Support Vector Machine - Regression (SVR) Support Vector Machine can also be used as a regression method, maintaining all the main features that characterize the algorithm (maximal margin). variate quantile estimation, the proposed approach can deal more broadly with multivariate data, for example, when the Box-Cox transformations fail to trans-form the data into multivariate Gaussian. Scikit-learn is a software machine learning library for the Python programming language that has a various classification, regression and clustering algorithms including support vector machines, random forests, gradient boosting, k-means and DBSCAN, and is designed to interoperate with the Python numerical and scientific libraries NumPy and SciPy. This function has a usage,where:x - the data pointsprob - the location to measurena. Moreover, there is a specific session dedicated to Time Series, and a specific topic about the Theory underlying the main algorithms. Quantile regression is not a regression estimated on a quantile, or subsample of data as the name may suggest. Hence our use of quantile regression is not justifiable for such quantiles. 95g are superimposed on the scatterplot. Below is the code for 1st, 2nd, and 3rd-order polynomial linear regression, confidence and prediction intervals, and quantile regression. Quantile Regression Roger Koenker and Kevin F. loss function to be optimized. The value of quantile regression. It is recommended in cases of regression where the number of explanatory variables is high, and where it is likely that the explanatory variables are correlated. As far as we know, there is no module for quantile adjustment normalization available in the biopython library; our attempt tries to fill this lack. Introduction to Quantile Regression CHUNG-MING KUAN Department of Finance National Taiwan University May 31, 2010 C. Python is a great language for doing data analysis, primarily because of the fantastic ecosystem of data-centric python packages. liquidSVM is an implementation of SVMs whose key features are: fully integrated hyper-parameter selection, extreme speed on both small and large data sets, full flexibility for experts, and. Quantile regression. This package implements the quantile regression neural network (QRNN) (Taylor, 2000; Cannon, 2011; Cannon, 2018), which is a flexible nonlinear form of quantile regression. Fourth, we analyze the rate of convergence of a two-step estimator that applies ordinary quantile regression to the selected model. I am predicting electricity usage for customers which is highly skewed. They are extracted from open source Python projects. Quantile Regression Forests Introduction. Second-order derivative of quantile regression loss is equal to 0 at every point except the one where it is not defined. Classification trees are nice. From my experience, one of point of confusion for new R users is how to handle exceptions, which is a bit more intuitive in Python. PROC QUANTREG. Thanks to Josef Perktold at StatsModels for assistance with the quantile regression code, and providing the creative "heteroscedastic" dataset that we will analyze. It's similar to minimizing the absolute error in this case, but it's much more general and work for other quantiles. Each tree in a decision forest outputs a Gaussian distribution by way of prediction. It appears we can make decent estimates of the 0. Hallock W e say that a student scores at the tth quantile of a standardized exam if he performs better than the proportion t of the reference group of students and worse than the proportion (1–t). Seven estimated quantile regression lines for ˝2f. plotly is a high-level interface to plotly. Total variation image in-painting. The default is to compute the quantile(s) along a flattened version of the array. Quantiles and percentiles are the same except for a factor of 100, e. Below, we present the standard quantile regression, copula and tail dependence, and then copula quantile regression. I created a quick and dirty quantile regression forest class as an extension of scikit learn's RandomForestRegressor. Dear all, I have a very difficult problem that I can't seem to produce using Stata. The project is aimed to mirror and aid the talk "Deep Learning series (session 2)" of the meetup Orlando Machine Learning and Data Science. The ALS minimisation is presented in expression (6). Whereas the method of least squares results in estimates of the conditional mean of the response variable given certain values of the predictor variables, quantile regression aims at estimating either the conditional median or other quantiles of the response variable. Find the 95 th percentile of the F distribution with (5, 2) degrees of freedom. I think I did not explain myself well. Quantile regression is an appropriate tool for accomplishing this task. Quantile or sequence of quantiles to compute, which must be between 0 and 1 inclusive. Quantile Regression. They are extracted from open source Python projects. Bear in mind that unlike OLS (or mean-regression), quantile regression is used to estimate a coefficient conditional on specific quantile. Micro Data. 