TPSRW - Thin-plate splines relative warp analysis
F. James Rohlf
17 Oct. 1994
Department of Ecology and Evolution
State University of New York at Stony Brook
Stony Brook, NY 11794-5245
Phone: 516-632-8580
ROHLF@life.bio.sunysb.edu
The purpose of this program is to facilitate the analysis of
variation in shape for landmark data among more than two
specimens. The program is designed to perform the equivalent of a
principal components analysis of the variation among specimens
within a single sample. It is possible, however, to output
information to a file so that more complex MANOVAs can be
performed using the projections of the specimens onto the
principal warps as variables.
The program reads a set of x,y-coordinates for a sample of
specimens and expresses their variation in terms of relative
warps. Relative warps are the principal component axes of a
multivariate space in which each point corresponds to a specimen
and the axes are the inversely weighted principal warps of the
bending energy matrix defined by a reference configuration of
landmarks. It can be thought of as a PCA of "the nonlinear part
of a sample of forms." The complementary PCA of the linear part
comes from a study of estimated uniform components (see the
Proceedings of the Michigan Morphometrics Workshop, pp. 243 and
274). The reference configuration can either be computed by this
program (as a sample mean) or can be furnished by the user
(perhaps corresponding to the output from the GRF or GRF_ND
programs).
A paper (Rohlf, 1993) has been published as part of
proceedings of the Valsain Morphometrics Workshop that explains in
detail the various computations performed by this program. The
symbols for the matrices in this program are as defined in that
paper. The reference is:
Rohlf, F. J. 1993 Relative warp analysis and an example of its
application to mosquito wings. Pp. 131-159 in
Marcus, L. F., Bello, E., and Garcia-Valedcasas, A. (eds.)
Contributions to Morphometrics. Museo Nacional de Ciencias Naturales.
The program is still under development -- please be patient!
There are two versions of the program. One for DOS real mode
and another for DOS protected mode (DPMI). Their use is identical
except that the protected mode version requires that the RTM.EXE
program be present (it will be loaded automatically in order to
switch TPSRW into protected mode). If software is not present
that provides DPMI services (e.g., Windows, 386Max, OS/2) then the
file DPMI16BI.OVL must be present. The protected mode version
requires an 80386 or 80486 computer and is able to use both
ordinary RAM and extended memory so that larger datasets can be
processed. It does not make use of overlay files.
Note: this version requires the set of BGI graphics driver
files whose names begin with the underscore character "_".
To use the program:
1. Type its name at the DOS prompt: TPSRW
2. A menu will be displayed. The legal options at a given time will
be shown highlighted.
3. First choose menu 1 to specify the input file and some
options. The program will ask a number of questions and then
will read the input data file. Several file formats are
supported. All sets of landmarks should have been aligned by
either careful digitizing or by processing (such as by the GRF
or GRF_ND programs). When a choice is indicated to be the
default, you may just press the Enter key to accept it.
"GRF" format:
The first line give an ID label for the first object, the
number of objects, and the number of landmarks. Subsequent
lines give the x,y-coordinates for the landmarks for the first
specimen. This is followed by a line giving the label for the
second specimen and then lines giving the coordinates for it,
etc.
"matrix" format:
The first line gives the number of objects and the number of
landmarks. The subsequent lines give the x,y-coordinates for
each specimen. If there are too many landmarks to place on a
single line you make continue onto an additional line. However
each new specimen must begin on a new line.
"NTSYSpc" format:
The format is the same as used by the Fourier program in
NTSYSpc, GRF-ND, and TPSREGR programs. There can be comment
lines, followed by a matrix header line, possibly followed by
label lines, and finally followed by x,y-coordinates as in the
"matrix" format described above.
" fake data for 4 specimens (identical) with 5 landmarks
1 4 10 0
1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 0.0
1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 0.0
1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 0.0
1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 0.0
The program will next ask for a name to be given to an output
listing file. Various numerical results will be stored in this
file. If a file already exists with the name you specify you
will be asked whether to overwrite (and hence destroy) the old
file or to append the new information to the end of the old
file.
