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**Quantile regression** is a type of regression analysis used in statistics and econometrics. Whereas the method of least squares estimates the conditional * mean * of the response variable across values of the predictor variables, quantile regression estimates the conditional median (or other quantiles) of the response variable. Quantile regression is an extension of linear regression used when the conditions of linear regression are not met.

- Advantages and applications
- History
- Quantiles
- Quantile of a random variable
- Sample quantile
- Conditional quantile and quantile regression
- Computation of estimates for regression parameters
- Asymptotic properties
- Equivariance
- Scale equivariance
- Shift equivariance
- Equivariance to reparameterization of design
- Invariance to monotone transformations
- Bayesian methods for quantile regression
- Machine learning methods for quantile regression
- Censored quantile regression
- Implementations
- References
- Further reading

One advantage of quantile regression relative to ordinary least squares regression is that the quantile regression estimates are more robust against outliers in the response measurements. However, the main attraction of quantile regression goes beyond this and is advantageous when conditional quantile functions are of interest. Different measures of central tendency and statistical dispersion can be useful to obtain a more comprehensive analysis of the relationship between variables.^{ [1] }

In ecology, quantile regression has been proposed and used as a way to discover more useful predictive relationships between variables in cases where there is no relationship or only a weak relationship between the means of such variables. The need for and success of quantile regression in ecology has been attributed to the complexity of interactions between different factors leading to data with unequal variation of one variable for different ranges of another variable.^{ [2] }

Another application of quantile regression is in the areas of growth charts, where percentile curves are commonly used to screen for abnormal growth.^{ [3] }^{ [4] }

The idea of estimating a median regression slope, a major theorem about minimizing sum of the absolute deviances and a geometrical algorithm for constructing median regression was proposed in 1760 by Ruđer Josip Bošković, a Jesuit Catholic priest from Dubrovnik.^{ [1] }^{: 4 }^{ [5] } He was interested in the ellipticity of the earth, building on Isaac Newton's suggestion that its rotation could cause it to bulge at the equator with a corresponding flattening at the poles.^{ [6] } He finally produced the first geometric procedure for determining the equator of a rotating planet from three observations of a surface feature. More importantly for quantile regression, he was able to develop the first evidence of the least absolute criterion and preceded the least squares introduced by Legendre in 1805 by fifty years.^{ [7] }

Other thinkers began building upon Bošković's idea such as Pierre-Simon Laplace, who developed the so-called "methode de situation." This led to Francis Edgeworth's plural median^{ [8] } - a geometric approach to median regression - and is recognized as the precursor of the simplex method.^{ [7] } The works of Bošković, Laplace, and Edgeworth were recognized as a prelude to Roger Koenker's contributions to quantile regression.

Median regression computations for larger data sets are quite tedious compared to the least squares method, for which reason it has historically generated a lack of popularity among statisticians, until the widespread adoption of computers in the latter part of the 20th century.

Quantile regression expresses the conditional quantiles of a dependent variable as a linear function of the explanatory variables. Crucial to the practicality of quantile regression is that the quantiles can be expressed as the solution of a minimization problem, as we will show in this section before discussing conditional quantiles in the next section.

Let be a real valued random variable with cumulative distribution function . The th quantile of Y is given by

where

Define the loss function as , where is an indicator function.

A specific quantile can be found by minimizing the expected loss of with respect to :^{ [1] }(pp. 5–6):

This can be shown by computing the derivative of the expected loss via an application of the Leibniz integral rule, setting it to 0, and letting be the solution of

This equation reduces to

and then to

If the solution is not unique, then we have to take the smallest such solution to obtain the th quantile of the random variable *Y*.

