-
摘要: 图实现(Graph realization)问题研究基于节点间全部或部分距离关系测量, 在$d$维空间中计算图的顶点坐标, 使得在所实现图中各节点之间实现距离与测量距离尽可能一致.图实现问题是一个典型的优化问题, 在传感器网络定位、蛋白质结构重建、数据可视化、社交网络分析、机器人同步定位与构图等领域有着广泛应用.图实现的研究同图刚性理论有着紧密的联系, 图的刚性与全局刚性决定图的可实现性.在可实现图中, 现有工作提出几类典型的代表性图实现算法, 包括: 1)基于三边测距类方法; 2)求解距离方程类方法; 3)基于全局优化类方法; 4)基于模块拼合类方法.本文对图实现的刚性理论, 四类图实现算法的设计思想、适用条件、算法流程等进行综述分析, 通过实验对算法进行准确性、计算复杂度、可靠性等方面的比较和分析.Abstract: The graph realization problem is to calculate vertex coordinates in d-dimensional space based on all or parts of distance measurements between the vertexes, so that the Euclidean distance between any two neighboring nodes in the realized graph is consistent with the measured distance. The graph realization problem is widely used in the fields of sensor network location, protein structure reconstruction, social network analysis and robot simultaneous localization and mapping. Graph realization is closely related to rigidity theory. The rigidity and global rigidity properties determine the realizability of the graph. A series of algorithms were presented to address the graph realization problem, including: 1) trilateration-based algorithm, 2) distance equation based, 3) global optimization based, and 4) module embedding based methods. This paper introduces the representative algorithms in details with their design ideas, applicable conditions. In particular, in sensor network localization applications, we conduct a series of experiments to compare and analyze these algorithms in terms of accuracy, computational complexity and robustness.
-
Key words:
- Graph realization /
- network localization /
- optimization /
- component stitching /
- survey /
- rigidity /
- global rigidity
-
高炉炼铁是钢铁工业的重要生产环节.由于工艺相对简单、产量大、劳动生产率高, 因而高炉炼铁仍是现代炼铁的最主要方式, 其产量约占世界生铁总产量的95%以上.如图 1所示, 现代化的高炉炼铁系统分为高炉本体、上料系统、送风系统、喷吹系统、煤气净化系统、渣铁处理系统和环保除尘系统等几个子系统[1-2].高炉炼铁时, 铁矿石、焦炭、溶剂按一定比例根据布料制度逐层从高炉顶部装载到炉喉位置.同时, 在高炉下部, 将预热的空气、氧气和煤粉通过热风口鼓入炉缸中.空气、氧气、煤粉和焦炭在高温作用下发生一系列复杂物理化学反应, 生成大量的高温还原性气体, 这些还原性气体不断向上运动将铁从铁矿石中还原出来.上行气体最终变为高炉煤气从炉顶回收, 而下行炉料则随着炉缸中焦炭的不断燃烧和铁水的不断滴落逐渐向下运动, 在下降过程中, 炉料经过加热、还原、熔化等一系列复杂的物理化学变化, 最终生成液态的铁水和炉渣从出铁口排出[1-4].高炉炼铁需要准确判断整个高炉运行态势, 及时调整相关操作制度(如布料制度、热风制度等)及工艺参数, 使炉内煤气分布合理、热量充分利用、渣铁顺利排放, 实现高炉生产的优质、高产、低耗和长寿运行.为了实现这一目标, 就应对高炉运行状态进行实时监测和有效控制.然而高炉内部冶炼环境极其严酷, 炉内温度高达2000多度, 压强高达标准大气压的4倍左右, 且伴随着固、液、气多相共存的状态, 使高炉内部状态的实时监测难以实现, 从而难以对高炉进行控制与操作优化[1, 5].目前, 被广泛用来间接反映高炉运行状态的指标为出铁口的铁水质量指标, 包括铁水温度(Molten iron temperature, MIT)、硅含量([Si])、磷含量([P])和硫含量([S]):
1) MIT是表征高炉热状态、能量消耗的重要指标. MIT过低不仅影响高炉顺行, 而且会影响后续转炉炼钢的运行性能, 增加生产成本.高炉生产操作时应及时掌握MIT及变化趋势, 预见性地采取调控措施, 这对于稳定高炉热制度、减少炉况波动以及提高生铁质量和降低焦比等具有重要意义.
2) 铁水[Si]是反映铁水化学热的重要指标.铁水[Si]过高, 渣量增加, 会使生铁变硬变脆, 收得率降低.另外, 高[Si]会使渣中SiO含量过高, 影响石灰渣化速度, 延长吹炼时间, 同时也会加剧对炉衬的冲蚀.因此, 必须对[Si]进行严格监视.
3) 硫和磷在钢材中均是有害元素, [S]过高会使铁硬脆, 产生热脆性和减低铁液流动性, 而[P]过高会加重炉衬的冲刷蚀损, 影响炉龄.因此, 在高炉冶炼过程中应严格监视铁水中的[S]和[P].
采用上述铁水质量指标作为高炉运行状态的评判指标可以较全面地了解高炉内部的运行状态, 为高炉日常操作与调节提供指导.然而, 现有检测技术难以对铁水质量指标进行直接在线测量, 且离线化验过程滞后时间较长, 通常为1小时左右.另外, 高炉炼铁是一个包含气、固、液三相混合和耦合交错, 物理化学反应极其复杂的过程, 其炉内环境极其恶劣, 导致操作人员难以对其运行状态的变化进行实时监测.因此有必要建立准确可靠的铁水质量模型来反映高炉当前和预期的铁水质量指标参数变化, 为现场操作人员提供炉况和铁水质量信息[1, 5-8].
目前常见的铁水质量模型有机理模型、知识推理模型和数据驱动模型三大类[5].机理模型在大量假设条件下, 依赖于高炉炼铁过程的冶炼机理, 从化学反应动力学和流体动力学的角度构建高炉内部反应场, 对反应场内的温度分布、物料运动状态等进行仿真[9-11].高炉炼铁是一个极为复杂、大时滞、高度耦合的非线性动态时变系统, 基于大量假设的机理模型难以用于实际高炉炼铁生产; 知识推理模型完全依赖于系统集成的有限专家知识, 无法应对复杂多样的实际高炉冶炼过程[12-13];数据驱动铁水质量智能建模不需要了解高炉内部发生的复杂机理, 仅通过数学工具和智能算法对运行数据进行处理而建立需要的铁水质量模型, 因而成为近年高炉炼铁过程建模研究的热点[1-5, 8].
现有数据驱动建模方法主要有多元统计分析方法[14-15]、神经网络(Neural network, NN)建模方法[8, 15-18]和支持向量回归(Support vector regression, SVR)建模方法[2, 3, 6-7, 19]等.其中, 神经网络是一种最常见的数据驱动建模方法, 它是对人脑神经系统进行抽象、简化和模仿所得到的复杂信息处理系统, 能够以任意精度逼近复杂非线性连续函数, 特别适用于处理高炉炼铁这种动态机理复杂、内部状态未知, 仅可获得过程的输入输出数据的复杂工业过程建模[16].常见的神经网络建模算法主要有BP-NN (BP neural network)和RBF-NN (Radial basis function neural network), 但是这些算法普遍存在"过拟合"、易陷入局部极小的问题.另外, 基于批学习的NN网络权值和偏差迭代算法容易造成网络训练时间长、收敛速度慢的问题[16-17, 20-21].
近年, 具有学习速度极快且泛化能力好的随机权神经网络(Random vector functional-link networks, RVFLNs)及其改进算法可有效克服传统神经网络的上述缺点, 被广泛用于高炉炼铁过程建模与控制[5, 20-21]. RVFLNs的最大特点是网络输入权值与偏置在给定范围随机选取, 然后采用最小二乘算法对输出权值进行计算, 因而学习速度极快, 而不像基于梯度的常规NN算法需要首先设置很多学习参数, 然后采用批学习策略并花费几分钟甚至几小时时间对众多网络参数进行学习[5, 21].文献[20]采用经典RVFLNs算法对铁水质量进行建模, 大大加快计算速度; 文献[21]进一步采用在线序贯学习型RVFLNs实现多元铁水质量的在线软测量建模.实际表明, RVFLNs比BP等常规NN具有更高的计算效率, 且RVFLNs及其改进算法都具有较高的模型精度.但是, 高炉炼铁生产过程中, 受检测仪表和变送器等装置的故障以及其他异常干扰的影响, 测量数据中经常存在离群点, 即远离数据一般水平的极端大值点或极端小值点.而常规RVFLNs的输出权值是由最小二乘估计得到, 因而鲁棒性不足, 建模时易受离群点干扰, 导致模型准确度下降, 不能为操作人员提供准确的指导.
针对上述问题, 也为了建立稳定、可靠的高炉铁水质量模型, 本文提出一种新型的数据驱动随机权神经网络鲁棒建模方法, 用于实现铁水质量指标的鲁棒估计.首先, 为了提高铁水质量建模效率, 采用数据驱动典型相关性分析方法提取与铁水质量指标相关性最强的最主要过程变量作为建模输入变量; 然后, 根据高炉炼铁过程数据特性, 采用基于Gaussian分布加权的M估计方法建立多元铁水质量指标的鲁棒RVFLNs模型; 同时, 在M估计鲁棒建模基础上, 进一步引入两个正则化项以防止模型过拟合, 稀疏鲁棒RVFLNs的输出权值矩阵.最后, 基于我国华南某大型高炉实际数据建立基于鲁棒正则化RVFLNs的多元铁水质量非线性自回归(Nonlinear autoregressive exogenous, NARX)模型, 并和其他几类铁水质量建模算法进行对比.结果表明, 相对于对比方法, 本文方法不仅具有更高的模型精度和更快的计算速度, 还解决了常规RVFLNs存在的数据建模鲁棒性差的问题.
