This section will show that the eigendirections of F, introduced in Section 3.3, are the same as PNDDTPN and also the eigenvalues of F are equal to the eigenvalues of
−DDT plus 4 for i= 1. . . d
N and 0 otherwise. Where the dimension of the null space is
dN. Let the eigendecomposition of PNDDTPN equal J M JT, where J is a matrix of the eigenvectors and M has the eigenvalues along the diagonal and 0’s off the diagonal.
Let the eigendecompostion of DDT = Γ
sΛsΓTs. This implies that the eigendecom- postion of −DDT = Γ
s(−Λs)ΓTs. We wish to add 4×Id to −DDT in order to have all eigenvalues be greater than 0. It was shown in Section A.1 thatFf ull = Γs(4Id−Λs)ΓTs. The matrix PN(−DDT)PN = PN[Γs(−Λs)ΓTs]PN = J(−M)JT, while the matrix
F =PNΓs(cId−Λs)ΓsTPN. We wish to show that F =J(Z +M)JT, where
Z = 4IdN 0 0 0
which implies that F has the eigenvalues 4Id−M for i = 1. . . dN and 0 otherwise and the eigenvectorsJ. This would mean thatPNDDTPN andF have the same eigenvectors, and the directions have simplicity scores which are shifted by a constant except for those that are 0.
We start with the characterization of F asPNΓs(−Λs+ 4Id)ΓTsPN.
PNΓs(−Λs+ 4Id)ΓTsPN
=PNΓs(−Λs)ΓTsPN +PNΓs(4Id)ΓTsPN =J(−M)JT +PN(4Id)PN
The last step in the above equations is done by noticing thatJ(−M)JT =PNΓs(−Λs)ΓTsPN by definition. Also notice that Γs(4Id)ΓTs = 4ΓsΓsT = 4Id. Next 4Id is replaced by J4JT, so
PNΓs(−Λs+ 4Id)ΓTsPN =J(−M)JT +PNJ4JTPN
By definition ofPN the firstdN eigenvectors ofJ will be in the null space and the others will not. Therefore the projection of Ji for i= 1. . . dN ontoN is equal to Ji. While the projection ofJi for i=dN + 1. . . d ontoN is equal to 0. Therefore
J(−M)JT +PNJ4JTPN
=J(−M)JT + [J1, J2, . . . , JdN,0, . . . ,0]4[J1, J2, . . . , JdN,0, . . . ,0]
T
=J(−M)JT + [J1, J2, . . . , JdN,0, . . . ,0]Z[J1, J2, . . . , JdN,0, . . . ,0]
T
In the above the zero vectors can be replace by [JdN+1. . . Jd] since the are being multiplied
by 0.
J(−M)JT + [J1, J2, . . . , JdN,0, . . . ,0]Z[J1, J2, . . . , JdN,0, . . . ,0]
T
=J(−M)JT +J ZJT
=J(−M+Z)JT
This shows the F = J(−M +Z)JT, so therefore F has eigenvectors J and eigenvalues which are the diagonal of (−M +Z). The value 4 can be replaced with any constant greater than 4 and the above would still hold.
BIBLIOGRAPHY
Anderson T.W. (1963). Asymptotic theory for principal component analysis. Annals of
Mathematical Statistics 34, 122–148.
Anderson T.W. (1984).An Introduction to Multivariate Statistical Analysis. Wiley Series in Probability and Mathematical Statistics. Wiley, New York.
Bookstein F.L. (1978). Measurement of Biological Shape and Shape Change. Lecture Notes in Biomathematics. Springer-Verlang, New York.
Dryden I.L. and Mardia K.V. (1998). Statistical Shape Analysis. Wiley Series in Proba- bility and Statistics. Wiley, New York.
Gaydos T.L. (2007). Additional web material. URL http://www.unc.edu/
˜tgaydos.
Gilbert L.I., Granger N.A. and Roe R.M. (2000). The juvenile hormone: historical facts and speculations on future research directions. Insect Biochemistry and Molecular
Biology 30, 617–644.
Hotelling H. (1936). Relations between two sets of variates. Biometrika 28, 321–377. Inselberg A. (1985). The plane with parallel coordinates. The Visual Computer 1, 69–91. Izem R. and Kingsolver J.G. (2005). Variation in continuous reaction norms: Quantifying
directions of biological interest. The American Naturalist 166, 277–289.
Kato T. (1966). Perturbation Theory for Linear Operators. Die Grundlehren der math- ematischen Wissenschaften in Einseldarstellungen ; Bd. 132. Spinger-Verlang, Berlin; New York.
Kendall D.G. (1999).Shape and Shape Theory. Wiley Series in Probability and Statistics. Wiley, New York.
Kingsolver J.G., Gomulkiewicz R. and Carter P.A. (2001). Variation, selection and evo- lution of function valued traits112-113, 87–104.
Kingsolver J.G., Ragland G.J. and Shlichta J.G. (2004). Quantitative genetics of con- tinuous reaction norms: Thermal sensitivity of caterpillar growth rates. Evolution 58, 1521–1529.
Kuss M. and Graepel T. (2003). The geometry of kernel canonical correlation analysis. Technical Report TR-108, Max Planck Institute for Biological Cybernetics, T¨ubingen, Germany. URLciteseer.ist.psu.edu/kuss03geometry.html.
Lynch M. and Walsh B. (1998). Genetics and Analysis of Quantitative Traits. Sinauer, Sunderland,Massachesetts.
Meyer K. (1988). Dfreml: a set of programs to estimate components uncer an individual animal model. Journal of Dairy Science 2, 33–34.
Meyer K. (1989). Restricted maximum likelihood to estimate variance components for animal models with several random effects using a derivative-free algorithm. Genetics
Selection and Evolution 21, 317–340.
Meyer K. (1998). DRREML, version3.0β user notes.
Muirhead R.J. (1982). Aspects of Multivariate Statistical Theory. Wiley Series in Prob- ability and Mathematical Statistics. Wiley, New York.
Nijhout H.F. (1994). Insect Hormones. Princeton University Press, Princeton, N.J. Ramsay J.O. and Silverman B.W. (2002). Applied Functional Data Analysis: methods
and case studies. Springer Series in Statistics. Springer, New York.
Ramsay J.O. and Silverman B.W. (2005). Functional Data Analysis. Springer Series in Statistics. Springer, New York.
Riddiford L.M., Hirune K., Zhou X. and Nelson C.A. (2003). Insights into the molecular basis of the hormonal control of molting and metamorphosis from manduca sexta and drosophila melanogaster. Insect Biochemistry and Molecular Biology 33, 1327–1338. Schatzman M. (2002). Numerical Analysis: A Mathematical Introduction. Claredon
Press, Oxford.
Scholkopf B., Smola A.J. and Muller K.R. (1998). Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 10, 1299–1319.
Searle S.R., Casella G. and McCulloch C.E. (1992). Variance Components. Wiley Series in Probability and Statistics. Wiley, New York.
Stewart G.W. and Sun J.g. (1990). Matrix Perturbation Theory. Computer Science and Scientific Computing. Academic Press, San Diego.
Tyler D.E. (1981). Asymptotic inference for eigenvectors. The Annals of Statistics 9, 725–736.
Tyler D.E. (1983). A class of asymptotic tests for principal component vectors. The
Annals of Statistics 11, 1243–1250.
van der Vaart A.W. (1998). Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, New York.
Vapnik V.N. (1995). The Nature of Statistical Learning. Statistics for engineering and information science. Springer, New York.
Wand M.P. and Jones M. (1995). Kernel Smoothing. Monographs on statistics and applied probability ; 60. Chapman and Hall, London.