Data Driven Reliability Assessment and Life-time Prognostics: A Review on Covariate Models
-
摘要: 作为保障工业过程可靠性和经济性的重要技术，可靠性评估与寿命预测在过去几十年得到了越来越广泛的关注和长足的发展.在实际应用中，由于难以获取复杂、高可靠性设备失效机理的物理模型，数据驱动的可靠性评估与寿命预测方法成为近年来的主流.同时，自动监测技术和传感器技术的快速发展，使得在工程实践中不仅能够获取系统的退化数据，还能得到大量的系统运行环境监测数据，从而使得数据驱动寿命预测中基于协变量的方法得到了广泛应用.本文根据系统运行环境中协变量数据的不同变化规律，将基于协变量方法的可靠性评估模型分为：固定协变量模型、时变协变量模型和随机协变量模型，并分别讨论了各模型的发展现状.最后，讨论了协变量处理中存在的一些挑战及未来的研究方向.Abstract: Reliability assessment and life-time prognostics have been widely concerned and developed fast in the past decades for their importance in industrial processes. Data driven approaches have been popular due to the complexity of failure mechanism about the reliable complex system. With the development of auto-monitoring and sensor technology, it is easy to obtain degradation data and environment information. A large number of methods based on hazard models with covariates have emerged. In this paper we review the state-of-the-art covariate models in the literature. We classify the approaches into three broad types of models, that is, constant models, time-dependent models and stochastic models. We systematically discuss these models and approaches and finally highlight future research challenges.
-
Key words:
- Life-time prognostics /
- data driven /
- reliability /
- covariate
-
现代工业过程中普遍具有强非线性和不确定性的特点, 难以用传统的数学机理建模方法描述过程动态变化, 因此这种形式的建模方法很难满足实际应用要求[1].在无数学机理模型的情况下, 人类需要依靠逻辑推理等智能对感知的信息进行处理, 形成经验、知识和规律, 进而完成复杂不确定系统的分析和控制.作为这种智能方法之一的模糊认知图(Fuzzy cognitive map, FCM)[2], 具有形式化描述、数值推理、模糊信息的表达等特点, 能够成为建立在系统的原始数据和研究者之间的一座桥梁. FCM是一种由节点、弧和权重组成的有向图[3-4].其中, 节点又被称为概念节点, 权值范围为[0, 1], 可以代表系统的某一种特性.两个概念节点之间的因果关系则用带符号和权重的弧表示, 取值范围为$[-1, 1]. $ FCM由于其良好的动态特性和学习能力, 因此被广泛应用在复杂系统建模、过程分析、决策制定等方面[5]. Anninou等通过分析影响人类患有帕金森综合征的相关因素, 使用FCM呈现了帕金森综合症的数学模型, 提供了一个非常有趣的研究方向[6].文献[7]对人的个性与情绪之间的关系进行了分析, 建立了一种基于FCM的数学模型, 这种模型对开发人性化系统具有非常可观的意义. Kreinovich采用Miller准则, 将多元神经网络简化为一元神经网络, 证明了仅用经验和主观意识也能让FCM有效地描述系统[8]. Obiedat等利用动力学系统与神经网络相结合的方法, 分析并建立了社会生态系统的FCM模型, 结果表明该方法能够为决策者提供合适的建议[9]. Christen使用FCM作为工具, 分析了苏格兰农业系统管理经验, 发现了农业制度不善的问题[10].张燕丽针对大型非线性系统, 提出了T-S建模方法, 并在倒车系统中进行了验证[11].
传统的FCM建模方法通过离线的方法模拟系统过程, 且对专家知识具有很强的依赖性.由于传统FCM的局限性, 在根据专家知识和实际系统数据建立了系统的模糊认知图的情况下, FCM就处于离线运作状态.因此, 如果专家的知识不够准确, 就会导致所建立的系统FCM模型无法正确地反应当前系统的实际状态.为了与实际系统保持密切实时联系, 并且对实时系统做出更准确的控制和决策[12], 一种基于FCM进行扩展的模糊认知网络(Fuzzy cognitive networks, FCN)[13-15]建模方法就应运而生了. FCN框架由一个模糊认知图、一个基于实际系统反馈的更新机制以及整个操作过程中获得知识的存储组成. FCN只在初始点使用或者完全脱离专家经验, 能够克服传统FCM建模方法对专家经验依赖性强的缺点.
灰色系统理论(Grey system theory, GST)在系统模型的建立过程中做出了巨大的贡献.该理论通过利用概率论与数理统计的方法取得随机性被弱化、规律性被增强的新数据序列, 由于新的序列既体现了原序列的变化又消除了波动性的特性, 灰色系统理论能够解决部分参数未知的系统问题.吉培荣提出了无偏差灰色预测模型, 该模型极大地消除了传统模型的固有偏差[16-17].文献[18]建立了灰色预测模型, 通过预测中国台湾人口中使用互联网的人数与实际值相比较, 验证了该模型的有效性.文献[19]提出了一种能应用于复杂工程的灰色模型控制器, 并验证了该控制器的优秀性能.
通过对FCN和灰色系统理论现状进行研究, 针对FCN和灰色系统理论的优点, 发现二者的结合对具有耦合性、非线性及高不确定性等特点的系统模型建立具有重要的意义. FCN模型参数设定很大程度上影响模型精度, 而FCN模型中初始权值就是一个非常重要的参数.权值学习方法通过一定的机制更新不断对系统权值进行更新, 从而达到提高FCN模型精度的目的.文献[20]提出了一种简单的微分Hebbian学习方法(Differential Hebbian learning, DHL), 而DHL是一种无监督的学习方法, 因此无法在实际系统中广泛应用.非线性Hebbian学习法(Nonlinear Hebbian learning, NHL)于2003年由Papageorgiou等[21]提出, 在系统权值学习具有广泛的应用.文献[22]提出了一种LASSOFCM的学习方法, 该方法能在没有任何先验知识的少量数据的情况下学习大规模的FCM系统, 在一些领域难以得到应用. Natarajan等对影响印度甘蔗产量的土壤和气候等各种因素进行了分析并建立了FCM模型, 采用结合数据驱动的非线性Hebbian学习方法(DDNHL)和遗传算法对模型的参数进行了辨识[23]. Baykasoglu等讨论了扩展的大洪水算法(Extended great deluge algorithm, EGDA)在训练FCM方面的应用, EGDA作为一种较好的算法, 主要存在的问题是参数唯一[24].
本文将灰色理论和网络反馈控制理论结合, 在FCM的基础上提出了模糊灰色认知网络建模方法和带终端约束的非线性Hebbian权值计算方法, 所建模型能做出接近人类智能的控制决策, 解决对数据存在不确定性或缺失的复杂系统建模的难题.本文安排如下:第1节介绍(Fuzzy grey cognitive networks, FGCN)模型建模机制及其特性; 第2节介绍传统NHL算法并提出了带终端约束的非线性Hebbian算法; 第3节利用一个水箱控制过程验证本文方法的有效性; 第4节对文章进行总结与展望.
1. 模糊灰色认知网络
模糊灰色认知网络是一种结合FCN和灰色系统理论(GST)的软计算技术. FGCN模型是在FCN的基础上, 通过对模糊概念和节点之间的灰色关系的不精确描述来表示非结构化知识, 从而提供一个抽象级的直观、详细的系统建模方法.
