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 f.james.rohlf at stonybrook.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. ---------------------------------------------------------------- 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. ---------------------------------------------------------------- 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. ---------------------------------------------------------------- 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. ---------------------------------------------------------------- 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. ---------------------------------------------------------------- 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%".