Regression analysis

Short description: Set of statistical processes for estimating the relationships among variables

Regression line for 50 random points in a Gaussian distribution around the line y=1.5x+2 Part of a series on

Regression analysis
Models

* Linear regression
* Simple regression
* Polynomial regression
* General linear model

* Generalized linear model
* Discrete choice
* Binomial regression
* Binary regression
* Logistic regression
* Multinomial logit
* Mixed logit
* Probit
* Multinomial probit
* Ordered logit
* Ordered probit
* Poisson

* Multilevel model
* Fixed effects
* Random effects
* Linear mixed-effects model
* Nonlinear mixed-effects model

* Nonlinear regression
* Nonparametric
* Semiparametric
* Robust
* Quantile
* Isotonic
* Principal components
* Least angle
* Local
* Segmented

* Errors-in-variables

Estimation

* Least squares
* Linear
* Non-linear

* Ordinary
* Weighted
* Generalized

* Partial
* Total
* Non-negative
* Ridge regression
* Regularized

* Least absolute deviations
* Iteratively reweighted
* Bayesian
* Bayesian multivariate

Background

* Regression validation
* Mean and predicted response
* Errors and residuals
* Goodness of fit
* Studentized residual
* Gauss–Markov theorem

* v
* t
* e

Machine learning and
data mining

Problems

* Classification
* Clustering
* Regression
* Anomaly detection
* AutoML
* Association rules
* Reinforcement learning
* Structured prediction
* Feature engineering
* Feature learning
* Online learning
* Semi-supervised learning
* Unsupervised learning
* Learning to rank
* Grammar induction

Supervised learning
(classification • regression)

* Decision trees
* Ensembles
* Bagging
* Boosting
* Random forest
* k-NN
* Linear regression
* Naive Bayes
* Artificial neural networks
* Logistic regression
* Perceptron
* Relevance vector machine (RVM)
* Support vector machine (SVM)

Clustering

* BIRCH
* CURE
* Hierarchical
* k-means
* Expectation–maximization (EM)

DBSCAN
* OPTICS
* Mean-shift

Dimensionality reduction

* Factor analysis
* CCA
* ICA
* LDA
* NMF
* PCA
* t-SNE

Structured prediction

* Graphical models
* Bayes net
* Conditional random field
* Hidden Markov

Anomaly detection

* k-NN
* Local outlier factor

Artificial neural networks

* Autoencoder
* Deep learning
* DeepDream
* Multilayer perceptron
* RNN
* LSTM
* GRU
* Restricted Boltzmann machine
* GAN
* SOM
* Convolutional neural network
* U-Net

Reinforcement learning

* Q-learning
* SARSA
* Temporal difference (TD)

Theory

* Bias–variance dilemma
* Computational learning theory
* Empirical risk minimization
* Occam learning
* PAC learning
* Statistical learning
* VC theory

Machine-learning venues

* NeurIPS
* ICML
* ML
* JMLR
* ArXiv:cs.LG

Glossary of artificial intelligence

* Glossary of artificial intelligence

Related articles

* List of datasets for machine-learning research
* Outline of machine learning

* v
* t
* e

In statistical modeling, regression analysis is a statistical method for estimating the relationship between a dependent variable (often called the outcome or response variable, or a label in machine learning parlance) and one or more independent variables (often called regressors, predictors, covariates, explanatory variables or features).[1][2]

The most common form of regression analysis is linear regression, in which one finds the line (or a more complex linear combination) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression), this allows the researcher to estimate the conditional expectation (or population average value) of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters (e.g., quantile regression or Necessary Condition Analysis[3]) or estimate the conditional expectation across a broader collection of non-linear models (e.g., nonparametric regression).

Regression analysis is primarily used for two conceptually distinct purposes. First, regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. Second, in some situations regression analysis can be used to infer causal relationships between the independent and dependent variables. Importantly, regressions by themselves only reveal relationships between a dependent variable and a collection of independent variables in a fixed dataset. To use regressions for prediction or to infer causal relationships, respectively, a researcher must carefully justify why existing relationships have predictive power for a new context or why a relationship between two variables has a causal interpretation. The latter is especially important when researchers hope to estimate causal relationships using observational data.[4][5]

## Contents

* 1 History
* 2 Regression model
* 3 Underlying assumptions
* 4 Linear regression
* 4.1 General linear model
* 4.2 Diagnostics
* 4.3 Limited dependent variables
* 5 Nonlinear regression
* 6 Prediction (interpolation and extrapolation)
* 6.1 Model selection
* 7 Power and sample size calculations
* 8 Other methods
* 9 Software
* 10 See also
* 11 References
* 12 Further reading
* 13 External links

## History

The earliest regression form was seen in Isaac Newton's work in 1700 while studying equinoxes, being credited with introducing "an embryonic linear regression analysis" as "not only did he perform the averaging of a set of data, 50 years before Tobias Mayer, but by summing the residuals to zero he forced the regression line to pass through the average point. He also distinguished between two inhomogeneous sets of data and might have thought of an optimal solution in terms of bias, though not in terms of effectiveness."[6] He previously used an averaging method in his 1671 work on Newton's rings, which was unprecedented at the time.[7]

The method of least squares was published by Legendre in 1805,[8] and by Gauss in 1809.[9] Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun (mostly comets, but also later the then newly discovered minor planets). Gauss published a further development of the theory of least squares in 1821,[10] including a version of the Gauss–Markov theorem.

