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时域流信号的多任务稀疏贝叶斯动态重构方法研究

董道广 芮国胜 田文飚

引用本文: 董道广, 芮国胜, 田文飚. 时域流信号的多任务稀疏贝叶斯动态重构方法研究[J]. 电子与信息学报, doi: 10.11999/JEIT190558 shu
Citation:  Daoguang DONG, Guosheng RUI, Wenbiao TIAN. Research on the Dynamic Sparse Bayesian Recovery of Multi-task Observed Streaming Signals in Time Domain[J]. Journal of Electronics and Information Technology, doi: 10.11999/JEIT190558 shu

时域流信号的多任务稀疏贝叶斯动态重构方法研究

    作者简介: 董道广: 男,1990年生,博士生,研究方向为贝叶斯统计学习、压缩感知和蒸发波导反演;
    芮国胜: 男,1968年生,教授,博士生导师,主要研究方向为混沌通信系统及现代滤波理论;
    田文飚: 男,1987年生,副教授,主要研究方向为压缩感知及蒸发波导反演
    通讯作者: 董道广,sikongyu@yeah.net
  • 基金项目: 国家自然科学基金(41606117, 41476089, 61671016)

摘要: 为了解决多任务观测条件下时域流信号动态重构面临的块效应问题,该文基于重叠正交变换(LOT)变换和稀疏贝叶斯学习的贪婪重构框架先后提出了一种流信号多任务稀疏贝叶斯学习算法及其鲁棒增强型的改进算法,前者将LOT时域滑窗推广到多任务条件下,通过贝叶斯概率建模将未知的噪声精度的估计任务从信号重构中解耦并省略,后者进一步引入了重构不确定性的度量,提高了算法的鲁棒性和抑制误差积累的能力。基于浮标实测数据的实验结果表明,相比多任务重构领域代表性较强的时间多稀疏贝叶斯学习(TMSBL)和多任务压缩感知(MT-CS)算法,本文算法在不同信噪比、观测数目和任务数目条件下具有显著更高的重构精度、成功率和效率。