1. 基于数据驱动典型相关分析的建模输入变量选择
高炉本体参数较多, 且变量之间存在较强相关性, 如果所有变量全部参与铁水质量建模, 会增加计算复杂度, 影响模型预测的准确性和有效性.现有数据建模研究中, 辅助变量的选取方法主要为主成分分析(Principle component analysis, PCA)方法[20-21].但是PCA技术只针对输入空间进行降维, 未考虑输入输出变量之间的关联性[1].为此, 本文将采用考虑输入输出空间关联关系的典型相关分析(Canonical correlation analysis, CCA)方法选择与铁水质量参数相关性最大的几个变量作为输入变量. CCA是研究多个输入变量与多个输出变量之间相关性的多元统计分析方法, 其基本原理是在两组输入输出关联变量中分别提取有代表性的综合指标, 也就是分别建立两个变量组中各变量的线性组合, 利用这两个综合指标之间的相关关系来反映两组变量之间的整体相关性[1, 22].基于CCA的数据降维过程简要描述如下:
基于CCA的数据降维需要在两组变量$ X = ({\pmb x}_1^{}, {\pmb x}_2^{}, \cdots, {\pmb x}_p^{}), {Y^{}} = ({\pmb y}_1^{}, {\pmb y}_2^{}, \cdots, {\pmb y}_q^{}) $中分别找出变量的一个线性组合, 即:
$$ \begin{equation} \left\{ \begin{array}{l} {U_i} = \alpha _1^{(i)}{\pmb x}_1^{} + \alpha _2^{(i)}{\pmb x}_2^{} + \cdots \alpha _p^{(i)}{\pmb x}_p^{} = X{\alpha ^{(i)}}\\ {V_i} = \beta _1^{(i)}{\pmb y}_1^{} + \beta _2^{(i)}{\pmb y}_2^{} + \cdots \beta _q^{(i)}{\pmb y}_q^{} = Y{\beta ^{(i)}} \end{array} \right. \end{equation} $$ (1) 式中, $ {\pmb\alpha} ^{(i)} = (\alpha _1^{(i)}, \alpha _2^{(i)}, \cdots, \alpha _p^{(i)})^{\rm T} $, $ {\pmb\beta} ^{(i)} = (\beta _1^{(i)}, \beta _2^{(i)}, \cdots, \beta _q^{(i)})^{\rm T} $分别为任意非零常系数向量, $ ({U_i}, {V_i}) $为典型变量组.
首先, 寻找通过计算$ {\pmb{\alpha} ^{(1)}}, {\pmb{\beta} ^{(1)}} $, 分别在每组变量中找到第一对典型变量$ ({U_1}, {V_1}) $, 使其具有最大相关性.然后, 在每组变量中找到第二对典型变量$ ({U_2}, {V_2}) $, 使其分别与本组内的第一对典型变量$ ({U_1}, {V_1}) $不相关, 并具有次大的相关性.如此继续下去, 直到两组变量的相关性被提取完为止.另外, 式(1)中典型变量$ {U_i}, {V_i} $之间的相关系数称为典型相关系数.可以看出, 典型相关分析研究的问题实际上就是如何选取典型变量的最优线性组合.
在进行变量选择时, 首先要对各对典型变量的典型相关系数进行显著性检验, 若某一对典型变量$ ({U_i}, {V_i}) $的相关程度不显著, 说明这对变量不具有代表性, 舍弃这一对典型变量$ ({U_i}, {V_i}) $.若相关程度显著, 则保留变量作为最后建模的输入变量.另外, 在高炉变量中, 各变量均具有一定相关性, 若两个变量相关性达到80选择其中一个变量作为建模输入变量即可.
2. 基于Gaussian分布加权M估计的鲁棒正则化RVFLNs建模算法
2.1 常规RVFLNs
随机权神经网络(RVFLNs)由Pao等于1992年提出[23-27].不同于传统基于梯度学习的NN需要通过误差的反向传播迭代寻优, 采用随机给定神经元输入权值和偏置, 通过正则化原则计算输出权值建立网络.实际上, 基于单隐层前馈神经网络的RVFLNs以全局逼近理论为基础, 保证隐含层节点参数的随机选取不会影响NN的学习能力, 同时网络采用Moore-Penrose广义逆矩阵方法一步求得输出权重, 使得算法相对于BP-NN、SVR等, 其计算速度提高了数千倍, 为实际工业过程实现在线软测量和在线控制奠定基础[23-27].
给定$ M $组任意样本集合$ ({{\pmb x}_i}, {{\pmb y}_i}) $, $ {{\pmb{y}}_i} = [{y_{i1}}, {y_{i2}}, \cdots, {y_{im}}]^{\rm{T}} \in {{\bf{R}}^m} $, $ {{\pmb{x}}_i} = [{x_{i1}}, {x_{i2}}, \cdots, {x_{in}}]^{\rm T} \in {{\bf{R}}^n} $, 一个带有$ L $个隐层节点, 以$ g(x) $为激活函数的标准单隐层前馈神经网络可以表示为:
$$ \begin{align} {f_{L, i}}({{\pmb{x}}_i}) = \sum\limits_{j = 1}^L {{{\pmb{\beta}} _j}} g( \langle {{\pmb{a}}_j}, {{\pmb{x}}_i} \rangle +& {b_j}), \\ &i = 1, 2, \cdots, M \end{align} $$ (2) 式中, $ {f_{L, i}}({{\pmb{x}}_i}) $是RVFLNs的输出, $ {{\pmb{a}}_j} \in {{\bf{R}}^n}, j = 1, \cdots, L $是连接输入层与隐层的权值矩阵, 且权值在给定确定范围内随机产生, $ {\pmb{\beta} _j} = {[{\beta _{j1}}, {\beta _{j2}}, \cdots, {\beta _{jm}}]^{\rm{T}}} $是连接第$ j $个隐含层节点与输出层神经元的输出权重向量, $ {b_j} $是第$ j $个隐含层节点的偏置, $ \langle {{\pmb{a}}_j}, {{\pmb{x}}_i} \rangle $表示$ {{\pmb{a}}_j} $与$ {{\pmb{x}}_i} $的内积.
当$ {f_L} $以0误差逼近连续系统的$ M $个样本时, 即$ \sum\nolimits_{i = 0}^M {\left\| {{f_L} - {y_i}} \right\|} = 0 $, 则式(2)可以转化为寻找$ {{\pmb{\beta }}_j}, {{\pmb{a}}_j} $和$ {b_j} $, 使得:
$$ \begin{align} \sum\limits_{j = 1}^L {{\pmb{\beta} _j}} g( \langle {{\pmb{a}}_j}, {{\pmb{x}}_i} \rangle +& {b_j}) = {{{y}}_i}, \\ & i = 1, 2, \cdots, M \end{align} $$ (3) 用矩阵表示为:
$$ \begin{equation} {{H}}\beta = {{Y}} \end{equation} $$ (4) 式中:
$$ \begin{array}{l} {{H}} = \left[ {\begin{array}{cc} {g( \langle {{\pmb{a}}_1}, {{\pmb{x}}_1} \rangle + {b_1})}&\cdots\\ \vdots &\cdots\\ {g( \langle {{\pmb{a}}_1}, {{\pmb{x}}_M} \rangle + {b_1})}&\cdots \end{array}} \right.\\ {\left. {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} &{g( \langle {{\pmb{a}}_L}, {{\pmb{x}}_1} \rangle + {b_L})} \end{array}}\\ {\begin{array}{*{20}{c}} & \vdots \end{array}}\\ {\begin{array}{*{20}{c}} &{g( \langle {{\pmb{a}}_L}, {{\pmb{x}}_M} \rangle + {b_L})} \end{array}} \end{array}} \right]_{M \times L}}, \\ {{Y}} = {\left[ {\begin{array}{*{20}{c}} {{{\pmb{y}}_1}}\\ \vdots \\ {{{\pmb{y}}_M}} \end{array}} \right]_{M \times m}}, \ \beta = {\left[ {\begin{array}{*{20}{c}} {{\pmb{\beta} _1}^{\rm{T}}}\\ \vdots \\ {{\pmb{\beta} _L}^{\rm{T}}} \end{array}} \right]_{L \times m}} \end{array} $$ 通常, 网络隐层节点数会比训练样本少很多, 从而$ {{H}} $不是方阵, 显然式(4)所示方程组没有解.为了选取最合适的$ {\beta } $尽量使式(4)成立, 那么就需要采用最小二乘方法求解上述方程组, 如式(5)所示:
$$ \begin{equation} \beta = {{{H}}^\dagger }{{Y}} \end{equation} $$ (5) 式中, $ {{{H}}^\dagger } $是根据Moore-Penrose方法计算得到的$ {{H}} $伪逆矩阵.
Igelnik等于1995年进一步提出RVFLNs中激活函数的约束条件以及随机权值和偏置的理论选取范围[24]. Igelnik等指出:对于满足万能逼近性的RVFLNs, 其激活函数$ g(x) $应当满足:
$$ \int\limits_R {{g^2}(x){\rm d}x < \infty }\ {\rm{ or }}\ \int\limits_R {{{[g'(x)]}^2}{\rm d}x < \infty } $$ 另外, 虽然Igelnik等给出了理论推导的概率空间作为权值$ {{\pmb{a}}_j} $和偏置$ {b_j} $的选取依据, 但是如何确定概率空间上的多个关键参数依然是个比较难的问题.在文献[28]中, Schmidt等依据实验结果确定了隐含层随机参数的选择范围并使用了三组数据进行验证, 其明确和简洁的$ [-1, 1] $区间确定还是为后续RVFLNs理论研究与实际应用提供非常有益的指导.