Let be a discrete random variable that takes values with with equal probabilities. The task is to find the median of Y, and hence the value is chosen. Then the expected loss of is

Since is a constant, it can be taken out of the expected loss function (this is only true if ). Then, at *u*=3,

Suppose that *u* is increased by 1 unit. Then the expected loss will be changed by on changing *u* to 4. If, *u*=5, the expected loss is

and any change in *u* will increase the expected loss. Thus *u*=5 is the median. The Table below shows the expected loss (divided by ) for different values of *u*.

u | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |

Expected loss | 36 | 29 | 24 | 21 | 20 | 21 | 24 | 29 | 36 |

Consider and let *q* be an initial guess for . The expected loss evaluated at *q* is

In order to minimize the expected loss, we move the value of *q* a little bit to see whether the expected loss will rise or fall. Suppose we increase *q* by 1 unit. Then the change of expected loss would be

The first term of the equation is and second term of the equation is . Therefore, the change of expected loss function is negative if and only if , that is if and only if *q* is smaller than the median. Similarly, if we reduce *q* by 1 unit, the change of expected loss function is negative if and only if *q* is larger than the median.

In order to minimize the expected loss function, we would increase (decrease) *L*(*q*) if *q* is smaller (larger) than the median, until *q* reaches the median. The idea behind the minimization is to count the number of points (weighted with the density) that are larger or smaller than *q* and then move *q* to a point where *q* is larger than % of the points.

The sample quantile can be obtained by solving the following minimization problem

- ,

where the function is the tilted absolute value function. The intuition is the same as for the population quantile.

The th conditional quantile of given is the th quantile of the Conditional probability distribution of given ,

- .

We use a capital to denote the conditional quantile to indicate that it is a random variable.

In quantile regression for the th quantile we make the assumption that the th conditional quantile is given as a linear function of the explanatory variables:

- .

Given the distribution function of , can be obtained by solving

Solving the sample analog gives the estimator of .

Note that when the loss function is proportional to the absolute value function and thus median regression is the same as linear regression by least absolute deviations.

The mathematical forms arising from quantile regression are distinct from those arising in the method of least squares. The method of least squares leads to a consideration of problems in an inner product space, involving projection onto subspaces, and thus the problem of minimizing the squared errors can be reduced to a problem in numerical linear algebra. Quantile regression does not have this structure, and instead the minimization problem can be reformulated as a linear programming problem

where

- ,

Simplex methods ^{ [1] }^{: 181 } or interior point methods ^{ [1] }^{: 190 } can be applied to solve the linear programming problem.

For , under some regularity conditions, is asymptotically normal:

where

- and

Direct estimation of the asymptotic variance-covariance matrix is not always satisfactory. Inference for quantile regression parameters can be made with the regression rank-score tests or with the bootstrap methods.^{ [9] }

See invariant estimator for background on invariance or see equivariance.

For any and

For any and

Let be any nonsingular matrix and

If is a nondecreasing function on '*R*, the following invariance property applies:

Example (1):

If and , then . The mean regression does not have the same property since

Because quantile regression does not normally assume a parametric likelihood for the conditional distributions of Y|X, the Bayesian methods work with a working likelihood. A convenient choice is the asymmetric Laplacian likelihood,^{ [10] } because the mode of the resulting posterior under a flat prior is the usual quantile regression estimates. The posterior inference, however, must be interpreted with care. Yang, Wang and He^{ [11] } provided a posterior variance adjustment for valid inference. In addition, Yang and He^{ [12] } showed that one can have asymptotically valid posterior inference if the working likelihood is chosen to be the empirical likelihood.

Beyond simple linear regression, there are several machine learning methods that can be extended to quantile regression. A switch from the squared error to the tilted absolute value loss function allows gradient descent based learning algorithms to learn a specified quantile instead of the mean. It means that we can apply all neural network and deep learning algorithms to quantile regression.^{ [13] }^{ [14] } Tree-based learning algorithms are also available for quantile regression (see, e.g., Quantile Regression Forests,^{ [15] } as a simple generalization of Random Forests).

