# isoreg: Isotonic / Monotone Regression

## Description

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

## Usage

 `1` ```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.

the plotting method `plot.isoreg` with more examples; `isoMDS()` from the MASS package internally uses isotonic regression.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21``` ```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)))) ```