Analysis of Correlation Matrices Using Scale-Invariant Common Principal Component Models and a Hierarchy of Relationships Between Correlation Matrices

Gu, Fei

doi:10.1080/10705511.2016.1207180

Cited by 5 publications

(7 citation statements)

References 22 publications

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“…Gu (2016) defined the scale‐invariant covariance structure model for PCA that has the following covariance structure:

{\boldsymbol{\Sigma}}_{xx}\left(\mathbf{V},\boldsymbol{\Lambda}, {\mathbf{D}}_x\right)={\mathbf{D}}_x\mathbf{V}\boldsymbol{\Lambda } {\mathbf{V}}^{\prime }{\mathbf{D}}_x,

where V is a p × p orthogonal matrix, and Λ and D x are p × p diagonal matrices. This scale‐invariant model has not only p ( p + 1)/2 constraints imposed on V (i.e.,

{\mathbf{V}}^{\prime}\mathbf{V}={\mathbf{I}}_p

) but also p additional constraints [i.e.,

\operatorname{diag}\left(\mathbf{V}\boldsymbol{\Lambda } {\mathbf{V}}^{\prime}\right)={\mathbf{I}}_p

] that guarantee a correlation structure embedded within the covariance structure in Equation (2).…”

Section: Covariance Structure Models For Pca and Mean And Covariance ...mentioning

confidence: 99%

“…To this end, we propose a model‐based approach to MPCR, in which we define two mean and covariance structure models for MPCR. The two models defined for MPCR are extensions of the regular and scale‐invariant covariance structure models for PCA (Dolan, 1996; Gu, 2016), separately. Specifically, the first model for MPCR is a regular mean and covariance structure model, whereas the second one is a scale‐invariant mean and covariance structure model.…”

Section: Introductionmentioning

confidence: 99%

“…Gu (2016) defined the scale-invariant common PC model that is essentially the multiple-group extension of the scale-invariant covariance structure model for PCA. For a single group, Gu's model is the scale-invariant covariance structure model for PCA.…”

mentioning

confidence: 99%

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A model‐based approach to multivariate principal component regression: Selecting principal components and estimating standard errors for unstandardized regression coefficients

Cheung

2023

Brit J Math & Statis

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Principal component regression (PCR) is a popular technique in data analysis and machine learning. However, the technique has two limitations. First, the principal components (PCs) with the largest variances may not be relevant to the outcome variables. Second, the lack of standard error estimates for the unstandardized regression coefficients makes it hard to interpret the results. To address these two limitations, we propose a model‐based approach that includes two mean and covariance structure models defined for multivariate PCR. By estimating the defined models, we can obtain inferential information that will allow us to test the explanatory power of individual PCs and compute the standard error estimates for the unstandardized regression coefficients. A real example is used to illustrate our approach, and simulation studies under normality and nonnormality conditions are presented to validate the standard error estimates for the unstandardized regression coefficients. Finally, future research topics are discussed.

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“…Gu (2016) defined the scale‐invariant covariance structure model for PCA that has the following covariance structure:

{\boldsymbol{\Sigma}}_{xx}\left(\mathbf{V},\boldsymbol{\Lambda}, {\mathbf{D}}_x\right)={\mathbf{D}}_x\mathbf{V}\boldsymbol{\Lambda } {\mathbf{V}}^{\prime }{\mathbf{D}}_x,

where V is a p × p orthogonal matrix, and Λ and D x are p × p diagonal matrices. This scale‐invariant model has not only p ( p + 1)/2 constraints imposed on V (i.e.,

{\mathbf{V}}^{\prime}\mathbf{V}={\mathbf{I}}_p

) but also p additional constraints [i.e.,

\operatorname{diag}\left(\mathbf{V}\boldsymbol{\Lambda } {\mathbf{V}}^{\prime}\right)={\mathbf{I}}_p

] that guarantee a correlation structure embedded within the covariance structure in Equation (2).…”

Section: Covariance Structure Models For Pca and Mean And Covariance ...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A model‐based approach to multivariate principal component regression: Selecting principal components and estimating standard errors for unstandardized regression coefficients

Cheung

2023

Brit J Math & Statis

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“…It extends the idea of principal component analysis by assuming all covariance matrices share the same set of eigenvectors. Since it was first introduced by Flury (1984), the CPCA has been extensively applied in various fields including statistics (Gu, 2016;Pepler et al, 2016), finance (Goyal et al, 2008;Xu et al, 2019), and computer science (Ye et al, 2012;Hadjipantelis et al, 2015).…”

