【作者】 周杰；

【导师】 朱允民；

【作者基本信息】 四川大学 ， 概率论与数理统计， 2003， 博士

【摘要】 现代信息处理、信息融合中存在着大量的优化问题，如通讯、控制、信号处理、多源信息融合、数据挖掘、人工智能、生物遗传信息处理、金融分析、智能决策支持系统等等。其显著特点是信息量大，要求处理的准确率高、速度快、可靠性强，因而研究求解这些优化问题的高效、精确且稳定的计算方法至关重要。对于存储量和计算量相对固定、计算复杂度低、稳定性高的递推算法，由于计算装置实现上的方便而在信息处理、融合的许多领域中广泛运用。所以，研究最优解的递推计算问题在理论和应用两方面都具有非常重要的意义。 论文研究了信息处理与融合中具有广泛而重要应用的某些递推算法，获得如下几方面研究成果： 1．给出一般条件下几种常用最小二乘估计精确初始化的递推公式。 众所周知，递推最小二乘算法在信息处理中有着广泛的应用。但是，在算法初始时，由于数据矩阵不满秩，多年来一直未能实现一般条件下的精确递推，只能人为给定一个满秩初始数据矩阵启动递推。虽然在渐近意义下，这样做影响不大，但由于人为初始数据矩阵实际上是添加的虚假数据，在要求初期处理效果的许多实际信息处理问题中却是不能容忍的。 在一般情形下，科学技术领域中许多问题的最优解中包含了矩阵的广义逆，其计算量随着矩阵阶数的增加而增大。为了给出这类最优解的递推算法，其关键是矩阵广义逆的递推计算问题。 论文首先改进了广义逆计算中经典的Greville阶数递推公式，每次递推所需存储量几乎减少一半，更为重要的是，新公式便于推导包含广义逆的最优解的递推计算公式。我们还进一步发展，分别得到正交投影变换下、遗忘因子加权情形下矩阵广义逆的阶数递推公式，它们具有统一的形式。摘要111 上述几个新的阶数递推公式的一个重要应用是直接导出了无约束、线性等式约束以及遗忘因子加权LS估计的精确初始化的递推公式.我们的整个推导过程比Alber七和Sittler在〔l]中的方法更加简明，且最终的递推公式也更加简洁.以往文献中在观测数据阵满列秩条件下相应LS估计的递推公式.均成为我们公式的特例.特别地，无约束和线性等式约束下的递推公式完全相同，只是初值不同，因而给递推算法的实际应用带来了方便.此外，我们还对精确初始化的递推最小二乘算法进行数值分析，研究了截断误差、累积误差对Ls估计的影响.该部分内容参见文【65，66}. 2.给出最好线性无偏佑计一个新的解析表达式二 Gauss一Markov模型是具有广泛应用的一种重要的线性模型.更一般地，设计矩阵不要求满列秩、误差方差阵也可以是奇异的，这种模型也称为广义Gauss一Markov模型.由于它更加符合实际情形，因而成为现代信息处理、信息融合中倍受关注的模型.该模型的最好线性无偏估计(BLUE)是观测数据的线性组合，著名统计学家Rao等在最近30多年里对BLUE的解析表达式进行了深人研究.但是，由于这些表达式包含了高阶矩阵的广义逆，给实际应用带来了困难. 我们注意到一个重要的事实:虽然广义Gauss一Markov模型的最好线性无偏估计(BLUE)几乎处处唯一，但是，将观测数据组合成为BLUE的方式却是无穷多的，这些组合系数在一个子空间中.由此，我们提出求解BLUE的一个新思路:根据实际应用问题的需要选择便于存储、计算量小且稳健性高的“适当”的组合系数. 为了解决一大类随机动态系统最优估计的快速计算问题，我们选择一种适当的组合系数，给出了计算BLUE的一个新解析表达式，其计算量比直接运用Albert所给公式的计算量小.特别地，我们给出该表达式的一种递推求解算法. 3.给出一般条件下分布式多传感器最优佑计融合的计算简单的算法. 1981年，国际信息融合界极负盛名的专家Bar一Shalom在假设传感器噪声不相关且估计误差方差阵非奇异的条件下，提出了计算简单的两传感器估计融合公式，这是该领域众人皆知的经典结果.但是，多年来一直未获得更一般条件下同样计算简单的多传感器估计融合公式.四川大学博士学位论文 我们用广义Gauss一Markov模型来描述多传感器最优估计融合问题，基于最好线性无偏估计这一优化准则给出新的融合公式，突破了Bar-Shalom公式只能处理两个传感器、传感器之间噪声不相关且估计误差方差阵可逆的限制，并且推导出估计融合的递推算法.该算法的一个突出优点是将结构复杂的矩阵的广义逆计算转化为若干个结构非常简单的矩阵的广义逆计算，每一步递推只涉及与单个传感器估计误差协方差阵阶数一致的矩阵的广义逆或逆.因此，当传感器数量较多时，我们的算法大大节约了计算量.我们还以三个和四个传感器系统为例，给出最优估计融合的详细计算过程，并且用模拟计算对新算法和直接计算所需时间进行比较，说明了我们所提出算法的优点.该部分内容参见文!68}. 4.研究了协方差矩阵广义逆的平方根分解问题. 对于随机系统的最优化问题，许多最优化准则下最优解表达式中包括协方差矩阵的逆或广义逆，其平方根分解在矩阵计算中具有较好的数值稳定性，因而有许多文献对更多还原