English

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  • 图 1  时域流信号的多任务在线滑窗观测

    图 2  不同算法的SER随观测数目变化的比较结果

    图 3  不同算法的SER随信噪比变化的比较结果

    图 4  不同算法的SR随观测数目变化的比较结果

    图 5  不同算法的SR随SNR变化的比较结果

    图 6  不同算法的运行时间随观测数目变化的比较结果

    图 7  不同算法的运行时间随SNR变化的比较结果

    表 1  目标函数、中间变量及超参数估计公式列表

    目标函数及其分解形式(其中${\tilde a_l} = 2{a_l} + M\left( {2d + 2} \right)$):
    $L\left( { { { {\bar{ \alpha } } }_t} } \right) = { { - \displaystyle\sum\limits_{l = 1}^L {\left\{ { { {\tilde a}_l}\ln \left( { {\bar{ y} }_{t,l}^{\rm{T} }{\bar{ C} }_l^{ - 1}{ { {\bar{ y} } }_{t,l} } + 2{b_l} } \right) + \ln \left| { { { {\bar{ C} } }_l} } \right|} \right\} } } / 2} \;\;(29)$
    $L\left( { { { {\bar{ \alpha } } }_t} } \right) = { { - \displaystyle\sum\limits_{l = 1}^L {\left\{ { { {\tilde a}_l}\ln \left( {1 - \frac{ { { {q_{j,l}^2} / { {g_{j,l} } } } } }{ { {\alpha _j} + {s_{j,l} } } }} \right) + \ln \left( {1 - \alpha _j^{ - 1}{s_{j,l} } } \right)} \right\} } } / 2}\;\; (30)$
    中间变量:
    ${{\bar{ C}}_l} = {{I}} + {{{\varPsi }}_l}{\bar{ A}}_t^{ - 1}{{\varPsi }}_l^{\rm{T}},_{}^{}{{\bar{ C}}_{ - j,l}} = {{I}} + \displaystyle\sum\limits_{k \ne j} {\alpha _k^{ - 1}{{{\psi }}_{k,l}}{{\psi }}_{k,l}^{\rm{T}}} \left( {{\rm{SMT - SBL}}} \right)\;\; (31)$
    ${{\bar{ C}}_l} = {{\hat{ \varOmega }}_{t,l}} + {{{\varPsi }}_l}{\bar{ A}}_t^{ - 1}{{\varPsi }}_l^{\rm{T}},{{\bar{ C}}_{ - j,l}} = {{\hat{ \varOmega }}_{t,l}} + \displaystyle\sum\limits_{k \ne j} {\alpha _k^{ - 1}{{{\psi }}_{k,l}}{{\psi }}_{k,l}^{\rm{T}}} \left( {{\rm{SMT - RSBL}}} \right)\;\;\;(32)$
    ${s_{j,l}} = {{\psi }}_{j,l}^{\rm{T}}{\bar{ C}}_{ - j,l}^{ - 1}{{{\psi }}_{j,l}},_{}^{}{q_{j,l}} = {{\psi }}_{j,l}^{\rm{T}}{\bar{ C}}_{ - j,l}^{ - 1}{{\bar{ y}}_{t,l}},_{}^{}{g_{j,l}} = {\bar{ y}}_{t,l}^{\rm{T}}{\bar{ C}}_{ - j,l}^{ - 1}{{\bar{ y}}_{t,l}} + 2{b_l}\;\;\; (33)$
    ${S_{j,l}} = {{\psi }}_{j,l}^{\rm{T}}{\bar{ C}}_l^{ - 1}{{{\psi }}_{j,l}},_{}^{}{Q_{j,l}} = {{\psi }}_{j,l}^{\rm{T}}{\bar{ C}}_l^{ - 1}{{\bar{ y}}_{t,l}},_{}^{}{G_l} = {\bar{ y}}_{t,l}^{\rm{T}}{\bar{ C}}_l^{ - 1}{{\bar{ y}}_{t,l}} + 2{b_l}\;\;\; (34)$
    ${\alpha _j}$更新公式:
    ${\alpha _j} = \left\{ \begin{aligned} & {L / { {\theta _j} } },{\theta _j} > 0\\ & \infty ,{\simfont\text{其他} } \end{aligned} \right.\;\;\;(35)$
    ${\theta _j} = \displaystyle\sum\limits_{l = 1}^L {\frac{{{{\tilde a}_l}({{q_{j,l}^2} / {{g_{j,l}}}}) - {s_{j,l}}}}{{{s_{j,l}}({s_{j,l}} - {{q_{j,l}^2} / {{g_{j,l}}}})}}} \;\;\;\;(36)$
    下载: 导出CSV

