THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

‡We describe the use of singular value decomposition in transforming genome-wide expression data from genes 3 arrays space to reduced diagonalized ‘‘eigengenes’’ 3 ‘‘eigenarrays’’ space, where the eigengenes (or eigenarrays) are unique orthonormal superpositions of the genes (or arrays). Normalizing the data by filtering out the eigengenes (and eigenarrays) that are inferred to represent noise or experimental artifacts enables meaningful comparison of the expression of different genes across different arrays in different experiments. Sorting the data according to the eigengenes and eigenarrays gives a global picture of the dynamics of gene expression, in which individual genes and arrays appear to be classified into groups of similar regulation and function, or similar cellular state and biological phenotype, respectively. After normalization and sorting, the significant eigengenes and eigenarrays can be associated with observed genome-wide effects of regulators, or with measured samples, in which these regulators are overactive or underactive, respectively.

Singular Value Decomposition for Genome-Wide Expression Data Processing and Modeling

Maximizing data information requires careful selection, termed design, of the points at which data are observed. Experimental design is reviewed here for broad classes of data collection and analysis problems, including: fractioning techniques based on orthogonal arrays, Latin hypercube designs and their variants for computer experimentation, efficient design for data mining and machine learning applications, and sequential design for active learning. © 2012 Wiley Periodicals, Inc. © 2012 Wiley Periodicals, Inc.

Experimental design

We introduce TestU01, a software library implemented in the ANSI C language, and offering a collection of utilities for the empirical statistical testing of uniform random number generators (RNGs). It provides general implementations of the classical statistical tests for RNGs, as well as several others tests proposed in the literature, and some original ones. Predefined tests suites for sequences of uniform random numbers over the interval (0, 1) and for bit sequences are available. Tools are also offered to perform systematic studies of the interaction between a specific test and the structure of the point sets produced by a given family of RNGs. That is, for a given kind of test and a given class of RNGs, to determine how large should be the sample size of the test, as a function of the generator's period length, before the generator starts to fail the test systematically. Finally, the library provides various types of generators implemented in generic form, as well as many specific generators proposed in the literature or found in widely used software. The tests can be applied to instances of the generators predefined in the library, or to user-defined generators, or to streams of random numbers produced by any kind of device or stored in files. Besides introducing TestU01, the article provides a survey and a classification of statistical tests for RNGs. It also applies batteries of tests to a long list of widely used RNGs.

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TestU01: A C library for empirical testing of random number generators

Prefaces Introduction What Is Computational Statistics? An Overview of the Book Probability Concepts Introduction Probability Conditional Probability and Independence Expectation Common Distributions Sampling Concepts Introduction Sampling Terminology and Concepts Sampling Distributions Parameter Estimation Empirical Distribution Function Generating Random Variables Introduction General Techniques for Generating Random Variables Generating Continuous Random Variables Generating Discrete Random Variables Exploratory Data Analysis Introduction Exploring Univariate Data Exploring Bivariate and Trivariate Data Exploring Multidimensional Data Finding Structure Introduction Projecting Data Principal Component Analysis Projection Pursuit EDA Independent Component Analysis Grand Tour Nonlinear Dimensionality Reduction Monte Carlo Methods for Inferential Statistics Introduction Classical Inferential Statistics Monte Carlo Methods for Inferential Statistics Bootstrap Methods Data Partitioning Introduction Cross-Validation Jackknife Better Bootstrap Confidence Intervals Jackknife-after-Bootstrap Probability Density Estimation Introduction Histograms Kernel Density Estimation Finite Mixtures Generating Random Variables Supervised Learning Introduction Bayes' Decision Theory Evaluating the Classifier Classification Trees Combining Classifiers Unsupervised Learning Introduction Measures of Distance Hierarchical Clustering K-Means Clustering Model-Based Clustering Assessing Cluster Results Parametric Models Introduction Spline Regression Models Logistic Regression Generalized Linear Models Nonparametric models Introduction Some Smoothing Methods Kernel Methods Smoothing Splines Nonparametric Regression-Other Details Regression Trees Additive Models Markov Chain Monte Carlo Methods Introduction Background Metropolis-Hastings Algorithms The Gibbs Sampler Convergence Monitoring Spatial Statistics Introduction Visualizing Spatial Point Processes Exploring First-Order and Second-Order Properties Modeling Spatial Point Processes Simulating Spatial Point Processes Appendix A: Introduction to Matlab What Is MATLAB? Getting Help in MATLAB File and Workspace Management Punctuation in MATLAB Arithmetic Operators Data Constructs in MATLAB Script Files and Functions Control Flow Simple Plotting Contact Information Appendix B: Projection Pursuit Indexes Indexes MATLAB Source Code Appendix C: Matlab Statistics Toolbox Appendix D: Computational Statistics Toolbox Appendix E: Exploratory Data Analysis Toolboxes Introduction EDA Toolbox EDA GUI Toolbox Appendix F: Data Sets Appendix G: NOTATION References INDEX MATLAB Code, Further Reading, and Exercises appear at the end of each chapter.

