lpc: Local principal curves

In LPCM: Local Principal Curve Methods

Local principal curves

Description

This is the main function which computes the actual local principal curve, i.e. a sequence of local centers of mass.

Usage

lpc(X, h, t0 = mean(h),  x0,  way = "two",  scaled = 1,
      weights=1, pen = 2, depth = 1, control=lpc.control())

Arguments

X	data matrix with N rows (observations) and d columns (variables).
h	bandwidth. May be either specified as a single number, then the same bandwidth is used in all dimensions, or as a d-dimensional bandwidth vector. If the data are scaled, then the bandwidth has to be specified in fractions of the data range or standard deviation, respectively, e.g. scaled=1 and h= c(0.2,0.1) gives 20 percent of the range of the first variable and 10 percent of the range of the second variable. If left unspecified, then default settings are invoked; see the ‘Notes’ section below.
t0	scalar step length. Default setting is t0=h, if h is a scalar, and t0=mean(h), if h is a vector.
x0	specifies the choice of starting points. The default choice x0=1 will select one suitable starting point automatically (in form of a local density mode). The second built-in option x0=0 will use all local density modes as starting points, hence produce as many branches as modes. Optionally, one can also set one or more starting points manually here. This can be done in form of a matrix, where each row corresponds to a starting point, or in form of a vector, where starting points are read in consecutive order from the entries of the vector. The starting point has always to be specified on the original data scale, even if scaled>0. A fixed number of starting points can be enforced through option mult in lpc.control.
way	"one": go only in direction of the first local eigenvector, "back": go only in opposite direction, "two": go from starting point in both directions.
scaled	if 1 (or TRUE), scales each variable by dividing through its range. If scaled=2, scaling is performed by dividing through the standard deviation (see also the Notes section below).
weights	a vector of observation weights (can also be used to exclude individual observations from the computation by setting their weight to zero.)
pen	power used for angle penalization (see [1]). If set to 0, the angle penalization is switched off.
depth	maximum depth of branches (\phi_{max} in [2]), restricted to the values 1,2 or 3 (The original LPC branch has depth 1. If, along this curve, a point features a high second local PC, this launches a new starting point, and the resulting branch has depth 2. If, along this branch, a point features a high second local PC, this launches a new starting point, and the resulting branch has depth 3. )
control	Additional parameters steering particularly the starting-, boundary-, and convergence behavior of the fitted curve. See lpc.control.

Value

A list of items:

LPC	The coordinates of the local centers of mass of the fitted principal curve.
Parametrization	Curve parameters and branch labels for each local center of mass.
h	The bandwidth used for the curve estimation.
to	The constant t_0 used for the curve estimation.
starting.points	The coordinates of the starting point(s) used.
data	The data frame used for curve estimation.
scaled	the user-supplied value, could be boolean or numerical
weights	The vector of weights used for curve estimation.
control	The settings used in lpc.control()
Misc	Miscellanea.

Note

All values provided in the output refer to the scaled data, unless scaled=0 or (equivalently) scaled=FALSE. Use unscale to convert the results back to the original data scale.

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). The setting scaled=2, and in fact 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 10 percent of the data range in each direction. If the data are scaled through the standard deviation, then the default setting is 40 percent of the standard deviation in each direction.

Author(s)

J. Einbeck and L. Evers. See LPCM-package for further acknowledgements.

References

[1] Einbeck, J., Tutz, G., & Evers, L. (2005). Local principal curves. Statistics and Computing 15, 301-313.

[2] Einbeck, J., Tutz, G., & Evers, L. (2005): Exploring Multivariate Data Structures with Local Principal Curves. In: Weihs, C. and Gaul, W. (Eds.): Classification - The Ubiquitous Challenge. Springer, Heidelberg, pages 256-263.

Examples

data(calspeedflow)
lpc1 <- lpc(calspeedflow[,3:4])
plot(lpc1)

data(mussels, package="dr")
 lpc2 <- lpc(mussels[,-3], x0=as.numeric(mussels[49,-3]),scaled=0)
 plot(lpc2, curvecol=2)

data(gaia)
s <- sample(nrow(gaia),200)
gaia.pc <-  princomp(gaia[s,5:20])
lpc3 <- lpc(gaia.pc$scores[,c(2,1,3)],scaled=0)
plot(lpc3, curvecol=2, type=c("curve","mass"))

# Simulated letter 'E' with branched LPC
ex<- c(rep(0,40), seq(0,1,length=20), seq(0,1,length=20), seq(0,1,length=20))
ey<- c(seq(0,2,length=40), rep(0,20), rep(1,20), rep(2,20))
sex<-rnorm(100,0,0.01); sey<-rnorm(100,0,0.01)
eex<-rnorm(100,0,0.1);  eey<-rnorm(100,0,0.1)
ex1<-ex+sex; ey1<-ey+sey
ex2<-ex+eex; ey2<-ey+eey
e1<-cbind(ex1,ey1); e2<-cbind(ex2,ey2)
lpc.e1 <- lpc(e1, h= c(0.1,0.1),  depth=2, scaled=0)
plot(lpc.e1, type=c("curve","mass", "start"))

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