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Monte Carlo Simulation and other random thoughts
Published in Alan R. Jones, Risk, Opportunity, Uncertainty and Other Random Models, 2018
To help us understand how LHS works, let’s first consider the simple model of a Latin Square which has only two variables. A Latin Square is an arrangement of symbols such that each symbol appears once, and only once, in each row and column. For instance, if each variable’s values are divided into tertile confidence intervals of nominally 0%–33%, 33%–67%, 67%–100%, we can draw 12 Latin Squares, as illustrated in Figure 3.56. For the two variables, each Latin Square allows us to draw three unique samples, although it can be argued that from a sampling perspective if we were to interchange B and C we would duplicate half the combinations. (Note that the left hand six sets of three samples are equivalent to the right hand six sets if we do this.) In this way we can select 18 unique Latin Square samples for two variables with three tertile confidence intervals.
Steiner Triple Systems
Published in C. C. Lindner, C. A. Rodger, Design Theory, 2017
A latin square of order n is an n × n array, each cell of which contains exactly one of the symbols in {1, 2,…, n}, such that each row and each column of the array contains each of the symbols in {1, 2,…, n} exactly once. A quasigroup of order n is a pair (Q,o) $ (Q, o) $ , where Q is a set of size n and “o” is a binary operation on Q such that for every pair of elements a, b ∊ Q, the equations a o x = b and y o a = b have unique solutions. As far as we are concerned a quasigroup is just a latin square with a headline and a sideline.
General linear codes
Published in Jürgen Bierbrauer, Introduction to Coding Theory, 2016
An OA of strength 1 is a very weak structure. All we demand is that in each column each entry shows up the same number of times. Most people have encountered OA of strength 2 and index 1, albeit in disguise. A popular structure, in discrete mathematics and statistics as well as in recreational math, are Latin squares. A Latin square of order n is by definition an (n, n)-array with entries from an alphabet of size n. Each of the n2 cells of the array is filled with an entry from the alphabet. The central property (making it Latin) is In each row and in each column each entry occurs precisely once, in other words: each row and each column is a permutation of the symbols.
Enumerating extreme points of the polytopes of stochastic tensors: an optimization approach
Published in Optimization, 2020
Let be the number of vertices of . Estimation of has been witnessed in three ways: (1). Combinatorial method using Latin squares. Ahmed, De Loera, and Hemmecke (see [22, Theorem 2.0.10] or [23, Theorem 0.1]) gave an explicit lower bound . This lower bound is immediately superiorized by the one obtained by Latin squares, because the number of Latin squares of order n, denoted by , is equal to the number of line-stochastic (0, 1)-tensors (see [20] or [24, p. 159–161]). Note that every (0-1)-stochastic tensor is an extreme point. (2). Analytic and topological approach by using hyperplane and induction. Chang, Paksoy, and Zhang [25] showed an upper bound (see Theorem 3.1 below). (3). Computational geometry approach using the known results, i.e. the Lower and Upper Bound Theorems, on polytopes. Li, Zhang and Zhang [16] presented that the upper bound obtained in this way is better (shaper) than the previous one in [25]. However, the lower bound is no better.
A dynamic geometry system approach to analyse distance geometry problems based on partial Latin squares
Published in International Journal of Mathematical Education in Science and Technology, 2020
A partial Latin square of order n is an array whose cells are either empty or contain an element from a finite set of n symbols so that no two identical symbols appear within any given row or column. This is a Latin square if all its cells are non-empty. It is said that a partial Latin square can be completed to a Latin square having the same order than P if , for all non-empty cell within P. Thus, for instance, the well-known puzzle called Sudoku consists of a partial Latin square of order nine that can be uniquely completed to a Latin square of the same order so that the 81 cells of the latter are partitioned into nine subsquares, all of them containing the nine distinct symbols under consideration.
Sequential Laplacian regularized V-optimal design of experiments for response surface modeling of expensive tests: An application in wind tunnel testing
Published in IISE Transactions, 2019
Adel Alaeddini, Edward Craft, Rajitha Meka, Stanford Martinez
Space-filling designs attempt to spread out the samples as evenly as possible to collect as much information about the entire design space as possible. Major space-filling methods include Orthogonal Arrays and various Latin hypercube designs. Latin hypercube is a statistical method for generating a distribution of plausible collections of parameter values from a multidimensional distribution (McKay et al., 1979; Tang, 1993; Kenny et al., 2000). In the context of statistical sampling, a square grid containing sample positions is a Latin square if and only if there is only one sample in each row and each column. A Latin hypercube is the generalization of this concept to an arbitrary number of dimensions, whereby each sample is the only one in each axis-aligned hyper-plane containing it. Orthogonal sampling adds the requirement that the entire sample space must be sampled evenly (Owen, 1992; Taguchi and Yokoyama, 1993). In other words, orthogonal sampling ensures that the ensemble of random numbers is a very good representative of the real variability. Although orthogonal sampling is generally more efficient, it is more difficult to implement, as all random samples must be generated simultaneously.