2.2 鲁棒正则化RVFLNs
上述常规RVFLNs采用最小二乘法计算网络输出权值, 当训练数据集中存在离群点时, 最小二乘法将"放大"离群点的影响, 使模型的整体特性发生很大改变.为了减小数据中离群点对RVFLNs数据建模的影响, 拟在RVFLNs网络中引入M估计技术, 提高模型的鲁棒性. Huber等于1964年提出的M估计技术是现在最常用的鲁棒估计方法[29-31].与使用误差平方和作为目标函数的最小二乘算法不同, M估计对残差定义了新的目标损失函数, 使其能够对数据中的离群点进行区别对待.提出的基于M估计的RVFLNs (M-RVFLNs)算法描述如下.
1) 单输出M-RVFLNs:当$ m = 1 $时, $ {\beta} $和$ {{Y}} $可以简化为: $ {\beta} = \begin{array}{*{20}{c}} {[{\pmb{\beta} _1}} & \cdots & {{\pmb{\beta} _L}} \end{array}]_{L \times 1}^{\rm T} $, $ {{Y}} = [\begin{array}{*{20}{c}} {{\pmb{y}_1}} & \cdots & {{\pmb{y}_L}} \end{array}]_{M \times 1}^{\rm T} $.此时, 式(4)的最小二乘解为:
$$ \begin{equation} \hat \beta = {{{H}}^{\bf{\dagger }}}{{Y}} = {({{{H}}^{{T}}}{{H}})^{ - 1}}{{{H}}^{{T}}}{{Y}} \end{equation} $$ (6) 对应的目标函数为:
$$ \begin{equation} Q = \sum\limits_{i = 1}^M {{r_i}^2} = \sum\limits_{i = 1}^M {{{\left( {{y_i} - {H_i}\beta } \right)}^2}} \end{equation} $$ (7) 引入M估计, 式(7)关于残差$ r $的目标函数变为:
$$ \begin{equation} Q = \sum\limits_{i = 1}^M {\rho ({r_i})} = \sum\limits_{i = 1}^M {\rho ({y_i} - H_i^{}\beta )} \end{equation} $$ (8) 式中, $ \rho $为M估计影响函数.
回归系数$ \beta $ (即RVFLNs网络输出权值)的求法如下:
$$ \begin{equation} \hat \beta = \arg \mathop {\min }\limits_\beta \sum\limits_{i = 1}^M {\rho ({y_i} - {H_i}\beta )} = \arg \mathop {\min }\limits_\beta \sum\limits_{i = 1}^M {\rho ({r_i}(\beta ))} \end{equation} $$ (9) 为保证估计结果具有尺度同变性, 引入残差的稳健尺度估计$ \hat \sigma $, 则求解回归系数$ \beta $形式变为:
$$ \begin{align} \hat \beta = & \arg \mathop {\min }\limits_\beta \sum\limits_{i = 1}^M {\rho \left( {\frac{{{y_i} - {H_i}\beta }}{{\hat \sigma }}} \right)} = \\ & \arg \mathop {\min }\limits_\beta \sum\limits_{i = 1}^M {\rho \left( {\frac{{{r_i}(\beta )}}{{\hat \sigma }}} \right)} \end{align} $$ (10) 式(10)中的残差稳健尺度估计$ \hat \sigma $对M估计很重要.根据鲁棒估计理论, $ \hat \sigma $取绝对离差中位数(Median absolute deviation, MAD)除以常数0.6745.数值0.6745是标准正态分布的MAD, 当残差服从高斯分布时, MAD除以0.6745可以保持估计的一致性.即:
$$ \begin{equation} \hat \sigma = \frac{{{\rm{MAD}}}}{{0.6745}} = \frac{{media{n_i}\left| {{r_i} - median({r_i})} \right|}}{{0.6745}} \end{equation} $$ (11) 令$ {{\partial \left({\sum\limits_{i = 1}^M {\rho \left({\frac{{{r_i}(\beta)}}{{\hat \sigma }}} \right)} } \right)} \mathord{\left/ {\vphantom {{\partial \left({\sum\limits_{i = 1}^M {\rho \left({\frac{{{r_i}(\beta)}}{{\hat \sigma }}} \right)} } \right)} {\partial \beta }}} \right. } {\partial \beta }} = 0 $, 可以得到:
$$ \begin{equation} \sum\limits_{i = 1}^M {\varphi \left( {\frac{{{r_i}(\beta )}}{{\hat \sigma }}} \right)} H_i^{\rm{T}} : = \sum\limits_{i = 1}^M {\rho '\left( {\frac{{{r_i}(\beta )}}{{\hat \sigma }}} \right)} H_i^{\rm{T}} = 0 \end{equation} $$ (12) 定义加权因子
$$ \begin{equation} w({r_i}) : = \frac{{\varphi ({r_i})}}{{{r_i}}} \end{equation} $$ (13) 那么等式(12)变为:
$$ \begin{equation} \sum\limits_{i = 1}^M {w\left( {\frac{{{r_i}(\beta )}}{{\hat \sigma }}} \right)} \times ({y_i} - {H_i}\beta )H_i^{\rm{T}} = 0 \end{equation} $$ (14) 进一步简化为:
$$ \begin{equation} {{{H}}^{\rm{T}}}{{WH}}\beta = {{{H}}^{\rm{T}}}{{WY}} \end{equation} $$ (15) 式中, $ W{\rm{ = diag\{ }}{w_i}{\rm{\} }} $, 每一个$ {w_i} $是等式(13)定义的权值因子, 约去等式两边$ {{H}^{\rm T}} $, 并进一步写为:
$$ \begin{equation} {{{H}}^{{'}}}\beta = {{{Y}}^{{'}}} \end{equation} $$ (16) 式中, $ {{{H}}^{'}} = {{WH}} $, $ {{{Y}}^{{'}}}{{ = WY}} $.
为了防止RVFLNs模型过拟合, 稀疏输出权值矩阵, 进一步引入$ {L_1} $和$ {L_2} $两个正则化项, 此时优化目标函数变为:
$$ \begin{align} \hat \beta = & \arg \mathop {\min }\limits_\beta \left\{ {\left\| {{{{H}}^{\bf{'}}}\beta - {{{Y}}^{\bf{'}}}} \right\|_2^2 + \lambda \left(\dfrac{{1 - {\lambda _0}}}{2}\right)||\beta ||_2^2 + } \right.\\ &\left. {{\lambda _0}||\beta |{|_1}} \right\} \end{align} $$ (17) 式中, $ {\lambda _0} \in [0, 1] $, $ \lambda \ge 0 $.
令$ {\lambda _1} = \lambda {\lambda _0} $, $ {\lambda _2} = {{\lambda (1 - {\lambda _0})} \mathord{\left/ {\vphantom {{\lambda (1 - {\lambda _0})} 2}} \right. } 2} $, 则式(17)变为:
$$ \begin{equation} \hat \beta = \arg \mathop {\min }\limits_\beta \left\{ {\left\| {{{{H}}^{\bf{'}}}\beta - {{{Y}}^{\bf{'}}}} \right\|_2^2 + {\lambda _2}||\beta ||_2^2 + {\lambda _1}||\beta |{|_1})} \right\} \end{equation} $$ (18) 进一步, 令$ (M + L) \times L $维矩阵$ {{{H}}^*}_{(M + L) \times L} = {(1 + {\lambda _2})^{ - \frac{1}{2}}}\left[{\begin{array}{*{20}{c}} {{{{H}}^{\bf{'}}}}\\ {\sqrt {{\lambda _2}} {{I}}} \end{array}} \right] $, $ {{Y}}_{(M + L)}^* = \left[{\begin{array}{*{20}{c}} {{{{Y}}^{\bf{'}}}}\\ {0} \end{array}} \right] $, 并使$ \gamma = \frac{{{\lambda _1}}}{{\sqrt {1 + {\lambda _2}} }} $, $ {\beta ^*} = \sqrt {1 + {\lambda _2}} \beta $.则上式变为:
$$ \begin{equation} \hat \beta = \frac{1}{{\sqrt {1 + {\lambda _2}} }}\arg \mathop {\min }\limits_{{\beta ^*}} \left\{ {\left\| {{{{H}}^*}{\beta ^*} - {{{Y}}^*}} \right\|_2^2 + \gamma ||{\beta ^*}|{|_1})} \right\} \end{equation} $$ (19) 式(19)乘上一个缩放因子$ 1 + {\lambda _2} $, 得到下式:
$$ \begin{equation} \hat \beta = \sqrt {1 + {\lambda _2}} \arg \mathop {\min }\limits_{{\beta ^*}} \left\{ {\left\| {{{{H}}^*}{\beta ^*} - {{{Y}}^*}} \right\|_2^2 + \gamma ||{\beta ^*}|{|_1})} \right\} \end{equation} $$ (20) 上述式(20)即可由坐标下降(Coordinate descent)算法求解[32].
评论1. 为了解决RVFLNs模型固有的多重共线性和过拟合问题, 在鲁棒加权后的最小二乘损失函数基础上后, 同时加入输出权值的$ {L_2} $范数项和输出权值的$ {L_1} $范数项作为惩罚, 构成式(17)所示优化目标函数的弹性网络(Elastic net)[33], 因而可有效稀疏化输出矩阵, 其效果比岭回归以及Lasso等任何单一范数作为正则化项的方法都要好[33].