Finally, the program will ask for the name of a file giving
a list of pairs of landmarks to connect in output plots.
This can sometimes make the plots easier to visualize.
This input is optional. The input format is that for a
graph matrix in NTSYS-pc. An example is as follows for
5 landmarks and 3 links (or edges of a graph) to be drawn.
The value for the length of each edge does not matter but
it must be provided.
" example of link matrix (type=7) for 5 landmark & 3 edgess
7 5 3 0
1 3 0.0
3 4 0.0
1 4 0.0
4. Next, select menu 2 to specify the reference configuration.
The program will ask which of four method should be used. (1)
Bookstein shape coordinates, (2) average position of each
landmark across all specimens, (3) center and scale each specimen
(mean = 0 and centroid size = 1) or read from a (4) GRF or (5)
NTSYSpc file. If you specify shape coordinates then the
program will ask for the numbers of the two landmarks to be
used as the baseline. Using the average position of the
landmarks makes sense only if the specimens have been aligned
by careful digitization or by a program such as GRF or GRF_ND.
An average consensus configuration can be computed by the GRF
or GRF-ND programs and then read as a file. It is probably
most convenient to output aligned specimens from the GRF or
GRF-ND program and then use the average of these data as a
reference (options 2 or 3, above).
In most cases you should use the
Generalized least-squares fitting method to compute the
consensus configuration for input to TPSRW. You should also
use the "output RES" option to output the coordinates of the
aligned specimens for input to TPSRW.
Note: the GRF-ND program outputs files in NTSYSpc format, not
the older GRF format. NTSYSpc compatible files must be of
NTSYSpc type 1 and can be arranged as dimensions x landmarks,
landmarks x dimensions, or strung out as a single row or column
in the order 1x, 1y, 2x, 2y, etc.
The program will ask for an exponential weight, alpha, for the
bending energy metric. The published accounts of relative warp
analysis (Bookstein, 1991, or PMMW pp. 246-248) are equivalent
to alpha = 1 (which corresponds to an inverse bending energy
metric. You can use a metric in which distances at all scales
are given the same weight (as in Procrustes analysis) by using
alpha=0. This is equivalent to a PCA of the residuals from GRF
least-squares affine. You can also specify an intermediate
value such as 0.5. It does make a difference! I would suggest
that a value of 0 is most reasonable for most applications
since variation along all principal warps are given equal
weight. However, by varying alpha closer to 1 you can give
greater weight to large-scale features and obtain smoother
looking patterns of variation.
If a value of alpha=0 was specified, then the program will
allow the addition of extra columns to the weight matrix in
order to allow the study of covariation with the affine or just
the uniform component.
A = project the specimens onto the 3 left eigenvectors of
the upper right submatrix of L inverse.
R = retain the last 3 eigenvectors of the E matrix and treat
them as if their eigenvalues were equal to 1.
X = append the x,y projections of each specimen from the
polar plot of ln(anisotropy) vs. 2*theta.
U = Estimate uniform component using new method proposed by
Bookstein in his ms for the 1993 NATO ASI in Italy.
N = none of the above (relative warp analysis as
defined by Bookstein, 1991). This is the default.
These options can be useful for statistical analyses to see
whether the uniform component covaries with the nonaffine
components. Plotting of relative warps as splines is not
supported when an option other than "N" is selected (I have not
yet figured out a good way to do it). Note also that the above
additions (especially U and X) have the problem that a PCA
analysis is sensitive to scale and thus the above additions to
the W matrix are somewhat arbitrary.
If the U option is not selected then the program will ask
whether the reference configuration should be rotated to align
it with its own principal axes. The reference will also be
centered on the origin and scaled to have centroid size = 1.
This rotation may be convenient so that the longest axis of the
reference is plotted along the x-axis. The question is not
asked for the U option since this transformation is required in
that case.
If the U option is not selected then
the program will ask whether the specimens are to be
superimposed in the first plot choice in the menu. Choices
are: (S) Shape coordinates, (I) use initial alignment as is,
(E) "minimum energy" (affine part of thin-plate spline), or (L)
least-squares Procrustes (the default). is. The initial
alignment is appropriate, for example, if the input is actually
the output of the GRF program so that the specimens have been
aligned using whichever criterion you prefer. The minimum
energy option will let you see the scatter that the relative
warps are "explaining."
