Isotonic / Monotone Regression

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

regression, smooth
isoreg(x, y = NULL)
x, y
coordinate vectors of the regression points. Alternatively a single plotting structure can be specified: see xy.coords.

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.


  • isoreg() returns an object of class isoreg which is basically a list with components
  • xoriginal (constructed) abscissa values x.
  • ycorresponding y values.
  • yffitted values corresponding to ordered x values.
  • yccumulative y values corresponding to ordered x values.
  • iKnotsinteger vector giving indices where the fitted curve jumps, i.e., where the convex minorant has kinks.
  • isOrdlogical indicating if original x values were ordered increasingly already.
  • ordif(!isOrd): integer permutation order(x) of original x.
  • callthe call to isoreg() used.


The code should be improved to accept weights additionally and solve the corresponding weighted least squares problem. Patches are welcome!


monotonic regression


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.

  • isoreg
library(stats) 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))))
Documentation reproduced from package stats, version 3.3, License: Part of R 3.3

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