2) 多输出M-RVFLNs:当$ m > 1 $时, M-RVFLNs为多输出, 此时$ \beta $和$ {{Y}} $表示如下:
$$ \begin{equation} \begin{array}{l} {\beta} = [{\pmb{\beta} _{j1}}, \cdots, {\pmb{\beta} _{jh}}, \cdots, {\pmb{\beta} _{jm}}] = \\ {\quad\left[ {\begin{array}{*{20}{c}} {{\beta _{11}}}& \cdots &{{\beta _{1m}}}\\ \vdots & \ddots & \vdots \\ {{\beta _{L1}}}& \cdots &{{\beta _{Lm}}} \end{array}} \right]_{L \times m}}, \begin{array}{*{20}{c}} {j = 1, \cdots, L}\\ {{\rm{ }}h = 1, \cdots, m} \end{array} \end{array} \end{equation} $$ (21) $$ \begin{equation} \begin{array}{l} {{Y}} = [{{\pmb{y}}_{i1}}, \cdots, {{\pmb{y}}_{ih}}, \cdots, {{\pmb{y}}_{im}}] = \\ {\quad\left[ {\begin{array}{*{20}{c}} {{y_{11}}}& \cdots &{{y_{1m}}}\\ \vdots & \ddots & \vdots \\ {{y_{M1}}}& \cdots &{{y_{M{\rm{m}}}}} \end{array}} \right]_{M \times m}}, \begin{array}{*{20}{c}} {i = 1, \cdots, M}\\ {h = 1, \cdots, m} \end{array} \end{array} \end{equation} $$ (22) 权值矩阵的维数$ {{w}}({{{r_i}(\beta)} \mathord{\left/ {\vphantom {{{r_i}(\beta)} {\hat \sigma }}} \right. } {\hat \sigma }}) $和$ {{Y}} $一致, 即为如下$ M \times m $维形式:
$$ \begin{equation} \begin{array}{l} {w} = [{\pmb{w}_{i1}}, \cdots, {\pmb{w}_{ih}}, \cdots, {\pmb{w}_{im}}] = \\ {\quad\left[ {\begin{array}{*{20}{c}} {{w_{11}}}& \cdots &{{w_{1m}}}\\ \vdots & \ddots & \vdots \\ {{w_{M1}}}& \cdots &{{w_{Mm}}} \end{array}} \right]_{M \times m}} \end{array} \end{equation} $$ (23) 此时, 多输出权值表示为:
$$ \begin{equation} {\hat \beta }_{}^{(k + 1)} = [\hat {\pmb{\beta}} _{j1}^{(k + 1)}, \cdots, \hat {\pmb{\beta}} _{jh}^{(k + 1)}, \cdots, \hat {\pmb{\beta}} _{jm}^{(k + 1)}] \end{equation} $$ (24) 这里的每一个$ \hat {\pmb{\beta}} _{jh}^{(k + 1)} $由等式(20)求得.
2.3 M估计Gaussian分布加权函数及参数选取
基于M估计的RVFLNs根据残差决定观测值对估计的贡献大小, 所以确定权重非常重要.通过研究随机变量的分布特性和统计特性, 我们采用统计学中应用较为广泛的高斯(Gaussian)分布进行M估计权值确定.设连续随机变量$ X $服从一个位置参数为$ {\mu} $、尺度(宽度)参数为$ {\sigma _1} $的概率分布, 且概率密度函数(Probability density function, PDF)为:
$$ f(x) = \frac{1}{{\sqrt {2\pi } {\sigma _1}}}\exp \left( { - \frac{{{{(x - \mu )}^2}}}{{2{\sigma _1}^2}}} \right) $$ 则$ X $服从高斯分布, 记作$ X\sim {\rm N}(\mu, {\sigma _1}^2) $.如果参数$ \mu, {\sigma _1} $已知, 就可以确定Gaussian分布的PDF曲线.
根据Gaussian分布的PDF特性, 在残差处于中间的样本点处取得最大权值, 在残差很小或者很大的样本处取得较小权值, 这样就能削弱离群点对建模带来的不良影响. Gaussian分布函数参数$ \mu $和$ {\sigma _1} $决定了曲线的特性, 因此如何确定这两个参数是该方法的重要内容:
1) $ \mu $值的确定: M估计加权的目的是调和"离群点"和"过拟合"的作用, 因此位置参数$ \mu $应取标准化残差$ {{{r_i}} \mathord{\left/ {\vphantom {{{r_i}} {\hat \sigma }}} \right. } {\hat \sigma }} $的中间值, 即$ \mu = {\rm{median}}({{{{r_i}} \mathord{\left/ {\vphantom {{{r_i}} {\hat \sigma }}} \right. } {\hat \sigma }}_i}) $.
2) $ {\sigma _1} $值的确定:参数$ {\sigma _1} $决定了Gaussian分布函数曲线的形状, 即样本点越靠近$ \mu $, 该点的权值也越大, 因此考虑误差的统计特性, 应遵循以下取值原则:如果误差分布较紧密, 则$ {\sigma _1} $取值应偏大; 如果误差分布较分散, 则$ {\sigma _1} $取值应偏小.标准差能反映出预测误差的离散程度, 误差分布越紧密时标准差越小, 误差分布越分散时标准差越大.为此, 采用标准化残差标准差的倒数来确定宽度参数$ {\sigma _1} $的取值, 即:
$$ \sigma _1 = \dfrac{1}{\sqrt{\dfrac{\sum\limits_{i = 1}^M\Big\| \frac{\frac{r_i}{\hat\sigma_i} -\sum\limits_{i = 1}^M \left(\frac{r_i}{\hat\sigma_i} \right) }{M} \Big\|^2 }{M}}} $$ 2.4 鲁棒建模算法实现步骤
所提M估计鲁棒RVFLNs算法的主要建模过程如下:
1) 根据式(6)计算$ {\hat \beta }_{}^{(0)} $;
2) 利用得到的$ {\hat \beta }_{}^{(0)} $, 计算残差向量$ {{\pmb{r}}^{(0)}} $并计算尺度估计值$ \hat \sigma $, 进而求出标准化残差向量$ {{\pmb{r}}^{(0)}}/\hat \sigma $;
3) 根据式(13)计算每个样本对应的权值, 求得权函数方阵$ {W^{(0)}} $;
4) 构造$ {{{H}}^{\bf{'}}} $和$ {{{Y}}^{\bf{'}}} $, 求解$ {{\beta}^{(1)}} $, 并重复迭代计算得到$ {{\beta}^{(2)}}, {{\beta}^{(3)}}, \cdots {{\beta}^{(k)}}, {{\beta}^{(k + 1)}} $, 直到每个参数估计值$ |\hat \beta _{j, h}^{(k + 1)} - \hat \beta _{j, h}^{(k)}| / \hat \beta _{j, h}^{(k)} $, $ j = 0, 1, \cdots, L, h = 1, \cdots, m $都小于给定收敛条件, 如$ {10^{ - 6}} $.
3. 工业数据测试
3.1 模型建立
采用我国华南某大型高炉的本体数据与铁水质量数据(采样间隔为1h)对所提方法进行数据测试, 根据该高炉炼铁工艺及相关仪器仪表设置, 确定影响Si含量、P含量、S含量、铁水温度等铁水质量指标的18个关键过程变量为:冷风流量、送风比、热风压力、压差、顶压风量比、透气性、阻力系数、热风温度、富氧流量、富氧率、设定喷煤量、鼓风湿度、理论燃烧温度、标准风速、实际风速、鼓风动能、炉腹煤气量、炉腹煤气指数等.考虑上述变量间具有很强的相关性, 并且过多的建模变量会加大建模复杂度, 影响模型性能, 因此首先采用典型相关分析(CCA)技术在原始高维的建模输入输出变量数据中提取出4对典型变量, 其相关系数分别为0.601、0.551、0.462、0.232.表 1表明在0.05显著性水平下, 前3对典型变量显著相关, 第4对典型变量相关性不显著, 因而在后续分析中不予考虑.表 2给出了各个高炉本体参数典型变量的标准化系数, 将每一个参数乘以对应的典型相关系数进行加权, 并求其绝对值之和, 可以得到每个参数变量的权值如表 2最后一列所示.最后, 确定选取综合权值较大的炉腹煤气量${u_1}$ (${{\rm{m}}^{\rm{3}}}$)、冷风流量${u_2}$ (${{\rm{m}}^{\rm{3}}}{\rm{/min}}$)、富氧流量${u_3}$ (${{\rm{m}}^{\rm{3}}}{\rm{/h}}$)、透气性${u_4}$ (${{\rm{m}}^{\rm{3}}}{\rm{/min}} \cdot {\rm{kPa}}$)、 富氧率${u_5}$ (${\rm{wt\% }}$)和理论燃烧温度${u_6}$ (℃)作为影响 [Si] ${y_1}$ (%)、[S] ${y_2}$ (%)、[P] ${y_3}$ (%)和铁水温度MIT ${y_4}$ (℃) 等铁水质量指标的最主要因素, 并将这6个变量组成新的样本集作为铁水质量建模输入变量.
