| BoundedAntiMeanTwo, BoundedIsoMeanTwo {OrdMonReg} | R Documentation |
Compute solution to the problem of two ordered isotonic or antitonic curves
Description
See details below.
Usage
BoundedIsoMeanTwo(g1, w1, g2, w2, K1 = 1000, K2 = 400,
delta = 10^(-4), errorPrec = 10, output = TRUE)
BoundedAntiMeanTwo(g1, w1, g2, w2, K1 = 1000, K2 = 400,
delta = 10^(-4), errorPrec = 10, output = TRUE)
Arguments
g1 |
Vector in |
w1 |
Vector in |
g2 |
Vector in |
w2 |
Vector in |
K1 |
Upper bound on number of iterations. |
K2 |
Number of iterations where step length is changed from the inverse of the norm of the subgradient to a diminishing function of the norm of the subgradient. |
delta |
Upper bound on the error, defines stopping criterion. |
errorPrec |
Computation of stopping criterion is expensive. Therefore, the stopping criterion is
only evaluated at every |
output |
Should intermediate results be output? |
Details
We consider the problem of estimating two isotonic (antitonic) regression curves g_1^\circ and
g_2^\circ under the
constraint that g_1^\circ \le g_2^\circ. Given two sets of n data points y_1, \ldots, y_n
and z_1, \ldots, z_n
that are observed at (the same) deterministic design points x_1, \ldots, x_n with weights
w_{1,i} and w_{2,i}, respectively, the estimates are obtained by
minimizing the Least Squares criterion
L_2(a, b) = \sum_{i=1}^n (y_i - a_i)^2 w_{1,i} + \sum_{i=1}^n (z_i - b_i)^2 w_{2,i}
over the class of pairs of vectors (a, b) such that a and b are isotonic (antitonic) and
a_i \le b_i for all i = {1, \ldots, n}. The estimates are computed with a projected
subgradient algorithm where the projection is calculated using a suitable version of the pool-adjacent-violaters
algorithm (PAVA).
The algorithm is implemented for antitonic curves in the function BoundedAntiMeanTwo.
The function BoundedIsoMeanTwo solves the same problem for isotonic curves, by simply invoking
BoundedAntiMeanTwo and suitably flipping some of the arguments.
Value
g1 |
The estimated function |
g2 |
The estimated function |
L |
Value of the least squares criterion at the minimum. |
error |
Value of error. |
k |
Number of iterations performed. |
tau |
Step length at final iteration. |
Author(s)
Fadoua Balabdaoui fadoua@ceremade.dauphine.fr
http://www.ceremade.dauphine.fr/~fadoua
Kaspar Rufibach (maintainer) kaspar.rufibach@gmail.com
http://www.kasparrufibach.ch
Filippo Santambrogio filippo.santambrogio@math.u-psud.fr
http://www.math.u-psud.fr/~santambr/
References
Balabdaoui, F., Rufibach, K., Santambrogio, F. (2009). Least squares estimation of two ordered monotone regression curves. Preprint.
See Also
The functions BoundedAntiMean and BoundedIsoMean for the problem of
estimating one antitonic (isotonic) regression
function bounded above and below by fixed functions. The function BoundedAntiMeanTwo depends
on the functions BoundedAntiMean, bstar_n,
LSfunctional, and Subgradient.