The term "regression" was coined by Francis Galton in the 19th century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average (a phenomenon also known as regression toward the mean).[11][12] For Galton, regression had only this biological meaning,[13][14] but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context.[15][16] In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925.[17][18][19] Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.

In the 1950s and 1960s, economists used electromechanical desk calculators to calculate regressions. Before 1970, it sometimes took up to 24 hours to receive the result from one regression.[20]

Regression methods continue to be an area of active research. In recent decades, new methods have been developed for robust regression, regression involving correlated responses such as time series and growth curves, regression in which the predictor (independent variable) or response variables are curves, images, graphs, or other complex data objects, regression methods accommodating various types of missing data, nonparametric regression, Bayesian methods for regression, regression in which the predictor variables are measured with error, regression with more predictor variables than observations, and causal inference with regression. Modern regression analysis is typically done with statistical and spreadsheet software packages on computers as well as on handheld scientific and graphing calculators.

## Regression model

In practice, researchers first select a model they would like to estimate and then use their chosen method (e.g., ordinary least squares) to estimate the parameters of that model. Regression models involve the following components:

* The unknown parameters, often denoted as a scalar or vector β.
* The independent variables, which are observed in data and are often denoted as a vector Xi (where i denotes a row of data).
* The dependent variable, which are observed in data and often denoted using the scalar Yi.
* The error terms, which are not directly observed in data and are often denoted using the scalar ei.

In various fields of application, different terminologies are used in place of dependent and independent variables.

Most regression models propose that Yi is a function (regression function) of Xi and β, with ei representing an additive error term that may stand in for un-modeled determinants of Yi or random statistical noise:

Yi=f(Xi,β)+ei

In the standard regression model, the independent variables Xi are assumed to be free of error. The errors-in-variables model can be used if the independent variables are assumed to contain errors. Other modifications to the standard regression model can be made to account for various scenarios, such as situations involving omitted variables, confounding variables or endogeneity.

The researchers' goal is to estimate the function f(Xi,β) that most closely fits the data. To carry out regression analysis, the form of the function f must be specified. Sometimes the form of this function is based on knowledge about the relationship between Yi and Xi that does not rely on the data. If no such knowledge is available, a flexible or convenient form for f is chosen. For example, a simple univariate regression may propose f(Xi,β)=β0+β1Xi, suggesting that the researcher believes Yi=β0+β1Xi+ei to be a reasonable approximation for the statistical process generating the data.

Once researchers determine their preferred statistical model, different forms of regression analysis provide tools to estimate the parameters β. For example, least squares (including its most common variant, ordinary least squares) finds the value of β that minimizes the sum of squared errors ∑i(Yi−f(Xi,β))2. A given regression method will ultimately provide an estimate of β, usually denoted β^ to distinguish the estimate from the true (unknown) parameter value that generated the data. Using this estimate, the researcher can then use the fitted value Yi^=f(Xi,β^) for prediction or to assess the accuracy of the model in explaining the data. Whether the researcher is intrinsically interested in the estimate β^ or the predicted value Yi^ will depend on context and their goals. As described in ordinary least squares, least squares is widely used because the estimated function f(Xi,β^) approximates the conditional expectation E(Yi|Xi).[9] However, alternative variants (e.g., least absolute deviations or quantile regression) are useful when researchers want to model other functions f(Xi,β).

It is important to note that there must be sufficient data to estimate a regression model. For example, suppose that a researcher has access to N rows of data with one dependent and two independent variables: (Yi,X1i,X2i). Suppose further that the researcher wants to estimate a bivariate linear model via least squares: Yi=β0+β1X1i+β2X2i+ei. If the researcher only has access to N=2 data points, then they could find infinitely many combinations (β^0,β^1,β^2) that explain the data equally well: any combination can be chosen that satisfies Y^i=β^0+β^1X1i+β^2X2i, all of which lead to ∑ie^i2=∑i(Y^i−(β^0+β^1X1i+β^2X2i))2=0 and are therefore valid solutions that minimize the sum of squared residuals. To understand why there are infinitely many options, note that the system of N=2 equations is to be solved for 3 unknowns, which makes the system underdetermined. Alternatively, one can visualize infinitely many 3-dimensional planes that go through N=2 fixed points.

More generally, to estimate a least squares model with k distinct parameters, one must have N≥k distinct data points. If N>k, then there does not generally exist a set of parameters that will perfectly fit the data. The quantity N−k appears often in regression analysis, and is referred to as the degrees of freedom in the model. Moreover, to estimate a least squares model, the independent variables (X1i,X2i,...,Xki) must be linearly independent: one must not be able to reconstruct any of the independent variables by adding and multiplying the remaining independent variables. As discussed in ordinary least squares, this condition ensures that XTX is an invertible matrix and therefore that a unique solution β^ exists.

## Underlying assumptions

By itself, a regression is just a computation performed on a set of data. In order to interpret the resultant regression as a meaningful statistical model that quantifies real-world relationships, researchers often rely on a number of classical assumptions. These assumptions often include:

* The sample is representative of the population at large.
* The independent variables are measured without error.
* Deviations from the model have an expected value of zero, conditional on covariates: E(ei|Xi)=0
* The variance of the residuals ei is constant across observations (homoscedasticity).
* The residuals ei are uncorrelated with one another. Mathematically, the variance–covariance matrix of the errors is diagonal.

