326 research outputs found

    UTILIZING DESIGN STRUCTURE FOR IMPROVING DESIGN SELECTION AND ANALYSIS

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    Recent work has shown that the structure for design plays a role in the simplicity or complexity of data analysis. To increase the knowledge of research in these areas, this dissertation aims to utilize design structure for improving design selection and analysis. In this regard, minimal dependent sets and block diagonal structure are both important concepts that are relevant to the orthogonality of the columns of a design. We are interested in finding ways to improve the data analysis especially for active effect detection by utilizing minimal dependent sets and block diagonal structure for design. We introduce a new classification criterion for minimal dependent sets to enhance existing criteria for design selection. The block diagonal structure of certain nonregular designs will also be discussed as a means of improving model selection. In addition, the block diagonal structure and the concept of parallel flats will be utilized to construct three-quarter nonregular designs. Based on the literature review on the effectiveness of the simulation study for slight the light on the success or failure of the proposed statistical method, in this dissertation, simulation studies were used to evaluate the efficacy of our proposed methods. The simulation results show that the minimal dependent sets can be used as a design selection criterion, and block-diagonal structure can also help to produce an effective model selection procedure. In addition, we found a strategy for constructing three-quarters of nonregular designs which depend on the orthogonality of the design columns. The results indicate that the structure of the design has an impact on developing data analysis and design selections. On this basis, it is recommended that analysts consider the structure of the design as a key factor in order to improve the analysis. Further research is needed to determine more concepts related to the structure of the design, which could help to improve data analysis

    Construction of nested space-filling designs

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    New types of designs called nested space-filling designs have been proposed for conducting multiple computer experiments with different levels of accuracy. In this article, we develop several approaches to constructing such designs. The development of these methods also leads to the introduction of several new discrete mathematics concepts, including nested orthogonal arrays and nested difference matrices.Comment: Published in at http://dx.doi.org/10.1214/09-AOS690 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    SLOPE - Adaptive variable selection via convex optimization

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    We introduce a new estimator for the vector of coefficients β\beta in the linear model y=Xβ+zy=X\beta+z, where XX has dimensions n×pn\times p with pp possibly larger than nn. SLOPE, short for Sorted L-One Penalized Estimation, is the solution to minbRp12yXb22+λ1b(1)+λ2b(2)++λpb(p),\min_{b\in\mathbb{R}^p}\frac{1}{2}\Vert y-Xb\Vert _{\ell_2}^2+\lambda_1\vert b\vert _{(1)}+\lambda_2\vert b\vert_{(2)}+\cdots+\lambda_p\vert b\vert_{(p)}, where λ1λ2λp0\lambda_1\ge\lambda_2\ge\cdots\ge\lambda_p\ge0 and b(1)b(2)b(p)\vert b\vert_{(1)}\ge\vert b\vert_{(2)}\ge\cdots\ge\vert b\vert_{(p)} are the decreasing absolute values of the entries of bb. This is a convex program and we demonstrate a solution algorithm whose computational complexity is roughly comparable to that of classical 1\ell_1 procedures such as the Lasso. Here, the regularizer is a sorted 1\ell_1 norm, which penalizes the regression coefficients according to their rank: the higher the rank - that is, stronger the signal - the larger the penalty. This is similar to the Benjamini and Hochberg [J. Roy. Statist. Soc. Ser. B 57 (1995) 289-300] procedure (BH) which compares more significant pp-values with more stringent thresholds. One notable choice of the sequence {λi}\{\lambda_i\} is given by the BH critical values λBH(i)=z(1iq/2p)\lambda_{\mathrm {BH}}(i)=z(1-i\cdot q/2p), where q(0,1)q\in(0,1) and z(α)z(\alpha) is the quantile of a standard normal distribution. SLOPE aims to provide finite sample guarantees on the selected model; of special interest is the false discovery rate (FDR), defined as the expected proportion of irrelevant regressors among all selected predictors. Under orthogonal designs, SLOPE with λBH\lambda_{\mathrm{BH}} provably controls FDR at level qq. Moreover, it also appears to have appreciable inferential properties under more general designs XX while having substantial power, as demonstrated in a series of experiments running on both simulated and real data.Comment: Published at http://dx.doi.org/10.1214/15-AOAS842 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

    The Effectiveness of Categorical Variables in Discriminant Function Analysis

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    A preliminary study of the feasibility of using categorical variables in discriminant function analysis was performed. Data including both continuous and categorical variables were used and predictive results examined. The discriminant function techniques were found to be robust enough to include the use of categorical variables. Some problems were encountered with using the trace criterion for selecting the most discriminating variables when these variables are categorical. No monotonic relationship was found to exist between the trace and the number of correct predictions. This study did show that the use of categorical variables does have much potential as a statistical tool in classification procedures. (50 pages
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