Research on the Dynamic Sparse Bayesian Recovery of Multi-task Observed Streaming Signals in Time Domain
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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)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。
Abstract:To eliminate the blocking effects in the dynamic recovery of the streaming signals observed from multiple tasks in time domain, a streaming multi-task sparse Bayesian learning based algorithm and its robust enhanced version are proposed in this paper, where the former extends Lapped Orthogonal Transform (LOT) sliding window in time domain to multi-task condition, and decouples the estimation of unknown noise accuracy from signal reconstruction by Bayesian probability modeling and omits it, the latter further introduces the measurement of reconstructed uncertainty, which improves the robustness of the algorithm and the ability to suppress the accumulation of errors. Experimental results based on measured meteorological data shows that the proposed algorithms have significantly higher reconstruction accuracy, success rate and running speed than the representative algorithms in the field of compressed sensing from multiple measurement vectors, namely, the Temporal Multiple Sparse Bayesian Learning (TMSBL) algorithm and the Multi-Task-Compressed Sensing (MT-CS) algorithm, under different conditions of Signal-to-Noise Ratios, number of observations and tasks.
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Key words:
- Signal processing /
- Streaming signals /
- Multi-task /
- Sparse Bayesian /
- Blocking effects
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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${\tilde a_l} = 2{a_l} + M\left( {2d + 2} \right)$)： $L\left( { { { {\bar{ \alpha } } }_t} } \right) = { { - \displaystyle\sum\limits_{l = 1}^L {\left\{ { { {\tilde a}_l}\ln \left( { {\bar{ y} }_{t,l}^{\rm{T} }{\bar{ C} }_l^{ - 1}{ { {\bar{ y} } }_{t,l} } + 2{b_l} } \right) + \ln \left| { { { {\bar{ C} } }_l} } \right|} \right\} } } / 2} \;\;\quad (29)$ $L\left( { { { {\bar{ \alpha } } }_t} } \right) = { { - \displaystyle\sum\limits_{l = 1}^L {\left\{ { { {\tilde a}_l}\ln \left( {1 - \frac{ { { {q_{j,l}^2} / { {g_{j,l} } } } } }{ { {\alpha _j} + {s_{j,l} } } }} \right) + \ln \left( {1 - \alpha _j^{ - 1}{s_{j,l} } } \right)} \right\} } } / 2}\;\; (30)$ 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： ${\alpha _j} = \left\{ \begin{aligned} & {L / { {\theta _j} } },{\theta _j} > 0\\ & \infty ,\quad {\simfont\text{其他} } \end{aligned} \right.\;\;\;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad (35)$ ${\theta _j} = \displaystyle\sum\limits_{l = 1}^L {\frac{ { { {\tilde a}_l}({ {q_{j,l}^2} / { {g_{j,l} } } }) - {s_{j,l} } } }{ { {s_{j,l} }({s_{j,l} } - { {q_{j,l}^2} / { {g_{j,l} } } })} } } \;\;\;\;\qquad\qquad\qquad\qquad\qquad\qquad\quad (36)$ 
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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{ \varSigma }}}}_{t,l}^{{\bar{ w}}} = [{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}} + {{\hat \varSigma }_{jj,l}}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{j,l}}{{\psi }}_{j,l}^{\rm{T}}{{{\varPsi }}_l}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}},\\ - {{\hat \varSigma }_{jj,l}}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{j,l}}; - {{\hat \varSigma }_{jj,l}}{{\psi }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}},{{\hat \varSigma }_{jj,l}}]\\ {\tilde{ \mu }}_{t,l}^{{\bar{ w}}} = [{{\mu }}_{t,l}^{{\bar{ w}}} + {\mu _{j,l}}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{j,l}};{\mu _{j,l}}]\\ {{{e}}_{j,l}} = {{{\psi }}_{j,l}} - {{{\varPsi }}_l}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{j,l}} \end{array}$ $ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} + {{{{({\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{k,l}})}^2}} / {{{\hat \varSigma }_{jj,l}}}}\\ {{\tilde Q}_{k,l}} = {Q_{k,l}} + {{{\mu _{j,l}}{\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{k,l}}} / {{{\hat \varSigma }_{jj,l}}}}\\ {{\tilde G}_l} = {G_l} + {{{{({\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\bar{ y}}}_{t,l}})}^2}} / {{{\hat \varSigma }_{jj,l}}}} \end{array}$ $ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} + {\gamma _{j,l}}{({\hat{ \varSigma }}_{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{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\psi }}_{k,l}}\\ {{\tilde G}_l} = {G_l} + {\gamma _{j,l}}{({\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{{{\bar{ y}}}_{t,l}})^2} \end{array}$ $ \begin{array}{l} {{\hat \varSigma }_{jj,l}} = {\left( {{{\tilde \alpha }_j} + {S_{j,l}}} \right)^{ - 1}},\\ {\mu _{j,l}} = {{\hat \varSigma }_{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{ \varSigma }}}}_{t,l}^{{\bar{ w}}} = [{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}} + {{\hat \varSigma }_{jj,l}}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\psi }}_{j,l}}{{\psi }}_{j,l}^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\varPsi }}_l}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}},\\ - {{\hat \varSigma }_{jj,l}}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\psi }}_{j,l}}; - {{\hat \varSigma }_{jj,l}}{{\psi }}_{j,l}^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{\varPsi }}_l^{}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}},{{\hat \varSigma }_{jj,l}}]\\ {\tilde{ \mu }}_{t,l}^{{\bar{ w}}} = [{{\mu }}_{t,l}^{{\bar{ w}}} + {\mu _{j,l}}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\psi }}_{j,l}};{\mu _{j,l}}]\\ {{{e}}_{j,l}} = ({\hat{ \varOmega }}_{t,l}^{ - 1} - {\hat{ \varOmega }}_{t,l}^{ - 1}{{{\varPsi }}_l}{\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}){{{\psi }}_{j,l}} \end{array}$ $ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} + {{{{({\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\psi }}_{k,l}})}^2}} / {{{\hat \varSigma }_{jj,l}}}}\\ {{\tilde Q}_{k,l}} = {Q_{k,l}} + {{{\mu _{j,l}}{\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\psi }}_{k,l}}} / {{{\hat \varSigma }_{jj,l}}}}\\ {{\tilde G}_l} = {G_l} + {{{{({\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\bar{ y}}}_{t,l}})}^2}} / {{{\hat \varSigma }_{jj,l}}}} \end{array}$ $ \begin{array}{l} {{\tilde S}_{k,l}} = {S_{k,l}} + {\gamma _{j,l}}{({\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\psi }}_{k,l}})^2}\\ {{\tilde Q}_{k,l}} = {Q_{k,l}} + {\gamma _{j,l}}{\mu _{j,l}}{\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\psi }}_{k,l}}\\ {{\tilde G}_l} = {G_l} + {\gamma _{j,l}}{({\hat{ \varSigma }}_{j,l}^{\rm{T}}{{\varPsi }}_l^{\rm{T}}{\hat{ \varOmega }}_{t,l}^{ - 1}{{{\bar{ y}}}_{t,l}})^2} \end{array}$ ${\hat \varSigma _{jj,l}}$是${\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}$的第$j$个对角元素，${{\hat{ \varSigma }}_{j,l}}$是${\hat{ \varSigma }}_{t,l}^{{\bar{ w}}}$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 \varSigma }_{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 \varSigma }_{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{ \varSigma }}}}_{t,l}^{{\bar{ w}}} = {\hat{ \varSigma }}_{t,l}^{{\bar{ w}}} - {{{{{\hat{ \varSigma }}}_{j,l}}{\hat{ \varSigma }}_{j,l}^{\rm{T}}} / {{{\hat \varSigma }_{jj,l}}}}\\ {\tilde{ \mu }}_{t,l}^{{\bar{ w}}} = {{\mu }}_{t,l}^{{\bar{ w}}} - {\mu _{j,l}}{{{{{\hat{ \varSigma }}}_{j,l}}} / {{{\hat \varSigma }_{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{ \varSigma }}}}_{t,l}^{{\bar{ w}}} = {\hat{ \varSigma }}_{t,l}^{{\bar{ w}}} - {\gamma _{j,l}}{{{\hat{ \varSigma }}}_{j,l}}{\hat{ \varSigma }}_{j,l}^{\rm{T}}\\ {\tilde{ \mu }}_{t,l}^{{\bar{ w}}} = {{\mu }}_{t,l}^{{\bar{ w}}} - {\gamma _{j,l}}{\mu _{j,l}}{{{\hat{ \varSigma }}}_{j,l}} \end{array}$ ${\hat \varSigma _{jj,l}},{{\hat{ \varSigma }}_{j,l}},{\mu _{j,l}}$与前述相同，${\gamma _{j,l}} = {\left[ {{{\hat \varSigma }_{jj,l}} + {{\left( {{{\tilde \alpha }_j} - {\alpha _j}} \right)}^{ - 1}}} \right]^{ - 1}}$。
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