If the response variable is subject to censoring, the conditional mean is not identifiable without additional distributional assumptions, but the conditional quantile is often identifiable. For recent work on censored quantile regression, see: Portnoy^{ [16] } and Wang and Wang^{ [17] }

Example (2):

Let and . Then . This is the censored quantile regression model: estimated values can be obtained without making any distributional assumptions, but at the cost of computational difficulty,^{ [18] } some of which can be avoided by using a simple three step censored quantile regression procedure as an approximation.^{ [19] }

For random censoring on the response variables, the censored quantile regression of Portnoy (2003)^{ [16] } provides consistent estimates of all identifiable quantile functions based on reweighting each censored point appropriately.

Numerous statistical software packages include implementations of quantile regression:

- Matlab function
`quantreg`

^{ [20] } - Eviews, since version 6.
^{[ citation needed ]} - gretl has the
`quantreg`

command.^{ [21] } - R offers several packages that implement quantile regression, most notably
`quantreg`

by Roger Koenker,^{ [22] }but also`gbm`

,^{ [23] }`quantregForest`

,^{ [24] }`qrnn`

^{ [25] }and`qgam`

^{ [26] } - Python, via
`Scikit-garden`

^{ [27] }and`statsmodels`

^{ [28] } - SAS through
`proc quantreg`

(ver. 9.2) and`proc quantselect`

(ver. 9.3).^{ [29] } - Stata, via the
`qreg`

command.^{ [30] }^{ [31] } - Vowpal Wabbit, via
`--loss_function quantile`

.^{ [32] } - Mathematica package
`QuantileRegression.m`

^{ [33] }hosted at the MathematicaForPrediction project at GitHub.

In statistics, an **expectation–maximization** (**EM**) **algorithm** is an iterative method to find (local) maximum likelihood or maximum a posteriori (MAP) estimates of parameters in statistical models, where the model depends on unobserved latent variables. The EM iteration alternates between performing an expectation (E) step, which creates a function for the expectation of the log-likelihood evaluated using the current estimate for the parameters, and a maximization (M) step, which computes parameters maximizing the expected log-likelihood found on the *E* step. These parameter-estimates are then used to determine the distribution of the latent variables in the next E step.

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The **polytomous Rasch model** is generalization of the dichotomous Rasch model. It is a measurement model that has potential application in any context in which the objective is to measure a trait or ability through a process in which responses to items are *scored* with successive integers. For example, the model is applicable to the use of Likert scales, rating scales, and to educational assessment items for which successively higher integer scores are intended to indicate increasing levels of competence or attainment.

In statistics, a **tobit model** is any of a class of regression models in which the observed range of the dependent variable is censored in some way. The term was coined by Arthur Goldberger in reference to James Tobin, who developed the model in 1958 to mitigate the problem of zero-inflated data for observations of household expenditure on durable goods. Because Tobin's method can be easily extended to handle truncated and other non-randomly selected samples, some authors adopt a broader definition of the tobit model that includes these cases.

In statistics, **M-estimators** are a broad class of extremum estimators for which the objective function is a sample average. Both non-linear least squares and maximum likelihood estimation are special cases of M-estimators. The definition of M-estimators was motivated by robust statistics, which contributed new types of M-estimators. The statistical procedure of evaluating an M-estimator on a data set is called **M-estimation**. 48 samples of robust M-estimators can be founded in a recent review study.

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The **Generalized Additive Model for Location, Scale and Shape (GAMLSS)** is an approach to statistical modelling and learning. GAMLSS is a modern distribution-based approach to (semiparametric) regression. A parametric distribution is assumed for the response (target) variable but the parameters of this distribution can vary according to explanatory variables using linear, nonlinear or smooth functions. In machine learning parlance, GAMLSS is a form of supervised machine learning.

**Quantile Regression Averaging (QRA)** is a forecast combination approach to the computation of prediction intervals. It involves applying quantile regression to the point forecasts of a small number of individual forecasting models or experts. It has been introduced in 2014 by Jakub Nowotarski and Rafał Weron and originally used for probabilistic forecasting of electricity prices and loads. Despite its simplicity it has been found to perform extremely well in practice - the top two performing teams in the *price track* of the Global Energy Forecasting Competition (GEFCom2014) used variants of QRA.

