Dictionary Definition
regress
Noun
1 the reasoning involved when you assume the
conclusion is true and reason backward to the evidence [syn:
reasoning
backward]
Verb
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]
User Contributed Dictionary
English
Pronunciation

 Rhymes: ɛs
Noun
 The act of passing back; passage back; return; retrogression.
 What is the progress or regress of man?
 The power or liberty of passing back.
Derived terms
Verb
 To move backwards to an earlier stage; to devolve.
Synonyms
Declension
References
Extensive Definition
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 timeseries
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 analysis
The 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".
Underlying assumptions
 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 errorfree. If this is not so, modeling should be done using errorsinvariables 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 variancecovariance matrix of the errors is diagonal and each nonzero 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 regression
In 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:
 SSE=\sum_^e_i^2
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 model
In 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\,
 \mathbf
Regression diagnostics
Once 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 Rsquared, analyses of the pattern of residuals and hypothesis testing. Statistical significance is checked by an Ftest of the overall fit, followed by ttests 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
ttest or
Ftest 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 noncontinuous. 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 extrapolation
Regression 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.Nonlinear regression
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 nonlinear least squares
Other methods
Although 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.
See also
 Segmented regression
 Confidence interval
 Confidence region
 Extrapolation
 Kriging (a linear least squares estimation algorithm)
 Forecasting
 Prediction interval
 Statistics
 Trend estimation
 Robust regression
 Multivariate normal distribution
 Important publications in regression analysis.
References
 Audi, R., Ed. (1996). "curve fitting problem," The Cambridge Dictionary of Philosophy. Cambridge, Cambridge University Press. pp.172173.
 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. 52341.
 Julian C. Stanley, "II. Analysis of Variance," pp. 541554.
 Lindley, D.V. (1987). "Regression and correlation analysis," New Palgrave: A Dictionary of Economics, v. 4, pp. 12023.
 Birkes, David and Yadolah Dodge, Alternative Methods of Regression. ISBN 0471568813
 Chatfield, C. (1993) "Calculating Interval Forecasts," Journal of Business and Economic Statistics, 11. pp. 121135.
 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 0521429501
 Meade, N. and T. Islam (1995) "Prediction Intervals for Growth Curve Forecasts," Journal of Forecasting, 14, pp. 413430.
 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, SpringerVerlag, 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. 6675
Software
All major statistical software packages perform the common types of regression analysis correctly and in a userfriendly 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.External links
 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 Yrange is much smaller than Xrange
 xuru.org Online regression tools
 Matlab SUrrogate MOdeling Toolbox  SUMO Toolbox  Matlab code for Active Learning + Model Selection + Surrogate Model Regression
regress in Bulgarian: Регресионен анализ
regress in Czech: Regresní analýza
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regress in Portuguese: Regressão
regress in Russian: Регрессионный анализ
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regress in Chinese: 迴歸分析
Synonyms, Antonyms and Related Words
aboutface, advance, ascend, back, back up, backing, backset, backslide, backsliding, backward
motion, backward step, budge, change, change place, circle, climb, cock, come after, come last,
degenerate, descend, deteriorate, disenchantment, disimprove, ebb, fall, fall again into, fall astern,
fall back, fall behind, fall from grace, flipflop, flow, follow, get behind, get over, get
worse, go, go around, go
back, go backwards, go behind, go round, go sideways, grow worse,
gyrate, have a relapse,
jerk back, lag behind, lapse, lapse back, let down, lose
ground, mount, move, move over, plunge, progress, pull back, reaction, recede, recession, recidivate, recidivation, recidivism, reclamation, reconversion, recur to,
reentry, refluence, reflux, regression, rehabilitation, reinstatement, relapse, restitution, restoration, retroaction, retrocede, retrocession, retroflex, retroflexion, retrogradation, retrograde, retrogress, retrogression, retroversion, retrovert, retrusion, return, return to, returning, reversal, reverse, reversion, revert, revert to, reverting, revulsion, rise, rollback, rotate, run, setback, shift, sicken, sink, sink back, slacken, slide back, slip back,
slipping back, soar,
spin, sternway, stir, stream, subside, throw back, throwback, trail, trail behind, travel, turn, turnabout, wane, whirl, worsen, yield again
to