'ls' refers to least squares regression. quantile() function return values at the given quantile over requested axis, a numpy. One could suspect that the weighted observations deliver not only a good approximation to the conditional, t t. Running a Regression (Using R Statistics Software) Step-by-step example of how to do a regression using R statistics software (including the models below). The only other dependency is on matplotlib, which is used to plot the data and the quantile estimates. In order to automatically select the relevant variable groups, we propose and study here the adaptive group LASSO quantile estimator. In the figure given above, Q2 is the median of the normally distributed data. It can be considered as a generalization of Poisson regression since it has the same mean structure as Poisson regression and it has an extra parameter to model the over. "3-step Censored Quantile Regression and Extramarital Affairs,"J. 75 quantile regression. Regression coefficients. 7 train Models By Tag. The CQIV estimator combines Powell (1986) censored quantile regression (CQR) to deal semiparametrically with censoring, with a control variable approach to incorporate endogenous regressors. This Python script demonstrates that one can perform quantile regression using only Python, NumPy, and SciPy. CARLIER, V. Replacement The Python or R software must be installed on the same machine as the compute server microservice. Koenker, Roger and Kevin F. I think I did not explain myself well. PseudoR2: in quatile regression another definition of R2 is used namely PseudoR2. Quantile Regression in Python 13 Mar 2017 In ordinary linear regression, we are estimating the mean of some variable y , conditional on the values of independent variables X. Censored survival outcomes should require coxph. This section contains basic information regarding the supported metrics for various machine learning problems. A linear transformation of the X variables is done so that the sum of squared deviations of the observed and predicted Y. The black dotted line are the quantile regression estimates and the gray area is the confidence interval for themfor various quantiles. 5 which is the median, but you can try whichever quantile that you are after. Their blossoming lives and selves bring me more joy than anything else in this world. The limits are the minimum and maximum values. 5, and so on. Its source code can easily be deployed to a PaaS. Stata can also perform simultaneous-quantile regression. minimizing absolute error), and quantile regression (for estimating percentiles of the conditional distribution of the outcome). “Quantile Regression”. Revised August 2005] Summary. I had a couple of queries regarding quantile regression: a. The idea is to estimate the errors by “median polishing” and then subtract the estimated errors to obtain adjusted probe summaries Expression value of probe set en el chip i Residuals of j-th probe on ith chip Probe. Coverage includes extensive, in-depth discussion of advanced statistical techniques, data visualization, predictive analytics, and SPSS programming, including automation and integration with other languages like R and Python. exogenous/explanatory variable(s) Notes. Comparison of Regression Splines with Polynomial Regression. Quantile regression helps you understand the distribution of the predicted value. Popular statistical methods such as t-test often focuses on calculating average treatment effects. m, which uses an interior point method of Koenker and Park (1996, J. For the sake of having them, it is beneficial to port quantile regression loss to xgboost. So make sure these dependencies are installed using pip: pip install setuptools numpy scipy scikit-learn cython After that Scikit-Garden can be installed. The weight file corresponds with data file line by line, and has per weight per line. September 16, 2014. Let's start with some dummy data , which we will enter using iPython. Quantile Regression has the advantage of targeting on a specific quantile of y. Quantile Regression lets us do just this. Statistics and Its Interface Volume 2 (2009) 299–310 Stepwise multiple quantile regression estimation using non-crossing constraints∗ Yichao Wu†,§ and Yufeng Liu‡. 4/11 Simple Regression and Multiple Regression. To my close friends David and Ron for their selfless. The algorithm is based on interior point ideas described in Koenker and Park (1994). "3-step Censored Quantile Regression and Extramarital Affairs,"J. A quantile is the value below which a fraction of observations in a group falls. Quantile regression is desired if conditional quantile functions are of interest. I have used the python package statsmodels 0. To construct a quantile-quantile plot for the residuals, we plot the quantiles of the residuals against the theorized quantiles if the residuals arose from a normal distribution. Parameter uncertainty and the predicted uncertainty is important for qualifying the confidence in the solution. That is, for the fit at point x, the fit is made using points in a neighbourhood of x, weighted by their distance from x (with differences in ‘parametric’ variables being ignored when computing the distance). The only other dependency is on matplotlib, which is used to plot the data and the quantile estimates. mquantiles The resulting quantile estimates are approximately median-unbiased regardless of the distribution of x. Quantile plays a very important role in Statistics when one deals with the Normal Distribution. Estimate a quantile regression model using