Split quaternion nonlinear adaptive filtering

Ujang, Bukhari Che; Took, Clive Cheong; Mandic, Danilo P.

doi:10.1016/j.neunet.2009.10.006

Cited by 40 publications

(48 citation statements)

References 22 publications

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“…8, observe that the QNGD outperformed the other algorithms considered. Also, observe that QMLP prediction gain was almost constant with the increase of the prediction horizon due to the structural richness of the feedforward multilayer neural network, which conforms to our earlier studies in [11].…”

Section: Wind Forecastingsupporting

confidence: 89%

“…Computational complexities of the QMLP-FIR is O(36L) and for the QMLP it is O(108L). The QNGD algorithm thus represents an improvement from our previous proposed algorithm AASQAFA [11] in terms of performance and simplicity, while maintaining similar computational complexity.…”

Section: Discussionmentioning

confidence: 93%

“…A comprehensive comparison of the performances is provided between the training algorithm for the feedforward QMLP [8], [38] and the nonlinear finite impulse response filters trained with the QMLP learning algorithm (QMLP-FIR) [11], adaptive amplitude split quaternion adaptive filtering algorithm (AASQAFA) [11], real-valued nonlinear gradient descent (NGD) [37], and the proposed algorithms based on fully quaternion nonlinear functions, i.e., QNGD and AQNGD. The quaternion multilayer perceptron-finite impulse response (QMLP-FIR), AASQAFA, NGD, QNGD, and AQNGD were implemented with a filter length L, whereas the QMLP had one hidden layer comprising of L input neurons, three hidden neurons, and one output neuron.…”

Section: Simulationsmentioning

confidence: 99%

“…However, the training of QMLP neglects the non-commutativity aspect of the quaternion algebra and thus does not exploit the full potential of the processing in the quaternion domain; this issue was addressed with the split quaternion nonlinear adaptive filtering algorithm (SQAFA) [11]. The nonlinearities used in SQAFA were still standard real activation functions applied componentwise, thus prohibiting a rigorous treatment of the cross-information across the data channels.…”

Section: Introductionmentioning

confidence: 99%

“…It is important to note that most practical learning algorithms are gradient-based [8]- [11], where the operating point moves at every sample interval, to build nonlinear adaptive models and we only require local analyticity at a point. In analogy to the complex domain, where the so-called fully complex nonlinearities (elementary transcendental functions) provide means for generic extensions of real neural networks [12], [13], our aim is to show that the class of elementary transcendental functions, such as tanh, are locally analytic in H and thus permit generalization of neural networks to the quaternion domain.…”

Section: Introductionmentioning

confidence: 99%

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Quaternion-Valued Nonlinear Adaptive Filtering

Ujang

Took

Mandic

2011

IEEE Trans. Neural Netw.

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183

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Abstract-A class of nonlinear quaternion-valued adaptive filtering algorithms is proposed based on locally analytic nonlinear activation functions. To circumvent the stringent standard analyticity conditions which are prohibitive to the development of nonlinear adaptive quaternion-valued estimation models, we use the fact that stochastic gradient learning algorithms require only local analyticity at the operating point in the estimation space. It is shown that the quaternion-valued exponential function is locally analytic, and, since local analyticity extends to polynomials, products, and ratios, we show that a class of transcendental nonlinear functions can serve as activation functions in nonlinear and neural adaptive models. This provides a unifying framework for the derivation of gradient-based learning algorithms in the quaternion domain, and the derived algorithms are shown to have the same generic form as their real-and complex-valued counterparts. To make such models second-order optimal for the generality of quaternion signals (both circular and noncircular), we use recent developments in augmented quaternion statistics to introduce widely linear versions of the proposed nonlinear adaptive quaternion valued filters. This allows full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances to cater rigorously for second-order noncircularity (improperness), and the corresponding power mismatch in the signal components. Simulations over a range of circular and noncircular synthetic processes and a real world 3-D noncircular wind signal support the approach.Index Terms-Augmented quaternion statistics, H-circularity, nonlinear adaptive filtering, quaternion least mean square, widely linear modeling, widely linear quaternion least mean square, wind prediction.

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Section: Wind Forecastingsupporting

confidence: 89%

Section: Discussionmentioning

confidence: 93%

Section: Simulationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quaternion-Valued Nonlinear Adaptive Filtering

Ujang

Took

Mandic

2011

IEEE Trans. Neural Netw.

Self Cite

183

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Algebraic foundations of split hypercomplex nonlinear adaptive filtering

Hitzer

2012

Math Methods in App Sciences

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A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed.

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Scaled Conjugate Gradient Learning for Quaternion-Valued Neural Networks

Popa

2016

Neural Information Processing

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Split quaternion nonlinear adaptive filtering

Cited by 40 publications

References 22 publications

Quaternion-Valued Nonlinear Adaptive Filtering

Quaternion-Valued Nonlinear Adaptive Filtering

Algebraic foundations of split hypercomplex nonlinear adaptive filtering

Scaled Conjugate Gradient Learning for Quaternion-Valued Neural Networks

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