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In machine learning, **Manifold regularization** is a technique for using the shape of a dataset to constrain the functions that should be learned on that dataset. In many machine learning problems, the data to be learned do not cover the entire input space. For example, a facial recognition system may not need to classify any possible image, but only the subset of images that contain faces. The technique of manifold learning assumes that the relevant subset of data comes from a manifold, a mathematical structure with useful properties. The technique also assumes that the function to be learned is *smooth*: data with different labels are not likely to be close together, and so the labeling function should not change quickly in areas where there are likely to be many data points. Because of this assumption, a manifold regularization algorithm can use unlabeled data to inform where the learned function is allowed to change quickly and where it is not, using an extension of the technique of Tikhonov regularization. Manifold regularization algorithms can extend supervised learning algorithms in semi-supervised learning and transductive learning settings, where unlabeled data are available. The technique has been used for applications including medical imaging, geographical imaging, and object recognition.

In statistics, the class of **vector generalized linear models** (**VGLMs**) was proposed to enlarge the scope of models catered for by generalized linear models (**GLMs**). In particular, VGLMs allow for response variables outside the classical exponential family and for more than one parameter. Each parameter can be transformed by a *link function*. The VGLM framework is also large enough to naturally accommodate multiple responses; these are several independent responses each coming from a particular statistical distribution with possibly different parameter values.

**Data-driven control systems** are a broad family of control systems, in which the identification of the process model and/or the design of the controller are based entirely on *experimental data* collected from the plant.