A handful of conditions are sufficient for the least-squares estimator to possess desirable properties: in particular, the Gauss–Markov assumptions imply that the parameter estimates will be unbiased, consistent, and efficient in the class of linear unbiased estimators. Practitioners have developed a variety of methods to maintain some or all of these desirable properties in real-world settings, because these classical assumptions are unlikely to hold exactly. For example, modeling errors-in-variables can lead to reasonable estimates independent variables are measured with errors. Heteroscedasticity-consistent standard errors allow the variance of ei to change across values of Xi. Correlated errors that exist within subsets of the data or follow specific patterns can be handled using clustered standard errors, geographic weighted regression, or Newey–West standard errors, among other techniques. When rows of data correspond to locations in space, the choice of how to model ei within geographic units can have important consequences.[21][22] The subfield of econometrics is largely focused on developing techniques that allow researchers to make reasonable real-world conclusions in real-world settings, where classical assumptions do not hold exactly.

## Linear regression

Main article: Linear regression

See simple linear regression for a derivation of these formulas and a numerical example

In linear regression, the model specification is that the dependent variable, yi 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: xi, and two parameters, β0 and β1:

straight line: yi=β0+β1xi+εi,i=1,…,n.

In multiple linear regression, there are several independent variables or functions of independent variables.

Adding a term in xi2 to the preceding regression gives:

parabola: yi=β0+β1xi+β2xi2+εi, i=1,…,n.

This is still linear regression; although the expression on the right hand side is quadratic in the independent variable xi, it is linear in the parameters β0, β1 and β2.

In both cases, εi is an error term and the subscript i indexes a particular observation.

Returning our attention to the straight line case: Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

y^i=β^0+β^1xi.

The residual, ei=yi−y^i, is the difference between the value of the dependent variable predicted by the model, y^i, and the true value of the dependent variable, yi. One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSR:

SSR=∑i=1nei2

Minimization of this function results in a set of normal equations, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators, β^0,β^1.

Illustration of linear regression on a data set

In the case of simple regression, the formulas for the least squares estimates are

β^1=∑(xi−x¯)(yi−y¯)∑(xi−x¯)2
β^0=y¯−β^1x¯

where x¯ is the mean (average) of the x values and y¯ is the mean of the y values.

Under the assumption that the population error term has a constant variance, the estimate of that variance is given by:

σ^ε2=SSRn−2

This is called the mean square error (MSE) of the regression. The denominator is the sample size reduced by the number of model parameters estimated from the same data, (n−p) for p regressors or (n−p−1) if an intercept is used.[23] In this case, p=1 so the denominator is n−2.

The standard errors of the parameter estimates are given by

σ^β1=σ^ε1∑(xi−x¯)2

σ^β0=σ^ε1n+x¯2∑(xi−x¯)2=σ^β1∑xi2n.

Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create confidence intervals and conduct hypothesis tests about the population parameters.

### General linear model

For a derivation, see linear least squares

For a numerical example, see linear regression

In the more general multiple regression model, there are p independent variables:

yi=β1xi1+β2xi2+⋯+βpxip+εi,

where xij is the i-th observation on the j-th independent variable. If the first independent variable takes the value 1 for all i, xi1=1, then β1 is called the regression intercept.

The least squares parameter estimates are obtained from p normal equations. The residual can be written as

εi=yi−β^1xi1−⋯−β^pxip.

The normal equations are

∑i=1n∑k=1pxijxikβ^k=∑i=1nxijyi, j=1,…,p.

In matrix notation, the normal equations are written as

(𝐗⊤𝐗)β^=𝐗⊤𝐘,

where the ij element of 𝐗 is xij, the i element of the column vector Y is yi, and the j element of β^ is β^j. Thus 𝐗 is n×p, Y is n×1, and β^ is p×1. The solution is

β^=(𝐗⊤𝐗)−𝟏𝐗⊤𝐘.

### Diagnostics

See also: Category:Regression diagnostics

Once a regression model has been constructed, it may be 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 can be 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's assumptions. Although examination of the residuals can be used to invalidate a model, the results of a t-test or F-test are sometimes more difficult to interpret if the model's assumptions are violated. For example, if the error term does not have a normal distribution, in small samples the estimated parameters will not follow normal distributions and complicate inference. With relatively large samples, however, a central limit theorem can be invoked such that hypothesis testing may proceed using asymptotic approximations.

### Limited dependent variables

Limited dependent variables, which are response variables that are categorical or constrained to fall only in a certain range, often arise in econometrics.

The response variable may be non-continuous ("limited" to lie on some subset of the real line). For binary (zero or one) variables, if analysis proceeds with least-squares linear regression, the model is called the linear probability model. Nonlinear models for binary dependent variables include the probit and logit model. The multivariate probit model is a standard method of estimating a joint 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. Censored regression models may be used when the dependent variable is only sometimes observed, and Heckman correction type models may be used when the sample is not randomly selected from the population of interest.

An alternative to such procedures is linear regression based on polychoric correlation (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, then count models like the Poisson regression or the negative binomial model may be used.

## Nonlinear regression

Main article: 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 non-linear least squares.

## Prediction (interpolation and extrapolation)

In the middle, the fitted straight line represents the best balance between the points above and below this line. The dotted straight lines represent the two extreme lines, considering only the variation in the slope. The inner curves represent the estimated range of values considering the variation in both slope and intercept. The outer curves represent a prediction for a new measurement.[24]

Regression models predict a value of the Y variable given known values of the X variables. Prediction within the range of values in the dataset used for model-fitting is known informally as interpolation. Prediction outside this range of the data is known as extrapolation. Performing extrapolation relies strongly on the regression assumptions. The further the extrapolation goes outside the data, the more room there is for the model to fail due to differences between the assumptions and the sample data or the true values.

