trajectory.analysis(f1, f2 = NULL, iter = 999, seed = NULL, traj.pts = NULL, data = NULL, print.progress = TRUE)geomorph.data.framelm function)Data input is specified by a two formulae (e.g., Y ~ X), where 'Y' specifies the response variables (trajectory data),
and 'X' contains one or more independent variables (discrete or continuous). The response matrix 'Y' can be either in the form of a two-dimensional data
matrix of dimension (n x [p x k]), or a 3D array (p x n x k).. The function
two.d.array can be used to obtain a two-dimensional data matrix from a 3D array of landmark
coordinates. It is assumed that the order of the specimens 'Y' matches the order of specimens in 'X'.
It is also assumed that the landmarks have previously been aligned using Generalized Procrustes
Analysis (GPA) [e.g., with gpagen]. Linear model
fits (using the lm function) can also be input in place of a formula. Arguments for lm
can also be passed on via this function. The first formula, f1, must contain the independent variable on the left-hand
side of the formula (e.g., Y ~) and either a single factor or a two factor interaction on the right-hand side. If
a single factor is provided, e.g., Y ~ A, it is assumed that groups to be described are the levels of factor A
and that the data in Y comprise trajectories. In this case, the traj.pts = NULL argument must be changed to a numeric value
to define the number of points in the trajectory. It is also assumed that the data are structured as variables within points.
For example, y11 y21 y31 y12 y22 y32 y13 y23 y33 y14 y24 y34 would be columns of a matrix, Y, describing a 4-point trajectory
in a data space defined by three variables. This is the proper arrangement; the following is an improper arrangement:
y11 y12 y13 y14 y21 y22 y23 y24 y31 y32 y33 y34, as it groups points within variables. This approach is typical when comparing
motion paths (see Adams and Cerney 2007).
If f1 is a two-factor factorial model, e.g., Y ~ A*B, it is assumed that the first factor defines groups, the second factor
defines trajectory points, and that trajectories are to be estimated from the linear model. In this case, the preceding example
would have a Y matrix comprised only of y1, y2, and y3, but the factor B would contain levels to define the four points (see Examples).
If one wishes to include other variables in the linear model, they should be indicated in the second formula, f2.
This formula can be simply a right-hand formula, e.g., ~ x1 + x2 + x3 +... Variables in this formula will typically
be covariates that one wishes to include to account for extraneous sources of shape variation. An analysis
of variance (ANOVA) will be performed with type I sums of squares (SS) and a randomized residual permutation procedure (RRPP).
The variables in f2 will be added prior to the trajectory defining variables in f1.
Once the function has performed the analysis, a plot can be generated of the trajectories as visualized in the
space of principal components (PC1 vs. PC2). The first point in each trajectory is displayed as white, the
last point is black, and any middle points on the trajectories are in gray. The colors of trajectories follow
the order in which they are found in the dataset as a default, using R's standard color palette: black, red, green,
blue, cyan, magenta, yellow, and gray. However, one can override these colors with the argument, "group.cols" in plots
using the function, plot. This will change the trajectory line colors.
One can also override default initial-middle-end point colors with the argument, "pt.seq.pattern". The default is
c("white", "gray", "black") for gray points, but with white initial points and black end points. If changed, the
pt.seq.pattern argument must be a vector with three color values.
One can also uniformly vary the size of points with the argument, "pt.scale". Examples are provided below.
The function, summary can be used to provide an ANOVA summary plus pairwise statistics of a
an object of class "trajectory.analysis". The argument, angle.type = c("r", "rad", "deg") can be used to
toggle between vector correlations, vector angles in radians, or vector angles in degrees, respectively.
Notes for geomorph 3.0
Previous versions of geomorph had two separate analytical approaches based on whether trajectories were estimated or
provided (as might be the case with motion trajectories; see Adams and Cerney 2007). Starting with geomorph 3.0,
commensurate analytical approaches are used. This involves converting 1 x vp vectors for trajectpries, were p is the
number of trajectory points and v is the number of variables in the data space, into p x v matrices, analogous to the
procedure for estimating trajectories. Thus, rather than providing separate ANOVAs for size, orientation, and shape
of trajectories, a general ANOVA is provided with pairwise statistics for the same attribute differences. This change
does not compromise any interpretations made with previous versions of geomorph, but enhances inferential capacity
by providing pairwise statistics and P-values.
Adams, D. C. 2010. Parallel evolution of character displacement driven by competitive selection in terrestrial salamanders. BMC Evol. Biol. 10:1-10.
Adams, D. C., and M. M. Cerney. 2007. Quantifying biomechanical motion using Procrustes motion analysis. J. Biomech. 40:437-444.
Adams, D. C., and M. L. Collyer. 2007. The analysis of character divergence along environmental gradients and other covariates. Evolution 61:510-515.
Adams, D. C., and M. L. Collyer. 2009. A general framework for the analysis of phenotypic trajectories in evolutionary studies. Evolution 63:1143-1154.
Collyer, M. L., and D. C. Adams. 2007. Analysis of two-state multivariate phenotypic change in ecological studies. Ecology 88:683-692.
# Estimate trajectories from LS means in 2-factor model
data(plethodon)
Y.gpa <- gpagen(plethodon$land)
gdf <- geomorph.data.frame(Y.gpa, species = plethodon$species, site = plethodon$site)
TA <- trajectory.analysis(coords ~ species*site, data=gdf, iter=199)
summary(TA, angle.type = "deg")
plot(TA)
# Change order of groups
site <- as.factor(plethodon$site)
levels(site) <- c("Symp", "Allo")
gdf <- geomorph.data.frame(Y.gpa, species = plethodon$species, site = site)
TA <- trajectory.analysis(coords ~ species*site, data=gdf, iter=199)
summary(TA, angle.type = "deg")
plot(TA)
attributes(TA) # list of extractable parts
# Add Centroid size as a covariate
TA <- trajectory.analysis(f1 = coords ~ species*site, f2 = ~ Csize, data=gdf, iter=199)
summary(TA, angle.type = "deg")
plot(TA)
# Change trajectory colors in plot
plot(TA, group.cols = c("dark red", "dark blue"))
# Change size of points and lines
plot(TA, group.cols = c("dark red", "dark blue"), pt.scale=1.5)
# Motion paths represented by 5 time points per motion
data(motionpaths)
gdf <- geomorph.data.frame(trajectories = motionpaths$trajectories,
groups = motionpaths$groups)
TA <- trajectory.analysis(f1 = trajectories ~ groups,
traj.pts = 5, data=gdf, iter=199)
summary(TA)
plot(TA)
plot(TA, group.cols = c("dark red", "dark blue", "dark green", "yellow"), pt.scale = 1.3)
plot(TA, group.cols = c("dark red", "dark blue", "dark green", "yellow"),
pt.seq.pattern = c("green", "gray30", "red"), pt.scale = 1.3)
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