ms.rep: Mean shift procedures.

In LPCM: Local Principal Curve Methods

Mean shift procedures.

Description

Functions for mean shift, iterative mean shift, and inverse mean shift.

Usage

meanshift(X, x, h)
ms.rep(X, x, h, thresh= 0.0001, iter=200)
ms.rep.min(X, x, h, thresh=0.000001, iter=200, adjust.convergence=FALSE, verbose=TRUE)

Arguments

X	data matrix or vector.
x	point from which we wish to shift to the local mean.
h	scalar or vector-valued bandwidth; see also description in ms.
thresh, iter	mean shift iterations are stopped when the mean shift length (relative to the distance of of x to the overall mean; see Note section) falls below thresh, or after iter iterations (whatever event happens first).
adjust.convergence	Only required for estimation of antimodes via ms.rep.min. See also Note section below.
verbose	Allows or suppresses text output messages.

Details

The function meanshift computes a single mean shift iteration, and ms.rep computes an iterative series of mean shift iterations. Both of these functions are rarely used on their own, but are typically called by the overarching function ms.

The function ms.rep.min implements an inverse version of the mean shift procedure which can be used for the computation of antimodes (minima of the density). The methodology has been presented in the univariate setting in Ameijeiras-Alonso and Einbeck (2023). The function has been tested to a moderate extent in two dimensions, and it can be expected to work here. It has not been tested for higher dimensions. That is, for data of dimension 3 or higher, the function may or may not produce a result which may or may not correspond to an antimode.

Value

The function meanshift delivers a single (vector-valued) value.

The functions ms.rep and ms.rep.min produce a list with the following items:

Meanshift.points	(called M for ms.rep.min); the trajectory of points found while proceeding from the starting value x to the mode (or antimode, respectively)
Threshold.values	These give the iteration-wise values of the relative length of the mean shift step (explained in the Note section) which are then compared to thresh.
start	The starting value.
final	The mode or antimode, respectively.

Note

The threshold thresh for stopping mean shift iterations works as follows. At each iteration, we compare the length of the mean shift, that is the Euclidean distance between the point x and its local mean m, to the Euclidean distance between the point x and the overall data mean. If this distance falls below thresh, the mean shift procedure is stopped.

When ms.rep is called by function ms, the relation of the thresholds thr and thresh is thresh = thr^2.

Convergence of the inverse mean shift algorithm is not mathematically guaranteed. Of course, no antimode will be found if there is none (i.e., if the value x is not situated between two modes), in which case the method will return a NA. But the algorithm may also fail to converge if the antimode is associated with a very small density value, which however rarely happens in practice unless the two distributions are fully separated. In this case the inverse mean shift algorithm will oscillate around the antimode. By reducing the step length successively once that such a situation is identified, convergence can still be ensured algorithmically. This adjustment is activated if the option check.convergence is set equal to TRUE. This functionality is experimental, and details are to be reported elsewhere.

Author(s)

J. Einbeck. See LPCM-package for further acknowledgements.

References

Ameijeiras-Alonso, J. and Einbeck, J. (2023). A fresh look at mean-shift based modal clustering. Advances in Data Analysis and Classification, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11634-023-00575-1")}.

See Also

ms

Examples

data(stamps, package="multimode")
h0 <- 0.005
hist(stamps, breaks=20)
# Take arbitrary starting value x=0.08, sitting between a mode and antimode
mode <- ms.rep(stamps, 0.08,h0)$final
antimode <- ms.rep.min(stamps, 0.08,h0, verbose=FALSE)$final
segments(mode, 0, mode, 100, col=2, lwd=3)
segments(antimode, 0, antimode,100, col=3, lwd=3)

LPCM documentation built on Sept. 11, 2024, 7:53 p.m.