma_=\hat\sigma_ \sqrt
- \hat\sigma_=\hat\sigma_ \sqrt
General linear data modelIn the more general multiple regression model, there are p independent variables: y_i = \beta_0 + \beta_1 x_ + \cdots + \beta_p x_ + \varepsilon_i , The least square parameter estimates are obtained by p normal equations. The residual can be written as
- e_i=y_i - \hat\beta_0 - \hat\beta_1 x_1 - \cdots - \hat\beta_p x_p
- \sum_^\sum_^ X_X_\hat \beta_k=\sum_^ X_y_i,\ j=1,p\,
Regression diagnosticsOnce a regression model has been constructed, it is important to confirm the goodness of fit of the model and the statistical significance of the estimated parameters. Commonly used checks of goodness of fit include the R-squared, analyses of the pattern of residuals and hypothesis testing. Statistical significance is checked by an F-test of the overall fit, followed by t-tests of individual parameters.
Interpretations of these diagnostic tests rest heavily on the model assumptions. Although examination of the residuals can be used to invalidate a model, the results of a t-test or F-test are meaningless unless the modeling assumptions are satisfied.
- The error term may not have a normal distribution. See generalized linear model.
- The response variable may be non-continuous. For binary (zero or one) variables, there are the probit and logit model. The multivariate probit model makes it possible to estimate jointly the relationship between several binary dependent variables and some independent variables. For categorical variables with more than two values there is the multinomial logit. For ordinal variables with more than two values, there are the ordered logit and ordered probit models. An alternative to such procedures is linear regression based on polychoric or polyserial correlations between the categorical variables. Such procedures differ in the assumptions made about the distribution of the variables in the population. If the variable is positive with low values and represents the repetition of the occurrence of an event, count models like the Poisson regression or the negative binomial model may be used
Interpolation and extrapolationRegression models predict a value of the y variable given known values of the x variables. If the prediction is to be done within the range of values of the x variables used to construct the model this is known as interpolation. Prediction outside the range of the data used to construct the model is known as extrapolation and it is more risky.
When the model function is not linear in the parameters the sum of squares must be minimized by an iterative procedure. This introduces many complications which are summarized in Differences between linear and non-linear least squares
Other methodsAlthough the parameters of a regression model are usually estimated using the method of least squares, other methods which have been used include:
- Bayesian methods, e.g. Bayesian linear regression
- Minimization of absolute deviations, leading to quantile regression
- Nonparametric regression. This approach requires a large number of observations, as the data are used to build the model structure as well as estimate the model parameters. They are usually computationally intensive.
- Audi, R., Ed. (1996). "curve fitting problem," The Cambridge Dictionary of Philosophy. Cambridge, Cambridge University Press. pp.172-173.
- William H. Kruskal and Judith M. Tanur, ed. (1978), "Linear Hypotheses," International Encyclopedia of Statistics. Free Press, v. 1,
- Evan J. Williams, "I. Regression," pp. 523-41.
- Julian C. Stanley, "II. Analysis of Variance," pp. 541-554.
- Lindley, D.V. (1987). "Regression and correlation analysis," New Palgrave: A Dictionary of Economics, v. 4, pp. 120-23.
- Birkes, David and Yadolah Dodge, Alternative Methods of Regression. ISBN 0-471-56881-3
- Chatfield, C. (1993) "Calculating Interval Forecasts," Journal of Business and Economic Statistics, 11. pp. 121-135.
- Draper, N.R. and Smith, H. (1998).Applied Regression Analysis Wiley Series in Probability and Statistics
- Fox, J. (1997). Applied Regression Analysis, Linear Models and Related Methods. Sage
- Hardle, W., Applied Nonparametric Regression (1990), ISBN 0-521-42950-1
- Meade, N. and T. Islam (1995) "Prediction Intervals for Growth Curve Forecasts," Journal of Forecasting, 14, pp. 413-430.
- Munro, Barbara Hazard (2005) "Statistical Methods for Health Care Research" Lippincott Williams & Wilkins, 5th ed.
- Gujarati, Basic Econometrics, 4th edition
- Sykes, A.O. "An Introduction to Regression Analysis" (Innaugural Coase Lecture)
- S. Kotsiantis, D. Kanellopoulos, P. Pintelas, Local Additive Regression of Decision Stumps, Lecture Notes in Artificial Intelligence, Springer-Verlag, Vol. 3955, SETN 2006, pp. 148 – 157, 2006
- S. Kotsiantis, P. Pintelas, Selective Averaging of Regression Models, Annals of Mathematics, Computing & TeleInformatics, Vol 1, No 3, 2005, pp. 66-75
SoftwareAll major statistical software packages perform the common types of regression analysis correctly and in a user-friendly way. Simple linear regression can be done in some spreadsheet applications. There are a number of software programs that perform specialized forms of regression, and experts may choose to write their own code to using statistical programming languages or numerical analysis software.
- Regression Analysis
- Curvefit: A complete guide to nonlinear regression - Online textbook
- Regression Made Easy with RM4Es
- Exegeses on Linear Models - Some comments on linear regression models by Bill Venables.
- Regression of Weakly Correlated Data - How linear regression mistakes can appear when Y-range is much smaller than X-range
- xuru.org Online regression tools
- Matlab SUrrogate MOdeling Toolbox - SUMO Toolbox - Matlab code for Active Learning + Model Selection + Surrogate Model Regression
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