Examples
## ========================================================
## The first example uses simulated data
## For the analysis of the mechIng dataset see below
## ========================================================
## --------------------------------------------------------
## initialization
## --------------------------------------------------------
set.seed(23041977)
n <- 100
x <- 1:n
g1 <- 1 / x^2 + 2
g1 <- g1 + 3 * rnorm(n)
g2 <- 1 / log(x+3) + 2
g2 <- g2 + 4 * rnorm(n)
w1 <- runif(n)
w1 <- w1 / sum(w1)
w2 <- runif(n)
w2 <- w2 / sum(w2)
## --------------------------------------------------------
## compute estimates
## --------------------------------------------------------
shor <- BoundedAntiMeanTwo(g1, w1, g2, w2, errorPrec = 20,
delta = 10^(-10))
## corresponding isotonic problem
shor2 <- BoundedIsoMeanTwo(-g2, w2, -g1, w1, errorPrec = 20,
delta = 10^(-10))
## the following vectors are equal
shor$g1 - -shor2$g2
shor$g2 - -shor2$g1
## --------------------------------------------------------
## for comparison, compute estimates via cyclical projection
## algorithm due to Dykstra (1983) (isotonic problem)
## --------------------------------------------------------
dykstra1 <- BoundedIsoMeanTwoDykstra(-g2, w2, -g1, w1,
delta = 10^(-10))
## the following vectors are equal
shor2$g1 - dykstra1$g1
shor2$g2 - dykstra1$g2
## --------------------------------------------------------
## Checking of solution
## --------------------------------------------------------
# This compares the first component of shor$g1 with a^*_1:
c(shor$g1[1], astar_1(g1, w1, g2, w2))
## --------------------------------------------------------
## plot original functions and estimates
## --------------------------------------------------------
par(mfrow = c(1, 1), mar = c(4.5, 4, 3, 0.5))
plot(x, g1, col = 2, main = "Original observations and estimates in problem
two ordered antitonic regression functions", xlim = c(0, max(x)), ylim =
range(c(shor$g1, shor$g2, g1, g2)), xlab = expression(x),
ylab = "measurements and estimates")
points(x, g2, col = 3)
lines(x, shor$g1 + 0.01, col = 2, type = 's', lwd = 2)
lines(x, shor$g2 - 0.01, col = 3, type = 's', lwd = 2)
legend("bottomleft", c(expression("upper estimated function g"[1]*"*"),
expression("lower estimated function g"[2]*"*")), lty = 1, col = 2:3,
lwd = 2, bty = "n")
## ========================================================
## Analysis of the mechIng dataset
## ========================================================
## --------------------------------------------------------
## input data
## --------------------------------------------------------
data(mechIng)
x <- mechIng$x
n <- length(x)
g1 <- mechIng$g1
g2 <- mechIng$g2
w1 <- rep(1, n)
w2 <- w1
## --------------------------------------------------------
## compute unordered estimates
## --------------------------------------------------------
g1_pava <- BoundedIsoMean(y = g1, w = w1, a = NA, b = NA)
g2_pava <- BoundedIsoMean(y = g2, w = w2, a = NA, b = NA)
## --------------------------------------------------------
## compute estimates via cyclical projection algorithm due to
## Dysktra (1983)
## --------------------------------------------------------
dykstra1 <- BoundedIsoMeanTwoDykstra(g1, w1, g2, w2,
delta = 10^-10, output = TRUE)
## --------------------------------------------------------
## compute smoothed versions
## --------------------------------------------------------
g1_mon <- dykstra1$g1
g2_mon <- dykstra1$g2
kernel <- function(x, X, h, Y){
tmp <- dnorm((x - X) / h)
res <- sum(Y * tmp) / sum(tmp)
return(res)
}
h <- 0.1 * n^(-1/5)
g1_smooth <- rep(NA, n)
g2_smooth <- g1_smooth
for (i in 1:n){
g1_smooth[i] <- kernel(x[i], X = x, h, g1_mon)
g2_smooth[i] <- kernel(x[i], X = x, h, g2_mon)
}
## --------------------------------------------------------
## plot original functions and estimates
## --------------------------------------------------------
par(mfrow = c(2, 1), oma = c(0, 0, 2, 0), mar = c(4.5, 4, 2, 0.5),
cex.main = 0.8, las = 1)
plot(0, 0, type = 'n', xlim = c(0, max(x)), ylim =
range(c(g1, g2, g1_mon, g2_mon)), xlab = "x", ylab =
"measurements and estimates", main = "ordered antitonic estimates")
points(x, g1, col = grey(0.3), pch = 20, cex = 0.8)
points(x, g2, col = grey(0.6), pch = 20, cex = 0.8)
lines(x, g1_mon + 0.1, col = 2, type = 's', lwd = 3)
lines(x, g2_mon - 0.1, col = 3, type = 's', lwd = 3)
legend(0.2, 10, c(expression("upper isotonic function g"[1]*"*"),
expression("lower isotonic function g"[2]*"*")), lty = 1, col = 2:3,
lwd = 3, bty = "n")
plot(0, 0, type = 'n', xlim = c(0, max(x)), ylim =
range(c(g1, g2, g1_mon, g2_mon)), xlab = "x", ylab = "measurements and
estimates", main = "smoothed ordered antitonic estimates")
points(x, g1, col = grey(0.3), pch = 20, cex = 0.8)
points(x, g2, col = grey(0.6), pch = 20, cex = 0.8)
lines(x, g1_smooth + 0.1, col = 2, type = 's', lwd = 3)
lines(x, g2_smooth - 0.1, col = 3, type = 's', lwd = 3)
legend(0.2, 10, c(expression("upper isotonic smoothed function "*tilde(g)[1]*"*"),
expression("lower isotonic smoothed function "*tilde(g)[2]*"*")),
lty = 1, col = 2:3, lwd = 3, bty = "n")
par(cex.main = 1)
title("Original observations and estimates in mechanical engineering example",
line = 0, outer = TRUE)