gpagen: Generalized Procrustes analyis of points, curves, and surfaces

Description

A general function to perform Procrustes analysis of two- or three-dimensional landmark data that can include both fixed landmarks and sliding semilandmarks

Usage

gpagen(A, Proj = TRUE, ProcD = TRUE, ShowPlot = TRUE,
    curves = NULL, surfaces = NULL, pointscale = 1)

Arguments

An array (p x k x n) containing landmark coordinates for a set of specimens

Proj

A logical value indicating whether or not the aligned Procrustes residuals should be projected into tangent space

ProcD

A logical value indicating whether or not Procrustes distance should be used as the criterion for optimizing the positions of semilandmarks

curves

An optional matrix defining which landmarks should be treated as semilandmarks on boundary curves, and which landmarks specify the tangent directions for their sliding

pointscale

An optional value defining the size of the points for all specimens

surfaces

An optional vector defining which landmarks should be treated as semilandmarks on surfaces

ShowPlot

A logical value indicating whether or not a plot of Procrustes residuals should be displayed

Value

Function returns a list with the following components:
coordsA (p x k x n) array of aligned Procrustes coordinates, where p is the number of landmark points, k is the number of landmark dimensions (2 or 3), and n is the number of specimens. The third dimension of this array contains names for each specimen if specified in the original input array
CsizeA vector of centroid sizes for each specimen, containing the names for each specimen if specified in the original input array

Details

The function performs a Generalized Procrustes Analysis (GPA) on two-dimensional or three-dimensional landmark coordinates. The analysis can be performed on fixed landmark points, semilandmarks on curves, semilandmarks on surfaces, or any combination. To include semilandmarks on curves, one must specify a matrix defining which landmarks are to be treated as semilandmarks using the "curves=" option. Likewise, to include semilandmarks on surfaces, one must specify a vector listing which landmarks are to be treated as surface semilandmarks using the "surfaces=" option. The "ProcD=TRUE" option will slide the semilandmarks along their tangent directions using the Procrustes distance criterion, while "ProcD=FALSE" will slide the semilandmarks based on minimizing bending energy. The aligned Procrustes residuals can be projected into tangent space using the "Proj=TRUE" option. NOTE: Large datasets may exceed the memory limitations of R. Generalized Procrustes Analysis (GPA: Gower 1975, Rohlf and Slice 1990) is the primary means by which shape variables are obtained from landmark data (for a general overview of geometric morphometrics see Bookstein 1991, Rohlf and Marcus 1993, Adams et al. 2004, Zelditch et al. 2004, Mitteroecker and Gunz 2009, Adams et al. 2012). GPA translates all specimens to the origin, scales them to unit-centroid size, and optimally rotates them (using a least-squares criterion) until the coordinates of corresponding points align as closely as possible. The resulting aligned Procrustes coordinates represent the shape of each specimen, and are found in a curved space related to Kendall's shape space (Kendall 1984). Typically, these are projected into a linear tangent space yielding Kendall's tangent space coordinates (Dryden and Mardia 1993, Rohlf 1999), which are used for subsequent multivariate analyses. Additionally, any semilandmarks on curves and are slid along their tangent directions or tangent planes during the superimposition (see Bookstein 1997; Gunz et al. 2005). Presently, two implementations are possible: 1) the locations of semilandmarks can be optimized by minimizing the bending energy between the reference and target specimen (Bookstein 1997), or by minimizing the Procrustes distance between the two (Rohlf 2010).

References

Adams, D. C., F. J. Rohlf, and D. E. Slice. 2004. Geometric morphometrics: ten years of progress following the 'revolution'. It. J. Zool. 71:5-16. Adams, D. C., F. J. Rohlf, and D. E. Slice. 2012. In Press. A field comes of age: Geometric morphometrics in the 21st century. Hystrix. Bookstein, F. L. 1991. Morphometric tools for landmark data: Geometry and Biology. Cambridge Univ. Press, New York. Bookstein, F. L. 1997. Landmark methods for forms without landmarks: morphometrics of group differences in outline shape. 1:225-243. Dryden, I. L., and K. V. Mardia. 1993. Multivariate shape analysis. Sankhya 55:460-480. Gower, J. C. 1975. Generalized Procrustes analysis. Psychometrika 40:33-51. Gunz, P., P. Mitteroecker, and F. L. Bookstein. 2005. semilandmarks in three dimensions. Pp. 73-98 in D. E. Slice, ed. Modern morphometrics in physical anthropology. Klewer Academic/Plenum, New York. Kendall, D. G. 1984. Shape-manifolds, Procrustean metrics and complex projective spaces. Bulletin of the London Mathematical Society 16:81-121. Mitteroecker, P., and P. Gunz. 2009. Advances in geometric morphometrics. Evol. Biol. 36:235-247. Rohlf, F. J., and D. E. Slice. 1990. Extensions of the Procrustes method for the optimal superimposition of landmarks. Syst. Zool. 39:40-59. Rohlf, F. J., and L. F. Marcus. 1993. A revolution in morphometrics. Trends Ecol. Evol. 8:129-132. Rohlf, F. J. 1999. Shape statistics: Procrustes superimpositions and tangent spaces. Journal of Classification 16:197-223. Rohlf, F. J. 2010. tpsRelw: Relative warps analysis. Version 1.49. Department of Ecology and Evolution, State University of New York at Stony Brook, Stony Brook, NY. Zelditch, M. L., D. L. Swiderski, H. D. Sheets, and W. L. Fink. 2004. Geometric morphometrics for biologists: a primer. Elsevier/Academic Press, Amsterdam.

Examples

Run this code

#Example 1: fixed points only
data(plethodon)
gpagen(plethodon$land)
points(mshape(gpagen(plethodon$land)$coords),pch=22,col="red",bg="red",cex=1.2)

#Example 2: points and semilandmarks on curves
data(hummingbirds)

#Matrix defining which points are semilandmarks (middle column) and in which directions they slide (columns 1 vs. 3)
hummingbirds$curvepts

gpagen(hummingbirds$land,curves=hummingbirds$curvepts,)   #Using Procrustes Distance for sliding

gpagen(hummingbirds$land,curves=hummingbirds$curvepts,ProcD=FALSE,)   #Using bending energy for sliding

#Example 3: points, curves and surfaces
data(scallops)
gpagen(A=scallops$coorddata, curves=scallops$curvslide, surfaces=scallops$surfslide) #Using Procrustes Distance for sliding

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