A prediction interval that represents the uncertainty may accompany the point prediction. Such intervals tend to expand rapidly as the values of the independent variable(s) moved outside the range covered by the observed data.

For such reasons and others, some tend to say that it might be unwise to undertake extrapolation.[25]

### Model selection

The assumption of a particular form for the relation between Y and X is another source of uncertainty. A properly conducted regression analysis will include an assessment of how well the assumed form is matched by the observed data, but it can only do so within the range of values of the independent variables actually available. This means that any extrapolation is particularly reliant on the assumptions being made about the structural form of the regression relationship. If this knowledge includes the fact that the dependent variable cannot go outside a certain range of values, this can be made use of in selecting the model – even if the observed dataset has no values particularly near such bounds. The implications of this step of choosing an appropriate functional form for the regression can be great when extrapolation is considered. At a minimum, it can ensure that any extrapolation arising from a fitted model is "realistic" (or in accord with what is known).

## Power and sample size calculations

There are no generally agreed methods for relating the number of observations versus the number of independent variables in the model. One method conjectured by Good and Hardin is N=mn, where N is the sample size, n is the number of independent variables and m is the number of observations needed to reach the desired precision if the model had only one independent variable.[26] For example, a researcher is building a linear regression model using a dataset that contains 1000 patients (N). If the researcher decides that five observations are needed to precisely define a straight line (m), then the maximum number of independent variables (n) the model can support is 4, because

log⁡1000log⁡5≈4.29.

## 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
* Percentage regression, for situations where reducing percentage errors is deemed more appropriate.[27]
* Least absolute deviations, which is more robust in the presence of outliers, leading to quantile regression
* Nonparametric regression, requires a large number of observations and is computationally intensive
* Scenario optimization, leading to interval predictor models

## Software

For a more comprehensive list, see Software:List of statistical software.

All major statistical software packages perform least squares regression analysis and inference. Simple linear regression and multiple regression using least squares can be done in some spreadsheet applications and on some calculators. While many statistical software packages can perform various types of nonparametric and robust regression, these methods are less standardized. Different software packages implement different methods, and a method with a given name may be implemented differently in different packages. Specialized regression software has been developed for use in fields such as survey analysis and neuroimaging.

## See also

* Anscombe's quartet
* Curve fitting
* Estimation theory
* Forecasting
* Fraction of variance unexplained
* Function approximation
* Generalized linear model
* Kriging (a linear least squares estimation algorithm)
* Local regression
* Modifiable areal unit problem
* Multivariate adaptive regression spline
* Multivariate normal distribution
* Pearson correlation coefficient
* Quasi-variance
* Prediction interval
* Regression validation
* Robust regression
* Segmented regression
* Signal processing
* Stepwise regression
* Taxicab geometry
* Linear trend estimation

