A two-stage method for the estimation of global sensitivity indices of non-parametric models

This article aims to propose a method to effectively estimate global sensitivity indices under non-parametric models. The new method involves two stages. First, all the non-influential sensitivity indices are filtered out by an adjustive W-statistic test process with low cost, and then the remaining significant sensitivity indices are precisely estimated by an orthogonal array (OA) with large number of levels and low strength. The method avoids complicated prototype building and shows a much lower experimental cost. The performance of this method as well as comparisons with polynomial regression method, Gaussian Process (GP) method, and component selection and smoothing operator (COSSO) method are tested on three numerical models that are widely used in engineering and statistical areas. Finally, a real data example is analyzed.     

Simultaneous one-sided tolerance intervals for sets of contrasts from normal populations

This paper considers the problem of constructing simultaneous prediction and tolerance intervals for sets of contrasts of normal variables in situations where simultaneous intervals are available. Tables are given with critical values used in simultaneous tolerance bounds for two classes of contrasts: pairwise many-one and profile type.     

A method for testing interaction in unreplicated two-way tables: using all pairwise interaction contrasts

Several methods exist for testing interaction in unreplicated two-way layouts. Some are based on specifying a functional form for the interaction term and perform well provided that the functional form is appropriate. Other methods do not require such a functional form to be specified but only test for the presence of non-additivity and do not provide a suitable estimate of error variance for a non-additive model. This paper presents a method for testing for interaction in unreplicated two-way tables that is based on testing all pairwise interaction contrasts. This method (i) is easy to implement, (ii) does not assume a functional form for the interaction term, (iii) can find a sub-table of data which may be free from interaction and to base the estimate of unknown error variance, and (iv) can be used for incomplete two-way layouts. The proposed method is illustrated using examples and its power is investigated via simulation studies. Simulation results show that the proposed method is competitive with existing methods for testing for interaction in unreplicated two-way layouts.     

On the relationship between the matrix operators,vech and vecd

We introduce a matrix operator, which we call “vecd” operator. This operator stacks up “diagonals” of a symmetric matrix. This operator is more convenient for some statistical analyses than the commonly used “vech” operator. We show an explicit relationship between the vecd and vech operators. Using this relationship, various properties of the vecd operator are derived. As applications of the vecd operator, we derive concise and explicit expressions of the Wald and score tests for equal variances of a multivariate normal distribution and for the diagonality of variance coefficient matrices in a multivariate generalized autoregressive conditional heteroscedastic (GARCH) model, respectively.     

从“烂版液”中回收硫酸铜

本文讨论了从印刷业制线路板时产生的“烂版液”中回收纯净硫酸铜的工艺流程，除杂质的条件及原理．以控制溶液ＰＨ值的简单方法除去杂质铁．回收到硫酸铜符合化学纯的标准．     

Theory of factorial designs of the parallel flats type. I: the coefficient matrix

Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and c_i(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[¹¹¹⁰₀₁₂₁], c₁=[¹₀], c₂=[⁰₂].) Let T_i (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the s^n?r solutions of the equations At=c_i, wheret′=(t₁,…,t_n) is a vector of variables.(Thus, EG(4, 3) consists of the 3⁴=81 points of the form (t₁,t₂,t₃,t₄), where t's take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=c_i is s^n?r, where r=Rank(A), and the set of such solutions is said to form an (n?r)-flat, i.e. a flat of (n?r) dimensions. In our example, both T₁ and T₂ are 2-flats consisting of 3^4?2=9 points each. The flats T₁,T₂,…,T_f are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T₁ are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T₂ consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sⁿ symmetric factorial experiment obtained by taking T₁,T₂,…,T_f together. (Thus, in the example, 3⁴=81 treatments of the 3⁴ factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T₁ and T₂ enumerated above.)In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X (n×v), occuring in the corresponding linear model, into components X._j(j=0,1,…,g), such that the information matrix X is the direct sum of the X′._jX._j. Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying (s?1) degrees of freedom as usual). For j≠0, we show that the spectrum of X′._jX._j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X._j into the X_ij (i=1,…,f). Furthermore, the X_ij are in turn partitioned into smaller matrices (which we shall here call the) X_ijh. We show that each X_ijh can be factored into a product of 3 matrices J, ζ (not depending on i,j, and h) and Q(j,h,i)where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF(s), and show that the Q(j,h,i) belong to a ring isomorphic to R. When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X._j and X′._jX._j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

The analysis of unreplicated factorial experiments from a geometric perspective

The author investigates the analysis of unreplicated factorial experiments from a geometric perspective. He considers more specifically a (k + 1)‐run experiment used to estimate k orthogonal contrasts. He observes that once centered and scaled to unit length, the response vector can be viewed as a point on the unit sphere in the vector space spanned by the contrasts. In this context, a model selection procedure is equivalent to a partition of the unit sphere into regions corresponding to the different models considered. The author exploits this approach to gain useful insights into the analysis of such experiments.     

A construction for generalized hadamard matrices GH(4q, EA(q))

We give a construction for a generalized Hadamard matrix GH(4q, EA(q)) as a 4 × 4 matrix of q × q blocks, for q an odd prime power other than 3 or 5. Each block is a GH(q, EA(q)) and certain combinations of 4 blocks form GH(2q, EA(q)) matrices. Hence a GH(4q, EA(q)) matrix exists for every prime power q.     

Some Methods of Construction of Gd-Rc and Rectangular-rc Designs

A number of methods of construction of GD-RC and RectanguIar-RC designs, based upon the series A of BIB-RC designs, using the method of Kronecker product, are developed.