landmarkreg: Landmark Registration of Functional Observations with...

In fda: Functional Data Analysis

Landmark Registration of Functional Observations with Differing Ranges

Description

It is common to see that among a set of functions certain prominent features such peaks and valleys, called $landmarks$, do not occur at the same times, or other argument values. This is called $phase variation$, and it can be essential to align these features before proceeding with further functional data analyses.

This function uses the timings of these features to align or register the curves. The registration involves estimating a nonlinear transformation of the argument continuum for each functional observation. This transformation is called a warping function. It must be strictly increasing and smooth.

Warning: As of March 2022, landmark registration cannot be done using function smooth.basis instead of function smooth.morph. The warping function must be strictly monotonic, and we have found that using smooth.basis too often violates this contraint. Function smooth.morph ensures monotonicity.

Usage

landmarkreg(unregfd, ximarks, x0marks, x0lim, 
             WfdPar=NULL, WfdPar0=NULL, ylambda=1e-10)

Arguments

unregfd	a functional data object containing the curves to be registered.
ximarks	A matrix containing the timings or argument values associated with the landmarks for the observations in fd to be registered. The number of rows N equals the number of observations, and the number of columns NL equals the number of landmarks. These landmark times must be in the interior of the interval over which the functions are defined. Object ximarks may also be a vector or a data.frame.
x0marks	a vector of length NL of times of landmarks for target curve. If not supplied, the mean of the landmark times in ximarks is used. Object x0marks may also be a vector.
x0lim	A vector of length 2 containing the lower and upper boundary of the interval containing x0marks.
WfdPar	a functional parameter object defining the warping functions that transform time in order to register the curves.
WfdPar0	a functional parameter object defining the inverse warping functions that transform time in order to register the curves.
ylambda	Smoothing parameter controlling the smoothness of the registered functions. It can happen with high dimensional bases that local wiggles can appear in the registered curves or their derivatives that are not seen in the unregistered versions. In this case, this parameter should be increased to the point where they disappear.

Details

A value of an arbitrary strictly monotone function at a point x can be defined as the indefinite integral from a fixed lower boundary to x of the exponential of an unconstrained function value W(x).

We use B-spline basis functions to define function W, and optimize the coefficients of its B-spline expansion with respect to the data we are fitting. The optimized core function W is output along with the registered functions, the warping function qnd the inverse warping function.

It is essential that the location of every landmark be clearly defined in each of the curves as well as the template function. If this is not the case, consider using the continuous registration function register.fd.

Not much curvature is usually required in the warping functions, so a rather lower power basis, usually B-splines, is suitable for defining the functional parameter argument WfdPar. A registration with a few prominent landmarks is often a good preliminary to using the more sophisticated but more lengthy process in register.fd.

Value

a named list of length 4 with components:

fdreg	a functional data object for the registered curves.
warpfd	a functional data object for the warping functions.
warpinvfd	a functional data object for the inverse warping functions.
Wfd	a functional data object for the core function defining the strictly monotone warping function.

References

Ramsay, James O., Hooker, Giles, and Graves, Spencer (2009), Functional data analysis with R and Matlab, Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2005), Functional Data Analysis, 2nd ed., Springer, New York.

Ramsay, James O., and Silverman, Bernard W. (2002), Applied Functional Data Analysis, Springer, New York.

See Also

register.fd, landmarkreg, smooth.morph

fda documentation built on Sept. 30, 2024, 9:19 a.m.