## References

1. ↑ Yan, Xin; Su, Xiaogang (2009). Linear Regression Analysis: Theory and Computing. World Scientific Publishing. pp. 2–3. ISBN 9789812834102. https://books.google.com/books?id=5pdpDQAAQBAJ.
2. ↑ Freund, Rudolf J.; Mohr, Donna L.; Wilson, William J. (2010). Statistical Methods. Elsevier Science. p. 323. ISBN 9780080961033. https://books.google.com/books?id=12LPWl6QxrsC.
3. ↑ Necessary Condition Analysis
4. ↑ David A. Freedman (27 April 2009). Statistical Models: Theory and Practice. Cambridge University Press. ISBN 978-1-139-47731-4. https://books.google.com/books?id=fW_9BV5Wpf8C&q="regression+analysis".
5. ↑ R. Dennis Cook; Sanford Weisberg Criticism and Influence Analysis in Regression, Sociological Methodology, Vol. 13. (1982), pp. 313–361
6. ↑ Belenkiy, Ari; Echague, Eduardo Vila (2008). "Groping Toward Linear Regression Analysis: Newton's Analysis of Hipparchus' Equinox Observations". arXiv:0810.4948 [physics.hist-ph].
7. ↑ Buchwald, Jed Z.; Feingold, Mordechai (2013). Newton and the Origin of Civilization. Princeton University Press. pp. 90–93, 101–103. ISBN 978-0-691-15478-7.
8. ↑ A.M. Legendre. Nouvelles méthodes pour la détermination des orbites des comètes, Firmin Didot, Paris, 1805. "Sur la Méthode des moindres quarrés" appears as an appendix.
9. ↑ 9.0 9.1 Chapter 1 of: Angrist, J. D., & Pischke, J. S. (2008). Mostly Harmless Econometrics: An Empiricist's Companion. Princeton University Press.
10. ↑ Theoria combinationis observationum erroribus minimis obnoxiae. 1821–1823. https://books.google.com/books?id=ZQ8OAAAAQAAJ&q=Theoria+combinationis+observationum+erroribus+minimis+obnoxiae.
11. ↑ Mogull, Robert G. (2004). Second-Semester Applied Statistics. Kendall/Hunt Publishing Company. p. 59. ISBN 978-0-7575-1181-3.
12. ↑ Galton, Francis (1989). "Kinship and Correlation (reprinted 1989)". Statistical Science 4 (2): 80–86. doi:10.1214/ss/1177012581.
13. ↑ Francis Galton. "Typical laws of heredity", Nature 15 (1877), 492–495, 512–514, 532–533. (Galton uses the term "reversion" in this paper, which discusses the size of peas.)
14. ↑ Francis Galton. Presidential address, Section H, Anthropology. (1885) (Galton uses the term "regression" in this paper, which discusses the height of humans.)
15. ↑ Yule, G. Udny (1897). "On the Theory of Correlation". Journal of the Royal Statistical Society 60 (4): 812-54. doi:10.2307/2979746. https://zenodo.org/record/1449703.
16. ↑ Pearson, Karl; Yule, G.U.; Blanchard, Norman; Lee, Alice (1903). "The Law of Ancestral Heredity". Biometrika 2 (2): 211–236. doi:10.1093/biomet/2.2.211. https://zenodo.org/record/1431601.
17. ↑ Fisher, R.A. (1922). "The goodness of fit of regression formulae, and the distribution of regression coefficients". Journal of the Royal Statistical Society 85 (4): 597–612. doi:10.2307/2341124.
18. ↑ Ronald A. Fisher (1970). Statistical Methods for Research Workers (Twelfth ed.). Edinburgh: Oliver and Boyd. ISBN 978-0-05-002170-5. https://archive.org/details/dli.scoerat.2986statisticalmethodsforresearchworkers/page/n7/mode/2up.
19. ↑ Aldrich, John (2005). "Fisher and Regression". Statistical Science 20 (4): 401-417. doi:10.1214/088342305000000331. https://eprints.soton.ac.uk/34871/1/088342305000000331.pdf.
20. ↑ Rodney Ramcharan. Regressions: Why Are Economists Obessessed with Them? March 2006. Accessed 2011-12-03.
21. ↑ Fotheringham, A. Stewart; Brunsdon, Chris; Charlton, Martin (2002). Geographically weighted regression: the analysis of spatially varying relationships (Reprint ed.). Chichester, England: John Wiley. ISBN 978-0-471-49616-8.
22. ↑ Fotheringham, AS; Wong, DWS (1 January 1991). "The modifiable areal unit problem in multivariate statistical analysis". Environment and Planning A 23 (7): 1025–1044. doi:10.1068/a231025. Bibcode: 1991EnPlA..23.1025F.
23. ↑ Steel, R.G.D, and Torrie, J. H., Principles and Procedures of Statistics with Special Reference to the Biological Sciences., McGraw Hill, 1960, page 288.
24. ↑ Rouaud, Mathieu (2013). Probability, Statistics and Estimation. p. 60. http://www.incertitudes.fr/book.pdf.
25. ↑ Chiang, C.L, (2003) Statistical methods of analysis, World Scientific. ISBN 981-238-310-7 - page 274 section 9.7.4 "interpolation vs extrapolation"
26. ↑ Good, P. I.; Hardin, J. W. (2009). Common Errors in Statistics (And How to Avoid Them) (3rd ed.). Hoboken, New Jersey: Wiley. p. 211. ISBN 978-0-470-45798-6.
27. ↑ Tofallis, C. (2009). "Least Squares Percentage Regression". Journal of Modern Applied Statistical Methods 7: 526–534. doi:10.2139/ssrn.1406472. https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1466&context=jmasm.

## Further reading

* 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 Dodge, Y., 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.; Smith, H. (1998). Applied Regression Analysis (3rd ed.). John Wiley. ISBN 978-0-471-17082-2.
* Fox, J. (1997). Applied Regression Analysis, Linear Models and Related Methods. Sage
* Hardle, W., Applied Nonparametric Regression (1990), ISBN 0-521-42950-1
* Meade, Nigel; Islam, Towhidul (1995). "Prediction intervals for growth curve forecasts". Journal of Forecasting 14 (5): 413–430. doi:10.1002/for.3980140502.
* A. Sen, M. Srivastava, Regression Analysis -- Theory, Methods, and Applications, Springer-Verlag, Berlin, 2011 (4th printing).
* T. Strutz: Data Fitting and Uncertainty (A practical introduction to weighted least squares and beyond). Vieweg+Teubner, ISBN 978-3-8348-1022-9.
* Stulp, Freek, and Olivier Sigaud. Many Regression Algorithms, One Unified Model: A Review. Neural Networks, vol. 69, Sept. 2015, pp. 60–79. https://doi.org/10.1016/j.neunet.2015.05.005.
* Malakooti, B. (2013). Operations and Production Systems with Multiple Objectives. John Wiley & Sons.
* Chicco, Davide; Warrens, Matthijs J.; Jurman, Giuseppe (2021). "The coefficient of determination R-squared is more informative than SMAPE, MAE, MAPE, MSE and RMSE in regression analysis evaluation". PeerJ Computer Science 7 (e623): e623. doi:10.7717/peerj-cs.623. PMID 34307865.

