| isoreg {stats} | R Documentation |
Isotonic / Monotone Regression
Description
Compute the isotonic (monotonically 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 |
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.
Value
isoreg() returns an object of class isoreg which is
basically a list with components
x |
original (constructed) abscissa values |
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 |
|
call |
the |
Note
The inputs can be long vectors, but iKnots will wrap around at
2^{31}.
The code should be improved to accept weights additionally and
solve the corresponding weighted least squares problem.
‘Patches are welcome!’
References
Barlow R. E., Bartholomew D. J., Bremner J. M., Brunk H. D. (1972). Statistical Inference Under Order Restrictions: Theory and Application of Isotonic Regression. Wiley, London. ISBN 978-0471049708.
Robertson T., Wright F. T., Dykstra R. L. (1988). Order Restricted Statistical Inference. Wiley, Chicester. ISBN 0471917877.
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\n",
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))))