Direct transformations yielding the knight's move pattern in 3x3x3 arrays

Three-way arrays (or tensors) can be regarded as extensions of the traditional two-way data matrices that have a third dimension. Studying algebraic properties of arrays is relevant, for example, for the Tucker three-way PCA method, which generalizes principal component analysis to three-way data. One important algebraic property of arrays is concerned with the possibility of transformations to simplicity. An array is said to be transformed to a simple form when it can be manipulated by a sequence of invertible operations such that a vast majority of its entries become zero. This paper shows how 3 × 3 × 3 arrays, whether symmetric or nonsymmetric, can be transformed to a simple form with 18 out of its 27 entries equal to zero. We call this simple form the “knight's move pattern” due to a loose resemblance to the moves of a knight in a game of chess. The pattern was examined by Kiers, Ten Berge, and Rocci. It will be shown how the knight's move pattern can be found by means of a numeric–algebraic procedure based on the Gröbner basis. This approach seems to work almost surely for randomly generated arrays, whether symmetric or nonsymmetric

First and second-order derivatives for CP and INDSCAL

In this paper we provide the means to analyse the second-order differential structure of optimization functions concerning CANDECOMP/PARAFAC and INDSCAL. Closed-form formulas are given under two types of constraint: unit-length columns or orthonormality of two of the three component matrices. Some numerical problems that might occur during the computation of the Jacobian and Hessian matrices are addressed. The use of these matrices is illustrated in three applications. (C) 2010 Elsevier B.V. All rights reserved

Polonaise

https://digitalcommons.library.umaine.edu/mmb-ps/3188/thumbnail.jp

Simplicity transformations for three-way arrays with symmetric slices, and applications to Tucker-3 models with sparse core arrays

AbstractTucker three-way PCA and Candecomp/Parafac are two well-known methods of generalizing principal component analysis to three way data. Candecomp/Parafac yields component matrices A (e.g., for subjects or objects), B (e.g., for variables) and C (e.g., for occasions) that are typically unique up to jointly permuting and rescaling columns. Tucker-3 analysis, on the other hand, has full transformational freedom. That is, the fit does not change when A,B, and C are postmultiplied by nonsingular transformation matrices, provided that the inverse transformations are applied to the so-called core array G̲. This freedom of transformation can be used to create a simple structure in A,B,C, and/or in G̲. This paper deals with the latter possibility exclusively. It revolves around the question of how a core array, or, in fact, any three-way array can be transformed to have a maximum number of zero elements. Direct applications are in Tucker-3 analysis, where simplicity of the core may facilitate the interpretation of a Tucker-3 solution, and in constrained Tucker-3 analysis, where hypotheses involving sparse cores are taken into account. In the latter cases, it is important to know what degree of sparseness can be attained as a tautology, by using the transformational freedom. In addition, simplicity transformations have proven useful as a mathematical tool to examine rank and generic or typical rank of three-way arrays. So far, a number of simplicity results have been attained, pertaining to arrays sampled randomly from continuous distributions. These results do not apply to three-way arrays with symmetric slices in one direction. The present paper offers a number of simplicity results for arrays with symmetric slices of order 2×2,3×3 and 4×4. Some generalizations to higher orders are also discussed. As a mathematical application, the problem of determining the typical rank of 4×3×3 and 5×3×3 arrays with symmetric slices will be revisited, using a sparse form with only 8 out of 36 elements nonzero for the former case and 10 out of 45 elements nonzero for the latter one, that can be attained almost surely for such arrays. The issue of maximal simplicity of the targets to be presented will be addressed, either by formal proofs or by relying on simulation results

Three-Dimensional Stereophotogrammetry Assessment of Facial Asymmetry in Facial Palsy

Three-dimensional stereophotogrammetry is not much used in assessing facial palsy and a comprehensive understanding of sources of variation in these measurements is lacking. The present study assessed intra- and interobserver reliability of a novel three-dimensional stereophotogrammetry measurement of facial asymmetry and examined sources of variation in these outcomes. Three photographs (rest, closed mouth smile, and maximum smile) were made of 60 participants, 30 facial palsy patients and 30 control subjects. All images were analyzed twice by 2 observers independently, to determine intra- and interobserver reliability. Variance component analysis was performed to investigate sources of variation in the outcomes. Intraobserver reliability was good with intraclass correlation coefficients ranging from 0.715 to 0.999. Interobserver reliability ranged from 0.442 to 0.929. Reliability of the smile image measurements was not clearly different from the rest images. Variation in measurement results was largely due to the status of a participant, facial palsy versus control. When splitting the sample, the facial expression was a major source of variation. Acceptable reliability of the proposed 3D facial asymmetry measurement was found, in facial palsy patients and control subjects. Interobserver reliability was marked less compared to intraobserver reliability. For follow-up data only one observer should assess 3D stereophotogrammetry measurements

Single-nucleotide polymorphism rs2070600 regulates AGER splicing and the sputum levels of the COPD biomarker soluble receptor for advanced glycation end-products.

The COPD susceptibility SNP rs2070600 affects the levels of the COPD biomarker sRAGE in sputum as well as splicing of AGER. Moreover, @PouwelsScience et al. demonstrate large differences in sRAGE levels between serum and sputum. https://bit.ly/3t0pJtK

Predicted values for the forced expiratory flow adjusted for forced vital capacity, a descriptive study

Background: The forced expiratory flows (FEFs) towards the end of the expiration may be more sensitive in detecting peripheral airways obstruction compared to the forced expiratory volume in 1 s and forced vital capacity (FVC). However, they are highly variable. A partial solution is to adjust the FEFs for FVC (FEF/FVC). Here we provide reference equations for these adjusted FEFs at 25%, 50%, 75% and 25-75% of FVC, which are currently lacking. Methods: We included pulmonary healthy, never-smoker adults; 14 472 subjects from Lifelines, a biobank for health research, and 338 subjects from the department's control cohorts (NORM and Fiddle). Reference equations were obtained by linear regression on 80% of the Lifelines dataset and validated on the remaining data. The best model was defined as the one with the highest adjusted R2-value. The difference in variability between adjusted and unadjusted FEFs was evaluated using the coefficient of variation. Results: For all adjusted FEFs, the best model contained age, height and weight. The adjustment improved the coefficient of variation of the FEF75 from 39% to 36% and from 43% to 40%, respectively, in males and females. The highest percentage of explained variance by the reference equation was obtained for FEF75/FVC, 32%-38% for males, and 41%-46% for females, depending on the validation set. Conclusion: We developed reference equations for FVC-adjusted FEF values. We demonstrated minimally yet significantly improved variability. Future studies in obstructive airway diseases should demonstrate whether it is worthwhile to use these (predicted) adjusted FEF values