## External links

* Hazewinkel, Michiel, ed. (2001), "Regression analysis", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/r080620
* Earliest Uses: Regression – basic history and references
* What is multiple regression used for? – Multiple regression
* Regression of Weakly Correlated Data – how linear regression mistakes can appear when Y-range is much smaller than X-range

* v
* t
* e

Artificial intelligence (AI)

* History
* timeline
* Glossary
* Companies
* Projects

Concepts|

* Parameter
* Hyperparameter
* Loss functions
* Regression
* Bias–variance tradeoff
* Double descent
* Overfitting
* Clustering
* Gradient descent
* SGD
* Quasi-Newton method
* Conjugate gradient method
* Backpropagation
* Attention
* Convolution
* Normalization
* Batchnorm
* Activation
* Softmax
* Sigmoid
* Rectifier
* Gating
* Weight initialization
* Regularization
* Datasets
* Augmentation
* Prompt engineering
* Reinforcement learning
* Q-learning
* SARSA
* Imitation
* Policy gradient
* Diffusion
* Latent diffusion model
* Autoregression
* Adversary
* RAG
* Uncanny valley
* RLHF
* Self-supervised learning
* Reflection
* Recursive self-improvement
* Hallucination
* Word embedding
* Vibe coding

Applications|

* Machine learning
* In-context learning
* Artificial neural network
* Deep learning
* Language model
* Large
* NMT
* Reasoning
* Model Context Protocol
* Intelligent agent
* Artificial human companion
* Humanity's Last Exam
* Lethal autonomous weapons (LAWs)
* Generative artificial intelligence (GenAI)
* (Hypothetical: Artificial general intelligence (AGI))
* (Hypothetical: Artificial superintelligence (ASI))
* Agent2Agent protocol

Implementations|

| Audio–visual|

* AlexNet
* WaveNet
* Human image synthesis
* HWR
* OCR
* Computer vision
* Speech synthesis
* 15.ai
* ElevenLabs
* Speech recognition
* Whisper
* Facial recognition
* AlphaFold
* Text-to-image models
* Aurora
* DALL-E
* Firefly
* Flux
* GPT Image
* Ideogram
* Imagen
* Midjourney
* Recraft
* Stable Diffusion
* Text-to-video models
* Dream Machine
* Runway Gen
* Hailuo AI
* Kling
* Sora
* Seedance
* Veo
* Music generation
* Riffusion
* Suno AI
* Udio

|

Text|

* Word2vec
* Seq2seq
* GloVe
* BERT
* T5
* Llama
* Chinchilla AI
* PaLM
* GPT
* 1
* 2
* 3
* J
* ChatGPT
* 4
* 4o
* o1
* o3
* 4.5
* 4.1
* o4-mini
* 5
* 5.1
* 5.2
* Claude
* Gemini
* Gemini (language model)
* Gemma
* Grok
* LaMDA
* BLOOM
* DBRX
* Project Debater
* IBM Watson
* IBM Watsonx
* Granite
* PanGu-Σ
* DeepSeek
* Qwen

Decisional|

* AlphaGo
* AlphaZero
* OpenAI Five
* Self-driving car
* MuZero
* Action selection
* AutoGPT
* Robot control

People|

* Alan Turing
* Warren Sturgis McCulloch
* Walter Pitts
* John von Neumann
* Christopher D. Manning
* Claude Shannon
* Shun'ichi Amari
* Kunihiko Fukushima
* Takeo Kanade
* Marvin Minsky
* John McCarthy
* Nathaniel Rochester
* Allen Newell
* Cliff Shaw
* Herbert A. Simon
* Oliver Selfridge
* Frank Rosenblatt
* Bernard Widrow
* Joseph Weizenbaum
* Seymour Papert
* Seppo Linnainmaa
* Paul Werbos
* Geoffrey Hinton
* John Hopfield
* Jürgen Schmidhuber
* Yann LeCun
* Yoshua Bengio
* Lotfi A. Zadeh
* Stephen Grossberg
* Alex Graves
* James Goodnight
* Andrew Ng
* Fei-Fei Li
* Alex Krizhevsky
* Ilya Sutskever
* Oriol Vinyals
* Quoc V. Le
* Ian Goodfellow
* Demis Hassabis
* David Silver
* Andrej Karpathy
* Ashish Vaswani
* Noam Shazeer
* Aidan Gomez
* John Schulman
* Mustafa Suleyman
* Jan Leike
* Daniel Kokotajlo
* François Chollet

Architectures|

* Neural Turing machine
* Differentiable neural computer
* Transformer
* Vision transformer (ViT)
* Recurrent neural network (RNN)
* Long short-term memory (LSTM)
* Gated recurrent unit (GRU)
* Echo state network
* Multilayer perceptron (MLP)
* Convolutional neural network (CNN)
* Residual neural network (RNN)
* Highway network
* Mamba
* Autoencoder
* Variational autoencoder (VAE)
* Generative adversarial network (GAN)
* Graph neural network (GNN)

Political|

* AI safety (Alignment)
* Ethics of AI
* EU AI Act
* Precautionary principle
* Regulation of AI
* Virtual politician

Social and economic|

* AI boom
* AI bubble
* AI literacy
* AI slop
* AI veganism
* AI winter
* Anthropomorphism
* In architecture
* In education
* In healthcare
* Chatbot psychosis
* Mental health
* In visual art

* Category

* v
* t
* e

Least squares and regression analysis

Computational statistics|

* Least squares
* Linear least squares
* Non-linear least squares
* Iteratively reweighted least squares

Correlation and dependence|

* Pearson product-moment correlation
* Rank correlation (Spearman's rho
* Kendall's tau)
* Partial correlation
* Confounding variable

Regression analysis|

* Ordinary least squares
* Partial least squares
* Total least squares
* Ridge regression

Regression as a
statistical model|

| Linear regression|

* Simple linear regression
* Ordinary least squares
* Generalized least squares
* Weighted least squares
* General linear model

|

Predictor structure|

* Polynomial regression
* Growth curve (statistics)
* Segmented regression
* Local regression

Non-standard|

* Nonlinear regression
* Nonparametric
* Semiparametric
* Robust
* Quantile
* Isotonic