iterative reweighted least squares. And good news! We’re open sourcing it. Quantile regression is an appropriate tool for accomplishing this task. Fernandez-Val. Econometrics). A topic we haven’t talked about yet is the commonly used quantile regression. "Quantile Regression". Fifth, we evaluate the performance of $\ell_1$-QR in a Monte-Carlo experiment, and illustrate its use on an international economic growth application. Statistical computations and models for Python. We'll skip Linear Regression as well because we have been using linear regression all along and you should be familiar with it. Alternative output array in which to place the result. Towards Data Science Sharing concepts, ideas, and codes. ) ﻿- Generalized Additive Models: An Introduction with R (Wood)﻿ - Quantile Regression (Koenker)﻿ - Handbook of Quantile Regression (Koenker et al. This Python script demonstrates that one can perform quantile regression using only Python, NumPy, and SciPy. median regression, is subsumed under the term quantile regression. Axis or axes along which the quantiles are computed. To my close friends David and Ron for their selfless. Koenker, Roger and Kevin F. Drawing them using ggvis and plotly is also possible, but demands way more code. set seed 1001. It allows the user to. Quantile regression. For quantile regression you can use the package quantregForest, which is based on the randomForest package. I see that when fitting the quantile regression model, there is an option to specify the significance level for confidence intervals of the regression coefficients, and the confidence interval result appears in the summary of the fit. When the x values are close to 0, linear regression is giving a good estimate of y, but we near end of x values the predicted y is far way from the actual values and hence becomes completely meaningless. A third distinctive feature of the LRM is its normality assumption. This gives me the normalized values. Similarly, in > regression settings the > optimization problem solved by the "br" version of the simplex > algorithm, > modified to do general quantile regression identifies cases where > there may > be non uniqueness of this type. Support Vector Machine - Regression (SVR) Support Vector Machine can also be used as a regression method, maintaining all the main features that characterize the algorithm (maximal margin). Censored survival outcomes should require coxph. So I have 3 set of predictions for the three models optimized at the quantiles specified above. Koenker, Roger and Kevin F. I have read counts from RNA seq data in row and columns. g: “Local Quantile Regression”). Such a problem. Find the 95 th percentile of the F distribution with (5, 2) degrees of freedom. Discover the new Lexile & Quantile Hub, a convenient online platform that provides you with easy access to more than a dozen new and enhanced reading and mathematics tools. ‘lad’ (least absolute deviation) is a highly robust loss function solely based on order information of the input variables. Updated 2019 October 11th to explain the index_to_mean function. 'huber' is a combination of the two. Fifth, we evaluate the performance of $\ell_1$-QR in a Monte-Carlo experiment, and illustrate its use on an international economic growth application. In this post, I want to approach two traditional problems: Quantile Regression and Ordinary Least Squares as convex problems and how to implement them in R using the package RMosek. See the user guide for more details on how the regression cutoff is used. Random Forest Regression and Classifiers in R and Python We've written about Random Forests a few of times before, so I'll skip the hot-talk for why it's a great learning method. We first describe Multiple Regression in an intuitive way by moving from a straight line in a single predictor case to a 2d plane in the case of two predictors. To do this, first we. 25 quantile regression, one with fitted values from the median regression and one with fitted values from the. where alpha is the required quantile (a value between 0 and 1) and. Quantile regression helps you understand the distribution of the predicted value. The YUIMA Project: A Computational Framework for Simulation and Inference of Stochastic Differential Equations: PDF: Alexandre Brouste, Masaaki Fukasawa, Hideitsu Hino, Stefano Iacus, Kengo Kamatani, Yuta Koike, Hiroki Masuda, Ryosuke Nomura, Teppei Ogihara, Yasutaka Shimuzu, Masayuki Uchida, Nakahiro Yoshida. Hierarchical mixed effects regression using maximum likelihood g02kac: 9 nag_regsn_ridge_opt Ridge regression, optimizing a ridge regression parameter g02kbc: 9 nag_regsn_ridge Ridge regression using a number of supplied ridge regression parameters g02lac: 9 nag_pls_orth_scores_svd Partial least-squares (PLS) regression using singular value. I am now trying quantile regression using vowpal wabbit. Word of caution, which applies to any quantile regression method; you may find that the quantile output might be extreme/ unexpected when you take extreme quantiles (eg. Quantile regression is a powerful technique and deserves the interest of financial decision makers given its remarkable capabilities for capturing and explaining the behavior of financial return series across a distribution more effectively than ordinary least squares regression methods which are the standard