If a reference option other than average is specified for the
computation of the bending energy matrix and not the U option
then the program will ask whether the weights should be based
on deviations from the means or deviations from the reference
object. This option influences how the scores (projections)
are computed. You should normally answer "R".
5. Next choose menu 3 to perform the computations. The
numerical results will be written to the listing file. Messages
will be displayed on the screen showing progress through the
computations. If the program runs out of memory you may only
get a message that says "Error!" or "Out of memory!".
6. The last choice, "C", should be specified before you try to
get hardcopy of the plots shown in the other menu choices.
See the section "Graphics hardcopy" below for more information.
7. Next you can choose any of the plotting or output options
(options 4 - 10). See below for information about each type of
plot.
4 - Plot aligned data & relwarps as vectors
The plot shows all of the specimens superimposed on the
reference configuration. The plot also shows the relative warp
loadings as vectors (as shown in Bookstein's publications). The
magnitude of the vectors is not on the same scale as the
x,y-coordinates of the specimens so that the "+" and "-" keys may
have to be used to get them to a convenient length.
The plot shows how well the objects have been superimposed by
the selected method (minimum energy, shape coordinates, the simple
mean, least-squares, or initial alignment). The affine component
of the variation among specimens is not shown. The difference
between this display and the standard Procrustes plot is that the
relative warps facilitate the study of the covariation of the
displacements at different landmarks.
You control this display by pressing various keys as described
below.
1. The default display shows the superimposition of all of the
specimens, the location of the landmarks in the reference
configuration, and the first relative warp as a vector attached
to each landmark in the reference configuration. In the
default display mode you can press the + or - keys to cycle
through the different relative warps. If you go past the last
one the program will "beep" and cycle back to the first one.
[The last is the minimum of (no. specimens-1) and 2*(no.
points -6).] After selecting one of the other display modes
described below you can return to this mode by pressing the "W"
key.
2. If you press "C" the landmarks for each specimen will be
connected by a series of lines in the order in which they were
entered. Press "C" again to turn this display off.
3. If you press "O" the display will highlight the landmarks for
one of the specimens. They will be shown as open circles. By
pressing the + and - keys you can cycle through the specimens.
If you go too far the program will "beep" and cycle back to the
first object. If the connect option is on the entire outline
for the selected specimen will be highlighted. The text at the
top of the screen will indicate the number of the selected
specimen.
4. If you press "L" the landmarks will be labelled (numbered).
Press "L" again to turn the labels off. After pressing "L"
you can press the "+" or "-" keys to highlight the points
corresponding to each landmark. This is useful if some of the
landmarks are close together so that the scatters overlap.
5. Often the lengths of the vectors showing the relative warps are
very short and hard to see. Press "M" and then the + and -
keys will allow you to change the magnification factor for
these vectors.
6. To print a copy of the graphics screen press "P". The screen
will clear until the print is complete. You should have
specified the hardcopy device in the Configuration menu
previously.
7. Press the "R" key to reset the display back to the default.
8. Press the "S" key to toggle the scatter of points off and on.
9. Press the "E" key to toggle the plot of the line segments between
pairs of landmarks on and off (the letter E is used since the plot
looks like the edges in a graph).
10. Press the ESC key to exit.
5 - Plot relative warps as splines
This plot shows each relative warp as a thin-plate spline
without any affine part. It will cycle through displays showing
the spline being stretched in the positive direction and then the
negative direction as an "animation." Uses information in the N'
matrix. Note the scale of the splines and the scale of the plot
of the reference configuration are in different units. You will
usually have to change the magnification ("M" key, see below) to
make the plot pleasent to look at). This option is not available
if the affine or uniform components are estimated.
1. In the default display mode you can select the relative warp
for viewing by pressing the + and - keys. After selecting
another mode (described below), you can return to this mode by
pressing the "W" key.
2. To control the amplitude of the deformation, press the "M" key
and then press the + or - keys to increase or decrease the
magnitude of the change in a + or - direction along a
relative warp axis.
