Augmented Statistics of Quaternion Random Variables: A lynchpin of quaternion learning machines

Sayed Talebi; Danilo Mandic; clive cheong took; Rosa  Maria Fernandez Alcala

doi:10.1109/MSP.2024.3384178

Augmented Statistics of Quaternion Random Variables: A lynchpin of quaternion learning machines

Sayed Talebi, Danilo Mandic, clive cheong took, Rosa Maria Fernandez Alcala

School of Arts, Humanities, and Social Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

Learning machines for vector sensor data are naturally developed in the quaternion domain and are underpinned by quaternion statistics. To this end, we revisit the “augmented” representation basis for discrete quaternion random variables (RVs) qa[n], i.e., [q[n]qı[n]qȷ[n]qκ[n]], and demonstrate its pivotal role in the treatment of the generality of quaternion RVs. This is achieved by a rigorous consideration of the augmented quaternion RV and by involving the additional second-order statistics, besides the traditional covariance E{q[n]q∗[n]} [1]. To illuminate the usefulness of quaternions, we consider their most well-known application—3D orientation—and offer an account of augmented statistics for purely imaginary (pure) quaternions. The quaternion statistics presented here can be exploited in the analysis of existing and the development of novel statistical machine learning methods, hence acting as a lynchpin for quaternion learning machines.

Original language	English
Journal	IEEE Signal Processing Magazine
DOIs	https://doi.org/10.1109/MSP.2024.3384178
Publication status	Published - 3 May 2024

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10.1109/MSP.2024.3384178

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title = "Augmented Statistics of Quaternion Random Variables: A lynchpin of quaternion learning machines",

abstract = "Learning machines for vector sensor data are naturally developed in the quaternion domain and are underpinned by quaternion statistics. To this end, we revisit the “augmented” representation basis for discrete quaternion random variables (RVs) qa[n], i.e., [q[n]qı[n]qȷ[n]qκ[n]], and demonstrate its pivotal role in the treatment of the generality of quaternion RVs. This is achieved by a rigorous consideration of the augmented quaternion RV and by involving the additional second-order statistics, besides the traditional covariance E{q[n]q∗[n]} [1]. To illuminate the usefulness of quaternions, we consider their most well-known application—3D orientation—and offer an account of augmented statistics for purely imaginary (pure) quaternions. The quaternion statistics presented here can be exploited in the analysis of existing and the development of novel statistical machine learning methods, hence acting as a lynchpin for quaternion learning machines.",

author = "Sayed Talebi and Danilo Mandic and {cheong took}, clive and {Maria Fernandez Alcala}, Rosa",

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T1 - Augmented Statistics of Quaternion Random Variables: A lynchpin of quaternion learning machines

AU - Talebi, Sayed

AU - Mandic, Danilo

AU - cheong took, clive

AU - Maria Fernandez Alcala, Rosa

PY - 2024/5/3

Y1 - 2024/5/3

N2 - Learning machines for vector sensor data are naturally developed in the quaternion domain and are underpinned by quaternion statistics. To this end, we revisit the “augmented” representation basis for discrete quaternion random variables (RVs) qa[n], i.e., [q[n]qı[n]qȷ[n]qκ[n]], and demonstrate its pivotal role in the treatment of the generality of quaternion RVs. This is achieved by a rigorous consideration of the augmented quaternion RV and by involving the additional second-order statistics, besides the traditional covariance E{q[n]q∗[n]} [1]. To illuminate the usefulness of quaternions, we consider their most well-known application—3D orientation—and offer an account of augmented statistics for purely imaginary (pure) quaternions. The quaternion statistics presented here can be exploited in the analysis of existing and the development of novel statistical machine learning methods, hence acting as a lynchpin for quaternion learning machines.

AB - Learning machines for vector sensor data are naturally developed in the quaternion domain and are underpinned by quaternion statistics. To this end, we revisit the “augmented” representation basis for discrete quaternion random variables (RVs) qa[n], i.e., [q[n]qı[n]qȷ[n]qκ[n]], and demonstrate its pivotal role in the treatment of the generality of quaternion RVs. This is achieved by a rigorous consideration of the augmented quaternion RV and by involving the additional second-order statistics, besides the traditional covariance E{q[n]q∗[n]} [1]. To illuminate the usefulness of quaternions, we consider their most well-known application—3D orientation—and offer an account of augmented statistics for purely imaginary (pure) quaternions. The quaternion statistics presented here can be exploited in the analysis of existing and the development of novel statistical machine learning methods, hence acting as a lynchpin for quaternion learning machines.

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DO - 10.1109/MSP.2024.3384178

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