BoundedAntiMeanTwo, BoundedIsoMeanTwo {OrdMonReg} | R Documentation |
See details below.
BoundedAntiMeanTwo(g1, w1, g2, w2, K1 = 1000, K2 = 400, delta = 10^(-4), errorPrec = 10, output = TRUE) BoundedIsoMeanTwo(g1, w1, g2, w2, K1 = 1000, K2 = 400, delta = 10^(-4), errorPrec = 10, output = TRUE)
g1 |
Vector in R^n, measurements of upper function. |
w1 |
Vector in R^n, weights for upper function. |
g2 |
Vector in R^n, measurements of lower function. |
w2 |
Vector in R^n, weights for lower function. |
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. See Table 1 in Balabdaoui et al. (2009). |
errorPrec |
Computation of stopping criterion is expensive. Therefore, the stopping criterion is
only evaluated at every errorPrec -th iteration of the algorithm. |
output |
Should intermediate results be output? |
We consider the problem of estimating two antitonic (isotonic) regression curves g_1^* and g_2^* under the constraint that g_1^* >= g_2^*. Given two sets of n data points g_1(x_1), ..., g_1(x_n) and g_2(x_1), ..., g_2(x_n) that are observed at (the same) deterministic points x_1, ..., x_n with weight functions w_1 and w_2, respectively, the estimates are obtained by minimizing the Least Squares criterion
L(f_1, f_2) = sum_{i=1}^n (g_1(x_i) - f_1(x_i))^2 w_1(x_i)+ sum_{i=1}^n (g_2(x_i) - f_2(x_i))^2 w_2(x_i)
over the class of pairs of functions (f_1, f_2) such that f_1 and f_2 are antitonic and f_1(x_i) >= f_2(x_i) for all i = {1, ..., n}. The estimates are computed with an projected subgradient algorithm where the projection is calculated using a suitable version of the pool-adjacent-violaters algorithm (PAVA).
The function BoundedIsoMeanTwo
solves the same problem for isotonic curves, by simply invoking
BoundedAntiMeanTwo
and suitably flipping some of the arguments.
g1 |
The estimated function g_1^*. |
g2 |
The estimated function g_2^*. |
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. |
Fadoua Balabdaoui fadoua@ceremade.dauphine.fr
http://www.ceremade.dauphine.fr/~fadoua
Kaspar Rufibach (maintainer) kaspar.rufibach@ifspm.uzh.ch
http://www.biostat.uzh.ch/aboutus/people/rufibach.html
Filippo Santambrogio filippo@ceremade.dauphine.fr
http://www.ceremade.dauphine.fr/~filippo
Balabdaoui, F., Rufibach, K., Santambrogio, F. (2009). Least squares estimation of two ordered antitonic regression curves. Preprint.
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
.
## ======================================================== ## 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 ## -------------------------------------------------------- res <- BoundedAntiMeanTwo(g1, w1, g2, w2, errorPrec = 20) ## corresponding isotonic problem res2 <- BoundedIsoMeanTwo(-g1, w1, -g2, w2, errorPrec = 20) ## the following vectors are equal res$g1 - -res2$g2 res$g2 - -res2$g1 ## -------------------------------------------------------- ## Checking of solution ## -------------------------------------------------------- # This compares the first component of res$g1 with a^*_1: c(res$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(res$g1, res$g2, g1, g2)), xlab = expression(x), ylab = "measurements and estimates") points(x, g2, col = 3) lines(x, res$g1 + 0.01, col = 2, type = 's', lwd = 2) lines(x, res$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) ## for quick analysis only use randomly chosen 200 observations set.seed(23041977) mechIng <- mechIng[sort(sample(1:dim(mechIng)[1])[1:200]), ] x <- mechIng$x n <- length(x) g1 <- mechIng$g1 g2 <- mechIng$g2 w1 <- rep(1, n) w2 <- w1 ## -------------------------------------------------------- ## compute unordered estimates ## -------------------------------------------------------- g1_pava <- BoundedIsoMean(y = g1, w = w1, a = NA, b = NA) g2_pava <- BoundedIsoMean(y = g2, w = w2, a = NA, b = NA) ## -------------------------------------------------------- ## compute estimates ## -------------------------------------------------------- res1 <- BoundedIsoMeanTwo(g1, w1, g2, w2, K1 = 1000, K2 = 400, delta = 10^-4, errorPrec = 20, output = TRUE) ## -------------------------------------------------------- ## compute smoothed versions ## -------------------------------------------------------- g1_mon <- res1$g1 g2_mon <- res1$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("lower 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)