3. If you press "C" the landmarks will be connected by a series of
lines in the order in which they were entered. Press "C" again
to turn this display off.
4. Press the "L" key to display labels for the points. Press it
again to turn off their display.
5. Press the "V" key to display displacement vectors. Press it
again to turn them off. These vectors are plotted only on the
untransformed grid. The end points of the vectors are the
locations of each point after the transformation that is about
to be applied. The vectors are usually similar to the relative
warp loading vectors. Sometimes they are quite different.
6. To print a copy of the graphics screen press "P". The program
will then prompt for you to press either the "+" or the "-"
keys. This allows you to specify whether to output the spline
based on the positive or negative warping of the space.
The screen will clear until the plot is complete.
7. Press the "E" key to toggle the plot of the line segments between
pairs of landmarks on and off.
8. Press the "R" key to reset the display back to the default.
9. Press the "D" key to select a delay time to slow down the display
on fast computers. Every time you press the "+" key a 0.5 sec.
delay will be added. You can press the "-" key to reduce the
delay. The program will beep when you reduce the delay to zero.
10. Press the ESC key to exit.
6 - Plot scores for objects on relative warps 1 & 2
In this mode the scores (projections) of the specimens onto the
first two relative warps are plotted against each other. If the
projection matrix is saved (see next menu choice) it can be read
into NTSYSpc and a 3-dimensional plot made. NTSYSpc can also be
used to compute distances among the objects and then perform a
cluster analysis. Note: the "distances" depend strongly on alpha
and should be used only in conjunction with affine distances.
1. Press the "L" key to display labels for the points. Press it
again to turn off their display.
2. To print a copy of the graphics screen press "P". The screen
will clear until the plot is complete.
3. Press the ESC key to exit.
7 - Save matrices to NTSYSpc files
Selecting this menu item will enable you to write the following
matrices to files in a format compatible with the NTSYSpc
program: Weights matrix, principal warps matrix, vector of
energy for each principal warp, a matrix of relative warp
scores, centroid size, and a matrix of log anisotropy and twice
the angle of the direction of maximum uniform shape change for
each specimen.
Within NTSYSpc you can make plots, compute distances
between pairs of objects and perform cluster analyses.
The weights matrix can be used as a description of each
specimen in terms of the principal warps. This matrix can be
used as input to other software that can analyze data for more
complex MANOVA designs with the principal warps for the x and y
coordinates used as variables. The principal warps matrix and
the vector of their energies can be used to express the results
of such analyses in terms of thin-plate splines for the
original landmarks. The program TPSREGR can be used to
plot an arbitrary linear combination of principal warps
(such as a discriminant function or set of partial regression
coefficients) or to regress the principal warps onto a
independent variable and to show the results as a spline.
Note: all of the principal warps are output including those
that have 0 energy (that correspond to the affine components).
For an analysis of deformations you will want to delete them.
In such cases the last 3 energy values should be deleted. If
the "retain last 3 eigenvectors of E" option was selected, then
all of the principal warps will be used in the construction of
the weight matrix (W). Depending upon the initial alignment of
the data, this matrix may be singular and will cause problems
in multivariate analysis packages that do not use generalized
inverses (the NTSYS-pc program CVA should work ok).
Note: the relative warp scores depend strongly on alpha and do
not contain any contribution due to affine variation (as seen
by a thin-plate spline) among the specimens unless you selected
one of the options about retaining affine or uniform
components. If the "retain" option is selected then the
ordination will be the same as if a SVD had been applied to the
initial specimens using their coordinates as variables. In
that case you haven't really accomplished anything by using
TPSRW if your goal was to just do this ordination!
8 - Plot scores against size
This menu item plots the scores for each specimen against the
square root of centroid size divided by the number of landmarks
for each specimen (a form of "average centroid size"). Trends
show how specimens change shape as they change size ("allometry").
Size is computed from the original data coordinates read by this
program. This assumes that the specimens were all digitized at
the same scale. It may be more interesting to regress the
principal warps onto centroid size using the TPSREGR program.