    表 2  相关的快速更新公式列表

    添加原子${{{\psi }}_{j,l}}$删除原子${{{\psi }}_{j,l}}$维持原子${{{\psi }}_{j,l}}$说明
    $ {\rm{SMT - SBL}}:$$ {\rm{SMT - SBL}}:$$ {\rm{SMT - SBL}}:$添加情形中:
    $ \begin{array}{l} {{\tilde{\hat{ \Sigma }}}}_{t,l}^{{\bar{ w}}} = [{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}} + {{\hat \Sigma }_{jj,l}}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{j,l}}{{\psi }}_{j,l}^{\rm{T}}{{{\varPsi }}_l}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}},\\ - {{\hat \Sigma }_{jj,l}}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{j,l}}; - {{\hat \Sigma }_{jj,l}}{{\psi }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}},{{\hat \Sigma }_{jj,l}}]\\ {\tilde{ \mu }}_{t,l}^{{\bar{ w}}} = [{{\mu }}_{t,l}^{{\bar{ w}}} + {\mu _{j,l}}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{j,l}};{\mu _{j,l}}]\\ {{{e}}_{j,l}} = {{{\psi }}_{j,l}} - {{{\varPsi }}_l}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{j,l}} \end{array}$$ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} + {{{{({\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{k,l}})}^2}} / {{{\hat \Sigma }_{jj,l}}}}\\ {{\tilde Q}_{k,l}} = {Q_{k,l}} + {{{\mu _{j,l}}{\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{k,l}}} / {{{\hat \Sigma }_{jj,l}}}}\\ {{\tilde G}_l} = {G_l} + {{{{({\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\bar{ y}}}_{t,l}})}^2}} / {{{\hat \Sigma }_{jj,l}}}} \end{array}$$ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} + {\gamma _{j,l}}{({\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{k,l}})^2}\\ {{\tilde Q}_{k,l}} = {Q_{k,l}} + {\gamma _{j,l}}{\mu _{j,l}}{\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{k,l}}\\ {{\tilde G}_l} = {G_l} + {\gamma _{j,l}}{({\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\bar{ y}}}_{t,l}})^2} \end{array}$$ \begin{array}{l} {{\hat \Sigma }_{jj,l}} = {\left( {{{\tilde \alpha }_j} + {S_{j,l}}} \right)^{ - 1}},\\ {\mu _{j,l}} = {{\hat \Sigma }_{jj,l}}{Q_{j,l}} \end{array}$
    $ {\rm{SMT - RSBL}}:$$ {\rm{SMT - RSBL}}:$$ {\rm{SMT - RSBL}}:$删除情形中:
    $ \begin{array}{l} {{\tilde{\hat{ \Sigma }}}}_{t,l}^{{\bar{ w}}} = [{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}} + {{\hat \Sigma }_{jj,l}}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\psi }}_{j,l}}{{\psi }}_{j,l}^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\varPsi }}_l}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}},\\ - {{\hat \Sigma }_{jj,l}}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\psi }}_{j,l}}; - {{\hat \Sigma }_{jj,l}}{{\psi }}_{j,l}^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{\varPsi }}_l^{}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}},{{\hat \Sigma }_{jj,l}}]\\ {\tilde{ \mu }}_{t,l}^{{\bar{ w}}} = [{{\mu }}_{t,l}^{{\bar{ w}}} + {\mu _{j,l}}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\psi }}_{j,l}};{\mu _{j,l}}]\\ {{{e}}_{j,l}} = ({\hat{ \Omega }}_{t,l}^{ - 1} - {\hat{ \Omega }}_{t,l}^{ - 1}{{{\varPsi }}_l}{\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}){{{\psi }}_{j,l}} \end{array}$$ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} + {{{{({\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\psi }}_{k,l}})}^2}} / {{{\hat \Sigma }_{jj,l}}}}\\ {{\tilde Q}_{k,l}} = {Q_{k,l}} + {{{\mu _{j,l}}{\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\psi }}_{k,l}}} / {{{\hat \Sigma }_{jj,l}}}}\\ {{\tilde