表 1 典型相关系数的显著性检验Table 1 Significance test of canonical correlation coefficient典型变量 显著性检验指标 Wilk's Chi-SQ DF Sig. 1 0.337 299.701 72 0 2 0.527 176.485 51 0 3 0.754 77.786 32 0 4 0.958 11.678 15 0.703 表 2 高炉本体参数典型变量的标准化系数Table 2 Standardized canonical coefficients of BF body variables影响变量 典型变量 变量权值 1 2 3 4 冷风流量 14.821 2.609 $-3.502$ $-24.047$ 11.95769 送风比 $-0.669 $ 1.803 0.658 3.633 1.695912 热风压力 0.724 2.664 $-2.139$ -0.209 2.885878 压差 2.749 1.384 $ -0.527$ 2.382 2.655439 顶压富氧率 $-0.06$ $ -0.712$ 0.95 -0.022 0.865848 透气性 5.292 5.441 $-6.701$ 10.201 9.263463 阻力系数 0.801 0.268 $ -4.063$ 8.006 2.505639 热风温度 0.587 $-0.469$ $ -1.356$ $-0.268$ 1.23674 富氧流量 11.697 $ -4.429$ 0.07 1.748 9.493758 富氧率 -5.751 3.556 $ -4.229$ $-4.556$ 7.362393 设定喷煤量 $-0.931$ 3.284 4.027 $ -0.233$ 4.222921 鼓风湿度 0.533 0.805 1.932 $-0.465$ 1.654862 理论燃烧温度 $-3.408$ 2.774 5.055 1.685 5.906544 标准风速 $-2.222$ $ -0.705$ 2.385 6.454 2.824337 实际风速 $ -0.224 $ 0.023 0.55 $-0.767$ 0.401351 鼓风动能 $-1.85 $ $-1.067$ $-0.255$ $-0.002$ 1.815443 炉腹煤气量 $ -14.106 $ $-6.874$ 0.208 15.053 12.34763 炉腹煤气指数 0.292 0.651 $-0.006$ $ -0.198$ 0.535663 为了更好地反映高炉非线性动态特性和输入输出变量的时序和时滞关系, 将当前时刻的输入$X(t)$、上一时刻的输入$X(t - 1)$和上一时刻的模型输出作为输入, 建立铁水质量的非线性自回归(NARX)模型.同时, 为了消除低幅值高频的随机噪声, 取连续输入变量在一段时间的平均值作为实际模型输入变量.在隐含层权值${{\pmb{a}}_j}$和偏差${b_j}$随机范围选取方面, 首先在$[-1, 1]$范围内依据高斯分布随机产生${{\pmb{a}}_j}$和${b_j}$[27].在此基础上, 通过增设倍数$M$并将$M$与权值范围$[-1, 1]$相乘以扩大${{\pmb{a}}_j}$与${b_j}$的选择范围, 即选择范围扩大为.由图 2可以看出, 当${{\pmb{a}}_j}$和${b_j}$的取值范围介于$[-1, 1] $时, 铁水质量指标测试均方根误差能够取得最小值, 因而${{\pmb{a}}_j}$与${b_j}$的最佳选择范围可缩小至由于RVFLNs的输入权值每次都是在一定范围随机产生, 导致每次运算结果具有随机性, 这里取20次运算结果的均值作为最终模型输出结果.为确定铁水质量指标模型的最优正则化参数${ \mathsf{ λ} _0}, \mathsf{ λ} $, 同样取模20次平均测试均方根误差的最小值对应的${ \mathsf{ λ} _0}, \mathsf{ λ} $作为最后结果, 如图 3所示, 结果为${ \mathsf{ λ} _0}{\rm{ = [0}}{\rm{.0563, 0}}{\rm{.0435, 0}}{\rm{.0515, 0}}{\rm{.0495]}}$和$\lambda$ = [3.475E-04, 1.8E-04, 2.675E-04, 2.475E-04].
图 2 建模误差RMSE与输入权值倍数和输入偏置倍数之间的关系Fig. 2 The relationship between the modeling RMSE and the input weight multiple and the input bias multiple
图 3 建模误差RMSE与正则化系数之间的关系曲线Fig. 3 The relation curve between the modeling RMSE and the regularization coefficient3.2 建模及其估计效果
数据中所含离群点比例以及离群点偏离正常取值范围的大小(即离群点幅值)对模型的精度有重要影响.为了更全面地验证所提方法的鲁棒性, 基于实际高炉炼铁数据进行鲁棒测试的数据设计.共设计两组数据, 分别从不同离群点比例时数据建模效果以及不同离群幅值时数据建模效果两个角度来评价模型的建模与鲁棒性能.
第一组数据集用来测试所提方法对不同离群点比例时数据建模的适用性.首先, 从原始数据中随机选取比例为0%, 5%, 10%, $\cdots$, 50%的样本点$ {y_{i, {\rm{Outlier}}}}$并对其进行处理, 令, 式中${y_{\max \min }} = \max ({y_i}) - \min ({y_i})$, $i = 1, 2, 3, 4$表示正常炉况下铁水质量参数波动范围内的最大值与最小值之差.为了使离群点更加不均衡, 在所选样本点中正向离群点与负向离群点的比例设置为2:1, 添加正向离群点时令$sign = 1$, 添加负向离群点时令$sign = -1$.
第二组数据集用来测试所提方法对相同离群点比例但离群点偏离幅值不同时的数据建模适用性.从原始数据中随机选取20%的样本点, 依据添加离群点, 式中${y_{\max \min }} = \max ({y_i}) - \min ({y_i})$, $a = 0, 0.5, 1, 1.5, \cdots, 5$用来调节幅值大小.同样为了使离群点更加不均衡, 在所选样本点中正向离群点与负向离群点的比例设置为2:1, 添加正向离群点时令$sign = 1$, 添加负向离群点时令$sign = -1$.
为了验证所提基于Gaussian分布加权M-估计的鲁棒正则化RVFLNs (Gaussian-M-RVFLNs)方法对多元铁水质量指标(即[Si]、[P]、[S]和MIT)的建模效果, 将其与基于最小二乘加权的常规RVFLNs (LS-RVFLNs)和基于Huber加权M-估计的鲁棒随机权神经网络(Huber-M-RVFLNs)进行比较, 如图 4和图 5所示.其中图 4为不同离群点比例时三种方法的多元铁水质量估计均方根误差箱形图, 图 5为不同离群幅值时三种方法的多元铁水质量估计均方根误差箱形图.三种方法的激活函数均使用Sigmoid函数, 隐层节点数均设置为30, 权值${{\pmb{a}}_j}$和偏差${{{b}}_j}$取值范围均确定为$[-1, 1]$. Huber加权算法中K通常取为1.345, 这样取值的估计方法既是稳健的, 又有较高的估计效率.另外, 为了避免不同数量级数据间的相互影响, 对所有数据进行归一化处理.由于随机权神经网络的${{\pmb{a}}_j}$和${{{b}}_j}$为随机生成, 每次结果不具有唯一性, 为更好地衡量模型的性能, 对每一个数据集分别进行30次实验, 通过30次数据实验的均方根误差(Root mean square error, RMSE)来比较不同方法的鲁棒性能.
图 4 不同离群点比例时铁水质量估计RMSE箱形图Fig. 4 The box diagram of the Estimation RMSE of MIQ indices with different methods at each of the outlier contamination rates
图 5 不同离群幅值时铁水质量估计RMSE箱形图Fig. 5 The box diagram of the Estimation RMSE of MIQ indices with different methods when the amplitudes of the outliers are increased从图 4和图 5所示的箱型图可以看出, 当没有离群点或者离群点比例和离群幅值较小时, 所提Gaussian-M-RVFLNs方法与常规LS-RVFLNs的铁水质量估计效果基本相当, 但要好于Huber-M-RVFLNs方法.随着离群点比例以及离群幅值的逐渐增大, 基于传统LS-RVFLs的铁水质量模型估计精度下降明显, 而所提Gaussian-M-RVFLNs建立的铁水质量模型能够始终保持很高的估计精度, 并且远好于对比的其他两种方法. Huber-M-RVFLNs方法虽然在离群点比例较低和离群幅值较小时可保持较高精度, 具有一定的鲁棒性, 但是当离群点比例较低和离群幅值持续增加时, 其估计性能下降明显, 这在图 4所示高离群点比例情况时尤其明显.
一个好的鲁棒模型要求在实际模型与理想分布模型差别微小时, 受离群点的影响较小, 接近正确估值, 更重要的是要求实际模型与理想分布模型差别较大时, 估计值也不会受大量离群点的破坏性影响, 依然能够得到接近正常模式下的正确估计.为此考虑离群比例为20%、离群幅值步长$a$为2的多元铁水质量建模与估计效果, 实验结果分别如图 6和图 7所示.为了说明问题, 同样将所提Gaussian-M-RVFLNs方法与LS-RVFLNs和Huber-M-RVFLNs进行比较分析.可以看出在存在较高离群点比例和较大离群幅值情况下, 所提方法的建模与估计性能最好, 能够根据实时输入数据, 对难测多元铁水质量指标进行准确估计, 且估计趋势基本与实际值一致.图 8为不同方法下铁水质量指标测试误差概率密度分布曲线, 可以看出所提Gaussian-M-RVFLNs模型和其他两种方法建立的模型相比, 其误差概率密度分布曲线基本为又瘦又高的围绕0均值的高斯分布曲线, 即大部分估计误差仅为实际工业数据中所包含的测量噪声.
图 6 离群比例为20%和离群幅值步长$a$为2时, 不同RVFLNs建模方法铁水质量指标建模效果Fig. 6 Modeling results of MIQ indices with different methods when outlier contamination rate is twenty percent and amplitude is two
图 7 离群比例为20%和离群幅值步长$a$为2时, 不同RVFLNs建模方法铁水质量指标估计效果Fig. 7 Estimation results of MIQ indices with different methods when outlier contamination rates is twenty percent and amplitude is two
图 8 不同RVFLNs建模方法铁水质量估计误差PDF曲线Fig. 8 PDF curve of MIQ estimation error with different methods图 9为不同离群点比例时三种不同方法铁水质量建模时RVFLNs网络中为0的输出权值数量曲线图.可以看出, 所提Gaussian-M-RVFLNs方法建立的铁水质量模型在4个铁水质量指标输出权值矩阵中权值为0的个数最多, Huber-M-RVFLNs网络模型次之, LS-RVFLNs网络模型最少且输出权值一直不为0.这说明, 所提方法由于在鲁棒建模基础上引入${L_1}$和${L_2}$两个正则化项范数, 可稀疏化输出权值矩阵, 提高模型泛化性能, 有效避免模型过拟合.
图 9 不同RVFLNs建模方法铁水质量估计时网络输出权值为0的数量曲线Fig. 9 The curve of the number of output weights with '0' value of MIQ estimation error with different methods4. 结论
本文针对高炉炼铁过程多元铁水质量指标难以在线检测, 以及采用常规方法难以进行有效建模与在线估计的难题, 基于Gaussian分布加权M-估计技术, 提出一种新型的数据驱动随机权神经网络(RVFLNs)鲁棒建模方法, 用于实现多元铁水质量指标的鲁棒估计.基于实际工业数据的研究表明:和常规最小二乘加权RVFLNs算法以及经典Huber加权M估计的RVFLNs算法相比, 所提Gaussian分布加权M-估计的鲁棒正则化RVFLNs算法可有效克服数据中不同离群点比例和不同离群幅值给数据建模带来的不利影响, 大大提高RVFLNs数据建模的鲁棒性能, 使得模型在复杂动态环境中持续有效, 并且估计精度较高.同时, 在鲁棒建模基础上, 在优化性能指标引入两个正则化项, 以最大化稀疏化RVFLNs网络的输出权值矩阵, 这不仅可显著降低模型复杂度, 还可有效避免模型过拟合, 进一步提高模型的适应性.
