Quaternion-valued Neural Network Research Articles

Exponential synchronization of a class of quaternion-valued neural network with time-varying delays: A Matrix Measure Approach

This article investigates the exponential synchronization (ES) of quaternion-valued neural networks (QVNNs) with time-varying delays. Global ES has been studied using matrix measure techniques. Due to the incompatibility of quaternion multiplication, the QVNN system is first separated into four real-valued systems. Then ES is established in the master-response systems of QVNN by constructing a state feedback controller. The stability criteria for the error system of QVNN is then provided using the existing lemmas and the Halanay inequality. Additionally, it is no longer assumed that the activation functions are differentiable, which make the analytical process more concise. The viability of our findings are demonstrated using two numerical examples. The present theoretical findings have been confirmed in the first case, while the second example includes the use of QVNNs connected to associative memory to demonstrate their ability to restore real color image patterns properly.

Stability and Hopf bifurcation for a quaternion-valued three-neuron neural network with leakage delay and communication delay

This paper proposes the stability and Hopf bifurcation for a quaternion-valued neural network(QVNN) with leakage delay and communication delay. Due to the higher order of the characteristic equation, the order of characteristic equation is reduced via the matrix block theory. Simultaneously, taking the leakage delay and communication delay as bifurcation parameter respectively, a number of conditions ensuring the stability of Hopf bifurcation periodic solutions are established. It lavishly illustrates that the stability of QVNN can be demolished more when considering the leakage delay as bifurcation parameter. Finally one simulations example is given to validate the theoretical results.

Nonseparation Approach to General-Decay Synchronization of Quaternion-Valued Neural Networks with Mixed Time Delays

In this paper, the general-decay synchronization issue of a class of quaternion-valued neural networks with mixed time delays is investigated. Firstly, unlike some previous works where the quaternion-valued model is separated into four real-valued networks or two complex-valued networks, we consider the mixed-delayed quaternion-valued neural network model as a whole and introduce a novel nonlinear feedback controller for the corresponding response system. Then, by introducing a suitable Lyapunov–Krasovskii functional and employing a novel inequality technique, some easily verifiable sufficient conditions are obtained to ensure the general-decay synchronization for the considered drive-response networks. Finally, the feasibility of the established theoretical results is verified by carrying out Matlab numerical simulations.

Multistability analysis of quaternion-valued neural networks with cosine activation functions

It is significant to design a system with high storage capacity for associative memory and pattern recognition. To address this issue, this paper first proposes a quaternion-valued neural network (QVNN) model with multiple equilibrium points in which the cosine function is used as the activation function of QVNN. Then, based on the Brouwer fixed point theorem and the geometric properties of the activation function, sufficient conditions for QVNN to have unique equilibrium points, finite equilibrium points, and countable infinite equilibrium points are obtained, respectively. Furthermore, sufficient conditions for the exponential stability of equilibrium points are derived, and the attraction basins of the stable equilibrium points are given. Finally, two numerical examples are given to confirm the validity of the proposed theoretical results.

Fixed-time and prescribed-time synchronization of quaternion-valued neural networks: A control strategy involving Lyapunov functions

A control strategy containing Lyapunov functions is proposed in this paper. Based on this strategy, the fixed-time synchronization of a time-delay quaternion-valued neural network (QVNN) is analyzed. This strategy is extended to the prescribed-time synchronization of the QVNN. Furthermore, an improved two-step switching control strategy is also proposed based on this flexible control strategy. Compared with some existing methods, the main method of this paper is a non-decomposition one, does not contain a sign function in the controller, and has better synchronization accuracy. Two numerical examples verify the above advantages.

Storage Capacity of Quaternion-Valued Hopfield Neural Networks With Dual Connections.

A complex-valued Hopfield neural network (CHNN) is a multistate Hopfield model. A quaternion-valued Hopfield neural network (QHNN) with a twin-multistate activation function was proposed to reduce the number of weight parameters of CHNN. Dual connections (DCs) are introduced to the QHNNs to improve the noise tolerance. The DCs take advantage of the noncommutativity of quaternions and consist of two weights between neurons. A QHNN with DCs provides much better noise tolerance than a CHNN. Although a CHNN and a QHNN with DCs have the samenumber of weight parameters, the storage capacity of projection rule for QHNNs with DCs is half of that for CHNNs and equals that of conventional QHNNs. The small storage capacity of QHNNs with DCs is caused by projection rule, not the architecture. In this work, the ebbian rule is introduced and proved by stochastic analysis that the storage capacity of a QHNN with DCs is 0.8 times as many as that of a CHNN.

