By Yasuko Chikuse
This booklet is anxious with statistical research at the specified manifolds, the Stiefel manifold and the Grassmann manifold, handled as statistical pattern areas which includes matrices. the previous is represented via the set of m x okay matrices whose columns are together orthogonal k-variate vectors of unit size, and the latter by way of the set of m x m orthogonal projection matrices idempotent of rank ok. The observations for the exact case k=1 are considered as directed vectors on a unit hypersphere and as axes or strains undirected, respectively. Statistical research on those manifolds is needed, specially for low dimensions in sensible purposes, within the earth (or geological) sciences, astronomy, medication, biology, meteorology, animal habit and plenty of different fields. The Grassmann manifold is a slightly new topic taken care of as a statistical pattern area, and the advance of statistical research at the manifold needs to make a few contributions to the comparable sciences. The reader could already understand the standard thought of multivariate research at the actual Euclidean house and intend to deeper or develop the study sector to stats on distinctive manifolds, which isn't taken care of typically textbooks of multivariate research.
The writer quite concentrates at the themes to which a large amount of own attempt has been committed. beginning with basic fabric of the particular manifolds and a few wisdom in multivariate research, the ebook discusses inhabitants distributions (especially the matrix Langevin distributions which are used for the main of the statistical analyses during this book), decompositions of the targeted manifolds, sampling distributions, and statistical inference at the parameters (estimation and checks for hypotheses). Asymptotic concept in sampling distributions and statistical inference is built for big pattern measurement, for big focus and for top measurement. extra investigated are Procrustes tools utilized at the distinctive manifolds, density estimation, and dimension of orthogonal organization.
This e-book is designed as a reference ebook for either theoretical and utilized statisticians. The ebook can be used as a textbook for a graduate path in multivariate research. it can be assumed that the reader understands the standard concept of univariate records and a radical historical past in arithmetic, specifically, wisdom of multivariate calculation innovations. To make the e-book self-contained, a short evaluation of a few of these features and similar subject matters is given.
Yasuko Chikuse is Professor of facts and arithmetic at Kagawa college, Japan. She earned a Ph.D. in data from Yale college and Sc.D. in arithmetic from Kyushu college, Japan. She is a member of the overseas Statistical Institute, the Institute of Mathematical information, the yankee Statistical organization, the Japan Statistical Society, and the Mathematical Society of Japan. She has held vacationing examine and/or educating appointments on the CSIRO, the college of Pittsburgh, the collage of California at Santa Barbara, York college, McGill collage, and the collage of St Andrews.
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This publication by means of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate historical past. In 1938-39, Nielsen gave a sequence of lectures on
discontinuous teams of motions within the non-euclidean aircraft, and this led him - in the course of
World conflict II - to put in writing the 1st chapters of the e-book (in German). whilst Fenchel,
who needed to get away from Denmark to Sweden end result of the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what grew to become recognized
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) was once comprehensive in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. despite the fact that, because of the speedy improvement of the topic, they felt
that giant adjustments needed to be made prior to book.
When Nielsen moved to Copenhagen college in 1951 (where he stayed until eventually
1955), he used to be a lot concerned with the overseas association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The documents of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity comprise unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and an entire manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The files additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes designated reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen college in 1956 (and stayed there
until 1974), used to be greatly concerned with a radical revision of the curriculum in al-
gebra and geometry, and centred his study within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally positioned a lot attempt into his task as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the
Fenchel-Nielsen undertaking the best way he desired to.
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eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - stumbled on time to
finish the publication trouble-free Geometry in Hyperbolic house, which used to be released by way of
Walter de Gruyter in 1989 presently after his demise. at the same time, and with a similar
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal the various imprecise issues that have been within the unique
manuscript. Fenchel informed me that he reflected removal elements of the introductory
Chapter I within the manuscript, considering this might be lined by way of the e-book pointed out above;
but to make the Fenchel-Nielsen e-book self-contained he eventually selected to not do
so. He did choose to omit
27, entitled Thefundamental team.
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I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed an excellent task of typing this manuscript in AMS- TEX. i've got additionally had
much aid from my colleague J0rn B0rling Olsson (himself a scholar of Kate Fenchel
at Aarhus collage) with the evidence interpreting of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a normal dialogue of the difference to the fashion
of TEX. In such a lot respects we determined to persist with Fenchel's intentions. even though, turning
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and the spelling of definite key-words, will be uniform through the publication. additionally,
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Studies in arithmetic. i'm so much thankful for this confident and fast response. One
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first agreed to convey those in ultimate digital shape, yet by way of 1997 it grew to become transparent that he
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whereby I may still bring targeted drawings of the figures (Fenchel didn't go away such
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a finished index has been additional. In either situations, all references are to sections,
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My ultimate and such a lot honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this e-book into life.
Statistics on Special Manifolds
This ebook is anxious with statistical research at the distinctive manifolds, the Stiefel manifold and the Grassmann manifold, handled as statistical pattern areas along with matrices. the previous is represented by means of the set of m x ok matrices whose columns are jointly orthogonal k-variate vectors of unit size, and the latter via the set of m x m orthogonal projection matrices idempotent of rank ok.
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Extra resources for Statistics on Special Manifolds
The matrix Langevin L(m, kj F) and L(P) (m, kj B) distributions are expon~n tial and modal distributions on Vk,m and Pk,m-k' respectively, and possess desirable properties and tractability corresponding to those possessed by the multivariate normal distributions on the real spaces Rm, Rm,k' and Sm. The special cases F = 0 and B = 0 yield the uniform distributions on Vk,m and Pk,m-k' respectively. Most of the statistical analyses on the manifolds Vk,m and Pk,m-k treated in later chapters are carried out for these matrix Langevin distributions.
The orientation of the orbit can be specified by the matrix X = (3: 1 3:2 ) in V;,3. Jupp (1995) gave applications of directional statistics to some astronomical problems on the orbits of comets using Marsden's catalogue of cometary orbits [see Marsden and Williams (1993)]; the same article considered directional problems on the orbits of visual binary stars and spin axes of asteroids as well. See Mardia (1975), Jupp and Mardia (1979), Waston (1983a), and Fisher, Lewis, and Embleton (1987) for analysis of related data.
9) on V,. 21) 38 2. 11). 17)(P). 15)(P). 13)(P), having the density function 1 1 F (lk' 1 . 22) is a slight modification of the Downs (1972) distribution on the Stiefel manifold, and may be called the matrix Langevin distribution on Pk,m-k' which is denoted by L