Construction of E (fNOD)-Optimal Supersaturated Designs VIA Room Squares

Supersaturated design is essentially a fractional factorial in which the number of potential effects is greater than the number of runs. And Room square is an important object in combinatorial design theory. We show a link between these two apparently unrelated kinds of designs. E ( fNOD) criterion for comparing supersaturated designs is proposed and a lower bound of E ( fNOD) is obtained as a benchmark of design optimality. It is shown that the E ( fNOD) criterion is an extension of the popular E( s2) and ave x2 criterion (for two- and three-level supersaturated designs respectively). A new construction method for multi-level supersaturated designs via Room squares is also proposed and some properties of the resulting designs are investigated.

A restricted occupancy problem

There are m urns each containing k cells; n balls are assigned to the m urns in such a way that each cell contains at most one ball. Let Mt be the number of urns containing exactly t balls (t = 0, 1, •••, k). In the paper, the distribution of Mt, its moments and moment-generating function are obtained. The derivation of the last two seems to be new.

Identification of Differentially Expressed Genes with Multivariate Outlier Analysis

DNA microarray offers a powerful and effective technology to monitor the changes in the gene expression levels for thousands of genes simultaneously. It is being widely applied to explore the quantitative alternation in gene regulation in response to a variety of aspects including diseases and exposure of toxicant. A common task in analyzing microarray data is to identify the differentially expressed genes under two different experimental conditions. Because of the large number of genes and small number of arrays, and higher signal-noise ratio in microarray data, many traditional approaches seem improper. In this paper, a multivariate mixture model is applied to model the expression level of replicated arrays, considering the differentially expressed genes as the outliers of the expression data. In order to detect the outliers of the multivariate mixture model, an effective and robust statistical method is first applied to microarray analysis. This method is based on the analysis of kurtosis coefficient (KC) of the projected multivariate data arising from a mixture model so as to identify the outliers. We utilize the multivariate KC algorithm to our microarray experiment with the control and toxic treatment. After the processing of data, the differential genes are successfully identified from 1824 genes on the UCLA M07 microarray chip. We also use the RT-PCR method and two robust statistical methods, minimum covariance determinant (MCD) and minimum volume ellipsoid (MVE), to verify the expression level of outlier genes identified by KC algorithm. We conclude that the robust multivariate tool is practical and effective for the detection of differentially expressed genes.

An Effective Algorithm for Generation of Factorial Designs with Generalized Minimum Aberration

Fractional factorial designs are popular and widely used for industrial experiments. Generalized minimum aberration is an important criterion recently proposed for both regular and non-regular designs. This paper provides a formal optimization treatment on optimal designs with generalized minimum aberration. New lower bounds and optimality results are developed for resolution-III designs. Based on these results, an effective computer search algorithm is provided for sub-design selection, and new optimal designs are reported.

The application of Kriging and empirical Kriging based on the variables selected by SCAD

The commonly used approach for building a structure-activity/property relationship consists of three steps. First, one determines the descriptors for the molecular structure, then builds a metamodel by using some proper mathematical methods, and finally evaluates the meta-model. Some existing methods only can select important variables from the candidates, while most metamodels just explore linear relationships between inputs and outputs. Some techniques are useful to build more complicated relationship, but they may not be able to select important variables from a large number of variables. In this paper, we propose to screen important variables by the smoothly clipped absolute deviation (SCAD) variable selection procedure, and then apply Kriging model and empirical Kriging model for quantitative structure-activity/property relationship (QSAR/QSPR) research based on the selected important variables. We demonstrate the proposed procedure retains the virtues of both variable selection and Kriging model.

Impersonality of the connectivity index and recomposition of topological indices according to different properties

The connectivity index chi can be regarded as the sum of bond contributions. In this article, boiling point (bp)-oriented contributions for each kind of bond are obtained by decomposing the connectivity indices into ten connectivity character bases and then doing a linear regression between bps and the bases. From the comparison of bp-oriented contributions with the contributions assigned by chi, it can be found that they are very similar in percentage, i.e. the relative importance of each particular kind of bond is nearly the same in the two forms of combinations (one is obtained from the regression with boiling point, and the other is decided by the constructor of the chi index). This coincidence shows an impersonality of chi on bond weighting and may provide us another interpretation of the efficiency of the connectivity index on many quantitative structure-activity/property relationship (QSAR or QSPR) results. However, we also found that chi's weighting formula may not be appropriate for some other properties. In fact, there is no universal weighting formula appropriate for all properties/activities. Recomposition of some topological indices by adjusting the weights upon character bases according to different properties/activities is suggested. This idea of recomposition is applied to the first Zagreb group index M(1) and a large improvement has been achieved.