-
表 1 算法分类表
Table 1 Algorithm classification
方法类型 算法 算法特点 三边测量类方法 Trilateration 基于几何性质, 增量式求解结果 距离方程类方法 MDS 求解完全图的实现方法 SDP 以半正定规划求解问题 全局优化类方法 $g^2o$ 基于最小二乘法的迭代搜索方法 SMACOF 基于构造辅助函数的迭代搜索方法 模块拼合类方法 AAAP 将部分模块变形, 以仿射变换拼合模块 ARAP 以迭代搜索的策略, 优化拼合后结果的形状偏差 ASAP 以矩阵特征向量求解模块拼合结果 
下载: 导出CSV
表 2 计算符号说明表
Table 2 Notation list
符号 定义 $n$ 节点数量, $|V|=n$ $m$ 图中边的数量, $|E|=m$ $t$ 迭代次数 $g $ 节点最大度数 $N $ 模块中节点数最大值
下载: 导出CSV
表 3 算法时间复杂度
Table 3 Time complexity of different algorithms
算法名称 时间复杂度 Trilateration ${\rm O}(n^2)$ MDS ${\rm O}(n^2)$ SDP ${\rm O}(n^2m^{2.5}+m^{3.5})$ $g^2o$ ${\rm O}(n^2t)$ SMACOF ${\rm O}(n^2t)$ AAAP ${\rm O}(mn^2)$ ARAP ${\rm O}(mn^2+n^3t)$ ASAP ${\rm O}(nN^2t+mn^2)$
下载: 导出CSV
表 4 算法比较表
Table 4 Algorithm comparison
方法类型 算法 准确性 计算效率 鲁棒性(稠密度) 鲁棒性(噪声) 三边测量类方法 Trilateration $\bigstar$ $\bigstar \bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar$ $\bigstar \bigstar$ 距离方程类方法 MDS $\bigstar \bigstar$ $\bigstar \bigstar \bigstar$ $\bigstar$ $\bigstar \bigstar$ SDP $\bigstar \bigstar$ $\bigstar \bigstar \bigstar$ $\bigstar \bigstar$ $\bigstar \bigstar \bigstar$ 全局优化类方法 $g^2o$ $\bigstar \bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar$ $\bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar \bigstar \bigstar$ SMACOF $\bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar$ $\bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar \bigstar \bigstar$ 模块拼合类方法 AAAP $\bigstar$ $\bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar$ $\bigstar$ ARAP $\bigstar \bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar \bigstar \bigstar \bigstar$ ASAP $\bigstar \bigstar \bigstar \bigstar \bigstar$ $\bigstar$ $\bigstar \bigstar \bigstar \bigstar \bigstar$ $\bigstar \bigstar \bigstar \bigstar$
下载: 导出CSV
-
[1] Bulusu N, Heidemann J, Estrin D. GPS-less low-cost outdoor localization for very small devices. IEEE Personal Communications, 2000, 7(5): 28-34 doi: 10.1109/98.878533 [2] Doherty L, Pister K S J, El Ghaoui L. Convex position estimation in wireless sensor networks. In: IEEE INFOCOM, Twentieth Annual Joint Conference of the IEEE Computer and Communications Societies. Anchorage, United States: IEEE, 2001. 1655-1663 https://www.researchgate.net/publication/3893900_Convex_position_estimation_in_wireless_sensor_networks [3] Hightower J, Borriello G. Location systems for ubiquitous computing. Computer, 2001, 34(8): 57-66 doi: 10.1109/2.940014 [4] Savvides A, Han C C, Strivastava M B. Dynamic fine-grained localization in Ad-Hoc networks of sensors. In: Proceedings of the 7th Annual International Conference on Mobile Computing and Networking. Rome, Italy: ACM, 2001. 166-179 https://www.researchgate.net/publication/220926176_Srivastava_Dynamic_Fine-grained_Localization_in_Ad-Hoc_Networks_of_Sensors [5] Savarese C, Rabaey J M, Langendoen K. Robust positioning algorithms for distributed Ad-Hoc wireless sensor networks. In: Proceedings of the General Track of the Annual Conference on USENIX Annual Technical Conference. Monterey, USA: USENIX Association, 2002. 317-327 https://www.researchgate.net/publication/2529202_Robust_Positioning_Algorithms_for_Distributed_Ad-Hoc_Wireless_Sensor_Networks [6] Hays P A. Proton nuclear magnetic resonance spectroscopy (NMR) methods for determining the purity of reference drug standards and illicit forensic drug seizures. Journal of Forensic Sciences, 2005, 50(6): 1342-1360 http://cn.bing.com/academic/profile?id=3517aaf1c734242ab4ed9eeb65736e5c&encoded=0&v=paper_preview&mkt=zh-cn [7] Kaufmann K W, Lemmon G H, DeLuca S L, Sheehan J H, Meiler J. Practically useful: what the Rosetta protein modeling suite can do for you. Biochemistry, 2010, 49(14): 2987-2998 doi: 10.1021/bi902153g [8] Sit A, Wu Z J, Yuan Y X. A geometric buildup algorithm for the solution of the distance geometry problem using least-squares approximation. Bulletin of Mathematical Biology, 2009, 71(8): 1914-1933 doi: 10.1007/s11538-009-9431-9 [9] Sit A, Wu Z J. Solving a generalized distance geometry problem for protein structure determination. Bulletin of Mathematical Biology, 2011, 73(12): 2809-2836 doi: 10.1007/s11538-011-9644-6 [10] 刘勘, 周晓峥, 周洞汝.数据可视化的研究与发展.计算机工程, 2002, 28(8): 1-2, 63 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=jsjgc200208001Liu Kan, Zhou Xiao-Zheng, Zhou Dong-Ru. Data visualization research and development. Computer Engineering, 2002, 28(8): 1-2, 63 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=jsjgc200208001 [11] Wilkinson L. SYSTAT: the System for Statistics. Evanston, IL: SYSTAT, Inc., 1986. [12] Kümmerle R, Grisetti G, Strasdat H, Konolige K, Burgard W. $G.2o$: a general framework for graph optimization. In: IEEE International Conference on Robotics and Automation. Shanghai, China: IEEE, 2011. 3607-3613 [13] Li Z, Trappe W, Zhang Y, Nath B. Robust statistical methods for securing wireless localization in sensor networks. In: Proceedings of the 4th International Symposium on Information Processing in Sensor Networks. Boise, USA: IEEE, 2005. 91-98 https://www.researchgate.net/publication/4149846_Robust_statistical_methods_for_securing_wireless_localization_in_sensor_networks [14] Saxe J B. Embeddability of Weighted Graphs in K-Space is Strongly NP-Hard. Qatar: Carnegie-Mellon University, 1980. [15] Yemini Y. Some theoretical aspects of position-location problems. In: Proceedings of the 20th Annual Symposium on Foundations of Computer Science. Washington D. C., USA: IEEE, 1979. 1-8 http://www.sciencedirect.com/science/article/pii/001457938680151X [16] Jackson B, Jordán T. Connected rigidity matroids and unique realizations of graphs. Journal of Combinatorial Theory, Series B, 2005, 94(1): 1-29 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=593b97de641132c7f040a84dc72841fa [17] Thomas F, Ros L. Revisiting trilateration for robot localization. IEEE Transactions on Robotics, 2005, 21(1): 93-101 doi: 10.1109/TRO.2004.833793 [18] Čapkun S, Buttyán L, Hubaux J P. SECTOR: secure tracking of node encounters in multi-hop wireless networks. In: Proceedings of the 1st ACM Workshop on Security of Ad Hoc and Sensor Networks. Fairfax, Virginia: ACM, 2003. 21-32 https://www.researchgate.net/publication/37402492_SECTOR_Secure_Tracking_of_Node_Encounters_in_Multi-hop_Wireless_Networks [19] Lazos L, Poovendran R. SeRLoc: secure range-independent localization for wireless sensor networks. In: Proceedings of the 3rd ACM Workshop on Wireless Security. Philadelphia, PA, USA: ACM, 2004. 21-30 https://www.researchgate.net/publication/221005650_SeRLoc_Secure_Range-Independent_Localization_for_Wireless_Sensor_Networks [20] Kusy B, Ledeczi A, Maroti M, Meertens L. Node-density independent localization. In: Proceedings of the 5th International Conference on Information Processing in Sensor Networks. Nashville, USA: IEEE, 2006. 441-448 https://www.researchgate.net/publication/224642430_Node-density_independent_localization [21] Hendrickson B. The molecule problem: exploiting structure in global optimization. Siam Journal on Optimization, 1995, 5(4): 835-857 doi: 10.1137/0805040 [22] Shang Y, Ruml W, Zhang Y, Fromherz M P J. Localization from mere connectivity. In: Proceedings of the 4th ACM International Symposium on Mobile Ad Hoc Networking and Computing. Annapolis, Maryland, USA: ACM, 2003. 201-212 https://www.researchgate.net/publication/2892241_Localization_from_Mere_Connectivity [23] Shang Y, Ruml W. Improved MDS-based localization. In: EEE INFOCOM 2004. Hong Kong, China: IEEE, 2004. 2640-2651 https://www.researchgate.net/publication/4102894_Improved_MDS-based_localization [24] Shang Y, Ruml W, Zhang Y, Fromherz M. Localization from connectivity in sensor networks. IEEE Transactions on Parallel and Distributed Systems, 2004, 15(11): 961-974 doi: 10.1109/TPDS.2004.67 [25] Biswas P, Lian T C, Wang T C, Ye Y Y. Semidefinite programming based algorithms for sensor network localization. ACM Transactions on Sensor Networks, 2006, 2(2): 188-220 doi: 10.1145/1149283.1149286 [26] Biswas P, Aghajan H, Ye Y Y. Semidefinite programming algorithms for sensor network localization using angle information. In: Proceedings of the Conference Record of the Thirty-Ninth Asilomar Conference on Signals, Systems and Computers. Pacific Grove, CA, USA: IEEE, 2005. 