一个非线性系统可以用FGCN表示成一个包括节点、权值、带系统反馈的有向图, 当有向图取不同初始值和权值时, 对应系统的不同工况.用$C=\{C_1, C_2, \cdots, C_n\}$表示构成有向图的顶点的概念集合, 每一个节点代表系统的特征, 如变量、状态、事件、目标等.节点状态值反应概念节点的取值(模糊的或确定的).灰数节点状态值用$\otimes{{\mathit{\boldsymbol A}}}$表示:
$ \otimes {{\mathit{\boldsymbol A}}} = ( \otimes{{A}}_1 \otimes{{A}}_2 , \cdots, \otimes{{A}}_n )=\\ {\rm{~~~~~~~}} ([\underline {A}_1 , \overline A_1 ], [\underline {A}_2 , \overline A_2 ], \cdots[\underline {A}_n , \overline A_n ]) $
(1) 其中, $n$是模型中节点总数目, $ \otimes \underline {A}_i^{} $可以是灰数也可以是白数(即$\underline {A}_i^{} = \overline A_i^{} $).用$A_i^k$表示节点$C_i$在$k$时刻的状态, 其取值由系统的实际取值转换而来, 值域为[0, 1].用$W_{ij}$表示节点$C_i$和节点$C_j$之间的因果影响程度, 值域为$[1, -1]. $如果$W_{ij}>0$, 则结果概念节点$C_j$的状态随原因概念节点$C_i$的状态值成正比例变化; 反之, 如果$W_{ij}<0$, 则结果概念节点$C_j$的状态随原因概念节点$C_i$的状态值成反比例变化; 如果$W_{ij}=0$, 则概念节点$C_j$的状态与概念节点$C_i$的状态值没有关联.
连接两个节点之间的边代表节点之间的因果关系, 值域为$[1, -1]. $如果取值为正, 则结果概念节点$C_j$的状态随原因概念节点$C_i$的状态值成正向变化; 反之, 如果取值为负, 则结果概念节点$C_j$的状态随原因概念节点$C_i$的状态值成反向变化; 如果取值为零, 则概念节点$C_j$的状态与概念节点$C_i$的状态值没有关联.由于FGCN是结合神经网络和灰色系统理论的方法, 两个节点之间带权弧的大小都表示为一个灰数权值.
$ \otimes W_{ij} \in [\underline {W}_{ij} , \overline W_{ij} ], \forall i, j \to - 1 \le \underline {W}_{ij} \le \overline W_{ij} \le 1 $
(2) 其中, $i$表示原因节点, $j$表示结果节点.
系统动态行为间的关系都存储于认知图的网络结构和节点间相互影响的因果关系中.每一时刻的节点状态值受该节点上一时刻以及与它有因果关系的节点取值影响.
在迭代计算节点下一个时刻的值时, 利用一个阈值函数将灰色状态值转换到一个规范化的范围内.通常使用S曲线函数来保证状态值取值在[0, 1]区间.
节点状态值的更新方程为
$ \otimes {{\mathit{\boldsymbol A}}}^k = f( \otimes {{\mathit{\boldsymbol A}}}^{k - 1} + \otimes {{\mathit{\boldsymbol A}}}^{k - 1} \cdot {{W}}( \otimes )) $
(3) 特别的, 稳定节点和输出节点的状态值应该取以系统实际测量值为基值的灰数:
$ \otimes{{A}}_i^k = A_i^{\rm system} + \varepsilon $
(4) 其中, $\otimes{{\mathit{\boldsymbol A}}}^k$是$k$时刻的灰色状态值矢量, $\otimes{{\mathit{\boldsymbol A}}}^{k-1}$是$k-1$时刻的灰色状态值矢量, ${{W}}(\otimes)$是该模型的灰数权值矩阵, $\otimes A_i^k$是第$i$个节点$k$时刻的灰数状态值, $A_i^{\rm system}$是稳定节点和输出节点的系统实际值, 通过实时测量或者事先给定, $\varepsilon$是振幅, $f = 1/(1 + {\rm e}^{ - cx} ) $是阈值函数, 用来保证状态值取值在[0, 1]区间.
由于引入了灰色理论, FGCN增加了一个灰度(用$\varphi(\otimes A_i)$表示)来衡量系统的不确定性.如果灰度取值大, 则结果的不确定性高; 反之, 则不确定性小.传统FCN没有使用不确定性的标准来评判结果.
$ \varphi ( \otimes{{A}}_i ) = \frac{{\left| {l( \otimes{{A}}_i )} \right|}}{{l( \otimes \phi )}} $
(5) 其中, $ \left| {l( \otimes{A}_i )} \right|$是灰色状态值$\otimes A_i$长度的绝对值, $l(\otimes\phi)$是状态值取值空间的长度绝对值.在FGCN中, 节点状态值的取值空间可以是[0, 1]或者$[-1, 1]$, 因此$l(\otimes\phi)$取值为
$ \begin{array}{l} {l}( \otimes \phi ) = \left\{ {\begin{array}{*{20}l} {2, \;\;\;\;\; \mbox{若} \otimes{{A}}_i \subseteq [ - 1, 1], \forall \otimes{{A}}_i } \\ {1, \;\;\;\;\; \mbox{若} \otimes{{A}}_i \subseteq [0, 1], \forall \otimes{{A}}_i } \\ \end{array}} \right. \\ \end{array} $
(6) 如果${l(\otimes{{A}}_i)}=0$, 则无论$l(\otimes\phi)$取值如何, ${l(\otimes{{A}}_i)}$为零, 即灰度为零, $\otimes A_i$是白数, 没有不确定性.如果$l(\otimes{{A}}_i )= l( \otimes \phi) $, 灰度将是无穷大, 即$\varphi(\otimes{{A}}_i ) = \infty $.
根据以上分析, FGCN与传统的FCN相比有以下优点:
1) FGCN作为一个广义概念, 包括了模糊和灰色的概念, 能够更好地处理信息不完整以及因素间关系不确定的系统, 比传统的FCN更加贴近人类的智能决策;
2) FGCN允许在建模过程中出现概念和概念节点之间因果关系的不确定性和多义性;
3) FGCN可以表示更多的节点之间的关系.比如, 它能够在系统节点之间关系只知道一部分或者完全未知的情况下建立准确的模型;
4) 在推理过程中用灰度表示输出节点的不确定性, 提高了输出结果的可靠性;
5) 当FGCN的灰色状态值和灰色权值的灰度全部取零时, 则输出结果与传统FCN一致, 而灰度的存在使其比FCN多了判断结果准确性的标准.
2. 基于FGCN的带终端约束的权值迭代算法
在已经建立系统的FGCN模型的前提下, 需要对模型中的权值进行辨识.非线性Hebbian算法是一种无监督权值学习方法[17], 它利用权值关联的原因节点状态值与结果节点状态值的乘积对连接权值进行无向修正, 没有其他约束条件来提高学习效率和模型准确度.由于对专家的依赖性较强, 自主学习能力比较差, 当系统发生变化时, 无法及时跟踪反应, 使学习效率和准确度都受到影响.
为使模糊认知网络实时精准反应系统状态并做出正确的控制, 必须改进NHL算法.由于离线学习模型对系统工况的变化不敏感, 要提高模型的跟踪能力, 需将权值的学习与系统实际工况相结合.考虑到每一次迭代学习得到的状态值作为控制量作用于原系统, 可以得到一个实时值并将其作为学习目标, 故本文引进系统实际测量值作为约束来更新FGCN的权值, 从而提出带终端约束的非线性Hebbian算法.
带终端约束的非线性Hebbian算法的权值更新引入了系统实际反馈作为约束, 将模型预测值与系统实际测量值之差作为调整权值的标准.由于差值大小直接反应预测值与真实值的差距, 故可由此实现对权值进行有向修正, 在提高收敛速率的同时也解决了传统的无监督NHL对初始值依赖性强的缺点.