tool. Drawing them under the ggplot2 package it's not challenging; it has a whole quantile dedicated function, geom_quantile(). Hallock W e say that a student scores at the tth quantile of a standardized exam if he performs better than the proportion t of the reference group of students and worse than the proportion (1-t). There entires in these lists are arguable. What is Logistic regression. Quantiles can be a very useful weapon in statistical research. Reich BJ, Fuentes M, Dunson DB (2012). Thus, half of students perform better than the median student and half perform worse. Fernandez-Val. mquantiles The resulting quantile estimates are approximately median-unbiased regardless of the distribution of x. exogenous/explanatory variable(s) Notes. Bubble charts are achieved by setting `marker. tab industry, or. We first describe Multiple Regression in an intuitive way by moving from a straight line in a single predictor case to a 2d plane in the case of two predictors. Quantile Regression and Healthcare Costs I thought this was a nice statement that speaks to the utility of quantile regression (which holds to any distribution with these issues not just cost data): The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost, and is. To construct a quantile-quantile plot for the residuals, we plot the quantiles of the residuals against the theorized quantiles if the residuals arose from a normal distribution. This example shows how quantile regression can be used to create prediction intervals. The quantile regression loss function solves this and similar problems by replacing a single value prediction by prediction intervals. Introduction to Quantile Regression CHUNG-MING KUAN Department of Finance National Taiwan University May 31, 2010 C. Quantile terdiri dari Quantile 0,25, Quantile 0,50, lalu Quantile 0,75. Quantile or sequence of quantiles to compute, which must be between 0 and 1 inclusive. • Programming languages: familiar with C++, Python and Fortran. A general method for finding confidence intervals for decision tree based methods is Quantile Regression Forests. Variable Selection. One quick use-case where this is useful is when there are a. For example, if we run a statistical analysis that assumes our dependent variable is Normally distributed, we can use a Normal Q-Q plot to check that assumption. Confidence intervals on linear regression. Quantile Regression - Theory Quantile regression - Predict the th percentile, instead of the mean, of the target variable against the covariates. , 2015) model in one of my studies. median regression, is subsumed under the term quantile regression. This gives me the normalized values. Kernel Quantile Regression with Mixed Data Types Description. Scikit-Garden depends on NumPy, SciPy, Scikit-Learn and Cython. We provide uniform convergence statements and guarantees on the quality of margins. But when you apply quantile regression to the data, you can determine which food expense can cover 90% of families (for 100 families with a given income). plotly is a high-level interface to plotly. Quantile Regression in Stata https://sites. 5/26 Quantile Regression. The idea behind quantile regression forests is simple: instead of recording the mean value of response variables in each tree leaf in the forest, record all observed responses in the leaf. My interest is to know the theoretical framework of the potential regression, since this regression model applied to the experimental data obtained in tests of metal fatigue, allows to obtain a better approximation of the variability of the data. Here are the examples of the python api statsmodels. Tree-based quantile regression models, such as the one used in this module, have the additional advantage that they can be used to predict non-parametric distributions. The only other dependency is on matplotlib, which is used to plot the data and the quantile estimates. You build logic to deal with the exceptions. This would require us to infer a nearly complete probability distribution for the ticket's fulfillment time. Using quantile regression to fit the clear sky signal in a daily solar energy data set. Quantile regression (QR) was first introduced by Roger Koenker and Gilbert Bassett in 1978. No doubt, it's fairly easy to implement. 25 quantile regression, one with fitted values from the median regression and one with fitted values from the. minimizing absolute error), and quantile regression (for estimating percentiles of the conditional distribution of the outcome). npqreg computes a kernel quantile regression estimate of a one (1) dimensional dependent variable on p-variate explanatory data, given a set of evaluation points, training points (consisting of explanatory data and dependent data), and a bandwidth specification using the method of Li and Racine (2008). In the figure given above, Q2 is the median of the normally distributed data. on the regression line. I am getting the same Convergence cycle detected warning running a quantile regression with statsmodels. LightGBM-Tutorial-and-Python-Practice On This Page. Figure 1: Basic Quantile-Quantile Plot in R. Interacts with other software such as, Python, Bioconductor, WinBUGS, JAGS etc Scope of functions, flexible, versatile etc. I composed a small Ipython notebook showing how the Python Etrics package can be used. Consider the