admm_lad: Fitting A Least Absolute Deviation Model Using ADMM Algorithm
In yixuan/ADMM: Solving Statistical Optimization Problems Using the ADMM Algorithm

Description Usage Arguments Additional Options Model Fitting Author(s) Examples

View source: R/20_admm_lad.R

Least Absolute Deviation (LAD) is similar to an OLS regression model, but it minimizes the absolute deviation ||y - X * β||_1 instead of the sum of squares ||y - X * β||_2^2. LAD is equivalent to the median regression, a special case of the quantile regression models. LAD is a robust regression technique in the sense that the estimated coefficients are insensitive to outliers.

This function will not directly conduct the computation, but rather returns an object of class "ADMM_LAD" that contains several memeber functions to actually constructs and fits the model.

Member functions that are callable from this object are listed below:

`$opts()`	Setting additional options. See section Additional Options for details.
`$fit()`	Fit the model and do the actual computation. See section Model Fitting for details.

1	admm_lad(x, y, intercept = TRUE, ...)

`x`	The data matrix.
`y`	The response vector.
`intercept`	Whether to include an intercept term. Default is `TRUE`.

Additional options related to ADMM algorithm can be set through the $opts() member function of an "ADMM_LAD" object. The usage of this method is

1 2	model$opts(maxit = 10000, eps_abs = 1e-4, eps_rel = 1e-4, rho = NULL)

Here model is the object returned by admm_lad(). Explanation of the arguments is given below:

maxit: Maximum number of iterations.
eps_abs: Absolute tolerance parameter.
eps_rel: Relative tolerance parameter.
rho: ADMM step size parameter. If set to NULL, the program will compute a default one.

This member function will implicitly return the "ADMM_LAD" object itself.

Model will be fit after calling the $fit() member function. This is no argument that needs to be set. The function will return an object of class "ADMM_LAD_fit", which contains the following fields:

x: The data matrix.
y: The response vector.
beta: The estimated regression coefficients, including the intercept.
niter: Number of ADMM iterations.

Class "ADMM_LAD_fit" also contains a $plot() member function, which plots the fitted values with observed values. See the examples below.

Yixuan Qiu <http://statr.me>

## Robust regression with LAD ##

## Generate data with an outlier
set.seed(123)
x = sort(rnorm(100))
y = x + rnorm(100, sd = 0.3)
y[1] = y[1] + 5

## Build an LAD model (median regression)
model = admm_lad(x, y)

## Lower down the precision for faster computation
model$opts(eps_rel = 1e-3)

## Fit the model
res = model$fit()

## Plot for the fitted values and observed values
res$plot()

## The steps above can be accomplished using a chainable call
admm_lad(x, y)$opts(eps_rel = 1e-3)$fit()$plot()

## Compare LAD with OLS
library(ggplot2)
ols = lm(y ~ x)$coefficients
d = data.frame(intercept = c(ols[1], res$beta[1], 0),
               slope = c(ols[2], res$beta[2], 1),
               method = c("OLS", "LAD", "Truth"))
ggplot(data.frame(x = x, y = y), aes(x = x, y = y)) +
    geom_point() +
    geom_abline(aes(intercept = intercept, slope = slope, color = method),
                data = d, show_guide = TRUE)