Keywords (Extracted from title, table of contents and
abstract of thesis) Contiguous, Selection, Designs,
Polygonal, Construct, Existence, Survey, Block, Generalized,
Polygonal, Designs, Identification |
Abstract Polygonal designs, a
class of partially balanced incomplete block designs (PBIBDs) with
regular polygons, are useful in survey sampling in terms of balanced
sampling plans excluding contiguous units (BSECs) and balanced
sampling plans to avoid the selection of adjacent units (BSAs), when
neighboring (contiguous or adjacent) units in a population provide
similar information. The reason for using such designs is that the
units that are physically close might be more similar than the
distant units. By the use of such designs or plans we can select the
units over the entire experimental region by avoiding the selection
of units that provide essentially redundant information. In other
words, these neighboring units are deliberately excluded from being
sampled under the idea that they provide little new information to
the sampling effort.
Searches for polygonal designs may be divided into two broad
categories: those which attempts to prove the existence of polygonal
designs with a given set of parameters (v; k; ¸; ®), and those which
attempts to construct (or enumerate) polygonal designs with a given
set of parameters (v; k; ¸; ®). In this thesis, the construction of
cyclic polygonal designs is generalized for the parameters: the
distance ® (orm), the concurrence (or index) parameter ¸ and the
treatments v.The major reasons for introducing generalized cyclic
polygonal designs in this thesis are that:
(i) the existing literature considers the existence and the
construction of cyclic polygonal designs only for the limited
distance ®, the concurrence parameter ¸ and the treatments v;
(ii) the existence and the construction of unequal block sized
cyclic polygonal designs for distance ® ¸ 1 has not been attempted
in literature.
In Chapter 1, an introduction to polygonal designs is given. A brief
review on the existing work on polygonal designs is presented, and
some limitations in the existing work are pointed out.
In Chapter 2, the method of cyclic shifts is briefly described, and
explained that how this method helps in the development of
concurrence matrix (or concurrence vector) which is the main tool
for the detection of the properties of cyclic polygonal designs. The
distinguishing feature of this method is that the properties of a
design can easily be obtained from the sets of shifts instead of
constructing the actual blocks of the design. The pattern of
off-diagonal zero elements (in bold form) from the main diagonal in
a concurrence matrix (or in a concurrence vector) is useful in the
identification of the distance in a cyclic polygonal design. In
Chapter 3, minimal cyclic polygonal designs with block size k = 3
and ¸ = 1 are constructed for distance ® = 1; 2; 3; : : : ; 16 and
for v < 100 treatments. In Chapter 4, the existence and construction
of cyclic polygonal designs with block size k = 3, for ¸ = 1; 2; 3;
4; 6; 12 and for ® = 1; 2; 3; 4; 5; 6 is considered, and complete
solutions for v · 100 treatments are presented.
In Chapter 5, the existence and construction of minimal cyclic
polygonal designs with unequal-sized blocks and ¸ = 1 is first ever
introduced for distance ® ¸ 1.In Chapter 6, the thesis is summarized
and future directions for the extension of cyclic polygonal designs
are proposed. |