New Approach by Kriging Models to Problems in QSAR

Most models in quantitative structure and activity relationship (QSAR) research, proposed by various techniques such as ordinary least squares regression, principal components regression, partial least squares regression, and multivariate adaptive regression splines, involve a linear parametric part and a random error part. The random errors in those models are assumed to be independently identical distributed. However, the independence assumption is not reasonable in many cases. Some dependence among errors should be considered just like Kriging. It has been successfully used in computer experiments for modeling. The aim of this paper is to apply Kriging models to QSAR. Our experiments show that the Kriging models can significantly improve the performances of the models obtained by many existing methods.

Structural Interpretation of the Topological Index. 2. The Molecular Connectivity Index, the Kappa Index, and the Atom-type E-State Index

The structural interpretation is extended to the topological indices describing cyclic structures. Three representatives of the topological index, such as the molecular connectivity index, the Kappa index, and the atom-type E-State index, are interpreted by mining out, through projection pursuit combining with a number theory method generating uniformly distributed directions on unit sphere, the structural features hidden in the spaces spanned by the three series of indices individually. Some interesting results, which can hardly be found by individual index, are obtained from the multidimensional spaces by several topological indices. The results support quantitatively the former studies on the topological indices, and some new insights are obtained during the analysis. The combinations of several molecular connectivity indices describe mainly three general categories of molecular structure information, which include degree of branching, size, and degree of cyclicity. The cyclicity can also be coded by the combination of chi cluster and path/cluster indices. The Kappa shape indices encode, in combination, significant information on size, the degree of cyclicity, and the degree of centralization/separation in branching. The size, branch number, and cyclicity information has also been mined out to interpret atom-type E-State indices. The structural feature such as the number of quaternary atoms is searched out to be an important factor. The results indicate that the collinearity might be a serious problem in the applications of the topological indices.

Structural Interpretation of a Topological Index. 1. External Factor Variable Connectivity Index (EFVCI)

The external factor variable connectivity index (EFVCI) is interpreted by mining out the structural features hidden in the space spanned by the EFVCI indices through projection pursuit combining with number-theory net (NT-net) on the unit sphere U(Us). Projection pursuit is concerned with "interesting" projections of high-dimensional data sets to machine-pick "interesting" low-dimensional projections of a high-dimensional point cloud by numerically maximizing a certain objective function or projection index. At first, the optimal EFVCI index reaches to -0.80 in the correlation with a retention index of 207 hydrocarbons produced by insects. The EFVCI indices, with regression results of R = 0.99998, s = 3.49, RMSECV = 3.90, and F = 7.9560e+005, obtain high regression quality. The model is proven valid by leave-one-out cross validation. Second, the EFVCI index is interpreted by the structure information, that is, size, branch number, graph center, and branching position of topological structures, which is searched out on the unit sphere U(Us) by projection pursuit. Finally, the interpretation information is used to discover some chemical knowledge concerning the variation of the retention index with the change in chemical structures.

Two-step multivariate adaptive regression splines for modeling a quantitative relationship between gas chromatography retention indices and molecular descriptors

The relationship between retention indices and molecular descriptors of alkanes is established by two-step multivariate adaptive regression splines (TMARS). TMARS combines linear regression with multivariate adaptive regression splines (MARS). It is demonstrated for the present data set that using linear regression or MARS modeling alone causes lack of fit. TMARS avoids lack of fit and appreciably improves the prediction ability for the model. The use of this combined approach permits the development of additional understanding of the adaptive nature in MARS modeling.

External factor variable connectivity index

A new variable index, external factor variable connectivity index (EFVCI), is proposed, in which the atomic attribute is divided into two parts. The innate part is denoted as outer-shell electrons and external part or perturbation by other atoms is represented as summation, multiplied by a variable x, of squared reciprocal matrix of i row (corresponds to atom A(i)). The division of atomic attribute in EFVCI is interpreted by using topological structure. In the correlation of boiling point of 149 acyclic alkanes, the optimal values will approach to a constant at -0.29 by using the zero to higher order indices of the same series. The new index, with high regression quality (R = 0.9986, s = 2.26, and F = 7088.4), is compared favorably with variable connectivity index and molecular connectivity index.