用符号$\otimes{{\mathit{\boldsymbol A}}}$表示FGCN中状态节点的状态值, 用$ \otimes {\mathop{W}\nolimits} _{}^k $代表$k$时刻节点间弧的灰数权值, 每一次迭代都利用模型的预测值与系统实际值之间的误差对权值进行修正.带终端约束的NHL算法如下:
$ \begin{align} & \otimes W_{ji}^{k + 1} = (1 - \gamma )( \otimes W_{ji}^k ) + \eta ( \otimes{{A}}_j^{\rm FGCN} ) \times \nonumber\\&\qquad( \otimes{{A}}_i^{\rm FGCN} {\rm{)+ }}\kappa \otimes p_i (1 - \otimes p_i ) \times ( \otimes A_j^{\rm FGCN} ) \end{align} $
(7) 其中, $ \otimes{{A}}_i^{\rm system} $是节点$ C_i $包含灰度的实际测量值, $\otimes{{A}}_i^{k, {\rm FGCN}}$是节点$ C_i $的FGCN计算值. $\otimes W_{ji}^{k + 1}$代表节点$ C_j $对节点$ C_i $在($k$+1)时刻的灰数权值, $\otimes W_{ji}^k$代表节点$ C_j $对节点$ C_i $在$k$时刻的灰数权值, $ \otimes{{A}}_i^{\rm FGCN} $代表节点$ C_i $的FGCN计算所得灰数, $ \otimes A_j^{\rm FGCN}$代表节点$C_j$的FGCN计算所得灰数, $\gamma$是衰减率, $ \eta $是学习率, $ \kappa $是修正率, 通常$ 0 < \gamma , \eta , \kappa < 0.1 $. $\otimes p_i $是节点$ C_i $的灰度误差, 其定义为
$ \begin{align} &\otimes p_i = \nonumber\\ &\ \otimes{{A}}_i^{\rm system}-\frac{1}{{1 + {\rm e}^{ - ( \otimes{A}_i^{\rm system} + \sum\limits_{j = 1,j \ne i}^N { \otimes{{A}}_j^{\rm system} } \otimes W_{ji} )} }}= \nonumber\\ &\ \otimes{{A}}_i^{\rm system}-\otimes{{A}}_i^{\rm FGCN} \end{align} $
(8) 为了避免$ \otimes {{W}} $的过度增长, 保证其取值在[$-1$, 1], 要求$ \otimes {{W}} $的取值满足$ \left\| { \otimes {{W}}} \right\|= 1 $, 故在每一步计算中均按式(9)对式(7)进行归一化.
在式(9)中, $n$是模型节点的个数, 分母包括了与节点$ C_i $有关的所有权值.当学习率$\eta$和修正率$\kappa$足够小时, 分母根号下表达式的泰勒展开式的高阶可以省略.假定权值矩阵的第$i$行已知, 则第$j$列的权值计算可简化为式(10).考虑到衰减率$\gamma$通常取为一个远小于1的正数, 即$(1-\gamma)^2\approx1$, 故式(10)可以继续近似化简为式(11).相应地, 权值更新公式(7)可以近似简化成式(14).
注1.在式(7)中, 前面两项$ (1 - \gamma )( \otimes W_{ji}^k ) + \eta ( \otimes{{A}}_j^{\rm FGCN} ) \times ( \otimes A_i^{\rm FGCN} {\rm{)}} $是Hebbian理论的权值修正, 计算模型中节点状态值对权值的修正; 后面一项$ \kappa \otimes p_i (1 - \otimes p_i ) \times ( \otimes{{A}}_j^{\rm FGCN} )$中, 因$\kappa$很小, $ \otimes p_i $和$ 1 - {\otimes p_i} $都小于1, 三者相乘小于等于$0.25 \kappa$, 因此可以保证每一个$ \otimes W_{ji}^k $的取值都在$[-1, 1]$范围内.
$ \otimes W_{ji}^{k + 1} = \frac{{(1 - \gamma )( \otimes W_{ji}^k ) + \eta ( \otimes A_j^{\rm FCN} ) \times ( \otimes A_i^k {\rm{) + }}\kappa ( \otimes p_i ) \cdot (1 - \otimes p_i )( \otimes A_j^{\rm FCN} )}}{{\sqrt {\sum\limits_{j = 1, j \ne i}^n {((1 - \gamma )( \otimes W_{ji}^k ) + \eta ( \otimes A_j^{\rm FGCN} ) \times ( \otimes A_i^{\rm FGCN} {\rm{) + }}\kappa ( \otimes p_i ) \cdot (1 - \otimes p_i )( \otimes A_j^{\rm FGCN} ))^2 } } }} $
(9) $ \sum\limits_{j = 1, j \ne i}^n {((1 - \gamma )( \otimes W_{ji}^k ) + \eta ( \otimes A_j^{\rm FGCN} ) \cdot ( \otimes A_i^{\rm FGCN} ) + \kappa ( \otimes p_i ) \cdot (1 - \otimes p_i ) \cdot A_j^{\rm FGCN} )^2 } \approx\\ \qquad \sum\limits_{j = 1}^n {[(1 - \gamma )^2 \cdot ( \otimes W_{ji}^k )^2 ) + 2\eta \cdot (1 - \gamma ) \cdot ( \otimes A_j^{\rm FGCN} ) \cdot ( \otimes A_i^{\rm FGCN} ) \cdot ( \otimes W_{ji}^k )} + \\ \qquad 2\kappa (1 - \gamma )( \otimes p_i ) \cdot (1 - \otimes p_i ) \cdot ( \otimes W_{ji}^k ]\approx \\ \qquad\qquad\qquad\qquad\qquad\qquad\vdots \\ \qquad (1 - \gamma )^2 + 2\eta \cdot (1 - \gamma ) \cdot ( \otimes A_j^{\rm FGCN} )^2 + 2\kappa (1 - \gamma )( \otimes p_i ) \cdot (1 - \otimes p_i ) \cdot ( \otimes A_j^{\rm FGCN} ) = \\ \qquad (1 - \gamma )^2 \left(1 + \dfrac{{2\eta \cdot ( \otimes A_j^{\rm FGCN} )^2 + 2\kappa ( \otimes p_i ) \cdot (1 - \otimes p_i ) \cdot \otimes A_j^{\rm FGCN} }}{{1 - \gamma }}\right) $
(10) $ \begin{align} &\dfrac{1}{{\sqrt {1 + \dfrac{{2\eta \cdot ( \otimes A_j^{\rm FGCN} )^2 + 2\kappa ( \otimes p_i )(1 - \otimes p_i ) \cdot ( \otimes A_j^{\rm FGCN} )}} {{1 - \gamma }}} }}\approx \nonumber\\ &\qquad \frac{1}{{1 + \dfrac{{\eta \cdot ( \otimes A_j^{\rm FGCN} )^2 + \kappa ( \otimes p_i )(1 - \otimes p_i ) \cdot ( \otimes A_j^{\rm FGCN} )}}{{1 - \gamma }}}}\approx \nonumber\\ &\qquad 1 - \dfrac{{\eta \cdot ( \otimes A_j^{\rm FGCN} )^2 + \kappa ( \otimes p_i )(1 - \otimes p_i ) \cdot ( \otimes A_j^{\rm FGCN} )}}{{1 - \gamma }} \end{align} $
(11) 迭代的两个终止标准为
标准1.
$ {\rm{ }}F_1 = |\hat A_i^{\rm FGCN} k + 1 - \hat A_i^{\rm FGCN} k| < e $
(12) 标准2.
$ F_2 = |\hat p_i | < v $
(13) 其中, $ \hat p_i $是节点的灰度误差白数化的取值, 一般取$ v = 0.001 $, 当所有节点的误差都满足式(13)时终止迭代.
$ \otimes W_{ji}^{k + 1} = (1 - \gamma ) \otimes W_{ji}^k + \\\qquad [\eta + \kappa ( \otimes p_i ) \cdot (1 - \otimes p_i )] ( \otimes A_j^{\rm FGCN} ) \cdot \\ \qquad( \otimes A_i^{\rm FGCN} ) + \kappa ( \otimes p_i ) \cdot (1 - \otimes p_i ) \cdot ( \otimes W_{ji}^k ) \cdot \\\qquad ( \otimes A_j^{\rm FGCN} ) - \eta \otimes W_{ji}^k ( \otimes A_j^{\rm FGCN} )^2 $
(14) 综上所述, 基于FGCN模型的带终端约束的非线性Hebbian算法流程图如图 1所示.
图 1 基于FGCN带终端约束的非线性Hebbian算法流程图Fig. 1 Flowchart of NHL with terminal constraints based on FGCN引进了系统反馈的带终端约束的NHL在每一次迭代中都根据实际测量值对权值进行修正, 故能快速收敛到准确反映系统的权值, 也克服了非线性Hebbian算法对初始值依赖性强的缺点, 提高了FGCN模型的灵活性和动态性.因此经过带终端约束的非线性Hebbian算法训练权值的FGCN能够更好地模拟实际系统.
