Wrap-Around L2-Discrepancy of Random Sampling, Latin Hypercube and Uniform Designs

For comparing random designs and Latin hypercube designs, this paper con- siders a wrap-around version of the L₂-discrepancy (WD). The theoretical expectation and variance of this discrepancy are derived for these two designs. The expectation and variance of Latin hypercube designs are significantly lower than those of the corresponding random designs. We also study construction of the uniform design under the WD and show that one-dimensional uniform design under this discrepancy can be any set of equidistant points. For high dimensional uniform designs we apply the threshold accepting heuristic for finding low discrepancy designs. We also show that the conjecture proposed by K. T. Fang, D. K. J. Lin, P. Winker, and Y. Zhang (2000, Technometrics) is true under the WD when the design is complete.     

A dimensional result for random self-similar sets

A very important property of a deterministic self-similar set is that its Hausdorff dimension and upper box-counting dimension coincide. This paper considers the random case. We show that for a random self-similar set, its Hausdorff dimension and upper box-counting dimension are equal

     

Domain functionals and exit times for Brownian motion

Two domain functionals describing the averaged expectation of exit times and averaged variance of exit times of Brownian motion from a domain, respectively, are studied. We establish the variational formulas for maximizing the functionals over domains with a volume constraint, and characterize the stationary points and maximizers.

     

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

New lower bounds for three- and four-level designs under the centered -discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered -discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.

     

投影均匀分片拉丁超立方体设计

空间填充设计是有效的计算机试验设计,比如均匀设计、最大最小距离拉丁超立方体设计等.虽然这些设计在整个试验空间中有较好的均匀性,但其低维投影均匀性可能并不理想.对于因子是定量的计算机试验,已有文献构造了诸如最大投影设计、均匀投影设计等相适应的设计;而对于同时含有定性因子和定量因子的计算机试验,尚未有投影均匀设计的相关文献.文章提出了综合投影均匀准则,利用门限接受算法构造了投影均匀的分片拉丁超立方体设计.在新构造设计中,整体设计与每一片设计均具有良好的投影均匀性.模拟结果显示,与随机分片拉丁超立方体设计相比,利用新构造设计进行试验而拟合的高斯过程模型具有更小的均方根预测误差.     

A new class of Latin hypercube designs with high-dimensional hidden projective uniformity

Latin hypercube design is a good choice for computer experiments. In this paper, we construct a new class of Latin hypercube designs with some high-dimensional hidden projective uniformity. The construction is based on a new class of orthogonal arrays of strength two which contain higher strength orthogonal arrays after their levels are collapsed. As a result, the obtained Latin hypercube designs achieve higher-dimensional uniformity when projected onto the columns corresponding to higher strength orthogonal arrays, as well as twodimensional projective uniformity. Simulation study shows that the constructed Latin hypercube designs are significantly superior to the currently used designs in terms of the times of correctly identifying the significant effects.     

Dimension of hyperbolic measures of random diffeomorphisms

We consider dynamics of compositions of stationary random diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.

     

一类正交空间填充设计的构造

空间填充设计在计算机试验中应用十分广泛,当拟合回归模型时,正交的空间填充设计保证了因子效应估计的独立性.基于广义正交设计,文章给出了构造二阶正交拉丁超立方体设计和列正交设计的方法,新构造的设计不仅满足任意两列之间相互正交,还能保证每一列与任一列元素平方组成的列以及任两列元素相乘组成的列都正交.当某些正交的空间填充设计不存在时,具有较小相关系数的近似正交设计可作为替代设计使用.设计构造的灵活性为计算机试验在实践中的广泛应用提供了必要的支持.     

Linear Perfect Codes and a Characterization of the Classical Designs

A new definition for the dimension of a combinatorial t-(v,k,) design over a finite field is proposed. The complementary designs of the hyperplanes in a finite projective or affine geometry, and the finite Desarguesian planes in particular, are characterized as the unique (up to isomorphism) designs with the given parameters and minimum dimension. This generalizes a well-known characterization of the binary hyperplane designs in terms of their minimum 2-rank. The proof utilizes the q-ary analogue of the Hamming code, and a group-theoretic characterization of the classical designs.     

