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

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

/pdf/tensor-decompositions-and-applications-46vl2ih5gn.pdf

Tensor Decompositions and Applications

https://imedea.uib-csic.es/master/cambioglobal/Modulo_V_cod101615/Theory/lit_support/pca_wold.pdf

Principal component analysis

https://libpls.net/publication/PLS_basic_2001.pdf

PLS-regression: a basic tool of chemometrics

http://home.mit.bme.hu/~horvath/IDA/1-s2.0-0003267086800289-main.pdf

Partial least-squares regression: a tutorial

Multi-Way Analysis with Applications in the Chemical Sciences: Smilde/Multi-Way Analysis with Applications in the Chemical Sciences

Foreword. Preface. Nomenclature and Conventions. 1 Introduction. 1.1 What is multi--way analysis? 1.2 Conceptual aspects of multi--way data analysis. 1.3 Hierarchy of multivariate data structures in chemistry. 1.4 Principal component analysis and PARAFAC. 1.5 Summary. 2 Array definitions and properties. 2.1 Introduction. 2.2 Rows, columns and tubes frontal, lateral and horizontal slices. 2.3 Elementary operations. 2.4 Linearity concepts. 2.5 Rank of two--way arrays. 2.6 Rank of three--way arrays. 2.7 Algebra of multi--way analysis. 2.8 Summary. Appendix 2.A. 3 Two--way component and regression models. 3.1 Models for two--way one--block data analysis: component models. 3.2 Models for two--way two--block data analysis: regression models. 3.3 Summary. Appendix 3.A: some PCA results. Appendix 3.B: PLS algorithms. 4 Three--way component and regression models. 4.1 Historical introduction to multi--way models. 4.2 Models for three--way one--block data: three--way component models. 4.3 Models for three--way two--block data: three--way regression models. 4.4 Summary. Appendix 4.A: alternative notation for the PARAFAC model. Appendix 4.B: alternative notations for the Tucker3 model. 5 Some properties of three--way component models. 5.1 Relationships between three--way component models. 5.2 Rotational freedom and uniqueness in three--way component models. 5.3 Properties of Tucker3 models. 5.4 Degeneracy problem in PARAFAC models. 5.5 Summary. 6 Algorithms. 6.1 Introduction. 6.2 Optimization techniques. 6.3 PARAFAC algorithms. 6.4 Tucker3 algorithms. 6.5 Tucker2 and Tucker1 algorithms. 6.6 Multi--linear partial least squares regression. 6.7 Multi--way covariates regression models. 6.8 Core rotation in Tucker3 models. 6.9 Handling missing data. 6.10 Imposing non--negativity. 6.11 Summary. Appendix 6.A: closed--form solution for the PARAFAC model. Appendix 6.B: proof that the weights in trilinear PLS1 can be obtained from a singular value decomposition. 7 Validation and diagnostics. 7.1 What is validation? 7.2 Test--set and cross--validation. 7.3 Selecting which model to use. 7.4 Selecting the number of components. 7.5 Residual and influence analysis. 7.6 Summary. 8 Visualization. 8.1 Introduction. 8.2 History of plotting in three--way analysis. 8.3 History of plotting in chemical three--way analysis. 8.4 Scree plots. 8.5 Line plots. 8.6 Scatter plots. 8.7 Problems with scatter plots. 8.8 Image analysis. 8.9 Dendrograms. 8.10 Visualizing the Tucker core array. 8.11 Joint plots. 8.12 Residual plots. 8.13 Leverage plots. 8.14 Visualization of large data sets. 8.15 Summary. 9 Preprocessing. 9.1 Background. 9.2 Two--way centering. 9.3 Two--way scaling. 9.4 Simultaneous two--way centering and scaling. 9.5 Three--way preprocessing. 9.6 Summary. Appendix 9.A: other types of preprocessing. Appendix 9.B: geometric view of centering. Appendix 9.C: fitting bilinear model plus offsets across one mode equals fitting a bilinear model to centered data. Appendix 9.D: rank reduction and centering. Appendix 9.E: centering data with missing values. Appendix 9.F: incorrect centering introduces artificial variation. Appendix 9.G: alternatives to centering. 10 Applications. 10.1 Introduction. 10.2 Curve resolution of fluorescence data. 10.3 Second--order calibration. 10.4 Multi--way regression. 10.5 Process chemometrics. 10.6 Exploratory analysis in chromatography. 10.7 Exploratory analysis in environmental sciences. 10.8 Exploratory analysis of designed data. 10.9 Analysis of variance of data with complex interactions. Appendix 10.A: an illustration of the generalized rank annihilation method. Appendix 10.B: other types of second--order calibration problems. Appendix 10.C: the multiple standards calibration model of the second--order calibration example. References. Index.

Multi-way Analysis: Applications in the Chemical Sciences

The Lohmoller–Wold decomposition of multi-way (three-way, four-way, etc.) data arrays is combined with the non-linear partial least squares (NIPALS) algorithms to provide multi-way solutions of principal components analysis (PCA) and partial least squares modelling in latent variables (PLS).

The decomposition of a multi-way array is developed as the product of a score vector and a loading array, where the score vectors have the same properties as those of ordinary two-way PCA and PLS. In image analysis, the array would instead be decomposed as the product of a loading vector and an image score matrix.

The resulting methods are equivalent to the method of unfolding a multi-way array to a two-way matrix followed by ordinary PCA or PLS analysis. This automatically proves the eigenvector and least squares properties of the multi-way PCA and PLS methods.

The methodology is presented; the algorithms are outlined and illustrated with a small chemical example.

Paul Geladi

Papers

Principal component analysis

Partial least-squares regression: a tutorial

Multi-Way Analysis with Applications in the Chemical Sciences: Smilde/Multi-Way Analysis with Applications in the Chemical Sciences

Multi-way Analysis: Applications in the Chemical Sciences

Multi‐way principal components‐and PLS‐analysis