Isotonic / Monotone Regression

Description

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

Usage

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isoreg(x, y = NULL)

Arguments

x, y

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

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 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.

Note

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., 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

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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))))

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