3. 仿真分析
图 2所示为包括两个水箱、三个开关、一个加热元件和两个温度计的水箱控制系统.每个水箱有一个入水阀和一个出水阀, 且第一个水箱的出水阀是第二个水箱的入水阀.其工作原理如下:
水箱1的温度通过调节加热元件控制, 水箱1的温度比水箱2的温度高; 水箱2的温度通过热量传递, 当水箱2温度过低时, 开关2打开, 热水从水箱1流进水箱2.水箱的液位高度通过开关1、开关2和开关3控制.当水箱1的水位过高时, 打开开关2放水, 当水箱1的水位过低时, 打开开关1进水; 当水箱2的水位过高时, 打开开关3放水, 当水箱2的水位过低时, 打开开关2进水.
为了确定描述水箱控制系统的FGCN对象, 必须考虑到系统的变量, 如两个水箱中液位的高度、温度的高低; 系统中影响变量的要素, 如阀门的状态(打开、关闭或者部分打开)、加热元件的状态等.
在控制过程中, 1号水箱的液位受开关1和开关2的影响; 2号水箱的液位受开关2和开关3的影响; 开关1受1号水箱温度和液位的影响; 开关2受1号水箱液位、2号水箱液位和温度的影响; 开关3受2号水箱的液位的影响; 1号水箱的温度与加热元件相互影响; 2号水箱的温度受开关2的影响.
针对上述水箱控制过程, 利用专家知识和历史数据, 通过一系列相互关联的节点来描述其相应的动态系统, 建立的FGCN模型如图 3所示.
该模型中包括的节点为水箱1的液面高度$C_1$, 水箱2的液面高度$C_2$, 开关1的状态$C_3$, 开关2的状态$C_4$, 开关3的状态$C_5$, 水箱1的温度$C_6$, 水箱2的温度$C_7$, 加热元件的状态$C_8$.两个节点间的权值$\otimes W_{ij} $表示节点$C_j$对节点$C_i$的影响.系统的控制目标有两个: 1)两个水箱的水面高度保持在一定的范围内, 即上限$H_{\max}$和下限$H_{\min}$之间; 2)两个水箱的温度保持在最大值$T_{\max}$和最小值$T_{\min}$之间.即控制目标变量是1号水箱液位$H^1$、2号水箱液位$H^2$、1号水箱的温度$T^1$、2号水箱的温度$T^2$, 控制目标的期望区间为
$ H_{\min }^1 \le H^1 \le H_{\max }^1 \\ H_{\min }^2 \le H^2 \le H_{\max }^2 \\ T_{\min }^1 \le T^1 \le T_{\max }^1 \\ T_{\min }^2 \le T^2 \le T_{\max }^2 $
(15) 水箱控制系统的FGCN模型包含8个节点, 权值矩阵维数为8 $\times$ 8, 节点1、节点2、节点6和节点7为输出节点, 输出值的目标值代表所模拟系统的期望输出.根据不同的需求, 水箱控制系统有不同的控制目标.在本次模拟中, 专家定义目标节点的控制目标为$ADC_1 = 0.72$、$ADC_2 = 0.66$、$ADC_6 = 0.78$、$ADC_7 =0.64.$由于实际系统不可能达到一个完全确定的平衡点, 故将控制目标扩展成一个适度的区间:
$ 0.65 \le ADC_1 \le 0.80 \\ 0.60 \le ADC_2 \le 0.75 \\ 0.70 \le ADC_6 \le 0.88 \\ 0.55 \le ADC_7 \le 0.75 $
(16) 系统中各个元素之间的影响程度各异, 根据历史数据和经验, 专家确定权值初始值矩阵为式(17).
$ {{W}}_0 = \left[ {\begin{array}{*{20}c} 0&0&{{\rm{ - }}0.5}&{0.2}&0&0&0&0 \\ 0&0&0&{{\rm{ - }}0.5}&{0.37}&0&0&0 \\ {0.5}&0&0&0&0&0&0&0 \\ { - 0.6}&{0.67}&0&0&0&0&{0.25}&0 \\ 0&{ - 0.82}&0&0&0&0&0&0 \\ 0&0&{0.53}&0&0&0&0&{{\rm{ - }}0.4} \\ 0&0&0&{{\rm{ - }}0.1}&0&0&0&0 \\ 0&0&0&0&0&{0.7}&0&0 \\ \end{array}} \right] $
(17) FGCN模型在灰度为零时, 灰色状态值退化为白数, 用灰色初始值表示为
$ \otimes {{\mathit{\boldsymbol A}}}_0 {\rm{ = [[0}}{\rm{.44, 0}}{\rm{.44}}], [{\rm{0}}{\rm{.55, 0}}{\rm{.55}}], \\ {\rm{~~~~~~~~~~~~}}[{\rm{0}}{\rm{.6, 0}}{\rm{.6}}], [{\rm{0}}{\rm{.7, 0}}{\rm{.7}}], [{\rm{0}}{\rm{.59, 0}}{\rm{.59}}], \\ {\rm{~~~~~~~~~~~~}}[{\rm{0}}{\rm{.61, 0}}{\rm{.61}}], [{\rm{0}}{\rm{.54, 0}}{\rm{.54}}], [{\rm{0}}{\rm{.52, 0}}{\rm{.52}}]{\rm{]}} $
(18) FGCN模型在灰度不为零时, 取初始值以$\pm0.1$为振幅, 即灰度$ \varphi ( \otimes{{\mathit{\boldsymbol A}}}_i ) = 0.2 $, 用灰色初始值表示为
$ \otimes {\pmb A}'_0 {\rm{ = [[0}}{\rm{.34, 0}}{\rm{.54}}], [{\rm{0}}{\rm{.45, 0}}{\rm{.65}}], [{\rm{0}}{\rm{.5, 0}}{\rm{.7}}], \\ {\rm{~~~~~~~~~~~~~}}[{\rm{0}}{\rm{.6, 0}}{\rm{.8}}], [{\rm{0}}{\rm{.49, 0}}{\rm{.69}}], [{\rm{0}}{\rm{.51, 0}}{\rm{.71}}], \\ {\rm{~~~~~~~~~~~~~}}[{\rm{0}}{\rm{.44, 0}}{\rm{.64}}], [{\rm{0}}{\rm{.42, 0}}{\rm{.62}}]{\rm{]}} $
(19) 在给定初始状态值和权值矩阵的基础上, 先后在灰度为零和灰度不为零的FGCN模型的基础上, 利用初值、NHL学习方法所得权值和用带终端约束的非线性Hebbian算法学习所得权值, 对达到平衡状态的各节点终值的准确度以及收敛速度进行对比, 分析FGCN模型的性能以及各学习算法的优劣.
在专家给定初始权值$ {{W}}_0 $的基础上, 基于NHL学习算法对权值进行训练, 得到一个如式(20)所示的最终权值矩阵.再分别取式(18)和式(19)所示的系统状态初始值, 将最终权值矩阵代入所建立的水箱系统FGCN模型中, 经过数次迭代达到平衡状态.系统状态稳态值如表 1所示, 迭代过程如图 4所示.
