This book offers a unique new look at the familiar quantification theory
from the point of view of mathematical symmetry and spatial symmetry.
Symmetry exists in many aspects of our life--for instance, in the arts
and biology as an ingredient of beauty and equilibrium, and more
importantly, for data analysis as an indispensable representation of
functional optimality. This unique focus on symmetry clarifies the
objectives of quantification theory and the demarcation of
quantification space, something that has never caught the attention of
researchers.
Mathematical symmetry is well known, as can be inferred from
Hirschfeld's simultaneous linear regressions, but spatial symmetry has
not been discussed before, except for what one may infer from
Nishisato's dual scaling. The focus on symmetry here clarifies the
demarcation of quantification analysis and makes it easier to understand
such a perennial problem as that of joint graphical display in
quantification theory. The new framework will help advance the frontier
of further developments of quantification theory.
Many numerical examples are included to clarify the details of
quantification theory, with a focus on symmetry as its operational
principle. In this way, the book is useful not only for graduate
students but also for researchers in diverse areas of data analysis.