| LDPC {RPointCloud} | R Documentation |
Local Dimension of Point Clouds
Description
Given a data set viewed as a point cloud in N-dimesnional space, compute the local estimate of the dimension of an underlying manifold, as defined by Ellis and McDermott, analogous to earlier related work by Takens.
Usage
takens(r, dists)
LDPC(CellID, dset, rg, quorum, samplesAreRows = TRUE)
Arguments
CellID |
An integer indexing one of the cells-samples in the data set. |
dset |
A data set. The usual orientation is that rows are cells and columns are features. |
rg |
A numerical vector of radial distances at which to compute Takens estimates of the local dimension. |
quorum |
The minimum number of neighboring cells required for the computation to be meaningful. |
samplesAreRows |
A logical value: do rows or columns represent samples at which to compute local dimensions. |
r |
A radial distances at which to compute the Takens estimate. |
dists |
A sorted vector of distances from one cell to all other cells. |
Details
"[T]he procedure is carried out as follows. A 'bin increment', $A$; a number, $m$, of bin increments; and a 'quorum', $q > 0$, are chosen and raw dimensions are calculated for $r=A$, $2A$, $...$, $m$. Next, for each observation, $x_i$, let $r_i$ be the smallest multiple of $A$ not exceeding $mA$ such that the ball with radius $r_i$ centered at $x_i$ contains at least $q$ observations, providing that there are at least $q$ observations within $mA$ of $x_i$. Otherwise let $r_i = mA$."
The takens function computes the Takens estimate of the local
dimension of a point cloud at radius $r$ around a data point. For each
cell-sample, we must compute and sort the distances from that cell to
all other cells. (For the takens function, these distances are
passed in as the second argument to the function.) Preliminary
histograms of distance distributions may be used to inform a good set
of radial distances. Note that the local dimension estimates are
infinite if the radius is so small that there are no neighbors. The
estmiates decrease as the radius increases os as the number of local
neighbors increases. The reference paper by Ellis and McDermott says:
The LDPC function iterates over all cells-samples int he data sets,
computes and sorts their distance to all other cells, and invokes teh
takens function to compute local estimates os dimension.
Value
The takens function returns a list with two items: the number
of neighbors $k$ and the dimension estimate $d$ at each value
of the radius from the input vector.
The LDPC function returns a list containing vectors $R$, $k$,
and $d$ values for each cell in the data set.
Author(s)
Kevin R. Coombes <krc@silicovore.com>
References
Ellis and McDermott
Examples
data(cytof)
localdim <- LDPC(1, AML10.node287, seq(1, 6, length=20), 30, TRUE)