Grad Pack Premium. Graphing the regression. Thus, half of students perform better than the median student and half perform worse. endogenous/response variable. So make sure these dependencies are installed using pip: pip install setuptools numpy scipy scikit-learn cython After that Scikit-Garden can be installed. Accordingly, this post provides a practical overview of how to handle exceptions in R by first illustrating the concept in Python. Model selection and estimation in regression with grouped variables MingYuan Georgia Institute of Technology, Atlanta, USA andYi Lin University of Wisconsin—Madison, USA [Received November 2004. About Statsmodels. Logistic regression is a frequently-used method as it enables binary variables, the sum of binary variables, or polytomous variables (variables with more than two categories) to be modeled (dependent variable). where alpha is the required quantile (a value between 0 and 1) and. The value range of τ is. Size of datasets analyzed is only limited by the machine Limitations Large online help community but no 'formal' tech support; Have to have a good understanding of different data types before real ease of use begins. ‘ls’ refers to least squares regression. I have used the python package statsmodels 0. Interacts with other software such as, Python, Bioconductor, WinBUGS, JAGS etc Scope of functions, flexible, versatile etc. Coverage includes extensive, in-depth discussion of advanced statistical techniques, data visualization, predictive analytics, and SPSS programming, including automation and integration with other languages like R and Python. Quantile Regression. In the example given, I don't see why using GB is anything but an inefficient routine for mimicking the process of a standard least-squares regression. 5 quantile, which means that the proportion 0. In addition to that,. Statsmodels is a Python package that provides a complement to scipy for statistical computations including descriptive statistics and estimation and inference for statistical models. At Instacart, quantile regression has been used to better understand and manage the risk of late deliveries. 1, so that the relationship between the. , ridge regression) for doing so. It estimates the mean value of the response variable for given levels of the predictor variables. Portfolio. Interpreting Coefficients in Regression with Log-Transformed Variables1 June 2012 Log transformations are one of the most commonly used transformations, but interpreting results of an analysis with log transformed data may be challenging. CARLIER, V. I see that when fitting the quantile regression model, there is an option to specify the significance level for confidence intervals of the regression coefficients, and the confidence interval result appears in the summary of the fit. The text: Provides an overview of several technical topics such as robustness of quantile regressions, bootstrap and elemental sets, treatment effect estimators Compares quantile regression with alternative estimators like expectiles, M-estimators and M-quantiles Offers a general introduction to linear programming focusing on the simplex method. The effectiveness of the application is however debatable. I want to quantile normalized them in R. Grows a quantile random forest of regression trees. It looks at all possible pairs of observations. Many of the problems we encounter in Econometrics can be formulated as a linear or a quadratic problem. where the big I in the formula is an indicator function, y is the residual, and τ means we want to estimate the τth quantile of y. There´s a section at the right of the Cook’s D table that calculates dfE (degrees of freedom of errors) substracting (k+1) instead of substracting k. 5 (half) will be below the median and 0. Kernel and bandwidth selection options are available for estimating the asymptotic covariance matrix using a kernel density estimator. Linear models extend beyond the mean to the median and other quantiles. 0 draft) Regression models: in text format. Hence, the corresponding variation tendencies of disparate quantile locations (containing median location) of the global ageing rate and population can be estimated by employing QRM. A new method is presented for flexible regression modeling of high dimensional data. This is likely due to a missing source distribution in PyPI, or improper metadata in this package. For a sample, you can find any quantile by sorting the sample. The Quantile Regression dialog box looks just like any native SPSS Statistics dialog box (Figure 4). out ndarray, optional. Let's return to the simplified, yet informative, example introduced above. At Instacart, quantile regression has been used to better understand and manage the risk of late deliveries. Constructive convex analysis and disciplined convex programming. Quantile regression is a type of regression analysis used in statistics and econometrics. Python code for the evaluation of linear regression and confidence intervals between two random variables x and y. geom_smooth. 