Research On The Theory And Application Of Sparse Bayesian Learning Algorithm

Sparse Bayesian learning(SBL)has been widely used in various fields,such as smart grid,networked system identification,signal processing.SBL imposes sparsity on the model to balance its fitting ability and generalization ability,and estimates model parameters based on Bayesian learning theory.On one hand,conventional SBL algorithms require a matrix inversion at each iteration,thereby leading to O(n~3)computational complexity.This hinders their practical applications with high-dimensional and large-scale data sets.On the other hand,due to the nonconvex objective function,conventional SBL algorithms would fall into a local optimal solution,which causes model quality to heavily depend on the initial values of the algorithm.This dissertation has conducted in-depth research on the above significant problems in SBL.The main research contents are as follows:An efficient SBL algorithm with low computational complexity is proposed.Aim-ing at the high computational complexity of SBL algorithms,the conditionally independent posterior distributions of model parameters are constructed by using the relaxed Gaussian likelihood function.Based on nonconvex optimization theory,a block coordinate descent method is used to solve the sparse Bayesian regression problem under an MM(majorization minimization)framework.Then,an SBL algorithm with O(n~2)computational complexity is developed,and it is proved that the algorithm converges to a stationary point of the opti-mization problem.The proposed algorithm is applied to the problem of high-dimensional sparse signal reconstruction.The results show that the proposed algorithm achieves better reconstruction performance than the classical SBL algorithms,while improving the compu-tational efficiency of nearly 10 times.An SBL algorithm with global convergence is proposed.Aiming at the problem of local convergence of the SBL algorithms developed in the MM framework,an SBL algo-rithm with global convergence is proposed by combining neurodynamic optimization and Bayesian learning theory.It employs multiple projection neural networks to collaboratively search global optimal solutions to the optimization problem.It is proved that the proposed algorithm is almost surely convergent to a global minimum to the nonconvex optimization problem.The proposed method is applied to sparse signal reconstruction and partial dif-ferential equation identification.The results show that the proposed algorithm consistently achieves a global optimal solution under different initial values of the algorithm.A group SBL algorithm with automatic parameter tuning is proposed.Aiming at the problem that the integral in the posterior distribution of model parameters is intractable in multi-class classification tasks,the classification problem is transformed into a joint discrim-inative dictionary and classifier learning problem based on sparse representation.Then,a Bayesian model is established,and a classification method based on group SBL is proposed.It is proved that the algorithm is convergent.To verify the effectiveness of the proposed method,we conducted extensive experiments on three benchmark data sets.The results show that compared with classical sparse representation methods,the proposed algorithm achieves automatic parameter adjustments and improves the classification accuracy.The proposed group SBL is used to solve the harmonic state estimation(HSE)problem in transmission systems that are not fully observable.Aiming at the partial observability of systems caused by the lack of harmonic measurements,the HSE problem is transformed into a sparse regression problem by virtue of the spatial sparsity and structural sparsity of harmonic sources.Then,a harmonic state estimator is proposed based on group SBL.To verify the effectiveness of the proposed method,extensive simulations were carried out on the IEEE 14-bus harmonic test system.The results show that the proposed method achieves high-precision positioning of harmonic sources and accurate estimation of harmonic cur-rents and voltages by using limited harmonic meters.A method is proposed to solve the HSE problem in time-varying distribution systems which are not fully observable.Aiming at the underdetermined HSE problem of a time-varying measurement matrix,a HSE method is developed based on long short-term memory recurrent neural networks and SBL.To verify the effectiveness of the proposed method,extensive simulations were carried out on the modified IEEE 13-node benchmark test feeder by using field harmonic data and load data.The results show that compared with traditional approaches,the proposed method improves the localization accuracy of harmonic sources and reduces the estimation errors of harmonic currents and voltages.