Note: if the input data are ouput from GRF or GRF-ND or have been
expressed as baseline-scaled shape coordinates then size will have
been removed.
The square root of the average centroid size for each specimen
and the statistics for the regression of each relative warp score
onto this measure of size are given in the output listing file.
1. Press the "+" and "-" keys to cycle through the relative warps.
2. Press "L" to toggle the labelling of the specimens.
3. Press "P" for graphics hardcopy.
4. Press the "ESC" key to exit.
9 - Plot uniform component
The form of the plot depends upon whether the X or the U option
was specified (x,y-projections from polar plot or Bookstein's
estimate of uniform component).
X Plot is of the log of the anisotropy vs. twice the angle of the
direction of maximum uniform shape change required to go from
the reference configuration to each specimen. The plot is in
polar coordinates and is of the form of that given to the right
of Figure 6.2.11b on page 213 of the "Orange book." Because
the angle has been doubled, points to the right correspond to
changes in the horizontal plane, those to the left correspond
to changes in the vertical direction, those above the origin
correspond to changes of the same sign both horizontally and
vertically, and points below the origin correspond to
horizontal and vertical changes having opposite signs.
The concentric rings are at intervals of 0.01 of log
anisotropy. If you set alpha=0 and select the "add uniform"
option then it is the x,y-coordinates in this plot that are
added to the weight matrix for statistical analysis.
U A bivariate plot of u1 versus u2. Note that directions are
relative to the reference specimen which has been rotated so as
to align with its principal axes.
1. Press the "L" key to display labels for the points. Press it
again to turn off their display.
2. To print a copy of the graphics screen press "P".
10 - Plot principal warps as splines
This plot shows each principal warp as a thin-plate spline
without any affine part. It will cycle through displays showing
the spline being stretched in the positive direction and then the
negative direction. The relative warps (in the x and the y
directions) are weighted linear combinations of these normalized
principal warps. The principal warps are shown with the same
displacement added to both the x and the y-coordinates of each
point. These plots look rather different from those of principal
warps given by the tpspline program. In that program the
principal warps were weighted in both the x and y-coordinates as
appropriate to fit to the second specimen.
1. In the default display mode you can select the principal warp
for viewing by pressing the + and - keys. After selecting
another mode (described below), you can return to this mode by
pressing the "W" key.
2. To control the amplitude of the deformation, press the "M" key
and then press the + or - keys to increase or decrease the
magnitude. This option must always be used since the scale for
each principal warp is arbitrary.
3. If you press "C" the landmarks for each specimen will be
connected by a series of lines in the order in which they were
entered. Press "C" again to turn this display off.
4. Press the "L" key to display labels for the points. Press it
again to turn off their display.
5. Press the "V" key to display displacement vectors. Press it
again to turn them off. These vectors are plotted only on the
untransformed grid. The end points of the vectors are the
locations of each point after the transformation that is about
to be applied.
6. To print a copy of the graphics screen press "P". The program
will then prompt for you to press either the "+" or the "-"
keys. This allows you to specify whether to output the spline
based on the positive or negative warping of the space.
The screen will clear until the plot is complete.
7. Press the "R" key to reset the display back to the default.
8. Press the ESC key to exit this display.
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Configuration for graphics hardcopy
A window will be displayed that lists the various devices and
their modes. Another window will then be displayed that asks for
a device or file name. If you would like the output written to a
file for later use then enter a valid file name. The name should
be short to allow for the fact that the program will append a
number so that each picture can be stored in a separate file.
To have the output sent directly to a printer attached to a
printer port enter LPT1 or LPT2. For output directly to a printer
or a plotter attached to a serial port enter COM1 or COM2. In the
later case you must also specify the baud rate, parity, number of
data bits, and whether or not to use XON/XOFF protocol.
The available baud rates are: 300, 1200, 2400, 4800, and 9600.
Parity can be N (none), E (even), or O (odd). The number of data
bits can be 7 or 8. Use the symbol "X" to indicate XON/XOFF.
