BoundedAntiMeanTwo: Compute solution to the problem of two ordered isotonic or...

In OrdMonReg: Compute least squares estimates of one bounded or two ordered isotonic regression 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 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?

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 ≤ 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) = ∑_{i=1}^n (y_i - a_i)^2 w_{1,i} + ∑_{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.

Value

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.

Author(s)

Fadoua Balabdaoui [e-mail]
http://www.ceremade.dauphine.fr/~fadoua

Kaspar Rufibach (maintainer) [e-mail]
http://www.kasparrufibach.ch

Filippo Santambrogio [e-mail]
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)

OrdMonReg documentation built on May 2, 2019, 1:45 p.m.