stats (version 3.3)

isoreg: Isotonic / Monotone Regression

Description

Compute the isotonic (monotonely increasing nonparametric) least squares regression which is piecewise constant.

Usage

isoreg(x, y = NULL)

Arguments

x, y

coordinate vectors of the regression points. Alternatively a single plotting structure can be specified: see xy.coords.

Value

• isoreg() returns an object of class isoreg which is basically a list with components
• x
  original (constructed) abscissa values x.
• y
  corresponding y values.
• yf
  fitted values corresponding to ordered x values.
• yc
  cumulative y values corresponding to ordered x values.
• iKnots
  integer vector giving indices where the fitted curve jumps, i.e., where the convex minorant has kinks.
• isOrd
  logical indicating if original x values were ordered increasingly already.
• ord
  if(!isOrd): integer permutation order(x) of original x.
• call
  the call to isoreg() used.

concept

monotonic regression

Details

The algorithm determines the convex minorant $m(x)$ of the cumulative data (i.e., cumsum(y)) which is piecewise linear and the result is $m'(x)$, a step function with level changes at locations where the convex $m(x)$ touches the cumulative data polygon and changes slope. as.stepfun() returns a stepfun object which can be more parsimonious.

References

Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D. (1972) Statistical inference under order restrictions; Wiley, London.

Robertson, T., Wright, F. T. and Dykstra, R. L. (1988) Order Restricted Statistical Inference; Wiley, New York.

See Also

the plotting method plot.isoreg with more examples; isoMDS() from the MASS package internally uses isotonic regression.

Examples

require(graphics)

(ir <- isoreg(c(1,0,4,3,3,5,4,2,0)))
plot(ir, plot.type = "row")

(ir3 <- isoreg(y3 <- c(1,0,4,3,3,5,4,2, 3))) # last "3", not "0"
(fi3 <- as.stepfun(ir3))
(ir4 <- isoreg(1:10, y4 <- c(5, 9, 1:2, 5:8, 3, 8)))
cat(sprintf("R^2 = %.2f
",
            1 - sum(residuals(ir4)^2) / ((10-1)*var(y4))))

## If you are interested in the knots alone :
with(ir4, cbind(iKnots, yf[iKnots]))

## Example of unordered x[] with ties:
x <- sample((0:30)/8)
y <- exp(x)
x. <- round(x) # ties!
plot(m <- isoreg(x., y))
stopifnot(all.equal(with(m, yf[iKnots]),
                    as.vector(tapply(y, x., mean))))