L1linreg {pracma}

L1 Linear Regression

Description

Solve the linear system A x = b in an Lp sense, that is minimize the term sum |b - A x|^p. The case p=1 is also called “least absolute deviation” (LAD) regression.

Usage

L1linreg(A, b, p = 1, tol = 1e-07, maxiter = 200)

Arguments

A

matrix of independent variables.

b

independent variables.

p

the p in L^p norm, p<=1.

tol

relative tolerance.

maxiter

maximum number of iterations.

Details

L1/Lp regression is here solved applying the “iteratively reweighted least square” (IRLS) method in which each step involves a weighted least squares problem.

If an intercept term is required, add a unit column to A.

Value

Returns a list with components x the linear coefficients describing the solution, reltol the relative tolerance reached, and niter the number of iterations.

Note

In this case of p=1, the problem would be better approached by use of linear programming methods.

References

Dasgupta, M., and S.K. Mishra (2004). Least absolute deviation estimation of linear econometric models: A literature review. MPRA Paper No. 1781.

See Also

lm, lsqnonlin, quantreg::rq

Examples

m <- 101; n <- 10       # no. of data points, degree of polynomial
x <- seq(-1, 1, len=m)
y <- runge(x)           # Runge's function
A <- outer(x, n:0, '^') # Vandermonde matrix
b <- y

( sol <- L1linreg(A, b) )
# $x
# [1] -21.93242   0.00000  62.91092   0.00000 -67.84854   0.00000
# [7]  34.14400   0.00000  -8.11899   0.00000   0.84533
# 
# $reltol
# [1] 6.712355e-10
# 
# $niter
# [1] 81

# minimum value of polynomial L1 regression
sum(abs(polyval(sol$x, x) - y))
# [1] 3.061811