These codes are entered after the port name. For example, for
a plotter attached to COM1 and working with 2400 baud, no parity,
8 data bits, and using XON/XOFF enter the following:
COM1,2400,N,8,X
The following printers are supported: Epson 9-pin printers
(including Epson FX and MX, IBM Graphics Printer and Proprinter,
and Panasonic and OkiData ["native" or with Epson or IBM
emulation]), Epson 24-pin printers (includes Epson LQ, NEC
Pinwriter, and Panasonic printers with Epson emulation), and
Toshiba P321 24-pin printer. The Epson 9-pin and 24-pin color dot
matrix printers are supported. The HP LaserJet (all models), HP
DeskJet (all models), and Canon LBP-8 laser and inkjet printers
are supported.
The following plotters are supported: HP7470, HP7475, and
HP7585. Many other plotters are compatible with these plotters.
If the plotting information is written to a file it can be read by
many word processors, desktop publishing programs, and by graphics
programs.
In addition, you can select output formats of CGM, GEM IMG,
PCX, WordPerfect WPG, and TIFF (both compressed and uncompressed).
MS Windows bitmap files (BMP) are also supported. These are
useful in order to import the graphics into various desktop
publishing and "paint" programs where you can add annotations,
delete unwanted details, etc.
BGI files
These are the files that provide the graphics support to the
program. You only need to have the BGI files on your disk for the
devices you expect to use. If you do not have the proper graphics
BGI file you will not be able to see a plot on the screen. The
menu choices for plotting will return directly back to the main
menu.
If the proper BGI file for graphics hardcopy is not present the
program will exit back to the main menu without any error message.
The correspondence between BGI files and devices is given below.
Graphics adapters:
ATT.BGI AT&T6300
CGA.BGI IBM CGA, MCGA
EGAVGA.BGI IBM EGA, VGA
HERC.BGI Hercules monochrome graphics
IBM8514.BGI IBM 8514
PC3270.BGI IBM 3270PC
Graphics printers and plotters:
_CANON.BGI Canon LBP-8 printer
_CFX.BGI 9-pin color dot matrix
_CLQ.BGI 24-pin color dot matrix
_DIC.BGI Kodax Diconic printer
_DJ.BGI HP DeskJet printer
_DJC.BGI HP Color DeskJet printer
_DMPL.BGI DM/PL plotters
_FX.BGI Epson 9-pin printers
_HP7470.BGI HP7470 plotter
_HP7475.BGI HP7475 plotter
_HP7550.BGI HP7550 plotter
_HP7585.BGI HP7585 plotter
_LQ.BGI Epson 24-pin printers
_LJ.BGI HP LaserJet printer
_LJ3R.BGI HP LaserJet III printer
_OKI92.BGI Okidata 92 native mode
_PJET.BGI HP paintjet
_PP24.BGI 24 pin dot matrix
_TJ.BGI HP ThinkJet printer
For the above devices you will need to know how it is attached
to your computer (printer or serial port). In the case of a
serial port you will also need to know the baud rate, parity,
number of data bits, and whether the XON/XOFF protocol is used.
Graphics file formats:
_AI.BGI Adobe Illustrator Postscript
_BMP.BGI MS Windows bitmap files
_CGM.BGI CGM files
_DXF.BGI AutoCad
_IMG.BGI GEM IMG files
_PCX.BGI PCX paint file format
_TIFF.BGI Compressed TIFF format
_UTIFF.BGI Uncompressed TIFF format
_WPG.BGI WordPerfect WPG files
The BGI files whose names begin with "_" are part of the
GRAF/DRIVE package from Flemming Software as is the GCOPY.EXE
program that can be used to copy graphic files to a printer or
plotter. Type GCOPY and instructions will be displayed.
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Sample data files
Two data files are provided as examples. The first file lets
you compare results with published descriptions of the method.
The two files also provide examples of what is meant by "matrix"
and GRF input formats.
RATS7.DTA: this is the 7-day rat data used in several of
Bookstein's publications (e.g., Cambridge book in press, 1991).
To duplicate his results, read the data in as type "matrix".
Specify alignment by shape coordinates using landmarks 1 and 6.