表 1 NHL算法学习的仿真结果Table 1 Simulation results trained by NHL概念节点 FGCN模型(灰度为零) FGCN模型(灰度不为零) 灰色稳态值$\otimes{{\mathit{\boldsymbol A}}}_i$ 灰度$ \varphi ( \otimes{{\mathit{\boldsymbol A}}}_i )$ 白化值$\hat A_i$ 灰色稳态值$\otimes{{\mathit{\boldsymbol A}}}_i$ 灰度$ \varphi ( \otimes{{\mathit{\boldsymbol A}}}_i )$ 白化值$\hat A_i$ 1 [0.728, 0.728] 0 0.728 [0.7279, 0.7280] 0.0001 0.72795 2 [0.6663, 0.6663] 0 0.6663 [0.6662, 0.6663] 0.0001 0.66625 3 [0.7523, 0.7523] 0 0.7523 [0.7522, 0.7523] 0.0001 0.75225 4 [0.6608, 0.6608] 0 0.6608 [0.6608, 0.6609] 0.0001 0.66085 5 [0.799, 0.799] 0 0.799 [0.7989, 0.7991] 0.0002 0.799 6 [0.835, 0.835] 0 0.835 [0.8349, 0.8351] 0.0002 0.835 7 [0.7826, 0.7826] 0 0.7826 [0.7826. 0.7827] 0.0001 0.75265 8 [0.6399, 0.6399] 0 0.6399 [0.6397, 0.6399] 0.0002 0.6398 当FGCN模型的初始值灰度为零时, 得到的稳定状态值满足所要求的控制区间, 利用灰度能判断所得结论的正确性; 当FGCN模型的初始值灰度不为零时, 由于引入灰色系统理论, 在动态迭代中能够减少系统的不确定性, 在达到稳定时能够得到一个灰度为零或灰度取值很小的输出结果, 且灰数区间与灰度为零时的结果很接近. NHL学习后新的权值$ W_{\rm NHL} $能够使FGCN模型达到平衡状态时保证所有输出节点$ A_1 , A_2 , A_6 , A_7 $的取值都满足式(16)所示的控制目标范围, 说明经过NHL学习后的系统模型, 能够让所有输出节点都达到目标的期望范围, 可以比较准确地模拟系统工况.由此可以得出结论, NHL对权值有一定的修正作用.但是NHL学习法也存在一定的缺点, 如收敛速度比较缓慢, 精度也不够高, 取值为零的权值在迭代过程中也会被修改成较小的非零值以及对初始权值依赖性大等.
$ {{W}}_{\rm NHL} = \left[ {\begin{array}{*{20}c} 0&{0.0555}&{ - 0.4649}&{0.2420}&{0.0657}&{0.0691}&{0.0641}&{0.0524} \\ {0.0577}&0&{0.0610}&{ - 0.4782}&{0.4136}&{0.0676}&{0.0672}&{0.0513} \\ {0.5304}&{0.0607}&0&{0.0593}&{0.0716}&{0.0753}&{0.0698}&{0.0571} \\ { - 0.5742}&{0.6881}&{0.0616}&0&{0.0649}&{0.0683}&{0.2985}&{0.0519} \\ {0.0672}&{ - 0.8157}&{0.0709}&{0.0619}&0&{0.0786}&{0.0729}&{0.0597} \\ {0.0706}&{0.0665}&{0.5611}&{0.0650}&{0.0785}&0&{0.0766}&{ - 0.3681} \\ {0.0651}&{0.0613}&{0.0687}&{ - 0.0453}&{0.0724}&{0.0761}&0&{0.0577} \\ {0.0542}&{0.0512}&{0.0573}&{0.0500}&{0.0604}&{0.7300}&{0.0589}&0 \\ \end{array}} \right] $
(20) 由于NHL是无监督学习算法, 仅仅依靠当前时刻原因节点状态值和结果节点状态值的乘积来对权值进行修正, 而原因节点状态值和结果节点状态值的取值又与前一时刻的权值密切相关.因此, 当权值的初始取值发生改变时, 其学习结果会跟着改变, 导致系统模型发生相应的变化, 即NHL存在对初始值的强依赖性.而带终端约束的NHL学习方法引入了系统反馈, 水箱系统中系统反馈为系统稳定时各节点的状态实际测量值.由实际系统测量可得, 在当水箱达到稳定时, 水箱的实际系统状态值为
$ A^{\rm system} = \\\quad \left[0.73~~0.67~~ 0.75~~ 0.66~~0.80~~0.83~~0.78~~ 0.63\right] $
(21) 在专家给定初始权值$ {{W}}_0 $的基础上, 基于带终端约束的NHL学习算法对权值进行训练, 得到一个如式(22)所示的最终权值矩阵.分别取式(18)和式(19)所示的系统状态初始值, 将最终权值矩阵代入所建立的水箱系统FGCN模型中, 经过数次迭代达到平衡状态.系统状态稳态值如表 2所示, 迭代过程如图 5所示.
表 2 带终端约束非线性Hebbian算法学习仿真结果Table 2 Simulation results trained by NHL with terminal constraints概念节点 FGCN模型(灰度为零) FGCN模型(灰度不为零) 灰色稳态值$\otimes{{\mathit{\boldsymbol A}}}_i$ 灰度$ \varphi ( \otimes{{\mathit{\boldsymbol A}}}_i )$ 白化值$\hat A_i$ 灰色稳态值$\otimes{{\mathit{\boldsymbol A}}}_i$ 灰度$ \varphi ( \otimes{{\mathit{\boldsymbol A}}}_i )$ 白化值$\hat A_i$ 1 [0.73, 0.73] 0 0.73 [0.7300, 0.7301] 0 0.73005 2 [0.67, 0.67] 0 0.67 [0.6700, 0.6701] 0.0001 0.67005 3 [0.75, 0.75] 0 0.75 [0.7500, 0.7500] 0.0001 0.75005 4 [0.66, 0.66] 0 0.66 [0.6599, 0.6600] 0.0001 0.65995 5 [0.80, 0.80] 0 0.8 [0.8000, 0.8001] 0.0001 0.80005 6 [0.8304, 0.8304] 0 0.8304 [0.8304, 0.8305] 0.0001 0.83045 7 [0.7799, 0.7799] 0 0.7799 [0.7798, 0.7799] 0.0001 0.77985 8 [0.6290, 0.6290] 0 0.629 [0.6290, 0.6291] 0.0001 0.62905 
图 5 带终端约束的NHL学习的FGCN仿真结果Fig. 5 FGCN simulation results trained by NHL with terminal constraints由表 2的结果可以看出, 当FGCN模型的初始值灰度为零时, 得到的稳定状态值满足所要求的控制区间, 利用灰度可以判断所得结论的准确性; 当FGCN模型的初始值灰度不为零时, 由于引入灰色系统理论, 在动态迭代中能够减少系统的不确定性, 在达到稳定时能够得到一个灰度为零或灰度取值很小的输出结果, 且灰数区间与灰度为零时的结果很接近.
对NHL和带终端约束的NHL算法学习后系统FGCN模型仿真结果, 以灰度为0.2下界为例, 如图 4 (b)和图 5 (b)所示.图 5 (b)中带终端约束的NHL迭代步数最少, 并且权值能收敛到工艺指标要求范围之内.带终端约束的NHL算法中, 由于引进系统的反馈, 在每一次的迭代中都根据实际测量值对权值进行修正, 故一般情况下迭代步数较少, 且达到稳定状态时会满足系统实际要求.图 4 (b)中的NHL算法是无监督机制, 满足标准1、标准2 (式(12)和式(13))才停止迭代, 故一般情况下迭代步数较多, 且达到稳定状态时是否满足系统实际要求不能得到保证.因此, 结果表明, 经过带终端约束的非线性Hebbian算法学习相对于非线性Hebbian算法学习, 不仅能保证系统状态值落入期望范围且精度较高, 而且收敛速度也明显加快.
$ {{W}}_{\rm NHL} = \left[ {\begin{array}{*{20}c} 0&0&{ - 0.4572}&{0.4013}&0&0&0&0 \\ 0&0&0&{ - 0.4736}&{0.8750}&0&0&0 \\ {0.8613}&0&0&0&0&0&0&0 \\ { - 0.5778}&{0.8981}&0&0&0&0&{0.7354}&0 \\ 0&{ - 0.9075}&0&0&0&0&0&0 \\ 0&0&{0.8217}&0&0&0&0&0 \\ 0&0&0&{0.0355}&0&0&0&{ - 0.1216} \\ 0&0&0&0&0&{0.9205}&0&0 \\ \end{array}} \right] $
(22) 4. 结论
1) 提出了非线性系统的模糊灰色认知网络模型及其建模方法.将模糊认知网络建模方法和灰色系统理论相结合, 把模糊认知网络的节点状态值和权值扩展为灰色区间, 引入了与FGCN模型中节点的系统实际测量值对应的灰数值来评判模型可靠性.仿真实验结果表明, 利用本文提出的建模方法能解决对数据存在不确定性或缺失的复杂系统建模的难题, 所建模型能做出接近人类智能的控制决策, 对不确定性系统的建模具有广泛适用性.
