自学习算法及其应用研究

发布时间：2018-04-28 00:45

本文选题：自学习 + 线性方程组　；参考：《长沙理工大学》2015年硕士论文

【摘要】：自学习算法是一种具有自学习能力的算法,不仅具有更高的适应性与精确性,而且还提供了解决一些疑难问题的新方法。该论文对实际应用中所涉及的线性方程组的求解、数值积分的计算和滤波器的优化设计等问题进行了深入的研究,具有较强的理论和实际意义。论文主要研究内容如下:(1)对基于自学习算法的线性方程组求解进行研究。采用梯度下降法、共轭梯度法和递推最小二乘法这三种算法分别对神经网络权值进行训练,得到的权值向量就是所求方程组的解。(2)对基于曲线拟合的数值积分进行研究。同样,论文采用梯度下降法、共轭梯度法和递推最小二乘法这三种多项式曲线拟合方法对定积分进行计算。用这三种算法取代传统算法从而得到多项式模型的待定系数。最后采用著名的牛顿-莱布尼兹公式获得以多项式为被积函数的原函数,从而达到求解数值积分的目的。(3)研究了三种FIR线性相位数字滤波器的优化设计算法。这三种算法都是使待设计的FIR线性相位数字滤波器的幅频特性尽可能地逼近理想滤波器的幅频特性,将幅度函数表示成余弦基函数的线性组合,因此,滤波优化问题就转化为求解余弦基函数的线性组合的系数问题,然后分别用梯度下降法和递推最小二乘法训练余弦基函数的神经网络系数,共轭梯度法计算余弦基函数的加权系数,从而获得FIR滤波器的单位脉冲响应。仿真结果表明,论文利用梯度下降法、共轭梯度法和递推最小二乘法等三种算法分别求解线性方程组、多项式曲线拟合以及FIR数字滤波器优化设计,取得了良好结果。特别是使用递推最小二乘法解决了病态方程组的难题,以及曲线拟合的噪声滤波问题。在随机噪声滤波与病态方程组求解领域具有重要的应用价值。
[Abstract]:Self-learning algorithm is an algorithm with self-learning ability, which not only has higher adaptability and accuracy, but also provides a new method to solve some difficult problems. In this paper, the solution of linear equations, the calculation of numerical integrals and the optimal design of filters are deeply studied, which is of great theoretical and practical significance. The main contents of this paper are as follows: (1) the solution of linear equations based on self-learning algorithm is studied. The weights of neural networks are trained by gradient descent method, conjugate gradient method and recursive least square method, respectively. The obtained weight vector is the solution of the equations. 2) the numerical integration based on curve fitting is studied. Similarly, three polynomial curve fitting methods, i.e. gradient descent method, conjugate gradient method and recursive least square method, are used to calculate the definite integral. The undetermined coefficients of the polynomial model are obtained by replacing the traditional algorithms with these three algorithms. Finally, by using the famous Newton-Leibniz formula, we obtain the original function with polynomial as the integral function, thus achieving the purpose of solving the numerical integral.) three optimal design algorithms for FIR linear phase digital filters are studied. These three algorithms make the amplitude-frequency characteristic of the FIR linear phase digital filter to be designed as close as possible to the ideal filter's amplitude-frequency characteristic. The amplitude function is expressed as a linear combination of the cosine basis function. The problem of filtering optimization is transformed into the problem of solving the linear combination coefficients of cosine basis functions, and then the neural network coefficients of cosine basis functions are trained by gradient descent method and recursive least square method, respectively. The conjugate gradient method is used to calculate the weighted coefficients of the cosine basis function, and the unit impulse response of the FIR filter is obtained. The simulation results show that three algorithms, I. e. Gradient descent method, conjugate gradient method and recursive least square method, are used to solve linear equations, polynomial curve fitting and FIR digital filter optimization, respectively, and good results are obtained. In particular, the recursive least square method is used to solve the problem of ill-conditioned equations and the noise filtering problem of curve fitting. It has important application value in the field of stochastic noise filtering and ill-conditioned equations.
【学位授予单位】：长沙理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TN713.7;TP183

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