Non-normal errors|

* Generalized linear model
* Binomial
* Poisson
* Logistic

Decomposition of variance|

* Analysis of variance
* Analysis of covariance
* Multivariate AOV

Model exploration|

* Stepwise regression
* Model selection
* Mallows's Cp
* AIC
* BIC
* Model specification
* Regression validation

Background|

* Mean and predicted response
* Gauss–Markov theorem
* Errors and residuals
* Goodness of fit
* Studentized residual
* Minimum mean-square error

Design of experiments|

* Response surface methodology
* Optimal design
* Bayesian design

Numerical approximation|

* Numerical analysis
* Approximation theory
* Numerical integration
* Gaussian quadrature
* Orthogonal polynomials
* Chebyshev polynomials
* Chebyshev nodes

Applications|

* Curve fitting
* Calibration curve
* Numerical smoothing and differentiation
* System identification
* Moving least squares

* Regression analysis category
* Statistics category
*   * Statistics outline
* Statistics topics

* v
* t
* e

Statistics

* Outline
* Index

Descriptive statistics

| Continuous data|

| Center|

* Mean
* arithmetic
* geometric
* harmonic
* Median
* Mode

|

Dispersion|

* Variance
* Standard deviation
* Coefficient of variation
* Percentile
* Range
* Interquartile range

Shape|

* Central limit theorem
* Moments
* Skewness
* Kurtosis
* L-moments

Count data|

* Index of dispersion

Summary tables|

* Grouped data
* Frequency distribution
* Contingency table

Dependence|

* Pearson product-moment correlation
* Rank correlation
* Spearman's rho
* Kendall's tau
* Partial correlation
* Scatter plot

Graphics|

* Bar chart
* Biplot
* Box plot
* Control chart
* Correlogram
* Fan chart
* Forest plot
* Histogram
* Pie chart
* Q–Q plot
* Run chart
* Scatter plot
* Stem-and-leaf display
* Radar chart
* Violin plot

Data collection

| Study design|

* Population
* Statistic
* Effect size
* Statistical power
* Optimal design
* Sample size determination
* Replication
* Missing data

|

Survey methodology|

* Sampling
* stratified
* cluster
* Standard error
* Opinion poll
* Questionnaire

Controlled experiments|

* Scientific control
* Randomized experiment
* Randomized controlled trial
* Random assignment
* Blocking
* Interaction
* Factorial experiment

Adaptive Designs|

* Adaptive clinical trial
* Up-and-Down Designs
* Stochastic approximation

Observational Studies|

* Cross-sectional study
* Cohort study
* Natural experiment
* Quasi-experiment

Statistical inference

| Statistical theory|

* Population
* Statistic
* Probability distribution
* Sampling distribution
* Order statistic
* Empirical distribution
* Density estimation
* Statistical model
* Model specification
* Lp space
* Parameter
* location
* scale
* shape
* Parametric family
* Likelihood (monotone)
* Location–scale family
* Exponential family
* Completeness
* Sufficiency
* Statistical functional
* Bootstrap
* U
* V
* Optimal decision
* loss function
* Efficiency
* Statistical distance
* divergence
* Asymptotics
* Robustness

|

Frequentist inference|

| Point estimation|

* Estimating equations
* Maximum likelihood
* Method of moments
* M-estimator
* Minimum distance
* Unbiased estimators
* Mean-unbiased minimum-variance
* Rao–Blackwellization
* Lehmann–Scheffé theorem
* Median unbiased
* Plug-in

|

Interval estimation|

* Confidence interval
* Pivot
* Likelihood interval
* Prediction interval
* Tolerance interval
* Resampling
* Bootstrap
* Jackknife

Testing hypotheses|

* 1- & 2-tails
* Power
* Uniformly most powerful test
* Permutation test
* Randomization test
* Multiple comparisons

Parametric tests|

* Likelihood-ratio
* Score/Lagrange multiplier
* Wald

Specific tests|

* Z-test (normal)
* Student's t-test
* F-test

Goodness of fit|

* Chi-squared
* G-test
* Kolmogorov–Smirnov
* Anderson–Darling
* Lilliefors
* Jarque–Bera
* Normality (Shapiro–Wilk)
* Likelihood-ratio test
* Model selection
* Cross validation
* AIC
* BIC

Rank statistics|

* Sign
* Sample median
* Signed rank (Wilcoxon)
* Hodges–Lehmann estimator
* Rank sum (Mann–Whitney)
* Nonparametric anova
* 1-way (Kruskal–Wallis)
* 2-way (Friedman)
* Ordered alternative (Jonckheere–Terpstra)

Bayesian inference|

* Bayesian probability
* prior
* posterior
* Credible interval
* Bayes factor
* Bayesian estimator
* Maximum posterior estimator

* Correlation
* Regression analysis

| Correlation|

* Pearson product-moment
* Partial correlation
* Confounding variable
* Coefficient of determination

|

Regression analysis|

* Errors and residuals
* Regression validation
* Mixed effects models
* Simultaneous equations models
* Multivariate adaptive regression splines (MARS)

Linear regression|

* Simple linear regression
* Ordinary least squares
* General linear model
* Bayesian regression

Non-standard predictors|

* Nonlinear regression
* Nonparametric
* Semiparametric
* Isotonic
* Robust
* Heteroscedasticity
* Homoscedasticity

Generalized linear model|

* Exponential families
* Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance|