5 (q argument of the. The closest thing I have been able to find is the quantregForest in R, so I wanted to bring it to python. bayesQR implements a Bayesian method for estimating quantile regression models (see references). Similarly, in > regression settings the > optimization problem solved by the "br" version of the simplex > algorithm, > modified to do general quantile regression identifies cases where > there may > be non uniqueness of this type. ~a 95% or 99% quantile) of the negative value of the returns. The model consists of an ensemble of decision trees. ‘lad’ (least absolute deviation) is a highly robust loss function solely based on order information of the input variables. Directional quantile regression in R 481 However, there exist some R packages on robust regression, data depth, regression depth, multiple-output regression, and single-response quantile regression, all of which touching. Ridge Regression. 95 quantile loss functions. I'm obtaining the models for 0. Quantile Regression lets us do just this. PROC QUANTREG. Whereas the method of least squares results in estimates of the conditional mean of the response variable given certain values of the predictor variables, quantile regression aims at estimating either the conditional median or other quantiles of the response variable. This is likely due to a missing source distribution in PyPI, or improper metadata in this package. Quantile regression is particularly useful when the rate of change in the conditional quantile, expressed by the regression coefﬁcients, depends on the quantile. Second-order derivative of quantile regression loss is equal to 0 at every point except the one where it is not defined. The median ˝=. Negative binomial regression - Negative binomial regression can be used for over-dispersed count data, that is when the conditional variance exceeds the conditional mean. Drawing them using ggvis and plotly is also possible, but demands way more code. For the sake of having them, it is beneficial to port quantile regression loss to xgboost. Plotting a graph of the regression coefficients 02 Jul 2014, 16:17. This function implements an R version of an interior point method for computing the solution to quantile regression problems which are nonlinear in the parameters. Quantile random forest is a quantile-regression method that uses a random forest of regression trees to model the conditional distribution of a response variable, given the value of predictor variables. Alternative output array in which to place the result. 0 for Quantile Regression. We'll skip Fast Forest Quantile Regression (pill) for this class because it's not a common problem that we need to solve. The survey included some statements regarding job satisfaction, some of which are shown below. Fernandez-Val. Quantile Regression. Quantile Regression lets us do just this. In this case, the regular squared norm penalty does not perform training data selection. Quantile regression is a type of regression analysis used in statistics and econometrics. 90 quantile for increasing values of x despite the increasing variability. This package also includes regression methods for least squares, absolute loss, t-distribution loss, quantile regression, logistic, multinomial logistic, Poisson, Cox proportional hazards partial likelihood, AdaBoost exponential loss, Huberized hinge loss, and Learning to Rank measures (LambdaMart). Quantile plays a very important role in Statistics when one deals with the Normal Distribution. So linear regression seem to be a nice place to start which should lead nicely on to logistic regression. Quantile regression forests give a non-parametric and accurate way of estimating conditional quantiles for high-dimensional predictor variables. Quantile regression¶ This example page shows how to use statsmodels ’ QuantReg class to replicate parts of the analysis published in. And good news! We’re open sourcing it. It appears we can make decent estimates of the 0. For Educators, Parents & Students. Quantile regression. The average loss over the entire dataset is shown below: Loss funtion. The closest thing I have been able to find is the quantregForest in R, so I wanted to bring it to python. Box whisker plots are used in stats to graphically view the spread of a data set, as well as to compare data sets. They carried out a survey, the results of which are in bank_clean. Quantile regression is an important tool for estimation of conditional quantiles of a response Y given a vector of covariates X. ‘quantile’ allows quantile regression (use alpha to specify the quantile). The only other dependency is on matplotlib, which is used to plot the data and the quantile estimates. A quantile is a fraction where certain values fall below that quantile. Quantile-Quantile Plot. Figure 1: Basic Quantile-Quantile Plot in R. regression, which is the least squares analogue of quantile regression. Conclusion. This page is intended to be a help in getting to grips with the powerful statistical program called R. By a quantile, we mean the fraction (or percent) of points below the given value. How is Quantile Regression of VaR is applied for a dynamic portfolio of 100+ instruments? How to apply autocorrelation to the model? How to apply GARCH volatility measure to the model? And most of all I am interested of which variables should be regressed? Carol Alexander gives an example of Linear Regression of VaR with:. axis {int, tuple of int, None}, optional. Similarly, the. Stata can also perform simultaneous-quantile regression. Convex optimization short course. median regression, is subsumed under the term quantile regression.