Exponential stability in the Lagrange sense for Clifford-valued recurrent neural networks with time delays

This paper considers the Clifford-valued recurrent neural network (RNN) models, as an augmentation of real-valued, complex-valued, and quaternion-valued neural network models, and investigates their global exponential stability in the Lagrange sense. In order to address the issue of non-commutative multiplication with respect to Clifford numbers, we divide the original n-dimensional Clifford-valued RNN model into 2^{m}n real-valued models. On the basis of Lyapunov stability theory and some analytical techniques, several sufficient conditions are obtained for the considered Clifford-valued RNN models to achieve global exponential stability according to the Lagrange sense. Two examples are presented to illustrate the applicability of the main results, along with a discussion on the implications.

Impulsive effects on Clifford-valued neural networks with time-varying delays: An asymptotic stability analysis

In this paper, we focus on the global asymptotic stability problem for Clifford-valued neural network models with time-varying delays as well as impulsive effects. By considering impulsive effects, a general class of network model is considered, which encompasses real-valued, complex-valued, and quaternion-valued neural network models as special cases. Firstly, the n-dimensional Clifford-valued model is decomposed into 2mn-dimensional real-valued model, which avoids non-commutativity of multiplication of Clifford numbers. Based on the Lyapunov stability theory, contraction mapping principle, and some mathematical concepts, we derive the existence, uniqueness of the equilibrium point with respect to the model. New sufficient conditions are also derived, in order to ensure the global asymptotic stability of the considered model. To illustrate the usefulness of the obtained results, a simulation example is presented.

Global exponential stability of Clifford-valued neural networks with time-varying delays and impulsive effects

In this study, we investigate the global exponential stability of Clifford-valued neural network (NN) models with impulsive effects and time-varying delays. By taking impulsive effects into consideration, we firstly establish a Clifford-valued NN model with time-varying delays. The considered model encompasses real-valued, complex-valued, and quaternion-valued NNs as special cases. In order to avoid the issue of non-commutativity of the multiplication of Clifford numbers, we divide the original n-dimensional Clifford-valued model into 2^{m}n-dimensional real-valued models. Then we adopt the Lyapunov–Krasovskii functional and linear matrix inequality techniques to formulate new sufficient conditions pertaining to the global exponential stability of the considered NN model. Through numerical simulation, we show the applicability of the results, along with the associated analysis and discussion.

Asymptotic behavior of Clifford-valued dynamic systems with D-operator on time scales

In this paper, a general class of Clifford-valued neutral high-order neural network (HNN) with D-operator on time scales is investigated. In this model, time-varying delays and continuously distributed delays are taken into account. As an extension of the real-valued neural network, the Clifford-valued neural network, which includes a familiar complex-valued neural network and a quaternion-valued neural network as special cases, has been an active research field recently. By utilizing this novel method, which incorporates the differential inequality techniques and the fixed point theorem and time-scale theory of computation, we derive a few sufficient conditions to ensure the existence, uniqueness, and exponential stability of the pseudo almost periodic (PAP) solution of the considered model. The results in this paper are new, even if time scale mathbb{T}=mathbb{R} or mathbb{T}=mathbb{Z}, and complementary to the previously existing works. Furthermore, an example and its numerical simulations are included to demonstrate the validity and advantage of the obtained results.

Stochastic stabilization of Markov jump quaternion-valued neural network using sampled-data control

This paper investigates the stochastic stabilization of Markov jump quaternion-valued neural networks (QVNNs) using a sampled-data control strategy. Firstly, Markov jump QVNNs are decomposed into two complex-valued systems using the plural decomposition method because the multiplication of quaternions is not commutative. Secondly, the existence and uniqueness of the equilibrium point of the Markov jump QVNNs is proved according to the theory of homeomorphism mapping. Thirdly, by choosing a suitable Lyapunov-Krasovskii functional and combining some inequality techniques, a new stochastic stability criterion is established for the Markov jump QVNNs. Based on this, several verifiable sufficient conditions for the stochastic stabilization of Markov jump QVNNs with sampled-data control are ensured. Finally, the correctness and effectiveness of the proposed method are verified by two numerical examples.

Wind Speed Forecasting Using the Stationary Wavelet Transform and Quaternion Adaptive-Gradient Methods

Accurate wind speed forecasting is a fundamental requirement for advanced and economically viable large-scale wind power integration. The hybridization of the quaternion-valued neural networks and stationary wavelet transform has not been proposed before. In this paper, we propose a novel wind-speed forecasting model that combines the stationary wavelet transform with quaternion-valued neural networks. The proposed model represents wavelet subbands in quaternion vectors, which avoid separating the naturally correlated subbands. The model consists of three main steps. First, the wind speed signal is decomposed using the stationary wavelet transform into sublevels. Second, a quaternion-valued neural network is used to forecast wind speed components in the stationary wavelet domain. Finally, the inverse stationary wavelet transform is applied to estimate the predicted wind speed. In addition, a softplus quaternion variant of the RMSProp learning algorithm is developed and used to improve the performance and convergence speed of the proposed model. The proposed model is tested on wind speed data collected from different sites in China and the United States, and the results demonstrate that it consistently outperforms similar models. In the meteorological terminal aviation routine (METAR) dataset experiment, the proposed wind speed forecasting model reduces the mean absolute error, and root mean squared error of predicted wind speed values by 26.5% and 33%, respectively, in comparison to several existing approaches.

