Optimal Row-Column Designs for Correlated Errors and Nested Row-Column Designs for Uncorrelated Errors

In this dissertation the design problems are considered in the row-column setting for second order autonormal errors when the treatment effects are estimated by generalized least squares, and in the nested row-column setting for uncorrelated errors when the treatment effects are estimated by ordinary least squares. In the former case, universal optimality conditions are derived separately for designs in the plane and on the torus using more general linear models than those considered elsewhere in the literature. Examples of universally optimum planar designs are given, and a method is developed for the construction of optimum and near optimum designs, that produces several infinite series of universally optimum designs on the torus and near optimum designs in the plane. Efficiencies are calculated for planar versions of the torus designs, which are found to be highly efficient with respect to some commonly used optimality criterion. In the nested row-column setting, several methods of construction of balanced and partially balanced incomplete block designs with nested rows and columns are developed, from which many infinite series of designs are obtained. In particular, 149 balanced incomplete block designs with nested rows and columns are listed (80 appear to be new) for the number of treatments, v \u3c 101, a prime power

Optimality and Construction of Designs with Generalized Group Divisible Structure

This thesis is an investigation of the optimality and construction problems attendant to the assignment of v treatments to experimental units in b blocks of size k, paying special attention to settings for which equal replication of the treatments is not possible. The model is that of one way elimination of heterogeneity, in which the expectation of an observation on treatment i in block j is Ti + βj (treatment effect + block effect), where Ti and βj are unknown constants, 1 ≤ i ≤ v and 1 ≤ j ≤ b. All observations are assumed to be uncorrelated with same variance. The generalized group divisible design with s groups, or GGDD(s), is defined in terms of the elements of the information matrix, instead of in terms of the elements of the concurrence matrix as done by Adhikary (1965) and extended by Jacroux (1982). This definition extends the class of designs to include non-binary members, and allows for broader optimality results. Some sufficient conditions are derived for GGDD(s) to be E- and MV-optimal. It is also shown how augmentation of addition blocks to certain GGDD(s)s produces other nonbinary, unequally replicated E- and MV-optimal block designs. Where nonbinary designs are found, they are generally preferable to binary designs in terms of interpretability, and often in terms of one or more formal optimality criteria as well. The class of generalized nearly balanced incomplete block designs with maximum concurrence range l, or NBBD(l), is defined. This class extends the nearly balance incomplete block designs as defined by Cheng & Wu (1981), and the semi-regular graph designs as defined by Jacroux (1985), to cases where off-diagonal entries of the concurrence matrix differ by at most the positive integer l. Sufficient conditions are derived for a NBBD(2) to be optimal under a given type-I criterion. The conditions are used to establish the A- and D-optimality of an infinite series of NBBD(2)s having unequal numbers of replicates. Also, a result from Jacroux (1985) is used to establish the A-optimality of a new series of NBBD(1)s. Several methods of construction of GGDD(s)s are developed from which many infinite series of designs are derived. Generally these designs satisfy the obtained sufficient conditions for E- and MV-optimality. Finally, in the nested row-column setting, the necessary conditions for existence of 2 x 2 balanced incomplete block designs with nested rows and columns (BIBRCs) are found to be sufficient. It is also shown that, sufficient for a BIBRC with p=q to generally balanced, is that the row and column classifications together form a balanced incomplete block design, as does the block classification. All of the 2 x 2 BIBRCs are constructed to have this property

Construction of some new three associate class partially balanced incomplete block designs in two replicates

Paper presented at the 2nd Strathmore International Mathematics Conference (SIMC 2013), 12 - 16 August 2013, Strathmore University, Nairobi, Kenya.Search for experimental designs which aid in research studies involving large number of treatments with minimal experimental units has been desired overtime. This paper constructs some new series of three associate Partially Balanced Incomplete Block (PBIB) designs having n(n - 2) /4 treatments with three associate classes in two replicates using the concept of triangular association scheme. The design is constructed from an even squared array of n rows and n columns (n _> 8) with its both diagonal entries bearing no treatment entries and that given the location of any treatment in the squared array, the other location of the same treatment in the array is predetermined. The design and association parameters for a general case of an even integer n >_8 are obtained with an illustrated case for n = 8. Efficiencies of the designs within the class of designs are obtained for a general case of even n >_8 with a listing of efficiencies of designs with blocks sizes in the interval [8,22]. The designs constructed have three associate classes and are irreducible to minimum number of associate classes.Search for experimental designs which aid in research studies involving large number of treatments with minimal experimental units has been desired overtime. This paper constructs some new series of three associate Partially Balanced Incomplete Block (PBIB) designs having n(n - 2) /4 treatments with three associate classes in two replicates using the concept of triangular association scheme. The design is constructed from an even squared array of n rows and n columns (n _> 8) with its both diagonal entries bearing no treatment entries and that given the location of any treatment in the squared array, the other location of the same treatment in the array is predetermined. The design and association parameters for a general case of an even integer n >_8 are obtained with an illustrated case for n = 8. Efficiencies of the designs within the class of designs are obtained for a general case of even n >_8 with a listing of efficiencies of designs with blocks sizes in the interval [8,22]. The designs constructed have three associate classes and are irreducible to minimum number of associate classes

Evaluation of Different Manual Placement Strategies to Ensure Uniformity of the V-FPGA

Virtual FPGA (V-FPGA) architectures are useful as both early prototyping testbeds for custom FPGA architectures, as well as to enable advanced features which may not be available on a given host FPGA. V-FPGAs use standard FPGA synthesis and placement tools, and as a result the maximum application frequency is largely determined by the synthesis of the V-FPGA onto the host FPGA. Minimal net delays in the virtual layer are crucial for applications, but due to increased routing congestion, these delays are often significantly worse for larger than for smaller designs. To counter this effect, we investigate three different placement strategies with varying amounts of manual intervention. Taking the regularity of the V-FPGA architecture into account, a regular placement of tiles can lead to an 37% improvement in the achievable clock frequency. In addition, uniformity of the measured net delays is increased by 39%, which makes implementation of user applications more reproducible. As a trade-off, these manual placement strategies increase area usage of the virtual layer up to 16%

Resolvable designs with large blocks

Resolvable designs with two blocks per replicate are studied from an optimality perspective. Because in practice the number of replicates is typically less than the number of treatments, arguments can be based on the dual of the information matrix and consequently given in terms of block concurrences. Equalizing block concurrences for given block sizes is often, but not always, the best strategy. Sufficient conditions are established for various strong optimalities and a detailed study of E-optimality is offered, including a characterization of the E-optimal class. Optimal designs are found to correspond to balanced arrays and an affine-like generalization.Comment: Published at http://dx.doi.org/10.1214/009053606000001253 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org