BoundedAntiMeanTwo, BoundedIsoMeanTwo {OrdMonReg} | R Documentation |
See details below.
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)
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. |
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 isotonic (antitonic) regression curves g_1^circ and g_2^circ under the constraint that g_1^circ <= g_2^circ. Given two sets of n data points y_1, ..., y_n and z_1, ..., z_n that are observed at (the same) deterministic design points x_1, ..., 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 <= b_i for all i = {1, ..., 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.
g1 |
The estimated function hat g_1^circ. |
g2 |
The estimated function hat g_2^circ. |
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 monotone 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 ## -------------------------------------------------------- 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("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)