Intercalates and discrepancy in random Latin squares

An intercalate in a Latin square is a 2 × 2 Latin subsquare. Let be the number of intercalates in a uniformly random n × n Latin square. We prove that asymptotically almost surely , and that (therefore asymptotically almost surely for any ). This significantly improves the previous best lower and upper bounds. We also give an upper tail bound for the number of intercalates in 2 fixed rows of a random Latin square. In addition, we discuss a problem of Linial and Luria on low‐discrepancy Latin squares.     

Dimension and randomness in groups acting on rooted trees

We explore the structure of the -adic automorphism group of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán.

We initiate the study of a general dimension theory of groups acting on rooted trees. We describe the relationship between dimension and other properties of groups such as solvability, existence of dense free subgroups and the normal subgroup structure. We show that subgroups of generated by three random elements are full dimensional and that there exist finitely generated subgroups of arbitrary dimension. Specifically, our results solve an open problem of Shalev and answer a question of Sidki.

     

Dimensional properties of the harmonic measure for a random walk on a hyperbolic group

This paper deals with random walks on isometry groups of Gromov hyperbolic spaces, and more precisely with the dimension of the harmonic measure associated with such a random walk. We first establish a link of the form between the dimension of the harmonic measure, the asymptotic entropy of the random walk and its rate of escape . Then we use this inequality to show that the dimension of this measure can be made arbitrarily small and deduce a result on the type of the harmonic measure.

     

Higher dimensional aposyndetic decompositions

Let be a homogeneous, decomposable continuum that is not aposyndetic. The Aposyndetic Decomposition Theorem yields a cell-like decomposition of into homogeneous continua with quotient space being an aposyndetic, homogeneous continuum.

Assume the dimension of is greater than one. About 20 years ago the author asked the following questions:

Can this aposyndetic decomposition raise dimension? Can it lower dimension? We answer these questions by proving the following theorem.

Theorem. The dimension of the quotient space is one.

     

低偏差OALHD的构造

本文给出了利用均匀设计和正交表构造低偏差OALH设计的方法，该方法构造的设计既有优良的均匀性具有正交设计的均衡性，一个更重要的优点是可以构造较大样本容量的设计点集，本文同时给出了某些参数的均匀设计表，这些设计优于现有的均匀设计，具有实用价值。     

The hereditary discrepancy is nearly independent of the number of colors

We investigate the discrepancy (or balanced coloring) problem for hypergraphs and matrices in arbitrary numbers of colors. We show that the hereditary discrepancy in two different numbers of colors is the same apart from constant factors, i.e.,

This contrasts the ordinary discrepancy problem, where no correlation exists in many cases.

     

Castelnuovo-Mumford regularity and extended degree

Our main result shows that the Castelnuovo-Mumford regularity of the tangent cone of a local ring is effectively bounded by the dimension and any extended degree of . From this it follows that there are only a finite number of Hilbert-Samuel functions of local rings with given dimension and extended degree.

     

Solution of Shannon's problem on the monotonicity of entropy

It is shown that if are independent and identically distributed square-integrable random variables, then the entropy of the normalized sum

is an increasing function of .

The result also has a version for non-identically distributed random variables or random vectors.

     

On the length of the spectral sequence of a Lie algebra extension

The length of the spectral sequence of a Lie algebra extension is at most the dimension of the quotient algebra. We show that this bound can be attained for arbitrarily large quotient algebras even when the algebra is nilpotent and the extension splits.

     

Construction of sliced (nearly) orthogonal Latin hypercube designs

Sliced Latin hypercube designs are very useful for running a computer model in batches, ensembles of multiple computer models, computer experiments with qualitative and quantitative factors, cross-validation and data pooling. However, the presence of highly correlated columns makes the data analysis intractable. In this paper, a construction method for sliced (nearly) orthogonal Latin hypercube designs is developed. The resulting designs have flexible sizes and most are new. With the orthogonality or near orthogonality being guaranteed, the space-filling property of the resulting designs is also improved. Examples are provided for illustrating the proposed method.