scholarly journals Convex combination of data matrices: PCA perturbation bounds for multi-objective optimal design of mechanical metafilters

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Giorgio Gnecco ◽  
Andrea Bacigalupo
Keyword(s):  
Principal Component Analysis ◽  
Optimization Problems ◽  
Convex Combination ◽  
Principal Component ◽  
Component Analysis ◽  
Data Matrix ◽  
Matrix Perturbation ◽  
Perturbation Bounds ◽  
Multi Objective ◽  
Data Matrices

<p style='text-indent:20px;'>In the present study, matrix perturbation bounds on the eigenvalues and on the invariant subspaces found by principal component analysis is investigated, for the case in which the data matrix on which principal component analysis is performed is a convex combination of two data matrices. The application of the theoretical analysis to multi-objective optimization problems – e.g., those arising in the design of mechanical metamaterial filters – is also discussed, together with possible extensions.</p>

scholarly journals Quantitative Analysis of a Weak Correlation between Complicated Data on the Basis of Principal Component Analysis

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Tao Pang ◽  
Haitao Zhang ◽  
Liliang Wen ◽  
Jun Tang ◽  
Bing Zhou ◽  
...  
Keyword(s):  
Principal Component Analysis ◽  
Principal Component ◽  
Component Analysis ◽  
New Method ◽  
Data Matrix ◽  
Chemical Compositions ◽  
Weak Correlation ◽  
The Difference ◽  
Correlation Information ◽  
Data Matrices

The mining of weak correlation information between two data matrices with high complexity is a very challenging task. A new method named principal component analysis-based multiconfidence ellipse analysis (PCA/MCEA) was proposed in this study, which first applied a confidence ellipse to describe the difference and correlation of such information among different categories of objects/samples on the basis of PCA operation of a single targeted data. This helps to find the number of objects contained in the overlapping and nonoverlapping areas of ellipses obtained from PCA runs. Then, a quantitative evaluation index of correlation between data matrices was defined by comparing the PCA results of more than one data matrix. The similarity and difference between data matrices was further quantified through comprehensively analyzing the outcomes. Complicated data of tobacco agriculture were used as an example to illustrate the strategy of the proposed method, which includes rich features of climate, altitude, and chemical compositions of tobacco leaves. The number of objects of these data reached 171,516 with 14, 4, and 5 descriptors of climate, altitude, and chemicals, respectively. On the basis of the new method, the complex but weak relationship between these independent and dependent variables were interestingly studied. Three widely used but conventional methods were applied for comparison in this work. The results showed the power of the new method to discover the weak correlation between complicated data.

Principal component analysis of substituent constants

1990 ◽  
Vol 55 (1) ◽  
pp. 55-62 ◽  
Author(s):  
Drahomír Hnyk
Keyword(s):  
Principal Component Analysis ◽  
Inductive Effect ◽  
Resonance Effect ◽  
Principal Component ◽  
Component Analysis ◽  
Total Variance ◽  
Data Matrix ◽  
Cumulative Proportion ◽  
Two Factors ◽  
Substituent Constants

The principal component analysis has been applied to a data matrix formed by 7 usual substituent constants for 38 substituents. Three factors are able to explain 99.4% cumulative proportion of total variance. Several rotations have been carried out for the first two factors in order to obtain their physical meaning. The first factor is related to the resonance effect, whereas the second one expresses the inductive effect, and both together describe 97.5% cumulative proportion of total variance. Their mutual orthogonality does not directly follow from the rotations carried out. With the help of these factors the substituents are divided into four main classes, and some of them assume a special position.

scholarly journals SVD-based principal component analysis of geochemical data

2005 ◽  
Vol 3 (4) ◽  
pp. 731-741 ◽  
Author(s):  
Petr Praus
Keyword(s):  
Principal Component Analysis ◽  
Euclidean Distance ◽  
Principal Component ◽  
Component Analysis ◽  
Data Matrix ◽  
Geochemical Data ◽  
Average Group ◽  
Testing Data ◽  
Value Decomposition ◽  
Geochemical Variables

AbstractPrincipal Component Analysis (PCA) was used for the mapping of geochemical data. A testing data matrix was prepared from the chemical and physical analyses of the coals altered by thermal and oxidation effects. PCA based on Singular Value Decomposition (SVD) of the standardized (centered and scaled by the standard deviation) data matrix revealed three principal components explaining 85.2% of the variance. Combining the scatter and components weights plots with knowledge of the composition of tested samples, the coal samples were divided into seven groups depending on the degree of their oxidation and thermal alteration.The PCA findings were verified by other multivariate methods. The relationships among geochemical variables were successfully confirmed by Factor Analysis (FA). The data structure was also described by the Average Group dendrogram using Euclidean distance. The found sample clusters were not defined so clearly as in the case of PCA. It can be explained by the PCA filtration of the data noise.

scholarly journals Principal Component Analysis Applied to Gradient Fields in Band Gap Optimization Problems for Metamaterials

2021 ◽  
Vol 2015 (1) ◽  
pp. 012047
Author(s):  
Giorgio Gnecco ◽  
Andrea Bacigalupo ◽  
Francesca Fantoni ◽  
Daniela Selvi
Keyword(s):  
Machine Learning ◽  
Principal Component Analysis ◽  
Band Gap ◽  
Optimization Problems ◽  
Principal Component ◽  
Component Analysis ◽  
Physical Models ◽  
Supervised Machine Learning ◽  
Iterative Optimization ◽  
Gradient Based

Abstract A promising technique for the spectral design of acoustic metamaterials is based on the formulation of suitable constrained nonlinear optimization problems. Unfortunately, the straightforward application of classical gradient-based iterative optimization algorithms to the numerical solution of such problems is typically highly demanding, due to the complexity of the underlying physical models. Nevertheless, supervised machine learning techniques can reduce such a computational effort, e.g., by replacing the original objective functions of such optimization problems with more-easily computable approximations. In this framework, the present article describes the application of a related unsupervised machine learning technique, namely, principal component analysis, to approximate the gradient of the objective function of a band gap optimization problem for an acoustic metamaterial, with the aim of making the successive application of a gradient-based iterative optimization algorithm faster. Numerical results show the effectiveness of the proposed method.

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