G}_l} = {G_l} + {{{{({\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\bar{ y}}}_{t,l}})}^2}} / {{{\hat \Sigma }_{jj,l}}}} \end{array}$$ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} + {\gamma _{j,l}}{({\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\psi }}_{k,l}})^2}\\ {{\tilde Q}_{k,l}} = {Q_{k,l}} + {\gamma _{j,l}}{\mu _{j,l}}{\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\psi }}_{k,l}}\\ {{\tilde G}_l} = {G_l} + {\gamma _{j,l}}{({\hat{ \Sigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \Omega }}_{t,l}^{ - 1}{{{\bar{ y}}}_{t,l}})^2} \end{array}$${\hat \Sigma _{jj,l}}$是${\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}$的第$j$个对角元素,${{\hat{ \Sigma }}_{j,l}}$是${\hat{ \Sigma }}_{t,l}^{{\bar{ w}}}$的第$j$列,${\mu _{j,l}}$是${{\mu }}_{t,l}^{{\bar{ w}}}$的第$j$个元素。
    通用公式:通用公式:通用公式:维持情形中:
    $ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} - {{\hat \Sigma }_{jj,l}}{({{\psi }}_{k,l}^{\rm{T}}{{{e}}_{j,l}})^2}\\ {{\tilde Q}_{k,l}} = {Q_{k,l}} - {\mu _{j,l}}{{\psi }}_{k,l}^{\rm{T}}{{{e}}_{j,l}}\\ {{\tilde G}_l} = {G_l} - {{\hat \Sigma }_{jj,l}}{({\bar{ y}}_{t,l}^{\rm{T}}{{{e}}_{j,l}})^2}\\ 2\Delta L = \sum\nolimits_{l = 1}^L {\ln \left[ {{{{{\tilde \alpha }_j}} / {\left( {{{\tilde \alpha }_j} + {s_{j,l}}} \right)}}} \right]} \\ \mathop {}\nolimits - \sum\nolimits_{l = 1}^L {{{\tilde a}_l}\ln \left[ {1 - {{\left( {{{q_{j,l}^2} / {{g_{j,l}}}}} \right)} / {\left( {{{\tilde \alpha }_j} + {s_{j,l}}} \right)}}} \right]} \end{array}$$ \begin{array}{l} 2\Delta L = - \sum\nolimits_{l = 1}^L {\ln \left( {1 - {{{S_{j,l}}} / {{\alpha _j}}}} \right)} \\ \mathop {}\nolimits \mathop {}\nolimits - \sum\nolimits_{l = 1}^L {{{\tilde a}_l}} \ln \left[ {1 + \frac{{{{Q_{j,l}^2} / {{G_l}}}}}{{{\alpha _j} - {S_{j,l}}}}} \right]\\ {{\tilde{\hat{ \Sigma }}}}_{t,l}^{{\bar{ w}}} = {\hat{ \Sigma }}_{t,l}^{{\bar{ w}}} - {{{{{\hat{ \Sigma }}}_{j,l}}{\hat{ \Sigma }}_{j,l}^{\rm{T}}} / {{{\hat \Sigma }_{jj,l}}}}\\ {\tilde{ \mu }}_{t,l}^{{\bar{ w}}} = {{\mu }}_{t,l}^{{\bar{ w}}} - {\mu _{j,l}}{{{{{\hat{ \Sigma }}}_{j,l}}} / {{{\hat \Sigma }_{jj,l}}}} \end{array}$$ \begin{array}{l} 2\Delta L = \sum\nolimits_{l = 1}^L {\left( {{{\tilde a}_l} - 1} \right)\ln \left( {1 + \frac{{{\alpha _j} - {{\tilde \alpha }_j}}}{{{\alpha _j}{{\tilde \alpha }_j}}}} \right)} \\ + \sum\nolimits_{l = 1}^L {{{\tilde a}_l}} \ln \frac{{\left[ {\left( {{\alpha _j} + {s_{j,l}}} \right){g_{j,l}} - q_{j,l}^2} \right]{{\tilde \alpha }_j}}}{{\left[ {\left( {{{\tilde \alpha }_j} + {s_{j,l}}} \right){g_{j,l}} - q_{j,l}^2} \right]{\alpha _j}}}\\ {{\tilde{\hat{ \Sigma }}}}_{t,l}^{{\bar{ w}}} = {\hat{ \Sigma }}_{t,l}^{{\bar{ w}}} - {\gamma _{j,l}}{{{\hat{ \Sigma }}}_{j,l}}{\hat{ \Sigma }}_{j,l}^{\rm{T}}\\ {\tilde{ \mu }}_{t,l}^{{\bar{ w}}} = {{\mu }}_{t,l}^{{\bar{ w}}} - {\gamma _{j,l}}{\mu _{j,l}}{{{\hat{ \Sigma }}}_{j,l}} \end{array}$${\hat \Sigma _{jj,l}},{{\hat{ \Sigma }}_{j,l}},{\mu _{j,l}}$与前述相同,${\gamma _{j,l}} = {\left[ {{{\hat \Sigma }_{jj,l}} + {{\left( {{{\tilde \alpha }_j} - {\alpha _j}} \right)}^{ - 1}}} \right]^{ - 1}}$。
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  • 通讯作者:  董道广, sikongyu@yeah.net
  • 收稿日期:  2019-07-25
  • 录用日期:  2020-03-26
  • 网络出版日期:  2020-04-24
通讯作者: 陈斌, bchen63@163.com
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    沈阳化工大学材料科学与工程学院 沈阳 110142

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