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

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lpc(X, h, t0 = mean(h),  x0,  way = "two",  scaled = TRUE,
      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. The default setting is 10 percent of the range in each direction. If scaled =TRUE then the bandwidth has to be specified in fractions of the data range, e.g. h= c(0.2,0.1), rather than absolute values.

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=TRUE. 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 TRUE, scales each variable by dividing through its range (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 t0 used for the curve estimation.

starting.points

The coordinates of the starting point(s) used.

data

The data frame used for curve estimation.

scaled

Logical.

weights

The vector of weights used for curve estimation.

control

The settings used in lpc.control()

Misc

Miscellannea.

Note

All values provided in the output refer to the scaled data, if scaled=TRUE. Use unscale to convert the results back to the original data scale.

The option scaled=TRUE scales the data by dividing each variable through their range. This differs from the usual scaling through the standard deviation as common for PCA, but we found the algorithm and the default bandwidth selection to work more reliably this way. If you wish to scale by the standard deviation, please do that by feeding the scaled data directly into the lpc function, i.e.

lpc(sweep(data, 2,sd(data), "/"),h, t0, ..., scaled=FALSE, ...).

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

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

data(mussels, package="dr")
 lpc2 <- lpc(mussels[,-3], x0=as.numeric(mussels[49,-3]),scaled=FALSE)
 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=FALSE)
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=FALSE)
plot(lpc.e1, type=c("curve","mass", "start"))