2) 采用了带终端约束的非线性Hebbian权值学习算法并将其用于辨识FGCN的模型参数.与普通的NHL学习方法相比, 本文算法将系统反馈引入到权值迭代过程中, 通过有监督的学习方法使权值的迭代趋近于系统真实值, 提高了学习质量.仿真实验结果表明, 该算法收敛速度快, 精确度高且不依赖于专家确定的初始值, 通过迭代过程减小了系统的灰度, 提高了模型精度和可靠性, 克服了传统NHL对初始值依赖性大的缺点.
-
[1] Gorjian N, Ma L, Mittinty M, Yarlagadda P, Sun Y. A review on reliability models with covariates. In: Proceedings of the 4th World Congress on Engineering Asset Management. Athens, Greece: Springer, 2009. 142-157 doi: 10.1007/978-0-85729-320-6_43 [2] 郑建飞, 胡昌华, 司小胜, 张正新, 张鑫.考虑不确定测量和个体差异的非线性随机退化系统剩余寿命估计.自动化学报, 2017, 43 (2):259-270 http://www.aas.net.cn/CN/abstract/abstract19004.shtmlZheng Jian-Fei, Hu Chang-Hua, Si Xiao-Sheng, Zhang Zheng-Xin, Zhang Xin. Remaining useful life estimation for nonlinear stochastic degrading systems with uncertain measurement and unit-to-unit variability. Acta Automatica Sinica, 2017, 43 (2):259-270 http://www.aas.net.cn/CN/abstract/abstract19004.shtml [3] 司小胜, 胡昌华, 周东华.带测量误差的非线性退化过程建模与剩余寿命估计.自动化学报, 2013, 39(5):530-541 http://www.aas.net.cn/CN/abstract/abstract17879.shtmlSi Xiao-Sheng, Hu Chang-Hua, Zhou Dong-Hua. Nonlinear degradation process modeling and remaining useful life estimation subject to measurement error. Acta Automatica Sinica, 2013, 39 (5):530-541 http://www.aas.net.cn/CN/abstract/abstract17879.shtml [4] Liao L X, Kottig F. Review of hybrid prognostics approaches for remaining useful life prediction of engineered systems, and an application to battery life prediction. IEEE Transactions on Reliability, 2014, 63 (1):191-207 doi: 10.1109/TR.2014.2299152 [5] Jardine A K S, Lin D M, Banjevic D. A review on machinery diagnostics and prognostics implementing condition-based maintenance. Mechanical Systems and Signal Processing, 2006, 20 (7):1483-1510 doi: 10.1016/j.ymssp.2005.09.012 [6] Hu C H, Zhou Z J, Zhang J X, Si X S. A survey on life prediction of equipment. Chinese Journal of Aeronautics, 2015, 28 (1):25-33 doi: 10.1016/j.cja.2014.12.020 [7] 周东华, 魏慕恒, 司小胜.工业过程异常检测、寿命预测与维修决策的研究进展.自动化学报, 2013, 39 (6):711-722 http://www.aas.net.cn/CN/abstract/abstract18097.shtmlZhou Dong-Hua, Wei Mu-Heng, Si Xiao-Sheng. A survey on anomaly detection, life prediction and maintenance decision for industrial processes. Acta Automatica Sinica, 2013, 39 (6):711-722 http://www.aas.net.cn/CN/abstract/abstract18097.shtml [8] Cox D R. Regression models and life-tables. Journal of the Royal Statistical Society, 1972, 34 (2):187-220 http://www.jstor.org/stable/2985181 [9] Kumar D, Westberg U. Some reliability models for analyzing the effect of operating conditions. International Journal of Reliability, Quality and Safety Engineering, 1996, 4 (2):133-148 doi: 10.1142/S0218539397000102?queryID=%24%7BresultBean.queryID%7D [10] Anderson J A, Senthilselvan A. A two-step regression model for hazard functions. Applied Statistics, 1982, 31 (1):44-51 doi: 10.2307/2347073 [11] Pijnenburg M. Additive hazards models in repairable systems reliability. Reliability Engineering and System Safety, 1991, 31 (3):369-390 doi: 10.1016/0951-8320(91)90078-L [12] Andersen P K, Vaeth M. Simple parametric and nonparametric models for excess and relative mortality. Biometrics, 1989, 45 (2):523-535 doi: 10.2307/2531494 [13] Shyur H J, Elsayed E A, Luxhoj J T. A general model for accelerated life testing with time-dependent covariates. Naval Research Logistics, 1999, 46 (3):303-321 doi: 10.1002/(ISSN)1520-6750 [14] Ciampi A, Etezadi-Amoli J. A general model for testing the proportional hazards and the accelerated failure time hypotheses in the analysis of censored survival data with covariates. Communications in Statistics-Theory and Methods, 1985, 14 (3):651-667 doi: 10.1080/03610928508828940 [15] McCullagh P. Regression models for ordinal data. Journal of the Royal Statistical Society:Series B (Methodological), 1980, 42 (2):109-142 http://www.jstor.org/stable/2984952 [16] Bennett S. Log-logistic regression models for survival data. Applied Statistics, 1983, 32 (2):165-171 doi: 10.2307/2347295 [17] Liao H T, Zhao W B, Guo H R. Predicting remaining useful life of an individual unit using proportional hazards model and logistic regression model. In: Proceedings of the 2006 Annual Reliability and Maintainability Symposium. Newport Beach, CA, USA: IEEE 2006. 127-132 http://ieeexplore.ieee.org/xpls/icp.jsp?arnumber=1677362 [18] Sun Y, Ma L, Mathew J, Wang W Y, Zhang S. Mechanical systems hazard estimation using condition monitoring. Mechanical Systems and Signal Processing, 2006, 20 (5):1189-1201 doi: 10.1016/j.ymssp.2004.10.009 [19] Jardine A K S, Anderson P M, Mann D S. Application of the Weibull proportional hazards model to aircraft and marine engine failure data. Quality and Reliability Engineering International, 1987, 3 (2):77-82 doi: 10.1002/(ISSN)1099-1638 [20] Aalen O O. Further results on the non-parametric linear regression model in survival analysis. Statistics in Medicine, 1993, 12(17):1569-1588 doi: 10.1002/(ISSN)1097-0258 [21] Gorjian N, Ma L, Mittinty M, Yarlagadda P, Sun Y. The explicit hazard model-Part 1: theoretical development. In: Proceedings of the Prognostics & System Health Management Conference. Macau, China: IEEE, 2010. 167-179 http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5414565 [22] Sheldon M R. Generalized Poisson shock models. The Annals of Probability, 1981, 9 (5):896-898 doi: 10.1214/aop/1176994318 [23] Jarrow R A, Lando D, Turnbull S M. A Markov model for the term structure of credit risk spreads. Review of Financial Studies, 1997, 10 (2):481-523 doi: 10.1093/rfs/10.2.481 [24] Zhao X J, Fouladirad M, Bérenguer C, Bordes L. Condition-based inspection replacement policies for non-monotone deteriorating systems with environmental covariates. Reliability Engineering and System Safety, 2010, 95 (8):921-934 doi: 10.1016/j.ress.2010.04.005 [25] Banjevic D, Jardine A K S. Calculation of reliability function and remaining useful life for a Markov failure time process. IMA Journal of Management Mathematics, 2006, 17 (2):115-130 doi: 10.1093/imaman/dpi029 [26] Myers L E. Survival functions induced by stochastic covariate processes. Journal of Applied Probability, 1981, 18 (2):523-529 doi: 10.2307/3213300 [27] Berman S M, Frydman H. Parametric estimation of hazard functions with stochastic covariate processes. Sankhyā:The Indian Journal of Statistics, 1999, 61 (2):174-188 http://www.ams.org/mathscinet-getitem?mr=1714869 [28] Yu J C, Liu Y Y, Cai J W, Sandler D P, Zhou H B. Outcome-dependent sampling design and inference for Cox's proportional hazards Model. Journal of Statistical Planning and Inference, 2016, 178 (5):24-36 http://www2.cscc.unc.edu/impact7/node/771 [29] Lin H Z, He Y, Huang J. A global partial likelihood estimation in the additive Cox proportional hazards model. Journal of Statistical Planning and Inference, 2016, 169 (9):71-87 https://www.researchgate.net/publication/282528853_A_global_partial_likelihood_estimation_in_the_additive_Cox_proportional_hazards_model [30] Cao Y X, Yu J C, Liu Y Y. Optimal generalized case-cohort analysis with Cox's proportional hazards model. Acta Mathematicae Applicatae Sinica, 2015, 31 (3):841-854 doi: 10.1007/s10255-015-0555-4 [31] Luo J, Su Z. A note on variance estimation in the Cox proportional hazards model. Journal of Applied Statistics, 2013, 40 (5):1132-1139 doi: 10.1080/02664763.2013.780161 [32] Cox D R. Partial likelihood. Biometrika, 1975, 62 (2):269-276 doi: 10.1093/biomet/62.2.269 [33] Kalbfleisch J D, Prentice R L. Marginal likelihoods based on Cox's regression and life model. Biometrika, 1973, 60 (2):267-278 doi: 10.1093/biomet/60.2.267 [34] Makis V, Jardine A K S. Optimal replacement policy for a general model with imperfect repair. Journal of the Operational Research Society, 1992, 43 (2):111-120 doi: 10.1057/jors.1992.17 [35] Banjevic D, Jardine A K S, Makis V, Ennis M. A Control-Limit Policy And Software For Condition-Based Maintenance Optimization. INFOR:Information Systems and Operational Research, 2001, 39(1):32-50 doi: 10.1080/03155986.2001.11732424 [36] Jardine A K S, Banjevic D, Makis V. Optimal replacement policy and the structure of software for condition-based maintenance. Journal of Quality in Maintenance Engineering, 1997, 3 (2):109-119 doi: 10.1108/13552519710167728 [37] Jardine A K S, Joseph T, Banjevic D. Optimizing condition-based maintenance decisions for equipment subject to vibration monitoring. Journal of Quality in Maintenance Engineering, 1999, 5 (3):192-202 doi: 10.1108/13552519910282647 [38] Vlok P J, Wnek M, Zygmunt M. Utilising statistical residual life estimates of bearings to quantify the influence of preventive maintenance actions. Mechanical Systems and Signal Processing, 2004, 18 (4):833-847 doi: 10.1016/j.ymssp.2003.09.003 [39] Kay R. Proportional hazard regression models and the analysis of censored survival data. Applied Statistics, 1977, 26 (3):227-237 doi: 10.2307/2346962 [40] Kumar D. Proportional hazards modelling of repairable systems. Quality and Reliability Engineering International, 1995, 11 (5):361-369 doi: 10.1002/(ISSN)1099-1638 [41] Hanson T, Jara A, Zhao L P. A Bayesian semiparametric temporally-stratified proportional hazards model with spatial frailties. Bayesian Analysis, 2012, 7 (1):147-188 doi: 10.1214/12-BA705 [42] Fibrinogen Studies Collaboration. Measures to assess the prognostic ability of the stratified Cox proportional hazards model. Statistics in Medicine, 2009, 28 (3):389-411 doi: 10.1002/sim.v28:3 [43] Mau J. On a graphical method for the detection of time-dependent effects of covariates in survival data. Applied Statistics, 1986, 35 (3):245-255 doi: 10.2307/2348023 [44] Aalen O O. A linear regression model for the analysis of life times. Statistics in Medicine, 1989, 8 (8):907-925 doi: 10.1002/(ISSN)1097-0258 [45] Newby M. Why no additive hazards models? IEEE Transactions on Reliability, 1994, 43 (3):484-488 doi: 10.1109/24.326450 [46] Newby M. A critical look at some point-process models for repairable systems. IMA Journal of Management Mathematics, 1992, 4 (4):375-394 doi: 10.1093/imaman/4.4.375 [47] Álvarez E E, Ferrario J. Robust estimation in the additive hazards model. Communications in Statistics:Theory & Methods, 2016, 45 (4):906-921 doi: 10.1080/03610926.2013.853790 [48] Wightman D, Bendell T. Comparison of proportional hazards modeling, additive hazards modeling and proportional intensity modeling when applied to repairable system reliability. International Journal of Reliability, Quality and Safety Engineering, 1995, 2 (1):23-34 doi: 10.1142/S0218539395000046 [49] Badía F G, Berrade M D, Campos C A. Aging properties of the additive and proportional hazard mixing models. Reliability Engineering & System Safety, 2002, 78 (2):165-172 https://www.sciencedirect.com/science/article/pii/S0951832002001564 [50] Cox D R, Oakes D. Analysis of Survival Data. New York:Chapman and Hall, 1984. 32-37 [51] Lin Z, Fei H. A nonparametric approach to progressive stress accelerated life testing. IEEE Transactions on Reliability, 1991, 40 (2):173-176 doi: 10.1109/24.87123 [52] Mettas A. Modeling and analysis for multiple stress-type accelerated life data. In: Proceedings of the 2000 Annual Reliability and Maintainability Symposium. Los Angeles, CA, USA: IEEE 2000. 138-143 http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=816297 [53] Gosho M, Maruo K, Sato Y. Effect of covariate omission in Weibull accelerated failure time model:a caution. Biometrical Journal, 2014, 56 (6):991-1000 doi: 10.1002/bimj.201300006 [54] Newby M. Accelerated failure time models for reliability data analysis. Reliability Engineering & System Safety, 1988, 20 (3):187-197 https://www.researchgate.net/publication/222366290_Accelerated_failure_time_models_for_reliability_data_analysis_Reliability_Engineering_System_Safety_203_187-197 [55] Galanova N S, Lemeshko B Y, Chimitova E V. Using nonparametric goodness-of-fit tests to validate accelerated failure time models. Optoelectronics, Instrumentation and Data Processing, 2012, 48 (6):580-592 doi: 10.3103/S8756699012060064 [56] Wang H, Dai H, Fu B. Accelerated failure time models for censored survival data under referral bias. Biostatistics, 2013, 14 (2):313-326 doi: 10.1093/biostatistics/kxs041 [57] Yang M G, Chen L H, Dong G H. Semiparametric Bayesian accelerated failure time model with interval-censored data. Journal of Statistical Computation and Simulation, 2015, 85 (10):2049-2058 doi: 10.1080/00949655.2014.915400 [58] Wang S Y, Hu T, Xiang L M, Cui H J. Generalized M-estimation for the accelerated failure time model. Statistics, 2016, 50 (1):114-138 doi: 10.1080/02331888.2015.1032970 [59] Etezadi-Amoli J, Ciampi A. Extended hazard regression for censored survival data with covariates:a spline approximation for the baseline hazard function. Biometrics, 1987, 43 (1):181-192 doi: 10.2307/2531958 [60] Louzada-Neto F. Extended hazard regression model for reliability and survival analysis. Lifetime Data Analysis, 1997, 3 (4):367-381 doi: 10.1023/A:1009606229786 [61] Tseng Y K, Su Y R, Mao M, Wang J L. An extended hazard model with longitudinal covariates. Biometrika, 2015, 102 (1):135-150 doi: 10.1093/biomet/asu058 [62] Tseng Y K, Hsu K N, Yang Y F. A semiparametric extended hazard regression model with time-dependent covariates. Journal of Nonparametric Statistics, 2014, 26 (1):115-128 doi: 10.1080/10485252.2013.836521 [63] Shyur H J, Keng H, Ia-Ka I, Huang C L. Using extended hazard regression model to assess the probability of aviation event. Applied Mathematics and Computation, 2012, 218 (21):10647-10655 doi: 10.1016/j.amc.2012.04.029 [64] Bennett S. Analysis of survival data by the proportional odds model. Statistics in Medicine, 1983, 2 (2):273-277 doi: 10.1002/(ISSN)1097-0258 [65] Zahid F M, Ramzan S, Heumann C. Regularized