78 research outputs found
Enumeration of regular fractional factorial designs with four-level and two-level factors
Designs for screening experiments usually include factors with two levels
only. Adding a few four-level factors allows for the inclusion of multi-level
categorical factors or quantitative factors with possible quadratic or
third-order effects. Three examples motivated us to generate a large catalog of
designs with two-level factors as well as four-level factors. To create the
catalog, we considered three methods. In the first method, we select designs
using a search table, and in the second method, we use a procedure that selects
candidate designs based on the properties of their projections into fewer
factors. The third method is actually a benchmark method, in which we use a
general orthogonal array enumeration algorithm. We compare the efficiencies of
the new methods for generating complete sets of non-isomorphic designs.
Finally, we use the most efficient method to generate a catalog of designs with
up to three four-level factors and up to 20 two-level factors for run sizes 16,
32, 64, and 128. In some cases, a complete enumeration was infeasible. For
these cases, we used a bounded enumeration strategy instead. We demonstrate the
usefulness of the catalog by revisiting the motivating examples.Comment: 37 pages, 1 figure, 13 table
A quasidouble of the affine plane of order 4 and the solution of a problem on additive designs
A 2-(v,k,λ) block design (P,B) is additive if, up to isomorphism, P can be represented as a subset of a commutative group (G,+) in such a way that the k elements of each block in B sum up to zero in G. If, for some suitable G, the embedding of P in G is also such that, conversely, any zero-sum k-subset of P is a block in B, then (P,B) is said to be strongly additive. In this paper we exhibit the very first examples of additive 2-designs that are not strongly additive, thereby settling an open problem posed in 2019. Our main counterexample is a resolvable 2-(16,4,2) design (F_4×F_4, B_2), which decomposes into two disjoint isomorphic copies of the affine plane of order four. An essential part of our construction is a (cyclic) decomposition of the point-plane design of AG(4,2) into seven disjoint isomorphic copies of the affine plane of order four. This produces, in addition, a solution to Kirkman's schoolgirl problem
On the seven non-isomorphic solutions of the fifteen schoolgirl problem
In this paper we give a simple and effective tool to analyze a given Kirkman triple system of order 15 and determine which of the seven well-known non-isomorphic KTS(15)s it is isomorphic to. Our technique refines and improves the lacing of distinct parallel classes introduced by F. N. Cole, by means of the notion of residual triple defined by G. Falcone and the present author in a previous paper. Unlike Cole's original lacing scheme, our algorithm allows one to distinguish two KTS(15)s also in the harder case where the two systems have the same underlying Steiner triple system. In the special case where the common STS is #19, an alternative method is given by means of the 1-factorizations of the complete graph K_8 associated to the two KTSs. Moreover, we present three new visual solutions to the schoolgirl problem, and we catalogue most of the classical (or interesting) solutions in the literature in terms of what KTS(15)s they are isomorphic to. This paper provides background on a classical topic, while shedding new light on the problem as well
Statistical Modelling
The book collects the proceedings of the 19th International Workshop on Statistical Modelling held in Florence on July 2004. Statistical modelling is an important cornerstone in many scientific disciplines, and the workshop has provided a rich environment for cross-fertilization of ideas from different disciplines. It consists in four invited lectures, 48 contributed papers and 47 posters. The contributions are arranged in sessions: Statistical Modelling; Statistical Modelling in Genomics; Semi-parametric Regression Models; Generalized Linear Mixed Models; Correlated Data Modelling; Missing Data, Measurement of Error and Survival Analysis; Spatial Data Modelling and Time Series and Econometrics
Binary Hamming codes and Boolean designs
In this paper we consider a finite-dimensional vector space P over the Galois field GF(2), and the family Bk (respectively, B 17k) of all the k-sets of elements of P (respectively, of P 17=P 16{0}) summing up to zero. We compute the parameters of the 3-design (P,Bk) for any (necessarily even) k, and of the 2-design (P 17,B 17k) for any k. Also, we find a new proof for the weight distribution of the binary Hamming code. Moreover, we find the automorphism groups of the above designs by characterizing the permutations of P, respectively of P 17, that induce permutations of Bk, respectively of B 17k. In particular, this allows one to relax the definitions of the permutation automorphism groups of the binary Hamming code and of the extended binary Hamming code as the groups of permutations that preserve just the codewords of a given Hamming weight
Groups (S_{n}times S_{m}) in construction of flag-transitive block designs
In this paper, we observe the possibility that the group (S_{n}times S_{m}) acts
as a flag-transitive automorphism group of a block design with point set ({1,ldots ,n}times {1,ldots ,m},4leq nleq mleq 70). We prove the
equivalence of that problem to the existence of an appropriately defined
smaller flag-transitive incidence structure. By developing and applying
several algorithms for the construction of the latter structure, we manage
to solve the existence problem for the desired designs with (nm) points in
the given range. In the vast majority of the cases with confirmed existence,
we obtain all possible structures up to isomorphism
An algebraic representation of Steiner triple systems of order 13
In this paper we construct an incidence structure isomorphic to a Steiner triple system of order 13 by defining a set B of twentysix vectors in the 13-dimensional vector space V = GF(5)^13, with the property that there exist precisely thirteen
6-subsets of B whose elements sum up to zero in V, which can also be characterized as the intersections of B with thirteen linear hyperplanes of V
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Time-varying networks: Measurement, Modeling, and Computation
Time-varying networks and techniques developed to study them have been used to analyze dynamic systems in social, computational, biological, and other contexts. Significant progress has been made in this area in recent years, resulting from a combination of statistical advances and improved computational resources, giving rise to a range of new research questions. This thesis addresses problems related to three lines of inquiry involving dynamic networks: data collection designs; the conditions needed for structural stability of an evolving network; and the computational scalability of statistical models for network dynamics. The first contribution involves a commonly neglected problem concerning data collection protocols for dynamic network data: the impact of in-design missingness. A systematic formalization is offered for the widely used class of retrospective life history designs, and it is shown that design parameters have nontrivial effects on both the quantity of missingness and the impact of such missingness on network modeling and reconstruction. Using a simulation study, we also show how the consequences of design parameters for inference vary as a function of look-back time relative to the time of measurement. The second contribution of this thesis is related to a fundamental question of network dynamics: when or where are changes in a network most likely to occur? A novel approach is taken to this question, by exploring its complement -- what factors stabilize a network (or subgraphs thereof) and make it resistant to change? For networks whose behavior can be parameterized in exponential family form, a formal characterization of the graph-stabilizing region of the parameter space is shown to correspond to a convex polytope in the parameter space. A related construction can be used to find subgraphs that are or are not stable with respect to a given parameter vector, and to identify edge variables that are most vulnerable to perturbation. Finally, the third contribution of this thesis is to scalable parameter estimation for a class of temporal exponential family random graph models (TERGM) from sampled data. An algorithm is proposed that allows accurate approximation of maximum likelihood estimates for certain classes of TERGMs from egocentrically sampled retrospective life history data, without requiring simulation of the underlying network (a major bottleneck when the network size is large). Estimation time for this algorithm scales with the data size, and not with the size of the network, allowing it to be employed on very large populations
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