Improperness Measures for Quaternion Random Vectors
J. Vía, D. Ramírez, I. Santamaría, L. Vielva, in: Proc.\ IEEE Int.\ Work. Machine Learning for Signal Process., Finland, 2010.
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Author
Vía, J.;
Ramírez, D.;
Santamaría, I.;
Vielva, L.
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Abstract
It has been recently proved that the two main kinds of quaternion improperness require two different kinds of widely linear process- ing. In this work, we show that these definitions satisfy some im- portant properties, which include the invariance to quaternion lin- ear transformations and right Clifford translations, as well as some clear connections with the case of proper complex vectors. More- over, we introduce a new kind of quaternion properness, which clearly relates the two previous definitions, and propose three mea- sures for the degree of improperness of a quaternion vector. The proposed measures are based on the Kullback-Leibler divergence between two zero-mean quaternion Gaussian distributions, one of them with the actual augmented covariance matrix, and the other with its closest proper version. These measures allow us to quan- tify the entropy loss due to the improperness of the quaternion vec- tor, and they admit an intuitive geometrical interpretation based on Kullback-Leibler projections onto sets of proper augmented co- variance matrices.
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Proc.\ IEEE Int.\ Work. Machine Learning for Signal Process.
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Vía J, Ramírez D, Santamaría I, Vielva L. Improperness Measures for Quaternion Random Vectors. In: Proc.\ IEEE Int.\ Work. Machine Learning for Signal Process. ; 2010. doi:10.1109/MLSP.2010.5589225
Vía, J., Ramírez, D., Santamaría, I., & Vielva, L. (2010). Improperness Measures for Quaternion Random Vectors. Proc.\ IEEE Int.\ Work. Machine Learning for Signal Process. https://doi.org/10.1109/MLSP.2010.5589225
@inproceedings{Vía_Ramírez_Santamaría_Vielva_2010, place={Finland}, title={Improperness Measures for Quaternion Random Vectors}, DOI={10.1109/MLSP.2010.5589225}, booktitle={Proc.\ IEEE Int.\ Work. Machine Learning for Signal Process.}, author={Vía, J. and Ramírez, D. and Santamaría, I. and Vielva, L.}, year={2010} }
Vía, J., D. Ramírez, I. Santamaría, and L. Vielva. “Improperness Measures for Quaternion Random Vectors.” In Proc.\ IEEE Int.\ Work. Machine Learning for Signal Process. Finland, 2010. https://doi.org/10.1109/MLSP.2010.5589225.
J. Vía, D. Ramírez, I. Santamaría, and L. Vielva, “Improperness Measures for Quaternion Random Vectors,” 2010, doi: 10.1109/MLSP.2010.5589225.
Vía, J., et al. “Improperness Measures for Quaternion Random Vectors.” Proc.\ IEEE Int.\ Work. Machine Learning for Signal Process., 2010, doi:10.1109/MLSP.2010.5589225.
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