# isoreg: Isotonic / Monotone Regression

 isoreg R Documentation

## Isotonic / Monotone Regression

### Description

Compute the isotonic (monotonely 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 `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.

### 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))))
```