- 1 2 3 4 5 Koenker, Roger (2005).
*Quantile Regression*. Cambridge University Press. pp. 146–7. ISBN 978-0-521-60827-5. - ↑ Cade, Brian S.; Noon, Barry R. (2003). "A gentle introduction to quantile regression for ecologists" (PDF).
*Frontiers in Ecology and the Environment*.**1**(8): 412–420. doi:10.2307/3868138. JSTOR 3868138. - ↑ Wei, Y.; Pere, A.; Koenker, R.; He, X. (2006). "Quantile Regression Methods for Reference Growth Charts".
*Statistics in Medicine*.**25**(8): 1369–1382. doi:10.1002/sim.2271. PMID 16143984. - ↑ Wei, Y.; He, X. (2006). "Conditional Growth Charts (with discussions)".
*Annals of Statistics*.**34**(5): 2069–2097 and 2126–2131. arXiv: math/0702634 . doi:10.1214/009053606000000623. - ↑ Stigler, S. (1984). "Boscovich, Simpson and a 1760 manuscript note on fitting a linear relation".
*Biometrika*.**71**(3): 615–620. doi:10.1093/biomet/71.3.615. - ↑ Koenker, Roger (2005).
*Quantile Regression*. Cambridge: Cambridge University Press. pp. 2. ISBN 9780521845731. - 1 2 Furno, Marilena; Vistocco, Domenico (2018).
*Quantile Regression: Estimation and Simulation*. Hoboken, NJ: John Wiley & Sons. pp. xv. ISBN 9781119975281. - ↑ Koenker, Roger (August 1998). "Galton, Edgeworth, Frisch, and prospects for quantile regression in economics" (PDF).
*UIUC.edu*. Retrieved August 22, 2018. - ↑ Kocherginsky, M.; He, X.; Mu, Y. (2005). "Practical Confidence Intervals for Regression Quantiles".
*Journal of Computational and Graphical Statistics*.**14**(1): 41–55. doi:10.1198/106186005X27563. - ↑ Kozumi, H.; Kobayashi, G. (2011). "Gibbs sampling methods for Bayesian quantile regression" (PDF).
*Journal of Statistical Computation and Simulation*.**81**(11): 1565–1578. doi:10.1080/00949655.2010.496117. - ↑ Yang, Y.; Wang, H.X.; He, X. (2016). "Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood".
*International Statistical Review*.**84**(3): 327–344. doi:10.1111/insr.12114. hdl: 2027.42/135059 . - ↑ Yang, Y.; He, X. (2010). "Bayesian empirical likelihood for quantile regression".
*Annals of Statistics*.**40**(2): 1102–1131. arXiv: 1207.5378 . doi:10.1214/12-AOS1005. - ↑ Petneházi, Gábor (2019-08-21). "QCNN: Quantile Convolutional Neural Network". arXiv: 1908.07978 [cs.LG].
- ↑ Rodrigues, Filipe; Pereira, Francisco C. (2018-08-27). "Beyond expectation: Deep joint mean and quantile regression for spatio-temporal problems". arXiv: 1808.08798 [stat].
- ↑ Meinshausen, Nicolai (2006). "Quantile Regression Forests" (PDF).
*Journal of Machine Learning Research*.**7**(6): 983–999. - 1 2 Portnoy, S. L. (2003). "Censored Regression Quantiles".
*Journal of the American Statistical Association*.**98**(464): 1001–1012. doi:10.1198/016214503000000954. - ↑ Wang, H.; Wang, L. (2009). "Locally Weighted Censored Quantile Regression".
*Journal of the American Statistical Association*.**104**(487): 1117–1128. CiteSeerX 10.1.1.504.796 . doi:10.1198/jasa.2009.tm08230. - ↑ Powell, James L. (1986). "Censored Regression Quantiles".
*Journal of Econometrics*.**32**(1): 143–155. doi:10.1016/0304-4076(86)90016-3. - ↑ Chernozhukov, Victor; Hong, Han (2002). "Three-Step Censored Quantile Regression and Extramarital Affairs".
*J. Amer. Statist. Assoc.***97**(459): 872–882. doi:10.1198/016214502388618663. - ↑ "quantreg(x,y,tau,order,Nboot) - File Exchange - MATLAB Central".
*www.mathworks.com*. Retrieved 2016-02-01. - ↑ "Gretl Command Reference" (PDF). April 2017.
- ↑ "quantreg: Quantile Regression".
*R Project*. 2018-12-18. - ↑ "gbm: Generalized Boosted Regression Models".
*R Project*. 2019-01-14. - ↑ "quantregForest: Quantile Regression Forests".
*R Project*. 2017-12-19. - ↑ "qrnn: Quantile Regression Neural Networks".
*R Project*. 2018-06-26. - ↑ "qgam: Smooth Additive Quantile Regression Models".
*R Project*. 2019-05-23. - ↑ "Quantile Regression Forests".
*Scikit-garden*. Retrieved 3 January 2019. - ↑ "Statsmodels: Quantile Regression".
*Statsmodels*. Retrieved 15 November 2019. - ↑ "An Introduction to Quantile Regression and the QUANTREG Procedure" (PDF).
*SAS Support*. - ↑ "qreg — Quantile regression" (PDF).
*Stata Manual*. - ↑ Cameron, A. Colin; Trivedi, Pravin K. (2010). "Quantile Regression".
*Microeconometrics Using Stata*(Revised ed.). College Station: Stata Press. pp. 211–234. ISBN 978-1-59718-073-3. - ↑ "JohnLangford/vowpal_wabbit".
*GitHub*. Retrieved 2016-07-09. - ↑ "QuantileRegression.m".
*MathematicaForPrediction*. Retrieved 3 January 2019.

The Wikibook R Programming has a page on the topic of: Quantile Regression |

- Angrist, Joshua D.; Pischke, Jörn-Steffen (2009). "Quantile Regression".
*Mostly Harmless Econometrics: An Empiricist's Companion*. Princeton University Press. pp. 269–291. ISBN 978-0-691-12034-8. - Koenker, Roger (2005).
*Quantile Regression*. Cambridge University Press. ISBN 978-0-521-60827-5.

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