R: Bijective Transformations of Time

timetrans {qualV} | R Documentation

## Bijective Transformations of Time

### Description

Various function models for isoton bijective transformation of a time interval to itself.

### Usage

    transBeta(x, p, interval = c(0, 1), inv = FALSE,
      pmin = -3, pmax = 3, p0 = c(0, 0))
    transSimplex(x, p, interval = c(0, 1), inv = FALSE,
      pmin = -2, pmax = 2, p0 = c(0, 0, 0, 0, 0))
    transBezier(x, p, interval = c(0, 1), inv = FALSE,
      pmin = 0, pmax = 1, p0 = c(0.25, 0.25, 0.75, 0.75))

### Arguments

`x` | a vector of values to be transformed,
`p` | the vector of parameters for the transformation,
`interval` | a vector of length 2 giving the minimum and maximum value in the transformation interval.
`inv` | a boolean, if true the inverse transform is computed.
`pmin` | a number or a vector giving the minimal useful value for the parameters. This information is not used by the function itself, but rather provides a meta information about the function used in `timeTransME`. The chosen values are quite restrictive to avoid stupid extreme transformation.
`pmax` | provides similar to `pmin` the upper useful bounds for the parameters.
`p0` | provides similar to `pmin` and `pmax` the parameterization for the identify transform.

### Details

`transBeta`

  The transformation provided is the distribution function of the Beta-Distribution with parameters `exp(p[1])` and `exp(p[2])` scaled to the given interval. This function is guaranteed to be strictly isotonic for every choice of p. p has length 2. The strength of the Beta transformation is the reasonable behavior for strong time deformations.

`transSimplex`

  The transformation provided a simple linear interpolation. The interval is separated into equidistant time spans, which are transformed to non-equidistant length. The length of the new time spans is the proportional to exp(c(p, 0)). This function is guaranteed to be strictly isotonic for every choice of p. p can have any length. The strength of the Simplex transformation is the possibility to have totally different speeds at different times.

`transBezier`

  The transformation is provided by a Bezier-Curve of order `length(p) / 2 + 1`. The first and last control point are given by `c(0, 0)` and `c(1, 1)` and the intermediate control points are given by `p[c(1, 2) + 2 * i - 2]`. This function is not guaranteed to be isotonic for `length(p) > 4`. However it seams useful. A major theoretical advantage is that this model is symmetric between image and coimage. The strength of the Bezier transformation is fine tuning of transformation.

### Value

The value is a vector of the same length as `x` providing the transformed values.

### See Also

`timeTransME`

### Examples

    t <- seq(0, 1, length.out = 101)
    par(mfrow = c(3, 3))
    plot(t, transBeta(t, c(0, 0)), type = "l")
    plot(t, transBeta(t, c(0, 1)), type = "l")
    plot(t, transBeta(t, c(-1,1)), type = "l")
    plot(t, transSimplex(t, c(0)), type = "l")
    plot(t, transSimplex(t, c(3, 2, 1)), type = "l")
    plot(t, transSimplex(t, c(0, 2)), type = "l")
    plot(t, transBezier(t, c(0, 1)), type = "l")
    plot(t, transBezier(t, c(0, 1, 1, 0)), type = "l")
    plot(t, transBezier(t, c(0.4, 0.6)), type = "l")

[Package qualV version 0.3-3 Index]