Quantile regression

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 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 leads to problems in linear programming that can be solved by the simplex method.

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.

Let ${\displaystyle Y}$ be a real valued random variable with cumulative distribution function ${\displaystyle F_{Y}(y)=P(Y\leq y)}$. The ${\displaystyle \tau }$th quantile of Y is given by

${\displaystyle Q_{Y}(\tau )=F_{Y}^{-1}(\tau )=\inf \left\{y:F_{Y}(y)\geq \tau \right\}}$

where ${\displaystyle \tau \in (0,1).}$

Define the loss function as ${\displaystyle \rho _{\tau }(y)=y(\tau -\mathbb {I} _{(y<0)})}$, where ${\displaystyle \mathbb {I} }$ is an indicator function.

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

${\displaystyle {\underset {u}{\min }}E(\rho _{\tau }(Y-u))}$${\displaystyle ={\underset {u}{\min }}{\biggl \{}}$${\displaystyle (\tau -1)}$${\displaystyle \int _{-\infty }^{u}}$${\displaystyle (y-u)}$${\displaystyle dF_{Y}(y)+\tau \int _{u}^{\infty }}$${\displaystyle (y-u)}$${\displaystyle dF_{Y}(y){\biggr \}}.}$

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 ${\displaystyle q_{\tau }}$ be the solution of

${\displaystyle 0=(1-\tau )\int _{-\infty }^{q_{\tau }}dF_{Y}(y)-\tau \int _{q_{\tau }}^{\infty }dF_{Y}(y).}$

This equation reduces to

${\displaystyle 0=F_{Y}(q_{\tau })-\tau ,}$

and then to

${\displaystyle F_{Y}(q_{\tau })=\tau .}$

Hence ${\displaystyle q_{\tau }}$ is ${\displaystyle \tau }$th quantile of the random variable Y.

Let ${\displaystyle Y}$ be a discrete random variable that takes values 1,2,..,9 with equal probabilities. The task is to find the median of Y, and hence the value ${\displaystyle \tau =0.5}$ is chosen. The expected loss, L(u), is

${\displaystyle L(u)={\frac {(\tau -1)}{9}}\sum _{y_{i}<u}}$${\displaystyle (y_{i}-u)}$${\displaystyle +{\frac {\tau }{9}}\sum _{y_{i}\geq u}}$${\displaystyle (y_{i}-u)}$${\displaystyle ={\frac {0.5}{9}}{\Bigl (}}$${\displaystyle -}$${\displaystyle \sum _{y_{i}<u}}$${\displaystyle (y_{i}-u)}$${\displaystyle +\sum _{y_{i}\geq u}}$${\displaystyle (y_{i}-u)}$${\displaystyle {\Bigr )}.}$

Since ${\displaystyle {0.5/9}}$ is a constant, it can be taken out of the expected loss function (this is only true if ${\displaystyle \tau =0.5}$). Then, at u=3,

${\displaystyle L(3)\propto \sum _{i=1}^{2}}$${\displaystyle -(i-3)}$${\displaystyle +\sum _{i=3}^{9}}$${\displaystyle (i-3)}$${\displaystyle =[(2+1)+(0+1+2+...+6)]=24.}$

Suppose that u is increased by 1 unit. Then the expected loss will be changed by ${\displaystyle (3)-(6)=-3}$ on changing u to 4. If, u=5, the expected loss is

${\displaystyle L(5)\propto \sum _{i=1}^{4}i+\sum _{i=0}^{4}i=20,}$

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 ${\displaystyle {0.5/9}}$) for different values of u.

Consider ${\displaystyle \tau =0.5}$ and let q be an initial guess for ${\displaystyle q_{\tau }}$. The expected loss evaluated at q is

${\displaystyle L(q)=-0.5\int _{-\infty }^{q}(y-q)dF_{Y}(y)+0.5\int _{q}^{\infty }(y-q)dF_{Y}(y).}$

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

${\displaystyle \int _{-\infty }^{q}1dF_{Y}(y)-\int _{q}^{\infty }1dF_{Y}(y).}$

The first term of the equation is ${\displaystyle F_{Y}(q)}$ and second term of the equation is ${\displaystyle 1-F_{Y}(q)}$. Therefore, the change of expected loss function is negative if and only if ${\displaystyle F_{Y}(q)<0.5}$, 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 ${\displaystyle 100\tau }$% of the points.

The ${\displaystyle \tau }$ sample quantile can be obtained by solving the following minimization problem

${\displaystyle {\hat {q}}_{\tau }={\underset {q\in \mathbb {R} }{\mbox{arg min}}}\sum _{i=1}^{n}\rho _{\tau }(y_{i}-q),}$

${\displaystyle ={\underset {q\in \mathbb {R} }{\mbox{arg min}}}\left[(\tau -1)\sum _{y_{i}<q}(y_{i}-q)+\tau \sum _{y_{i}\geq q}(y_{i}-q)\right]}$, where the function ${\displaystyle \rho _{\tau }}$ is the tilted absolute value function. The intuition is the same as for the population quantile.

Suppose the ${\displaystyle \tau }$th conditional quantile function is ${\displaystyle Q_{Y|X}(\tau )=X\beta _{\tau }}$. Given the distribution function of ${\displaystyle Y}$, ${\displaystyle \beta _{\tau }}$ can be obtained by solving

${\displaystyle \beta _{\tau }={\underset {\beta \in \mathbb {R} ^{k}}{\mbox{arg min}}}E(\rho _{\tau }(Y-X\beta )).}$

Solving the sample analog gives the estimator of ${\displaystyle \beta }$.

${\displaystyle {\hat {\beta _{\tau }}}={\underset {\beta \in \mathbb {R} ^{k}}{\mbox{arg min}}}\sum _{i=1}^{n}(\rho _{\tau }(Y_{i}-X_{i}\beta )).}$

The minimization problem can be reformulated as a linear programming problem

${\displaystyle {\underset {\beta ,u^{+},u^{-}\in \mathbb {R} ^{k}\times \mathbb {R} _{+}^{2n}}{\min }}\left\{\tau 1_{n}^{'}u^{+}+(1-\tau )1_{n}^{'}u^{-}|X\beta +u^{+}-u^{-}=Y\right\},}$

where

${\displaystyle u_{j}^{+}=\max(u_{j},0)}$ ,    ${\displaystyle u_{j}^{-}=-\min(u_{j},0).}$

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

For ${\displaystyle \tau \in (0,1)}$, under some regularity conditions, ${\displaystyle {\hat {\beta }}_{\tau }}$ is asymptotically normal:

${\displaystyle {\sqrt {n}}({\hat {\beta }}_{\tau }-\beta _{\tau }){\overset {d}{\rightarrow }}N(0,\tau (1-\tau )D^{-1}\Omega _{x}D^{-1}),}$

where

${\displaystyle D=E(f_{Y}(X\beta )XX^{\prime })}$ and ${\displaystyle \Omega _{x}=E(X^{\prime }X).}$

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.

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 ${\displaystyle Y^{c}=\max(0,Y)}$ and ${\displaystyle Q_{Y|X}=X\beta _{\tau }}$. Then ${\displaystyle Q_{Y^{c}|X}(\tau )=\max(0,X\beta _{\tau })}$. 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.
• 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]
• Statsmodels package for Python, via QuantReg[33]
• Mathematica package QuantileRegression.m[34] hosted at the MathematicaForPrediction project at GitHub.

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.