220-224 http://www.researchgate.net/publication/4225543_Semidefinite_Programming_Algorithms_for_Sensor_Network_Localization_using_Angle_Information [27] Biswas P, Liang T C, Toh K C, Ye Y, Wang T C. Semidefinite programming approaches for sensor network localization with noisy distance measurements. IEEE Transactions on Automation Science and Engineering, 2006, 3(4): 360-371 doi: 10.1109/TASE.2006.877401 [28] Zhu Z S, So A M C, Ye Y Y. Universal rigidity: towards accurate and efficient localization of wireless networks. In: Proceedings IEEE INFOCOM. San Diego, USA: IEEE, 2010. 1-9 [29] Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear Matrix Inequalities in System and Control Theory. Philadelphia: Society for Industrial and Applied Mathematics, 1994. http://library.wur.nl/WebQuery/clc/1669445 [30] Liang T C, Wang T C, Ye Y Y. A Gradient Search Method to Round the Semidefinite Programming Relaxation Solution for Ad Hoc Wireless Sensor Network Localization, Technical Report, Department of Management Science and Engineering, Stanford University, 2004. [31] Cox T F, Cox M A A. Multidimensional Scaling (Second Edition). London: Chapman Hall/CRC, 2000. [32] Savvides A, Park H, Srivastava M B. The bits and flops of the n-hop multilateration primitive for node localization problems. In: Proceedings of the 1st ACM International Workshop on Wireless Sensor Networks and Applications. Atlanta, USA: ACM, 2002. 112-121 [33] Costa J A, Patwari N, Hero A O Ⅲ. Distributed weighted-multidimensional scaling for node localization in sensor networks. ACM Transactions on Sensor Networks, 2006, 2(1): 39-64 http://d.old.wanfangdata.com.cn/OAPaper/oai_doaj-articles_1757e346002855db37f97718c6d6e3b1 [34] Groenen P J F. The Majorization Approach to Multidimensional Scaling: Some Problems and Extensions[Ph.D. dissertation], Leiden University, Leiden, The Netherlands, 1993 https://www.researchgate.net/publication/243785955_The_majorization_approach_to_multidimensional_scaling_some_problems_and_extensions [35] Press W H, Teukolsky S A, Vetterling W T, Flannery B P. Numerical Recipes (Second Edition). Cambridge: Cambridge University Press, 1992. [36] Lourakis M I A, Argyros A A. SBA: a software package for generic sparse bundle adjustment. ACM Transactions on Mathematical Software, 2009, 36(1): Article No. 2 http://d.old.wanfangdata.com.cn/Periodical/wjclxb200304024 [37] Endres F, Hess J, Engelhard N, Sturm J, Cremers D, Burgard W. An evaluation of the RGB-D SLAM system. In: Proceedings of the 2012 IEEE International Conference on Robotics and Automation. Saint Paul, MN, USA: IEEE, 2012. 1691-1696 https://www.researchgate.net/publication/254041427_An_evaluation_of_the_RGB-D_SLAM_system [38] Sturm J, Engelhard N, Endres F, Burgard W, Cremers D. A benchmark for the evaluation of RGB-D SLAM systems. In: IEEE/RSJ International Conference on Intelligent Robots and Systems. Vilamoura, Portugal: IEEE, 2012. 573-580 https://www.researchgate.net/publication/261353760_A_benchmark_for_the_evaluation_of_RGB-D_SLAM_systems?ev=auth_pub [39] Engel J, Schōps T, Cremers D. LSD-SLAM: large-scale direct monocular SLAM. In: European Conference on Computer Vision. Zurich, Switzerland: Springer, 2014. 834-849 http://www.researchgate.net/publication/290620817_LSD-SLAM_Large-Scale_Direct_Monocular_SLAM [40] Forster C, Pizzoli M, Scaramuzza D. SVO: fast semi-direct monocular visual odometry. In: Proceedings of the 2014 IEEE International Conference on Robotics and Automation. Hong Kong, China: IEEE, 2014. 15-22 https://www.researchgate.net/publication/262378002_SVO_Fast_Semi-Direct_Monocular_Visual_Odometry [41] Fraundorfer F, Heng L, Honegger D, Lee G H, Meier L, Tanskanen P, et al. Vision-based autonomous mapping and exploration using a quadrotor MAV. In: Proceedings of the 2012 IEEE/RSJ International Conference on Intelligent Robots and Systems. Vilamoura, Portugal: IEEE, 2012. 4557-4564 http://www.researchgate.net/publication/261353707_Vision-based_autonomous_mapping_and_exploration_using_a_quadrotor_MAV [42] De Leeuw J. Applications of convex analysis to multidimensional scaling. Department of Statistics, UCLA, 2005. http://www.researchgate.net/publication/239417876_APPLICATIONS_OF_CONVEX_ANALYSIS_TO_MULTIDIMENSIONAL_SCALING [43] Borg I, Groenen P J F. Modern Multidimensional Scaling: Theory and Applications (Second Edition). New York, NY: Springer, 2005. [44] Moore D C. Robust distribution sensor network localization with noisy range measurements. Massachusetts Institute of Technology, 2005. 50-61 https://www.researchgate.net/publication/35943813_Robust_distribution_sensor_network_localization_with_noisy_range_measurements [45] Koren Y, Gotsman C, Ben-Chen M. PATCHWORK: efficient localization for sensor networks by distributed global optimization. Technical Report, 2005. https://www.researchgate.net/publication/254039773_PATCHWORK_Efficient_Localization_for_Sensor_Networks_by_Distributed_Global_Optimization [46] Roweis S T, Saul L K. Nonlinear dimensionality reduction by locally linear embedding. Science, 2000, 290(5500): 2323-2326 doi: 10.1126/science.290.5500.2323 [47] Shi J B, Malik J. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000, 22(8): 888-905 doi: 10.1109/34.868688 [48] Zhang L, Liu L G, Gotsman C, Gortler S J. An as-rigid-as-possible approach to sensor network localization. ACM Transactions on Sensor Networks, 2010, 6(4): Article No. 35 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=fa857a82ca6af6ecf09b9cba6458a2e5 [49] Sorkine O, Alexa M. As-rigid-as-possible surface modeling. In: Proceedings of the 5th Eurographics Symposium on Geometry Processing. Barcelona, Spain, 2007. 109-116 https://www.researchgate.net/publication/221316589_As-Rigid-As-Possible_Surface_Modeling [50] Alexa M, Cohen-Or D, Levin D. As-rigid-as-possible shape interpolation. In: Proceedings of the 27th Annual Conference on Computer Graphics and Interactive Techniques. New Orleans, USA: ACM, 2000. 157-164 https://www.researchgate.net/publication/2538876_As-Rigid-As-Possible_Shape_Interpolation [51] Press W H, Teukolsky S A, Vetterling W T, Flannery B P. Numerical Recipes: The Art of Scientific Computing (3rd Edition). Cambridge: Cambridge University Press, 2007. [52] Cornilescu G, Marquardt J L, Ottiger M, Bax A. Validation of protein structure from anisotropic carbonyl chemical shifts in a dilute liquid crystalline phase. Journal of the American Chemical Society, 1998, 120(27): 6836-6837 doi: 10.1021/ja9812610 [53] Hendrickson B. Conditions for unique graph realizations. SIAM Journal on Computing, 1992, 21(1): 65-84 doi: 10.1137/0221008 [54] Gower J C, Dijksterhuis G B. Procrustes Problems. Oxford: Oxford University Press, 2004. [55] Golub G H, Van Loan C F. Matrix Computations (4th Edition). Baltimore, MD: Johns Hopkins University Press, 2012. [56] Gortler S J, Healy A D, Thurston D P. Characterizing generic global rigidity. American Journal of Mathematics, 2010, 132(4): 897-939 doi: 10.1353/ajm.0.0132 [57] Cucuringu M, Lipman Y, Singer A. Sensor network localization by eigenvector synchronization over the euclidean group. ACM Transactions on Sensor Networks, 2012, 8(3): Article No. 19 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=51a43c69302e35adee96e330ee6450b8 [58] Sobeih A, Hack M, Liu Z, Zhang L. Almost peer-to-peer clock synchronization. In: Proceedings of the 2007 IEEE International Parallel and Distributed Processing Symposium. Rome: IEEE, 2007. 1-10 https://www.researchgate.net/publication/220951841_Almost_Peer-to-Peer_Clock_Synchronization?ev=prf_cit [59] Mainwaring A, Culler D, Polastre J, Szewczyk R, Anderson J. Wireless sensor networks for habitat monitoring. In: Proceedings of the 1st ACM International Workshop on Wireless Sensor Networks and Applications. Atlanta, USA: ACM, 2002. 