pcaLocalDimEst {intrinsicDimension}

Local Dimension Estimation with PCA

Description

Estimates local manifold dimension using the largest singular values of the covariance matrix.

Usage

pcaLocalDimEst(data, ver, alphaFO = .05, alphaFan = 10, betaFan = .8, PFan = .95,
     ngap = 5, maxdim = min(dim(data)), verbose = TRUE)

Arguments

data

a local data set for which dimension should be estimated.

ver

possible values: 'FO', 'fan', 'maxgap', 'cal'. 'cal' is often very slow.

alphaFO

only for ver = 'FO'. An eigenvalue is considered significant if it is larger than alpha times the largest eigenvalue.

alphaFan

only for ver = 'Fan'. The alpha parameter (large gap threshold).

betaFan

only for ver = 'Fan'. The beta parameter (total covariance threshold).

PFan

only for ver = 'Fan'. Total covariance in non-noise.

ngap

only for ver = 'cal'. How many of the largest gaps that should be considered.

maxdim

only for ver = 'cal'. The maximal manifold dimension of the data.

verbose

should information about the process be printed out?

Details

Version 'FO' is the method by Fukunaga-Olsen, version 'fan' is the method by Fan et al..

Version 'maxgap' returns the position of the largest relative gap in the sequence of singular values.

Version 'cal' considers the positions of the ngap largest relative gaps in the sequence of singular values and generates calibration data to determine which one of them is most likely.

All versions assume that the data is local, i.e. that it is a neighborhood taken from a larger data set, such that the curvature and the noise within the neighborhood is relatively small. In the ideal case (no noise, no curvature) this is equivalent to the data being uniformly distributed over a hyper ball.

Value

A DimEst object with slots:

dim.est

the dimension estimate

gap.size

if ver is not 'cal', the size of the gap in singular values corresponding to the estimated dimension

likelihood

if ver is cal, the likelihood of the estimated dimension.

Author(s)

Kerstin Johnsson, Lund University

References

Fukunaga, K. and Olsen, D. R. (1971). An algorithm for finding intrinsic dimensionality of data. IEEE Trans. Comput., c-20(2):176-183.

Fan, M. et al. (2010). Intrinsic dimension estimation of data by principal component analysis. arXiv preprint 1002.2050.

See Also

pcaOtpmPointwiseDimEst

Examples

data <- cutHyperPlane(100, 4, 10, .05)
pcaLocalDimEst(data, 'fan')
pcaLocalDimEst(data, 'FO')
pcaLocalDimEst(data, 'maxgap')