* Analysis of variance (ANOVA, anova)
* Analysis of covariance
* Multivariate ANOVA
* Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

| Categorical|

* Cohen's kappa
* Contingency table
* Graphical model
* Log-linear model
* McNemar's test

|

Multivariate|

* Regression
* Manova
* Principal components
* Canonical correlation
* Discriminant analysis
* Cluster analysis
* Classification
* Structural equation model
* Factor analysis
* Multivariate distributions
* Elliptical distributions
* Normal

Time-series|

| General|

* Decomposition
* Trend
* Stationarity
* Seasonal adjustment
* Exponential smoothing
* Cointegration
* Structural break
* Granger causality

|

Specific tests|

* Dickey–Fuller
* Johansen
* Q-statistic (Ljung–Box)
* Durbin–Watson
* Breusch–Godfrey

Time domain|

* Autocorrelation (ACF)
* partial (PACF)
* Cross-correlation (XCF)
* ARMA model
* ARIMA model (Box–Jenkins)
* Autoregressive conditional heteroskedasticity (ARCH)
* Vector autoregression (VAR)

Frequency domain|

* Spectral density estimation
* Fourier analysis
* Wavelet
* Whittle likelihood

Survival|

| Survival function|

* Kaplan–Meier estimator (product limit)
* Proportional hazards models
* Accelerated failure time (AFT) model
* First hitting time

|

Hazard function|

* Nelson–Aalen estimator

Test|

* Log-rank test

Applications

| Biostatistics|

* Bioinformatics
* Clinical trials / studies
* Epidemiology
* Medical statistics

|

Engineering statistics|

* Chemometrics
* Methods engineering
* Probabilistic design
* Process / quality control
* Reliability
* System identification

Social statistics|

* Actuarial science
* Census
* Crime statistics
* Demography
* Econometrics
* Jurimetrics
* National accounts
* Official statistics
* Population statistics
* Psychometrics

Spatial statistics|

* Cartography
* Environmental statistics
* Geographic information system
* Geostatistics
* Kriging

* Category
* '
* Commons
* WikiProject

* v
* t
* e

Quantitative forecasting methods

Historical data forecasts

* Moving average
* Exponential smoothing
* Trend analysis
* Decomposition of time series
* Naïve approach

Associative (causal) forecasts
* Moving average
* Simple linear regression
* Regression analysis
* Econometric model

* v
* t
* e

Public health

General|

* Auxology
* Biological hazard
* Chief Medical Officer
* Cultural competence
* Deviance
* Environmental health
* Euthenics
* Genomics
* Globalization and disease
* Health economics
* Health literacy
* Health policy
* Health system
* Health care reform
* Public health law
* Maternal health
* Medical anthropology
* Medical sociology
* Mental health
* Pharmaceutical policy
* Public health intervention
* Public health laboratory
* Reproductive health
* Social psychology
* Sociology of health and illness

Preventive healthcare|

* Behavior change
* Theories
* Family planning
* Health promotion
* Human nutrition
* Preventive nutrition
* Hygiene
* Food safety
* Hand washing
* Infection control
* Oral hygiene
* Occupational safety and health
* Human factors and ergonomics
* Hygiene
* Injury prevention
* Medicine
* Nursing
* Patient safety
* Organization
* Pharmacovigilance
* Safe sex
* Sanitation
* Emergency
* Fecal–oral transmission
* Open defecation
* Sanitary sewer
* Waterborne diseases
* Smoking cessation
* Vaccination
* Vector control

Population health|

* Biostatistics
* Child mortality
* Community health
* Epidemiology
* Global health
* Health impact assessment
* Health system
* Infant mortality
* Open-source healthcare software
* Public health informatics
* Social determinants of health
* Health equity
* Race and health
* Social medicine

Biological and
epidemiological statistics|

* Case–control study
* Randomized controlled trial
* Relative risk
* Statistical hypothesis testing
* Analysis of variance (ANOVA)
* Regression analysis
* ROC curve
* Student's t-test
* Z-test
* Statistical software

Infectious and epidemic
disease prevention|

* Asymptomatic carrier
* Epidemics
* List
* Notifiable diseases
* List
* Public health surveillance
* Disease surveillance
* Quarantine
* Sexually transmitted infection
* Tropical disease
* Vaccine trial

Food hygiene and
safety management|

* Food
* Additive
* Chemistry
* Engineering
* Microbiology
* Processing
* Safety
* Safety scandals
* Genetically modified food
* Good agricultural practice
* Good manufacturing practice
* HACCP
* ISO 22000

Health behavioral
sciences|

* Diffusion of innovations
* Health belief model
* Health communication
* Health psychology
* Positive deviance
* PRECEDE-PROCEED model
* Social cognitive theory
* Social norms approach
* Theory of planned behavior
* Transtheoretical model

Organizations,
education
and history|

| Organizations|

* Europe
* Centre for Disease Prevention and Control
* Committee on the Environment, Public Health and Food Safety
* India
* Ministry of Health and Family Welfare
* U.S.
* Centers for Disease Control and Prevention
* City and county health departments
* Council on Education for Public Health
* Public Health Service
* World Health Organization
* World Toilet Organization

|

Education|

* Health education
* Higher education
* Bachelor of Science in Public Health
* Doctor of Public Health
* Professional degrees of public health
* Schools of public health

History|

* Sara Josephine Baker
* Samuel Jay Crumbine
* Carl Rogers Darnall
* Joseph Lister
* Margaret Sanger
* John Snow
* Typhoid Mary
* Germ theory of disease
* Social hygiene movement

* Category
* Commons
* WikiProject

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