frame_fn_obs: Visualization of quantile regression model fitting: interior...

In quokar: Quantile Regression Outlier Diagnostics with K Left Out Analysis

Description

observations used in quantile regression fitting

min_{b \in R^{p}}∑_{i=1}^{n}ρ_{τ}(y_i-x_{i}^{'}b)

where ρ_{τ}(r)=r[τ-I(r<0)]$ for $τ \in (0,1). This yields the modified linear program

min(τ e^{'}u+(1-τ)e^{'}v|y=Xb+u-v,(u,v) \in R_{+}^{2n})

Adding slack variables, s, satisfying the constrains a+s=e, we obtain the barrier function

B(a, s, u) = y^{'}a+μ ∑_{i=1}^{n}(loga_{i}+logs_{i})

which should be maximized subject to the constrains X^{'}a=(1-τ)X^{'}e and a+s=e. The Newton step δ_{a} solving

max{y^{'}δ_{a}+μ δ^{'}_{a}(A^{-1}-S^{-1})e- \frac{1}{2}μ δ_{a}^{'}(A^{-2}+S^{-2})δ_{a}}

subject to X{'}δ_{a}=0, satisfies

y+μ(A^{-1}-S^{-1})e-μ(A^{-2}+S^{-2})δ_{a}=Xb

for some b\in R^{p}, and δ_{a} such that X^{'}δ_{a}=0. Using the constraint, we can solve explicitly for the vector b,

b=(X^{'}WX)^{-1}X^{'}W[y+μ(A^{-1}-S^{-1})e]

where W=(A^{-2}+S^{-2})^{-1}. This is a form of the primal log barrier algorithm described above. Setting μ=0 in each step yields an affine scaling variant of the algorithm. The basic linear algebra of each iteration is essentially unchanged, only the form of the diagonal weighting matrix W has chagned.

Usage

frame_fn_obs(object, tau)

Arguments

object	quantile regression model using interior point method for estimating
tau	quantile

Details

This function used to illustrate data used in fitting process of quantile regression based on interior point method. Koenker and Bassett(1978) introduced asymmetric weight on positive and negative residuals, and solves the slightly modified l1-problem.

Value

Weighted observations in quantile regression fitting using interior point algorithm

Author(s)

Wenjing Wang [e-mail]

References

Portnoy S, Koenker R. The Gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators. Statistical Science, 1997, 12(4): 279-300.

Examples

library(ggplot2)
library(quantreg)
library(tidyr)
library(dplyr)
library(gridExtra)
data(ais)
tau <- c(0.1, 0.5, 0.9)
object <- rq(BMI ~ LBM + Ht, data = ais, tau = tau, method = 'fn')
fn <- frame_fn_obs(object, tau)
##For tau = 0.1, plot the observations used in quantile regression
##fitting based on interior point method
fn1 <- fn[ ,1]
case <- 1:length(fn1)
fn1 <- cbind(case, fn1)
m <- data.frame(y = ais$BMI, x1 = ais$LBM, x2 = ais$Ht, fn1)
p <- length(attr(object$coefficients, "dimnames")[[1]])
m_f <- m %>% gather(variable, value, -case, -fn1, -y)
mf_a <- m_f %>%
 group_by(variable) %>%
 arrange(variable, desc(fn1)) %>%
 filter(row_number() %in% 1:p )
p1 <- ggplot(m_f, aes(x = value, y = y)) +
 geom_point(alpha = 0.1) +
 geom_point(data = mf_a, size = 3) +
 facet_wrap(~variable, scale = "free_x")
## For tau = 0.5, plot the observations used in quantile regression
##fitting based on interior point method
fn2 <- fn[,2]
case <- 1: length(fn2)
fn2 <- cbind(case, fn2)
m <- data.frame(y = ais$BMI, x1 = ais$LBM, x2 = ais$Ht, fn2)
p <- length(attr(object$coefficients, "dimnames")[[1]])
m_f <- m %>% gather(variable, value, -case, -fn2, -y)
mf_a <- m_f %>%
   group_by(variable) %>%
   arrange(variable, desc(fn2)) %>%
   filter(row_number() %in% 1:p )
p2 <- ggplot(m_f, aes(x = value, y = y)) +
   geom_point(alpha = 0.1) +
   geom_point(data = mf_a, size = 3) +
   facet_wrap(~variable, scale = "free_x")
## For tau = 0.9
fn3 <- fn[,3]
case <- 1: length(fn3)
fn3 <- cbind(case, fn3)
m <- data.frame(y = ais$BMI, x1 = ais$LBM, x2 = ais$Ht, fn3)
p <- length(attr(object$coefficients, "dimnames")[[1]])
m_f <- m %>% gather(variable, value, -case, -fn3, -y)
mf_a <- m_f %>%
  group_by(variable) %>%
  arrange(variable, desc(fn3)) %>%
  filter(row_number() %in% 1:p )
p3 <- ggplot(m_f, aes(x = value, y = y)) +
  geom_point(alpha = 0.1) +
  geom_point(data = mf_a, size = 3) +
  facet_wrap(~variable, scale = "free_x")
grid.arrange(p1, p2, p3, ncol = 1)

quokar documentation built on May 2, 2019, 6:39 a.m.