Isometric Feature Mapping Ordination

The function performs isometric feature mapping which consists of three simple steps: (1) retain only some of the shortest dissimilarities among objects, (2) estimate all dissimilarities as shortest path distances, and (3) perform metric scaling (Tenenbaum et al. 2000).

Usage

isomap(dist, ndim=10, ...)
isomapdist(dist, epsilon, k, path = "shortest", fragmentedOK =FALSE, ...)
# S3 method for class 'isomap'
summary(object, ...)
# S3 method for class 'isomap'
plot(x, net = TRUE, n.col = "gray", type = "points", ...)

Arguments

dist

Dissimilarities.

ndim

Number of axes in metric scaling (argument k in cmdscale).

epsilon

Shortest dissimilarity retained.

k

Number of shortest dissimilarities retained for a point. If both epsilon and k are given, epsilon will be used.

path

Method used in stepacross to estimate the shortest path, with alternatives "shortest" and "extended".

fragmentedOK

What to do if dissimilarity matrix is fragmented. If TRUE, analyse the largest connected group, otherwise stop with error.

x, object

An isomap result object.

net

Draw the net of retained dissimilarities.

n.col

Colour of drawn net segments. This can also be a vector that is recycled for points, and the colour of the net segment is a mixture of joined points.

type

Plot observations either as "points", "text" or use "none" to plot no observations. The "text" will use ordilabel if net = TRUE and ordiplot if net = FALSE, and pass extra arguments to these functions.

...

Other parameters passed to functions.

Details

The function isomap first calls function isomapdist for dissimilarity transformation, and then performs metric scaling for the result. All arguments to isomap are passed to isomapdist. The functions are separate so that the isompadist transformation could be easily used with other functions than simple linear mapping of cmdscale.

Function isomapdist retains either dissimilarities equal or shorter to epsilon, or if epsilon is not given, at least k shortest dissimilarities for a point. Then a complete dissimilarity matrix is reconstructed using stepacross using either flexible shortest paths or extended dissimilarities (for details, see stepacross).

De'ath (1999) actually published essentially the same method before Tenenbaum et al. (2000), and De'ath's function is available in function xdiss in non-CRAN package mvpart. The differences are that isomap introduced the k criterion, whereas De'ath only used epsilon criterion. In practice, De'ath also retains higher proportion of dissimilarities than typical isomap.

The plot function uses internally ordiplot, except that it adds text over net using ordilabel. The plot function passes extra arguments to these functions. In addition, vegan3d package has function rgl.isomap to make dynamic 3D plots that can be rotated on the screen.

Value

Function isomapdist returns a dissimilarity object similar to dist. Function isomap returns an object of class isomap with plot and summary methods. The plot function returns invisibly an object of class ordiplot. Function scores can extract the ordination scores.

References

De'ath, G. (1999) Extended dissimilarity: a method of robust estimation of ecological distances from high beta diversity data. Plant Ecology 144, 191–199

Tenenbaum, J.B., de Silva, V. & Langford, J.C. (2000) A global network framework for nonlinear dimensionality reduction. Science 290, 2319–2323.

Author

Jari Oksanen

Note

Tenenbaum et al. (2000) justify isomap as a tool of unfolding a manifold (e.g. a 'Swiss Roll'). Even with a manifold structure, the sampling must be even and dense so that dissimilarities along a manifold are shorter than across the folds. If data do not have such a manifold structure, the results are very sensitive to parameter values.

See also

The underlying functions that do the proper work are stepacross, distconnected and cmdscale. Function metaMDS may trigger stepacross transformation, but usually only for longest dissimilarities. The plot method of vegan minimum spanning tree function (spantree) has even more extreme way of isomapping things.

Examples

## The following examples also overlay minimum spanning tree to
## the graphics in red.
op <- par(mar=c(4,4,1,1)+0.2, mfrow=c(2,2))
data(BCI)
dis <- vegdist(BCI)
tr <- spantree(dis)
pl <- ordiplot(cmdscale(dis), main="cmdscale")
#> species scores not available
lines(tr, pl, col="red")
ord <- isomap(dis, k=3)
ord
#> 
#> Isometric Feature Mapping (isomap)
#> 
#> Call:
#> isomap(dist = dis, k = 3) 
#> 
#> Distance method: bray shortest isomap 
#> Criterion: k = 3 
pl <- plot(ord, main="isomap k=3")
lines(tr, pl, col="red")
pl <- plot(isomap(dis, k=5), main="isomap k=5")
lines(tr, pl, col="red")
pl <- plot(isomap(dis, epsilon=0.45), main="isomap epsilon=0.45")
lines(tr, pl, col="red")

par(op)
## colour points and web by the dominant species
dom <- apply(BCI, 1, which.max)
## need nine colours, but default palette  has only eight
op <- palette(c(palette("default"), "sienna"))
plot(ord, pch = 16, col = dom, n.col = dom) 

palette(op)