An Algorithm for Simultaneous Orthogonal Transformation of Several Positive Definite Symmetric Matrices to Nearly Diagonal Form

Flury, Bernhard; Gautschi, Walter

doi:10.1137/0907013

Cited by 163 publications

(116 citation statements)

References 2 publications

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“…We can minimize (11) by taking an approach similar to the one introduced in [4]. Given the sets {BC k B H } and {Λ k }, the idea is to find a matrix A such that the set {AΛ k A H } gets closer (according to (5)) to the set {BC k B H }. This way, matrices…”

Section: Riemannian Diagonality Measurementioning

confidence: 99%

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Approximate Joint Diagonalization According to the Natural Riemannian Distance

Bouchard¹,

Malick²,

Congedo³

2017

Latent Variable Analysis and Signal Separation

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International audienceIn this paper, we propose for the first time an approximate joint diagonalization (AJD) method based on the natural Riemannian distance of Hermitian positive definite matrices. We turn the AJD problem into an optimization problem with a Riemannian criterion and we developp a framework to optimize it. The originality of this criterion arises from the diagonal form it targets. We compare the performance of our Riemannian criterion to the classical ones based on the Frobe-nius norm and the log-det divergence, on both simulated data and real electroencephalographic (EEG) signals. Simulated data show that the Riemannian criterion is more accurate and allows faster convergence in terms of iterations. It also performs well on real data, suggesting that this new approach may be useful in other practical applications

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Section: Riemannian Diagonality Measurementioning

confidence: 99%

“…Further, note that when the best possible matrix A is the identity matrix I n , {BC k B H } contains matrices as diagonal as possible according to (5). To find A, we define the optimization subproblem with cost function f R as…”

Section: Riemannian Diagonality Measurementioning

confidence: 99%

Approximate Joint Diagonalization According to the Natural Riemannian Distance

Bouchard¹,

Malick²,

Congedo³

2017

Latent Variable Analysis and Signal Separation

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show abstract

“…For the CPC and CPC estimators, the Flury-Gautschi algorithm (Flury and Gautschi, 1986) was used to estimate the common eigenvector matrices. The only exception was for the p = 2 case, where the Σ similar scenario is excluded as it is by definition the same as Σ opposite .…”

Section: Simulation Studymentioning

confidence: 99%

Discriminant analysis under the common principal components model

Pepler

Uys

Nel

2017

Communications in Statistics - Simulation and Computation

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For two or more populations of which the covariance matrices have a common set of eigenvectors, but different sets of eigenvalues, the common principal components (CPC) model is appropriate. Pepler et al. (2015) proposed a regularised CPC covariance matrix estimator and showed that this estimator outperforms the unbiased and pooled estimators in situations where the CPC model is applicable. This paper extends their work to the context of discriminant analysis for two groups, by plugging the regularised CPC estimator into the ordinary quadratic discriminant function.Monte Carlo simulation results show that CPC discriminant analysis offers significant improvements in misclassification error rates in certain situations, and at worst performs similar to ordinary quadratic and linear discriminant analysis. Based on these results, CPC discriminant analysis is recommended for situations where the sample size is small compared to the number of variables, in particular for cases where there is uncertainty about the population covariance matrix structures.

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“…, k. Note that, in contrast with the formulation (8), no canonical ordering of the columns of ˜i s specified here, since the rank order of the diagonal elements of the h is not necessarily the same for all groups. In order to apply Flury's proposal, the F-G algorithm, developed by Flury and Gautschi (1986) and coded in FORTRAN by Flury and Constantine (1985), can be considered to estimate the sample counterpart G of ˜( see Flury 1984). Under H CPC 0 , the following transformation…”

Section: Testing Homotroposcedasticitymentioning

confidence: 99%

Assessing the pattern of covariance matrices via an augmentation multiple testing procedure

2011

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This paper extends the scedasticity comparison among several groups of observations, usually complying with the homoscedastic and the heteroscedastic cases, in order to deal with data sets laying in an intermediate situation. As is well known, homoscedasticity corresponds to equality in orientation, shape and size of the group scatters. Here our attention is focused on two weaker requirements: scatters with the same orientation, but with different shape and size, or scatters with the same shape and size but different orientation. We introduce a multiple testing procedure that takes into account each of the above conditions. This approach discloses a richer information on the data underlying structure than the classical method only based on homo/heteroscedasticity. At the same time, it allows a more parsimonious parametrization, whenever the patterned model is appropriate to describe the real data. The new inferential methodology is then applied to some well-known data sets, chosen in the multivariate literature, to show the real gain in using this more informative approach. Finally, a wide simulation study illustrates and compares the performance of the proposal using data sets with gradual departure from homoscedasticity.

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An Algorithm for Simultaneous Orthogonal Transformation of Several Positive Definite Symmetric Matrices to Nearly Diagonal Form

Cited by 163 publications

References 2 publications

Approximate Joint Diagonalization According to the Natural Riemannian Distance

Approximate Joint Diagonalization According to the Natural Riemannian Distance

Discriminant analysis under the common principal components model

Assessing the pattern of covariance matrices via an augmentation multiple testing procedure

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