https://www.jstatsoft.org/article/view/v011b06/v11b06.pdf

Computational Statistics Handbook with MATLAB

Resolution has been the most widely used criterion for comparing regular fractional factorials since it was introduced in 1961 by Box and Hunter. In this pa- per, we examine how a generalized resolution criterion can be defined and used for assessing nonregular fractional factorials, notably Plackett-Burman designs. Our generalization is intended to capture projection properties, complementing that of Webb (1964) whose concept of resolution concerns the estimability of lower order ef- fects under the assumption that higher order effects are negligible. Our generalized resolution provides a fruitful criterion for ranking different designs while Webb's resolution is mainly useful as a classification rule. An additional advantage of our approach is that the idea leads to a natural generalization of minimum aberration. Examples are given to illustrate the usefulness of the new criteria.

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Generalized resolution and minimum aberration criteria for plackett-burman and other nonregular factorial designs

Deng and Tang (1999) and Tang and Deng (1999) proposed and justified two criteria of generalized minimum aberration for general two-level fractional factorial designs. The criteria are defined using a set of values called J characteristics. In this article, we examine the practical use of the criteria in design selection. Specifically, we consider the problem of classifying and ranking designs that are based on Hadamard matrices. A theoretical result on J characteristics is developed to facilitate the computation. The issue of design selection is further studied by linking generalized aberration with the criteria of efficiency and estimation capacity. Our studies reveal that generalized aberration performs quite well under these familiar criteria.

Design Selection and Classification for Hadamard Matrices Using Generalized Minimum Aberration Criteria

We present a new reseeding-mixing method to extend the system period length and to enhance the statistical properties of a chaos-based logistic map pseudo random number generator (PRNG). The reseeding method removes the short periods of the digitized logistic map and the mixing method extends the system period length to 2253 by “xoring” with a DX generator. When implemented in the TSMC 0.18- μm 1P6M CMOS process, the new reseeding-mixing PRNG (RM-PRNG) attains the best throughput rate of 6.4 Gb/s compared with other nonlinear PRNGs. In addition, the generated random sequences pass the NIST SP 800-22 statistical tests including ratio test and U-value test.

Period Extension and Randomness Enhancement Using High-Throughput Reseeding-Mixing PRNG

Use of empirical studies based on computer-generated random numbers has become a common practice in the development of statistical methods, particularly when the analytical study of a statistical procedure becomes intractable. The quality of any simulation study depends heavily on the quality of the random number generators. Classical uniform random number generators have some major defects—such as the (relatively) short period length and the lack of higher-dimension uniformity. Two recent uniform pseudo-random number generators (MRG and MCG) are reviewed. They are compared with the classical generator LCG. It is shown that MRG/MCG are much better random number generators than the popular LCG. Special forms of MRG/MCG are introduced and recommended as the random number generators for the new century. A step-by-step procedure for constructing such random number generators is also provided.

Random Number Generation for the New Century

We propose a system of multiple recursive generators of modulus p and order k where all nonzero coefficients of the recurrence are equal. The advantage of this property is that a single multiplication is needed to compute the recurrence, so the generator would run faster than the general case. For p = 231 − 1, the most popular modulus used, we provide tables of specific parameter values yielding maximum period for recurrence of order k = 102 and 120. For p = 231 − 55719 and k = 1511, we have found generators with a period length approximately 1014100.5.

Lih-Yuan Deng

Papers

Generalized resolution and minimum aberration criteria for plackett-burman and other nonregular factorial designs

Design Selection and Classification for Hadamard Matrices Using Generalized Minimum Aberration Criteria

Period Extension and Randomness Enhancement Using High-Throughput Reseeding-Mixing PRNG

Random Number Generation for the New Century

A system of high-dimensional, efficient, long-cycle and portable uniform random number generators