proportional odds models. Journal of Statistical Computation and Simulation, 2015, 85 (2):251-168 doi: 10.1080/00949655.2013.814133 [66] Sinha S, Ma Y Y. Analysis of proportional odds models with censoring and errors-in-covariates. Journal of the American Statistical Association, 2016, 111 (515):1301-1312 doi: 10.1080/01621459.2015.1093943 [67] Zahid F M, Tutz G. Proportional odds models with high-dimensional data structure. International Statistical Review, 2013, 81 (3):388-406 doi: 10.1111/insr.v81.3 [68] Landers T L, Soroudi H E. Robustness of a semi-parametric proportional intensity model. IEEE Transactions on Reliability, 1991, 40 (2):161-164 doi: 10.1109/24.87120 [69] Lugtigheid D, Jardine Andrew K S, Jiang X Y. Optimizing the performance of a repairable system under a maintenance and repair contract. Quality and Reliability Engineering International, 2007, 23 (8):943-960 doi: 10.1002/(ISSN)1099-1638 [70] Kumar D, Westberg U. Proportional hazards modeling of time-dependent covariates using linear regression:a case study. IEEE Transactions on Reliability, 1996, 45 (3):386-392 doi: 10.1109/24.536990 [71] Jiang S T, Landers T L, Rhoads T R. Proportional intensity models robustness with overhaul intervals. Quality and Reliability Engineering International, 2006, 22 (3):251-263 doi: 10.1002/(ISSN)1099-1638 [72] Prentice R L, Williams B J, Peterson A V. On the regression analysis of multivariate failure time data. Biometrika, 1981, 68 (2):373-379 doi: 10.1093/biomet/68.2.373 [73] Yuan F Q, Kumar U. Proportional Intensity Model considering imperfect repair for repairable systems. International Journal of Performability Engineering, 2013, 9(2):163-174 http://paris.utdallas.edu/IJPE/Vol09/Issue02/pp.163-174%20Paper%204%20IJPE%20407.12%20Fuqing.pdf [74] Percy D F, Kobbacy K A H, Ascher H E. Using proportional-intensities models to schedule preventive-maintenance intervals. IMA Journal of Management Mathematics, 1998, 9 (3):289-302 doi: 10.1093/imaman/9.3.289 [75] Alkali B. Evaluation of generalised proportional intensities models with application to the maintenance of gas turbines. Quality and Reliability Engineering International, 2012, 28 (6):577-584 https://www.deepdyve.com/lp/wiley/evaluation-of-generalised-proportional-intensities-models-with-fZ2VbgUKzF [76] Syamsundar A, Achutha Naikan V N. Imperfect repair proportional intensity models for maintained systems. IEEE Transactions on Reliability, 2011, 60 (4):782-787 doi: 10.1109/TR.2011.2161110 [77] Wu J B. Modified restricted Liu estimator in logistic regression model. Computational Statistics, 2016, 31 (4):1557-1567 doi: 10.1007/s00180-015-0609-3 [78] Li C S. A test for the linearity of the nonparametric part of a semiparametric logistic regression model. Journal of Applied Statistics, 2016, 43 (3):461-475 doi: 10.1080/02664763.2015.1070803 [79] Sun Y, Ma L. Notes on "mechanical systems hazard estimation using condition monitoring"-Response to the letter to the editor by Daming Lin and Murray Wiseman. Mechanical Systems and Signal Processing, 2007, 21 (7):2950-2955 doi: 10.1016/j.ymssp.2007.06.004 [80] Cai G G, Chen X F, Li B, Chen B J, He Z J. Operation reliability assessment for cutting tools by applying a proportional covariate model to condition monitoring information. Sensors, 2012, 12 (10):12964-12987 http://www.mdpi.com/1424-8220/12/10/12988 [81] Pettitt A N, Daud I B. Investigating time dependence in Cox's proportional hazards model. Applied Statistics, 1990, 39 (3):313-329 doi: 10.2307/2347382 [82] Jardine A K S. Component and system replacement decisions. Image Sequence Processing and Dynamic Scene Analysis. Berlin:Springer-Verlag, 1983. 647-654 [83] Jardine A K S, Ralston P, Reid N, Stafford J. Proportional hazards analysis of diesel engine failure data. Quality and Reliability Engineering International, 1989, 5 (3):207-216 doi: 10.1002/(ISSN)1099-1638 [84] Jóźwiak I J. An introduction to the studies of reliability of systems using the Weibull proportional hazards model. Microelectronics Reliability, 1997, 37 (6):915-918 doi: 10.1016/S0026-2714(96)00285-5 [85] Jardine A K S, Anders M. Use of concomitant variables for reliability estimation. Maintenance Management International, 1985, 5 (4):135-140 https://www.researchgate.net/publication/291839844_USE_OF_CONCOMITANT_VARIABLES_FOR_RELIABILITY_ESTIMATION [86] Newby M. Perspective on Weibull proportional-hazards models. IEEE Transactions on Reliability, 1994, 43 (2):217-223 doi: 10.1109/24.294993 [87] Sha N J, Pan R. Bayesian analysis for step-stress accelerated life testing using Weibull proportional hazard model. Statistical Papers, 2014, 55 (3):715-726 doi: 10.1007/s00362-013-0521-2 [88] Zhang Q, Hua C, Xua G H. A mixture Weibull proportional hazard model for mechanical system failure prediction utilising lifetime and monitoring data. Mechanical Systems and Signal Processing, 2014, 43 (1-2):103-112 doi: 10.1016/j.ymssp.2013.10.013 [89] Gorjian N, Mittinty M, Ma L, Yarlagadda P, Sun Y. The explicit hazard model-Part 2: applications. In: Proceedings of the 2010 Prognostics and Health Management Conference. Macau, China: IEEE, 2010. 189-202 http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=5413493 [90] Cha J H, Mi J. Study of a stochastic failure model in a random environment. Journal of Applied Probability, 2007, 44 (1):151-163 doi: 10.1239/jap/1175267169 [91] Cha J H, Lee E Y. An extended stochastic failure model for a system subject to random shocks. Operations Research Letters, 2010, 38 (5):468-473 doi: 10.1016/j.orl.2010.06.004 [92] Cha J H, Mi J. On a stochastic survival model for a system under randomly variable environment. Methodology and Computing in Applied Probability, 2011, 13 (3):549-561 doi: 10.1007/s11009-010-9171-1 [93] Junca M, Sanchez Silva M. Optimal maintenance policy for a compound Poisson shock model. IEEE Transactions on Reliability, 2013, 62 (1):66-72 doi: 10.1109/TR.2013.2241193 [94] Jiang H Y. Parameter estimation of the Poisson shock model using masked data. International Journal of Pure and Applied Mathematics, 2011, 71 (4):559-569 https://www.researchgate.net/publication/264993406_Parameter_estimation_of_the_Poisson_shock_model_using_masked_data [95] Pandey A, Mitra M. Poisson shock models leading to new classes of non-monotonic aging life distributions. Microelectronics and Reliability, 2011, 51 (12):2412-2415 doi: 10.1016/j.microrel.2011.04.001 [96] Ghasemi A, Yacout S, Salah-Ouali M. Evaluating the reliability function and the mean residual life for equipment with unobservable states. IEEE Transactions on Reliability, 2010, 59 (1):45-54 doi: 10.1109/TR.2009.2034947 [97] Çinlar E. Markov additive processes. I. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1972, 24 (2):85-93 http://en.cnki.com.cn/Article_en/CJFDTOTAL-WHDY404.003.htm -
下载:
下载:
计量
- 文章访问数: 3138
- HTML全文浏览量: 753
- PDF下载量: 1879
- 被引次数: 0