Novel bifurcation results for a delayed fractional-order quaternion-valued neural network

This paper reports the innovative results on the stability and bifurcation for a delayed fractional-order quaternion-valued neural network(FOQVNN). Delay-stimulated bifurcation criteria of the developed FOQVNN are attained. Then, the bifurcation diagrams are perfectly exhibited to authenticate the veracity of the bifurcation results. Besides, the stability zone is more larger of the addressed FOQVNN in comparison with its counterpart if other parameters are intercalated. It further witnesses that the amplitudes of bifurcation oscillation get bigger with the augmentation of time delay. It discloses that the bifurcation phenomena engender earlier as the order incrementally magnifies. The exactness and merits of the achieved analytic results are eventually substantiated by a simulation example.

Fixed-time synchronization of quaternion-valued neural networks

In this paper, the fixed-time synchronization problem of quaternion-valued neural networks is investigated. Different from the finite-time synchronization related to the initial conditions, the settling time of fixed-time synchronization can be changed properly according to the actual needs. The fixed-time synchronization of quaternion-valued neural networks is firstly studied by adding the new nonlinear feedback controllers and employing the theorem of fixed-time stability. The quaternion-valued neural network is addressed by dividing the system under investigation into four real-valued parts. Then based on related theorem of fixed-time stability, and selecting the appropriate controllers and Lyapunov function, sufficient conditions of fixed-time synchronization for quaternion-valued neural networks are obtained. In the same way, the results of finite-time synchronization can also be got. Finally, one numerical example is provided to illustrate the effectiveness of the proposed results.

Boundedness and periodicity for linear threshold discrete-time quaternion-valued neural network with time-delays

In this paper, the discrete-time quaternion-valued neural network with linear threshold activation functions is investigated. The sufficient conditions to the boundedness and global exponential periodicity of the neural network are obtained by using characteristic equation, Lyapunov functional and M-matrix. Simulation results illustrative the effectiveness of the conclusions obtained in this paper.

Cognitive Quaternion Valued Neural Network and some applications

A Meta-cognitive Quaternion Valued Neural Network (Mc-QVNN) learning algorithm and its forecasting applications is proposed in this paper. The Mc-QVNN has two parts, the cognitive part that contains the QVNN and a meta-cognitive part, which self-regulates the learning algorithm. At each epoch, when the Mc-QVNN receives a new sample, the meta-cognitive part makes a decision about the manner, the time and the need to learn this sample or not. In this case, the algorithm deletes the unneeded samples and keeps just the necessary ones for learning. The meta-cognitive component makes the decision according to the quaternion magnitude and phases. Three forecasting problems, which are Mackey–Glass time series, Lorenz attractor and the real home's power in the city of Honolulu in Hawaii, USA, are taken to test the performance of the proposed algorithm. Comparison with other existing methods shows that the Mc-QVNN is promising for forecasting chaotic systems.

A novel method to forecast 24 h of global solar irradiation

Forecasting solar irradiation is a primordial task for planning and sizing photovoltaic power systems. In this paper, the global solar irradiation is forecasted using the quaternion-valued neural network (QVNN). A new method to represent the meteorological values to the quaternion domain in the daily or the hourly indexes is proposed. This method specifies each location in the world with one parameter contains at the same time the latitude, the longitude, and the time indexes. The resulting network has two advantages which are: the input/output layers are reduced three times and the learning algorithm increased three times naturally. The measured data of Tamanrasset city, Algeria is used to validate the developed model. The 1–24 h ahead solar irradiation are forecasted using the mean quaternion values (i.e. the meteorological data combined with time indexes, the longitude, and the latitude in one quaternion value). To test the performance and the feasibility of the QVNN for forecasting 24 h ahead global solar irradiation, the meteorological parameters are combined. Results show that the QVNN technique is a promising method for forecasting solar irradiation.

Global stability of Clifford-valued recurrent neural networks with time delays

In this paper, we study an issue of stability analysis for Clifford-valued recurrent neural networks (RNNs) with time delays. As an extension of real-valued neural network, the Clifford-valued neural network, which includes familiar complex-valued neural network and quaternion-valued neural network as special cases, has been an active research field recently. To the best of our knowledge, the stability problem for Clifford-valued systems with time delays has still not been solved. We first explore the existence and uniqueness for the equilibrium of delayed Clifford-valued RNNs, based on which some sufficient conditions ensuring the global asymptotic and exponential stability of such systems are obtained in terms of a linear matrix inequality (LMI). The simulation result of a numerical example is also provided to substantiate the effectiveness of the proposed results.