時(shí)域流信號(hào)的多任務(wù)稀疏貝葉斯動(dòng)態(tài)重構(gòu)方法研究
doi: 10.11999/JEIT190558 cstr: 32379.14.JEIT190558
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海軍航空大學(xué)信號(hào)與信息處理山東省重點(diǎn)實(shí)驗(yàn)室 煙臺(tái) 264001
Research on the Dynamic Sparse Bayesian Recovery of Multi-task Observed Streaming Signals in Time Domain
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Signal and Information Processing Key Laboratory in Shandong, Navy Aviation University, Yantai 264001, China
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摘要:
為了解決多任務(wù)觀測(cè)條件下時(shí)域流信號(hào)動(dòng)態(tài)重構(gòu)面臨的塊效應(yīng)問題,該文基于重疊正交變換(LOT)和稀疏貝葉斯學(xué)習(xí)的貪婪重構(gòu)框架先后提出了一種流信號(hào)多任務(wù)稀疏貝葉斯學(xué)習(xí)算法及其魯棒增強(qiáng)型的改進(jìn)算法,前者將LOT時(shí)域滑窗推廣到多任務(wù)條件下,通過貝葉斯概率建模將未知的噪聲精度的估計(jì)任務(wù)從信號(hào)重構(gòu)中解耦并省略,后者進(jìn)一步引入了重構(gòu)不確定性的度量,提高了算法的魯棒性和抑制誤差積累的能力?;诟?biāo)實(shí)測(cè)數(shù)據(jù)的實(shí)驗(yàn)結(jié)果表明,相比多任務(wù)重構(gòu)領(lǐng)域代表性較強(qiáng)的時(shí)間多稀疏貝葉斯學(xué)習(xí)(TMSBL)和多任務(wù)壓縮感知(MT-CS)算法,本文算法在不同信噪比、觀測(cè)數(shù)目和任務(wù)數(shù)目條件下具有顯著更高的重構(gòu)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 目標(biāo)函數(shù)、中間變量及超參數(shù)估計(jì)公式列表
目標(biāo)函數(shù)及其分解形式(其中${\tilde a_l} = 2{a_l} + M\left( {2d + 2} \right)$): $L\left( { { { {\bar{ \alpha } } }_t} } \right) = { { - \displaystyle\sum\limits_{l = 1}^L {\left\{ { { {\tilde a}_l}\ln \left( { {\bar{ y} }_{t,l}^{\rm{T} }{\bar{ C} }_l^{ - 1}{ { {\bar{ y} } }_{t,l} } + 2{b_l} } \right) + \ln \left| { { { {\bar{ C} } }_l} } \right|} \right\} } } / 2} \;\;\quad (29)$ $L\left( { { { {\bar{ \alpha } } }_t} } \right) = { { - \displaystyle\sum\limits_{l = 1}^L {\left\{ { { {\tilde a}_l}\ln \left( {1 - \frac{ { { {q_{j,l}^2} / { {g_{j,l} } } } } }{ { {\alpha _j} + {s_{j,l} } } }} \right) + \ln \left( {1 - \alpha _j^{ - 1}{s_{j,l} } } \right)} \right\} } } / 2}\;\; (30)$ 中間變量: ${{\bar{ C}}_l} = {{I}} + {{{\varPsi }}_l}{\bar{ A}}_t^{ - 1}{{\varPsi }}_l^{\rm{T}},_{}^{}{{\bar{ C}}_{ - j,l}} = {{I}} + \displaystyle\sum\limits_{k \ne j} {\alpha _k^{ - 1}{{{\psi }}_{k,l}}{{\psi }}_{k,l}^{\rm{T}}} \left( {{\rm{SMT - SBL}}} \right)\;\; (31)$ ${{\bar{ C}}_l} = {{\hat{ \varOmega }}_{t,l}} + {{{\varPsi }}_l}{\bar{ A}}_t^{ - 1}{{\varPsi }}_l^{\rm{T}},{{\bar{ C}}_{ - j,l}} = {{\hat{ \varOmega }}_{t,l}} + \displaystyle\sum\limits_{k \ne j} {\alpha _k^{ - 1}{{{\psi }}_{k,l}}{{\psi }}_{k,l}^{\rm{T}}} \left( {{\rm{SMT - RSBL}}} \right)\;\;\;(32)$ ${s_{j,l}} = {{\psi }}_{j,l}^{\rm{T}}{\bar{ C}}_{ - j,l}^{ - 1}{{{\psi }}_{j,l}},_{}^{}{q_{j,l}} = {{\psi }}_{j,l}^{\rm{T}}{\bar{ C}}_{ - j,l}^{ - 1}{{\bar{ y}}_{t,l}},_{}^{}{g_{j,l}} = {\bar{ y}}_{t,l}^{\rm{T}}{\bar{ C}}_{ - j,l}^{ - 1}{{\bar{ y}}_{t,l}} + 2{b_l}\;\;\; (33)$ ${S_{j,l}} = {{\psi }}_{j,l}^{\rm{T}}{\bar{ C}}_l^{ - 1}{{{\psi }}_{j,l}},_{}^{}{Q_{j,l}} = {{\psi }}_{j,l}^{\rm{T}}{\bar{ C}}_l^{ - 1}{{\bar{ y}}_{t,l}},_{}^{}{G_l} = {\bar{ y}}_{t,l}^{\rm{T}}{\bar{ C}}_l^{ - 1}{{\bar{ y}}_{t,l}} + 2{b_l}\;\;\; (34)$ ${\alpha _j}$更新公式: ${\alpha _j} = \left\{ \begin{aligned} & {L / { {\theta _j} } },{\theta _j} > 0\\ & \infty ,\quad {\simfont\text{其他} } \end{aligned} \right.\;\;\;\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad (35)$ ${\theta _j} = \displaystyle\sum\limits_{l = 1}^L {\frac{ { { {\tilde a}_l}({ {q_{j,l}^2} / { {g_{j,l} } } }) - {s_{j,l} } } }{ { {s_{j,l} }({s_{j,l} } - { {q_{j,l}^2} / { {g_{j,l} } } })} } } \;\;\;\;\qquad\qquad\qquad\qquad\qquad\qquad\quad (36)$ 下載: 導(dǎo)出CSV
表 2 相關(guān)的快速更新公式列表
添加原子${{{\psi }}_{j,l}}$ 刪除原子${{{\psi }}_{j,l}}$ 維持原子${{{\psi }}_{j,l}}$ 說明 $ {\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}}:$ 刪除情形中: $ \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$j$個(gè)對(duì)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$j$個(gè)元素。 通用公式: 通用公式: 通用公式: 維持情形中: $ \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}$ ${\hat \varSigma _{jj,l}},{{\hat{ \varSigma }}_{j,l}},{\mu _{j,l}}$與前述相同,${\gamma _{j,l}} = {\left[ {{{\hat \varSigma }_{jj,l}} + {{\left( {{{\tilde \alpha }_j} - {\alpha _j}} \right)}^{ - 1}}} \right]^{ - 1}}$。 下載: 導(dǎo)出CSV
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