isoreg {stats}R Documentation

Isotonic / Monotone Regression


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


isoreg(x, y = NULL)


x, y in isoreg, 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

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.


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


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.


(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("R^2 =", formatC(sum(residuals(ir4)^2) / (9*var(y4)), dig=2),"\n")

[Package stats version 2.5.0 Index]