Enabling quaternion derivatives: the generalized HR calculus

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis

Detection of interictal discharges with convolutional neural networks using discrete ordered multichannel intracranial EEG

Detection algorithms for electroencephalography (EEG) data, especially in the field of interictal epileptiform discharge (IED) detection, have traditionally employed handcrafted features which utilised specific characteristics of neural responses. Although these algorithms achieve high accuracy, mere detection of an IED holds little clinical significance. In this work, we consider deep learning for epileptic subjects to accommodate automatic feature generation from intracranial EEG data, while also providing clinical insight. Convolutional neural networks are trained in a subject independent fashion to demonstrate how meaningful features are automatically learned in a hierarchical process. We illustrate how the convolved filters in the deepest layers provide insight towards the different types of IEDs within the group, as confirmed by our expert clinicians. The morphology of the IEDs found in filters can help evaluate the treatment of a patient. To improve the learning of the deep model, moderately different score classes are utilised as opposed to binary IED and non-IED labels. The resulting model achieves state of the art classification performance and is also invariant to time differences between the IEDs. This study suggests that deep learning is suitable for automatic feature generation from intracranial EEG data, while also providing insight into the dat

Quaternion Common Spatial Patterns

A novel quaternion-valued common spatial patterns (QCSP) algorithm is introduced to model co-channel coupling of multi-dimensional processes. To cater for the generality of quaternion-valued non-circular data, we propose a generalized QCSP (G-QCSP) which incorporates the information on power difference between the real and imaginary parts of data channels. As an application, we demonstrate how G-QCSP can be used to provide high classification rates, even at a signal-to-noise ratio (SNR) as low as -10 dB. To illustrate the usefulness of our method in EEG analysis, we employ G-QCSP to extract features for discriminating between imagery left and right hand movements. The classification accuracy using these features is 70%. Furthermore, the proposed method is used to distinguish between Parkinson's disease (PD) patients and healthy control subjects, providing an accuracy of 87%

Almost commuting unitary matrices related to time reversal

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either the matrices over A for A the real numbers, A the complex numbers or A the algebra of quaternions. There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple. We consider two-dimensional geometries and so this obstruction lives in KO_{-2}(A). This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.Comment: 33 pages, 2 figures. In version 2 some formulas have been corrected and some proofs have been rewritten to improve the expositio

Characteristics associated with reported CAM use in patients attending six GP practices in the Tayside and Grampian regions of Scotland: a survey.

OBJECTIVES: To study the nature of CAM use in primary care attenders, the involvement of their NHS healthcare professionals in their CAM care and differences in characteristics between CAM users and non-users. DESIGN: Postal questionnaire for primary care attenders and analysis of practice leaflets. SETTING: Six Scottish GP practices with a range of practice size, CAM provision within practice, deprivation and rurality. RESULTS: Five hundred and fourteen primary care attenders described 1194 incidences of CAM use and gave details about their main therapy. 37% had contact with a practitioner, the rest mainly self-prescribed. The perceived effectiveness of CAM was high. Patients used CAM for a variety of health problems, mainly as an adjuvant to orthodox medicine rather than an alternative. The involvement of the NHS in CAM delivery was small but there is a significant role to ensure patient safety, especially regarding herb-drug interactions. Disclosure rate of CAM use was low. CAM offered options in areas where the provision in the NHS is difficult, including musculo-skeletal and mental health problems. Provision of CAM by the GP is associated with higher CAM use in primary care attenders. CONCLUSIONS: It is recommended that healthcare professionals include patients' use of CAM in history taking and clinical decision making

Enabling quaternion derivatives: the generalized HR calculus

Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis