A few months ago, I posted a note with some home made codes for quantile regression… there was something odd on the output, but it was because there was a (small) mathematical problem in my equation. So since I should teach those tomorrow, let me fix them.

Median

Consider a sample \(\{y_1,\cdots,y_n\}\). To compute the median, solve\(\min_\mu \left\lbrace\sum_{i=1}^n|y_i-\mu|\right\rbrace\)which can be solved using linear programming techniques. More precisely, this problem is equivalent to\(\min_{\mu,\mathbf{a},\mathbf{b}}\left\lbrace\sum_{i=1}^na_i+b_i\right\rbrace\)with \(a_i,b_i\geq 0\) and \(y_i-\mu=a_i-b_i\), \(\forall i=1,\cdots,n\). Heuristically, the idea is to write \(y_i=\mu+\varepsilon_i\), and then define \(a_i\)‘s and \(b_i\)‘s so that \(\varepsilon_i=a_i-b_i\) and \(|\varepsilon_i|=a_i+b_i\), i.e. \(a_i=(\varepsilon_i)_+=\max\lbrace0,\varepsilon_i\rbrace=|\varepsilon|\cdot\boldsymbol{1}_{\varepsilon_i>0}\)and\(b_i=(-\varepsilon_i)_+=\max\lbrace0,-\varepsilon_i\rbrace=|\varepsilon|\cdot\boldsymbol{1}_{\varepsilon_i<0}[/latex]denote respectively the positive and the negative parts.

Unfortunately (that was the error in my previous post), the expression of linear programs is[latex display="true"]\min_{\mathbf{z}}\left\lbrace\boldsymbol{c}^\top\mathbf{z}\right\rbrace\text{ s.t. }\boldsymbol{A}\mathbf{z}=\boldsymbol{b},\mathbf{z}\geq\boldsymbol{0}\)In the equation above, with the \(a_i\)‘s and \(b_i\)‘s, we’re not far away. Except that we have \(\mu\in\mathbb{R}\), while it should be positive. So similarly, set \(\mu=\mu^+-\mu^-\) where \(\mu^+=(\mu)_+\) and \(\mu^-=(-\mu)_+\).

Thus, let\(\mathbf{z}=\big(\mu^+;\mu^-;\boldsymbol{a},\boldsymbol{b}\big)^\top\in\mathbb{R}_+^{2n+2}\)and then write the constraint as \(\boldsymbol{A}\mathbf{z}=\boldsymbol{b}\) with \(\boldsymbol{b}=\boldsymbol{y}\) and \(\boldsymbol{A}=\big[\boldsymbol{1}_n;-\boldsymbol{1}_n;\mathbb{I}_n;-\mathbb{I}_n\big]\)And for the objective function\(\boldsymbol{c}=\big(\boldsymbol{0},\boldsymbol{1}_n,-\boldsymbol{1}_n\big)^\top\in\mathbb{R}_+^{2n+2}\)

To illustrate, consider a sample from a lognormal distribution,

n = 101 
set.seed(1)
y = rlnorm(n)
median(y)
[1] 1.077415

For the optimization problem, use the matrix form, with \(3n\) constraints, and \(2n+1\) parameters,

library(lpSolve) 
X = rep(1,n) 
A = cbind(X, -X, diag(n), -diag(n))
b = y
c = c(rep(0,2), rep(1,n),rep(1,n))
equal_type = rep("=", n) 
r = lp("min", c,A,equal_type,b)
head(r$solution,1)
[1] 1.077415

It looks like it’s working well…

Quantile

Of course, we can adapt our previous code for quantiles

tau = .3
quantile(y,tau)
      30% 
0.6741586

The linear program is now\(\min_{q^+,q^-,\mathbf{a},\mathbf{b}}\left\lbrace\sum_{i=1}^n\tau a_i+(1-\tau)b_i\right\rbrace\)with \(a_i,b_i,q^+,q^-\geq 0\) and \(y_i=q^+-q^-+a_i-b_i\), \(\forall i=1,\cdots,n\). The R code is now

c = c(rep(0,2), tau*rep(1,n),(1-tau)*rep(1,n))
r = lp("min", c,A,equal_type,b)
head(r$solution,1)
[1] 0.6741586

So far so good…

Quantile Regression

Consider the following dataset, with rents of flat, in a major German city, as function of the surface, the year of construction, etc.

base=read.table("http://freakonometrics.free.fr/rent98_00.txt",header=TRUE)

The linear program for the quantile regression is now\(\min_{\boldsymbol{\beta}^+,\boldsymbol{\beta}^-,\mathbf{a},\mathbf{b}}\left\lbrace\sum_{i=1}^n\tau a_i+(1-\tau)b_i\right\rbrace\)with \(a_i,b_i\geq 0\) and \(y_i=\boldsymbol{x}^\top[\boldsymbol{\beta}^+-\boldsymbol{\beta}^-]+a_i-b_i\)\(\forall i=1,\cdots,n\) and \(\beta_j^+,\beta_j^-\geq 0\) \(\forall j=0,\cdots,k\). So use here

require(lpSolve) 
tau = .3
n=nrow(base)
X = cbind( 1, base$area)
y = base$rent_euro
K = ncol(X)
N = nrow(X)
A = cbind(X,-X,diag(N),-diag(N))
c = c(rep(0,2*ncol(X)),tau*rep(1,N),(1-tau)*rep(1,N))
b = base$rent_euro
const_type = rep("=",N)
r = lp("min",c,A,const_type,b)
beta = r$sol[1:K] -  r$sol[(1:K+K)]
beta
[1] 148.946864   3.289674

Of course, we can use R function to fit that model

library(quantreg)
rq(rent_euro~area, tau=tau, data=base)
Coefficients:
(Intercept)        area 
 148.946864    3.289674

Here again, it seems to work quite well. We can use a different probability level, of course, and get a plot

plot(base$area,base$rent_euro,xlab=expression(paste("surface (",m^2,")")),
     ylab="rent (euros/month)",col=rgb(0,0,1,.4),cex=.5)
sf=0:250
yr=r$solution[2*n+1]+r$solution[2*n+2]*sf
lines(sf,yr,lwd=2,col="blue")
tau = .9
r = lp("min",c,A,const_type,b)
tail(r$solution,2)
[1] 121.815505   7.865536
yr=r$solution[2*n+1]+r$solution[2*n+2]*sf
lines(sf,yr,lwd=2,col="blue")

And we can adapt the later to multiple regressions, of course,

X = cbind(1,base$area,base$yearc)
K = ncol(X)
N = nrow(X)
A = cbind(X,-X,diag(N),-diag(N))
c = c(rep(0,2*ncol(X)),tau*rep(1,N),(1-tau)*rep(1,N))
b = base$rent_euro
const_type = rep("=",N)
r = lp("min",c,A,const_type,b)
beta = r$sol[1:K] -  r$sol[(1:K+K)]
beta
[1] -5542.503252     3.978135     2.887234

to be compared with

library(quantreg)
rq(rent_euro~ area + yearc, tau=tau, data=base)

Coefficients:
 (Intercept)         area        yearc 
-5542.503252     3.978135     2.887234 

Degrees of freedom: 4571 total; 4568 residual