【Abstract】 There are many optimization problems in modern information processing and fusion, for example, communication, control, signal processing, multi-source information fusion, data mining, artificial intelligence, biological genetic information processing, financial analysis, and decision support systems. The notable characters of these problems are that the amount of information is large, and accuracy, high speed and reliability are required in the processing of information. Therefore, it is necessary to study the algorithms in various optimization problems. Especially, recursive algorithms are used in various engineering and technical problems since they can be implemented easily in the actual computational equipments. It is significant to investigate the recursive computational methods of the optimal solutions of various problems in information processing and fusion.In this thesis, we studied some recursive algorithms with wide and important applications to information processing and fusion.1. We proposed the exactly initialized recursive formulas of the least squares problems.It is well-known that the recursive least squares (LS) have very wide applications in various practical fields. In the past, the recursive LS algorithm must start at approximate initialization since the initial observed matrix does not have full column rank. Although the approximate initialization makes a small impact on the whole recursive process in asymptotic sense, it can not be neglected in some practical applications.Moore-Penrose generalized inverses of matrices are often involved in the optimal solutions of various scientific and engineering problems. Their computationinvolves an increasing number of variables with a corresponding increase in the matrix order. To find a recursive version of such an optimal solution, a key technique is an order-recursive version of the generalized inverse of a matrix.We improved the classical order-recursive algorithm proposed by Greville, and obtained new recursive formulas for three different matrix structures. Not only do the proposed formulas reduce the required memory locations of the Greville formula at each recursion by almost half, but they are also very useful to derive the recursive formulas for the optimal solutions involving the generalized inverses of matrices.By using the new formulas, we studied the recursive computation for the LS estimation when the observed data are linearly correlated, and derived recursive LS procedures which coincide exactly with the batch LS solutions for the unconstrained LS, forgetting factor weighted LS, and LS with linear equality constraints, respectively, including their simple and exact initializations. New findings include that the linear equality constrained LS and the unconstrained LS can have an identical recursion ?their only difference is the initial conditions. In addition, some robustness issues, during the exact initialization of the recursive LS, are studied. See [65, 66].2. We proposed a new formula of the best linear unbiased estimation (BLUE) of the generalized Gauss-Markov model.The generalized Gauss-Markov model is an important linear statistical model, in which the designed matrix can have no full column rank and the error covariance matrix can be singular. The best linear unbiased estimation (BLUE) of this model can be regarded as the linear combination of the observed data. The generalized inverses of high dimensional matrices are involved in the BLUE, therefore, it is necessary to study the computational methods for the BLUE.Although the BLUE is unique with probability 1, the coefficients of combination have infinite choices within a subspace. Based on this viewpoint, we selected an appropriate coefficient for a kind of models in dynamic stochastic systems, and proposed a new formula for the BLUE which can be computed recursively.3. We proposed a simple algorithm for the distributed multi-sensor optimal estimation fusion under the general conditions.Under the assumption of independence cross sensor更多还原