timetrans {qualV}

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")