R: Isotonic / Monotone Regression

isoreg {stats} | 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.

### 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))))

[Package stats version 3.5.1 Index]