coneA {coneproj} | R Documentation |
Cone Projection – Polar Cone
Description
This routine implements the hinge algorithm for cone projection to minimize
||y - \theta||^2
over the cone C
of the form \{\theta: A\theta \ge 0\}
.
Usage
coneA(y, amat, w = NULL, face = NULL, msg = TRUE)
Arguments
y |
A vector of length |
amat |
A constraint matrix. The rows of amat must be irreducible. The column number of amat must equal the length of |
w |
An optional nonnegative vector of weights of length |
face |
A vector of the positions of edges, which define the initial face for the cone projection. For example, when there are |
msg |
A logical flag. If msg is TRUE, then a warning message will be printed when there is a non-convergence problem; otherwise no warning message will be printed. The default is msg = TRUE |
Details
The routine coneA dynamically loads a C++ subroutine "coneACpp". The rows
of - A
are the edges of the polar cone \Omega^o
. This routine first projects y
onto \Omega^o
to get the residual of the projection onto the constraint cone C
, and then uses the fact that y
is equal to the sum of the projection of y
onto C
and the projection of y
onto \Omega^o
to get the estimation of \theta
. See references cited in this section for more details about the relationship between polar cone and constraint cone.
Value
df |
The dimension of the face of the constraint cone on which the projection lands. |
thetahat |
The projection of |
steps |
The number of iterations before the algorithm converges. |
xmat |
The rows of the matrix are the edges of the face of the polar cone on which the residual of the projection onto the constraint cone lands. |
face |
A vector of the positions of edges, which define the face on which the final projection lands on. For example, when there are |
Author(s)
Mary C. Meyer and Xiyue Liao
References
Meyer, M. C. (1999) An extension of the mixed primal-dual bases algorithm to the case of more constraints than dimensions. Journal of Statistical Planning and Inference 81, 13–31.
Meyer, M. C. (2013b) A simple new algorithm for quadratic programming with applications in statistics. Communications in Statistics 42(5), 1126–1139.
Liao, X. and M. C. Meyer (2014) coneproj: An R package for the primal or dual cone projections with routines for constrained regression. Journal of Statistical Software 61(12), 1–22.
See Also
Examples
# generate y
set.seed(123)
n <- 50
x <- seq(-2, 2, length = 50)
y <- - x^2 + rnorm(n)
# create the constraint matrix to make the first half of y monotonically increasing
# and the second half of y monotonically decreasing
amat <- matrix(0, n - 1, n)
for(i in 1:(n/2 - 1)){
amat[i, i] <- -1; amat[i, i + 1] <- 1
}
for(i in (n/2):(n - 1)){
amat[i, i] <- 1; amat[i, i + 1] <- -1
}
# call coneA
ans1 <- coneA(y, amat)
ans2 <- coneA(y, amat, w = (1:n)/n)
# make a plot to compare the unweighted fit and the weighted fit
par(mar = c(4, 4, 1, 1))
plot(y, cex = .7, ylab = "y")
lines(fitted(ans1), col = 2, lty = 2)
lines(fitted(ans2), col = 4, lty = 2)
legend("topleft", bty = "n", c("unweighted fit", "weighted fit"), col = c(2, 4), lty = c(2, 2))
title("ConeA Example Plot")