ms {LPCM}

Mean shift clustering.

Description

Function for mean shift clustering, which, for a given bandwidth, detects the local modes and performs the clustering.

Usage

ms(X, h, subset,  thr=0.01, scaled= 1, iter=200, plot=TRUE, ...)

Arguments

X

data matrix or vector.

h

scalar or vector-valued bandwidth (by default, 5 percent of the data range, or 20 percent of the standard deviation, respectively, in each direction). If set manually and scaled>0, this bandwidth needs to be set on the scaled scale; for instance setting scale; for instance scaled=1 and h=0.10 will use a bandwidth of 10 percent of the data range in either direction.

subset

vector specifying a subset of 1:n, where n is the sample size. This allows to run the iterative mean shift procedure only from a subset of points (if unspecified, 1:n is used here, i.e. each data point serves as a starting point).

thr

adjacent mean shift clusters are merged if their relative distance falls below this threshold (see Note section).

scaled

if equal to 1 (default), each variable is divided by its range, and if equal to 2 (or any other positive value other than 1), each variable is divided by its standard deviation. If equal to 0, then no scaling is applied.

iter

maximum mean shift iterations (passed to ms.rep).

plot

if equal to 0, then no plotted output. For bivariate data, plot=1 gives by default a dynamically created color plot showing the mean shift trajectories and the resulting clustering.

...

further graphical parameters.

Details

The methods implemented here can be used for density mode estimation, clustering, and the selection of starting points for the LPC algorithm. They are based on Almeijeiras-Alonso and Einbeck (2023).

It can be shown (Chen, 1995, Comaniciu & Meer, 2002, Li, 2005) that, if the mean shift is computed iteratively, the resulting sequence of local means converges to a mode of the estimated density function. By assigning each data point to the mode to which it has converged, this turns into a clustering technique.

Value

The function ms produces an object of class ms, with components:

cluster.center

a matrix which gives the coordinates of the estimated density modes (i.e., of the mean-shift based cluster centers).

cluster.label

assigns each data point to the cluster center to which its mean shift trajectory has converged.

closest.label

assigns each data point to the closest cluster center in terms of Euclidean distance.

data

the data frame (scaled if scaled=TRUE).

scaled

the user-supplied value, could be boolean or numerical.

scaled.by

the data were scaled by dividing each variable through the values provided in this vector.

Note

All values provided in the output refer to the scaled data, unless scaled=0 or (equivalently) scaled=FALSE.

The default option scaled=1 or scaled=TRUE scales the data by dividing each variable through their range (differing from the scaling through the standard deviation as common e.g. for PCA). All other settings scaled>0 will scale the data by their standard deviation.

If scaled=1 or if no scaling is applied, then the default bandwidth setting is 5 percent of the data range in each direction. If the data are scaled through the standard deviation, then the default setting is 20 percent of the standard deviation in each direction.

The threshold thr for merging cluster centers works as follows: After identification of a new cluster center, we compute the Euclidean distance of the new center to (each) existing center, relative to the Euclidean distance of the existing center to the overall mean. If this distance falls below thr, then the new center is deemed identical to the old one.

The goodness-of-fit measure Rc can also be applied in this context. For instance, a value of R_C=0.8 means that, after the clustering, the mean absolute residual length has been reduced by 80\% (compared to the distances to the overall mean).

Author(s)

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

References

Almeijeiras-Alonso, J. and Einbeck, J. (2023). A fresh look at mean-shift based modal clustering, Advances in Data Analysis and Classification, doi:10.1007/s11634-023-00575-1.

Chen, Y. (1995). Mean Shift, Mode Seeking, and Clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 17, 790-799.

Comaniciu, D. and Meer,P. (2002). Mean shift: a robust approach toward feature space analysis, IEEE Transactions on Pattern Analysis and Machine Intelligence 24, 603-619.

Li, X, Hu, Z, and Wu, F. (2007). A note on the convergence of the mean shift, Pattern Recognition 40, 1756 - 1762.

See Also

ms.rep, Rc, plot.ms

Examples

data(faithful)
# Mean shift clustering with default bandwidth (5 percent of data range)
ms(faithful)