88-97 [60] Brust M R, Akbaș M I, Turgut D. Multi-hop localization system for environmental monitoring in wireless sensor and actor networks. Concurrency and Computation: Practice and Experience, 2013, 25(5): 701-717 doi: 10.1002/cpe.1812 [61] Wang X P, Liu Y H, Yang Z, Lu K, Luo J. Robust component-based localizationin sparse networks. IEEE Transactions on Parallel and Distributed Systems, 2014, 25(5): 1317-1327 doi: 10.1109/TPDS.2013.85 [62] Korniienko A, Khan U A, Scorletti G. Robust sensor localization with locally-computed, global $H_\infty$-design. In: Proceedings of the 2016 American Control Conference. Boston, USA: IEEE, 2016. 6375-6380 [63] Khan U A, Kar S, Moura J M F. DILAND: an algorithm for distributed sensor localization with noisy distance measurements. IEEE Transactions on Signal Processing, 2010, 58(3): 1940-1947 doi: 10.1109/TSP.2009.2038423 [64] Lin Z Y, Fu M Y, Diao Y F. Distributed self localization for relative position sensing networks in 2D space. IEEE Transactions on Signal Processing, 2015, 63(14): 3751-3761 doi: 10.1109/TSP.2015.2432739 [65] Hubbell W L, Cafiso D S, Altenbach C. Identifying conformational changes with site-directed spin labeling. Nature Structural and Molecular Biology, 2000, 7(9): 735-739 doi: 10.1038/78956 [66] Oh K J, Altenbach C, Collier R J, Hubbell W L. Site-directed spin labeling of proteins. Bacterial Toxins: Methods and Protocols. New York: Humana Press, 2000. 147-169 [67] Dong Q F, Wu Z J. A linear-time algorithm for solving the molecular distance geometry problem with exact inter-atomic distances. Journal of Global Optimization, 2002, 22(1-4): 365-375 https://www.researchgate.net/publication/227120408_A_linear-time_algorithm_for_solving_the_molecular_distance_geometry_problem_with_exact_inter-atomic_distances [68] Wu D, Wu Z J. An updated geometric build-up algorithm for solving the molecular distance geometry problems with sparse distance data. Journal of Global Optimization, 2007, 37(4): 661-673 doi: 10.1007/s10898-006-9080-6 [69] Lavor C, Liberti L, Donald B, Worley B, Bardiaux B, Malliavin T E, Nilges M. Minimal NMR distance information for rigidity of protein graphs. Discrete Applied Mathematics, 2019, 256: 91-104 doi: 10.1016/j.dam.2018.03.071 [70] Blumenthal L M. Theory and Applications of Distance Geometry. Oxford: Clarendon Press, 1953. [71] Moré J J, Wu Z J. Distance geometry optimization for protein structures. Journal of Global Optimization, 1999, 15(3): 219-234 doi: 10.1023/A:1008380219900 [72] Cleveland W S, McGill M E. Dynamic Graphics for Statistics. Belmont, California: Wadsworth, 1988. [73] Littlefield R J. Using the glyph concept to create user-definable display formats. Pacific Northwest Lab, Richland, USA, 1983. [74] 柳萌萌, 赵书良, 韩玉辉, 苏东海, 李晓超, 陈敏.多尺度数据挖掘方法.软件学报, 2016, 27(12): 3030-3050 http://d.old.wanfangdata.com.cn/Periodical/jsjkx201904009Liu Meng-Meng, Zhao Shu-Liang, Han Yu-Hui, Su Dong-Hai, Li Xiao-Chao, Chen Min. Research on multi-scale data mining method. Journal of Software, 2016, 27(12): 3030-3050 http://d.old.wanfangdata.com.cn/Periodical/jsjkx201904009 [75] Levy S. A cross-cultural analysis of the structure and levels of attitudes towards acts of political protest. Social Indicators Research, 1983, 12(3): 281-309 doi: 10.1007/BF00319806 [76] Castellanos J A, Montiel J M M, Neira J, Tardos J D. The SPmap: a probabilistic framework for simultaneous localization and map building. IEEE Transactions on Robotics and Automation, 1999, 15(5): 948-952 doi: 10.1109/70.795798 [77] Paz L M, Tardos J D, Neira J. Divide and conquer: EKF SLAM in O($n$). IEEE Transactions on Robotics, 2008, 24(5): 1107-1120 doi: 10.1109/TRO.2008.2004639 [78] Montemerlo M, Thrun S, Koller D, Wegbreit B. FastSLAM: a factored solution to the simultaneous localization and mapping problem. In: Proceedings of the AAAI National Conference on Artificial Intelligence. San Francisco: AAAI, 2002. 593-598 [79] Hahnel D, Burgard W, Fox D, Thrun S. An efficient fastSLAM algorithm for generating maps of large-scale cyclic environments from raw laser range measurements. In: IEEE/RSJ International Conference on Intelligent Robots and Systems. Las Vegas, USA: IEEE, 2003. 206-211 [80] Eustice R M, Singh H, Leonard J J. Exactly sparse delayed-state filters. In: Proceedings of the 2005 IEEE International Conference on Robotics and Automation. Barcelona, Spain: IEEE, 2005. 2417-2424 https://www.researchgate.net/publication/221077974_Exactly_Sparse_Delayed-State_Filters [81] Thrun S, Liu Y F, Koller D, Ng A Y, Ghahramani Z, Durrant-Whyte H. Simultaneous localization and mapping with sparse extended information filters. International Journal of Robotics Research, 2004, 23(7-8): 693-716 doi: 10.1177/0278364904045479 [82] Grisetti G, Kummerle R, Stachniss C, Burgard W. A tutorial on graph-based SLAM. IEEE Intelligent Transportation Systems Magazine, 2010, 2(4): 31-43 doi: 10.1109/MITS.2010.939925 [83] Saez J M, Hogue A, Escolano F, Jenkin M. Underwater 3D SLAM through entropy minimization. In: Proceedings of the 2006 IEEE International Conference on Robotics and Automation. Orlando, USA: IEEE, 2006. 3562-3567 http://www.researchgate.net/publication/221069355_Underwater_3D_SLAM_through_Entropy_Minimization [84] Kümmerle R, Steder B, Dornhege C, Kleiner A, Grisetti G, Burgard W. Large scale graph-based SLAM using aerial images as prior information. Autonomous Robots, 2011, 30(1): 25-39 doi: 10.1007/s10514-010-9204-1 [85] 张毅, 汪龙峰, 余佳航.基于深度信息的移动机器人室内环境三维地图创建.计算机应用, 2014, 34(12): 3438-3440 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=jsjyy201412015Zhang Yi, Wang Long-Feng, Yu Jia-Hang. Depth-image based 3D map reconstruction of indoor environment for mobile robots. Journal of Computer Applications, 2014, 34(12): 3438-3440 http://www.wanfangdata.com.cn/details/detail.do?_type=perio&id=jsjyy201412015 [86] Levi Z, Gotsman C. Smooth rotation enhanced As-Rigid-As-Possible mesh animation. IEEE Transactions on Visualization and Computer Graphics, 2015, 21(2): 264-277 doi: 10.1109/TVCG.2014.2359463 [87] Parashar S, Pizarro D, Bartoli A, Collins T. As-rigid-as-possible volumetric shape-from-template. In: Proceedings of the 2015 IEEE International Conference on Computer Vision. Santiago, Chile: IEEE, 2015. 891-899 https://www.researchgate.net/publication/300408305_As-Rigid-as-Possible_Volumetric_Shape-from-Template?ev=auth_pub [88] Jacobs D J, Hendrickson B. An algorithm for two-dimensional rigidity percolation: the pebble game. Journal of Computational Physics, 1997, 137(2): 346-365 doi: 10.1006/jcph.1997.5809 [89] Connelly R. Generic global rigidity. Discrete & Computational Geometry, 2005, 33(4): 549-563 http://www.researchgate.net/profile/R_Connelly/publication/220453469_Generic_Global_Rigidity/links/5447b0890cf2f14fb8121c28?ev=pub_ext_doc_dl_meta 期刊类型引用(12)
1. 李诗航，孙玉洁，崔建峰，曾建潮，卢志刚. 军用无人系统协同感知与分布式滤波技术. 火力与指挥控制. 2024(03): 88-94 . 
百度学术2. 李颖，杨晓坤，李思嘉，熊洪樟. 基于无迹卡尔曼滤波的反窃电自动化检验系统. 电子设计工程. 2024(13): 104-107+112 . 百度学术3. 胡晓工，苏志坚，贾晓亮，吴光平，刘鹏. 基于分布式容积信息卡尔曼滤波的锂电池SOC估计. 自动化技术与应用. 2024(12): 1-7 . 百度学术4. 符太东，刘松旭，李育强. 基于盲均衡算法的网络大数据异常节点检测. 计算机仿真. 2023(07): 462-466 . 百度学术5. 刘宇，袁正，陈燕苹，彭慧. 基于鲁棒增强的因子图多源信息融合算法. 电子质量. 2023(08): 27-32 . 百度学术6. 刘鹏，徐昕恒，曾建潮，张鹏，卢志刚，马跃东. 机器人单兵室内定位理论与关键技术分析. 火力与指挥控制. 2022(03): 1-7 . 百度学术7. 李旭妍，刘长明，陈燕，畅彦祥. 基于EKF算法的微型AHRS模块设计. 国外电子测量技术. 2022(04): 93-98 . 百度学术8. 刘鹏，于镇滔，曾建潮，张鹏，卢志刚，马跃东. 机器人班组协同定位理论与技术研究. 火力与指挥控制. 2022(04): 5-10+18 . 百度学术9. 李孟委，徐文武，张鹏. 基于四元数互补滤波算法的车载MIMU. 河南理工大学学报(自然科学版). 2021(01): 111-117 . 百度学术10. 李维刚，张鹏，李孟委，王树森. 基于卡尔曼滤波的主子惯组匹配标定方法设计. 河南理工大学学报(自然科学版). 2021(02): 118-126 . 百度学术11. 张鹏，王树森，李孟委. SINS/GPS/PDR室内外无缝导航定位算法. 河南理工大学学报(自然科学版). 2021(03): 113-119 . 百度学术12. 曹红燕，刘长明，沈小林，牛兴龙，李大威，陈燕. 基于GPS/INS的自适应无迹Kalman滤波算法. 河南理工大学学报(自然科学版). 2021(06): 149-156 . 百度学术其他类型引用(23)
-
下载:
计量
- 文章访问数: 3485
- HTML全文浏览量: 1366
- PDF下载量: 874
- 被引次数: 35