Section: Introductionmentioning

confidence: 99%

Semiparametric partial common principal component analysis for covariance matrices

et al. 2020

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We consider the problem of jointly modeling multiple covariance matrices by partial common principal component analysis (PCPCA), which assumes a proportion of eigenvectors to be shared across covariance matrices and the rest to be individual-specific. This paper proposes consistent estimators of the shared eigenvectors in PCPCA as the number of matrices or the number of samples to estimate each matrix goes to infinity. We prove such asymptotic results without making any assumptions on the ranks of eigenvalues that are associated with the shared eigenvectors.When the number of samples goes to infinity, our results do not require the data to be Gaussian distributed. Furthermore, this paper introduces a sequential testing procedure to identify the number of shared eigenvectors in PCPCA. In simulation studies, our method shows higher accuracy in estimating the shared eigenvectors than competing methods. Applied to a motor-task functional magnetic resonance imaging data set, our estimator identifies meaningful brain networks that are consistent with current scientific understandings of motor networks during a motor paradigm.

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“…Data for independent variables were subjected to Factor analysis to quantify trait relationships within independent factors; Partial Least Squares Regression (PLSR) was used to model the relationship between dependent and independent variables; a validation step (using 30% of data) was included to verify the utility of the developed PLSR model [53]. The resulting model coefficients for centered and scaled data were used to construct a correlation matrix between the independent variables for each sub-factor, which were then subjected to a Mantel test to quantify and test the significance of relationships at the matrix level [55,56]. Similarly, correlation matrices were developed for each sub-factor while using independent variables, and then were subjected to a Mantel test, as described before.…”

Section: Statistical Analyses and Modelingmentioning

confidence: 99%

Statistical Modeling of Phenotypic Plasticity under Abiotic Stress in Triticum durum L. and Triticum aestivum L. Genotypes

Jaradat

2018

Agronomy

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Future challenges to the role of durum and bread wheat in global food security will be shaped by their potential to produce larger yields and better nutritional quality, while increasingly adapting to multiple biotic and abiotic stresses in the view of global climate change. There is a dearth of information on comparative assessment of phenotypic plasticity in both wheat species under long-term multiple abiotic stresses. Phenotypic plasticities of two durum and bread wheat genotypes were assessed under increasing abiotic and edaphic stresses for six years. Combinations of normal and reduced length of growing season and population density, with or without rotation, generated increasing levels of competition for resources and impacted phenotypic plasticity of several plant and yield attributes, including protein and micronutrients contents. All the phenotypic plasticity (PPs) estimates, except for the C:N ratio in both genotypes and grain protein content in T. aestivum genotype, were impacted by abiotic stresses during the second stress phase (PS II) compared with the first (PS I); whereas, covariate effects were limited to a few PPs (e.g., biomass, population density, fertile tillers, grain yield, and grain protein content). Discrimination between factor levels decreased from abiotic phases > growth stages > stress treatments and provided selection criteria of trait combinations that can be positively resilient under abiotic stress (e.g., spike harvest and fertility indices combined with biomass and grain yield in both genotypes). Validation and confirmatory factor models and multiway cluster analyses revealed major differences in phenotypic plasticities between wheat genotypes that can be attributed to differences in ploidy level, length of domestication history, or constitutive differences in resources allocation. Discriminant analyses helped to identify genotypic differences or similarities in the level of trait decoupling in relation to the strength of their correlation and heritability estimates. This information is useful in targeted improvement of traits directly contributing to micronutrient densities, yield components, and yield. New wheat ideotype(s) can be designed for larger grain yield potential under abiotic stress by manipulating yield components that affect kernels m −2 (e.g., number of tillers, number of florets per spikelet, and eventually spike fertility and harvest indices) without impacting nutrient densities and kernel weight, thus raising harvest index beyond its current maximum.

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Analysis of Correlation Matrices Using Scale-Invariant Common Principal Component Models and a Hierarchy of Relationships Between Correlation Matrices

Cited by 5 publications

References 22 publications

A model‐based approach to multivariate principal component regression: Selecting principal components and estimating standard errors for unstandardized regression coefficients

A model‐based approach to multivariate principal component regression: Selecting principal components and estimating standard errors for unstandardized regression coefficients

Semiparametric partial common principal component analysis for covariance matrices

Statistical Modeling of Phenotypic Plasticity under Abiotic Stress in Triticum durum L. and Triticum aestivum L. Genotypes

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