Specify alpha = 1 and to use deviations from the mean in computing
the weight matrix. You can superimpose by any of the three
methods since he does not show a plot of the specimens
superimposed on the reference configuration. Since the landmarks
were digitized in order around the skull the "connect" option can
be used for plotted output.
MOSQ18R.GRF: this is a file with 18 wing landmark locations
for 8 species of mosquitoes. This file is the output of the GRF
program from running affine generalized resistant fit on these 8
species. Specify that the data file is in GRF format and use the
"average" option to compute the target configuration since the
specimens have already been aligned using the generalized
resistant fit procedure of Rohlf and Slice (1990). A link file,
MOSQ18R.LNK is also provided with a list of 23 edges so that the
plot of the wing looks more realistic. MOSQ18R.NTS is the same
data file but in a format compatible with NTSYS-pc.
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Output listing
The program produces a rather large listing file containing
printouts of most of the matrices involved in a relative warps
analysis. If you have a large number of variables or specimens
you may not wish to print the entire file (especially not until
you are certain the various options are set correctly for your
data).
One of the first things to try is a run with alpha=0 to
see if any of the singular values are significant. If they are
not, then there may not be much of interest to investigate in the
data.
If the above test is significant and you wish to use a value of
alpha>0 then you should try that and see if the singular values
are still significant. The test in this case is rather ad hoc so
you may wish be rather conservative in how you interpret it. The
matrix of singular values (D) gives the values of the standard
deviations for the relative warps. (the These give the amount of
variation along each relative warp axis (relative to bending
energy if alpha>0). When alpha > 0 the adjusted D-values are
computed as follows. First the energy values (eigenvalue matrix
Lambda) are raised to the 1+alpha/2 power. If there are p
landmarks, then D(1) and D(2) are both multiplied by the (p-3)
term. D(3) and D(4) are multiplied by the p-4 term, etc. The
"reversal" in order is due to the fact that weighting is by the
inverse of bending energy.
The purpose of the adjustment is to get rid of the differences
among the singular values caused simply by the fact that different
principal warps are given different weights if alpha is not equal
to zero. The next column gives the squares of the adjusted values
and these are tested for equality using the usual test for
equality of eigenvalues of a variance-covariance matrix (Chi
square, degrees of freedom, and probability values are also
shown). If the data consist of nothing but normally distributed
uncorrelated random variation at each landmark (like digitizing
error), then there should not be any significant relative warps.
If alpha=0 then the average singular value should be an estimate
of the digitizing error variance in this case.
The output listing also contains comments about the various
matrices. These may be useful for understanding what is going on.
You should examine the polar lot of the log anisotropy versus
the doubled angle of the direction of maximum uniform shape
change. See the Orange book for hints on interpretation.
It is not clear what the best strategy is for studing the
relationship (if any) between uniform and nonuniform components of
shape variation. You can examine the relative warp loadings
matrix to see if the first few relative warps have high
coefficients on the added columns of the W matrix. Perhaps the
easiest to try to interpret is the option to add the
x,y-projections to the W matrix (but remember that the angle used
is 2*theta).
Next, it may be of interest to relate shape variation (as
captured by the weight matrix) to external variables such as
longitude, temperature, age, etc. depending upon the study. The
TPSREGR program can be used for this.
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Changes from previous version
7/6/93 Recompiled for BP7 and added support for DOS DPMI mode.
Increased the size of datasets that could be processed.
9/27/93 Added Bookstein's 1993 NATO ASI method for estiation of
the uniform component. A option was also added to allow
rotation of reference to its principal axes.
10/2/93 Added center, scale, and then average option for construcing
a reference. Fixed a scale bug for the Bookstein NATO estimate
of the uniform component.
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Corrections to Rohlf (1993) "Relative warp analysis and an example of
its application to mosquito wings" In Marcus et al. "Contributions to
morphometrics"
Page 135, para 3:
You should now FTP to the morphmet directory at
life.bio.sunysb.edu to obtain copies of the TPSRW program.
Page 138, equations 8 and 9:
The "[0/1]" should be "[1|0]" in equation 8 and "[0|1]" in
equation 9.
Page 155, line 10:
The "50.8%" should be "68.8%".