coneproj (version 1.18)

coneA: 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)

Value

df

The dimension of the face of the constraint cone on which the projection lands.

thetahat

The projection of \(y\) on the constraint cone.

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 \(m\) cone edges, then face is a subset of \(1,\ldots,m\).

Arguments

y

A vector of length \(n\).

amat

A constraint matrix. The rows of amat must be irreducible. The column number of amat must equal the length of \(y\).

w

An optional nonnegative vector of weights of length \(n\). If w is not given, all weights are taken to equal 1. Otherwise, the minimization of \((y - \theta)'w(y - \theta)\) over \(C\) is returned. The default is w = NULL.

face

A vector of the positions of edges, which define the initial face for the cone projection. For example, when there are \(m\) cone edges, then face is a subset of \(1,\ldots,m\). The default is face = NULL.

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

Author

Mary C. Meyer and Xiyue Liao

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.

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

coneB, constreg, qprog

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")