查看“︁協方差交叉”︁的源代码

{{orphan|time=2018-12-02T14:31:00+00:00}}
{{technical|date=2018年12月}}
{{NoteTA|G1=Math}}
'''協方差交叉'''（Covariance intersection）是在[[卡尔曼滤波]]中，二個[[狀態變數]]之間不確定其協方差時，合併其估測值的[[算法]]<ref name="uthesis">{{cite thesis |degree=Ph.D. |first=Jeffrey |last=Uhlmann |title=Dynamic Map Building and Localization: New Theoretical Foundations |publisher=University of Oxford |date=1995}}</ref><ref>{{cite conference | first = Sonia | last = Marques | url = http://islab.isr.ist.utl.pt/htdocs/workshop4/presentations/sonia.pdf | title = Covariance intersection algorithm for formation flying spacecraft navigation from RF measurements | conference = 4 ISLAB workshop | date = 12 November 2007 | access-date = 2018-12-02 | archive-date = 2020-08-14 | archive-url = https://web.archive.org/web/20200814224118/http://islab.isr.ist.utl.pt/htdocs/workshop4/presentations/sonia.pdf }}</ref><ref>{{cite journal | first1 = Simon J. | last1 = Julier | first2 = Jeffrey K. | last2 = Uhlmann | year = 2007 | citeseerx = 10.1.1.106.8515 | title = Using covariance intersection for SLAM | journal = Robotics and Autonomous Systems | volume = 55 | issue = 7 | pages = 3–20 | doi=10.1016/j.robot.2006.06.011}}</ref><ref>{{cite conference | first1 = Lingji | last1 = Chen | first2 = Pablo O. | last2 = Arambel | first3 = Raman K. | last3 = Mehra | year = 2002 | url = http://www.isif.org/fusion/proceedings/fusion02CD/pdffiles/papers/W1A02.pdf | title = Fusion under unknown correlation - Covariance intersection as a special case | conference = International Conference on Information Fusion 2002 | access-date = 2018-12-02 | archive-date = 2013-11-09 | archive-url = https://web.archive.org/web/20131109075749/http://www.isif.org/fusion/proceedings/fusion02CD/pdffiles/papers/W1A02.pdf }}</ref>。

==規格==
資訊項'''a'''及'''b'''已知，要融合成資訊項'''c'''。已知'''a'''和'''b'''的[[平均数]]/協方差 <math>\hat a</math>, <math>A</math>及<math>\hat b</math>, <math>B</math>，但是交叉[[相關 (概率論)|相關]]未知。協方差交叉可以更新'''c'''的平均数/協方差為

: <math>C^{-1} = \omega A^{-1} + (1-\omega) B^{-1} \, ,</math>

: <math>\hat c = C(\omega A^{-1} \hat a + (1-\omega)B^{-1} \hat b) \, .</math>

其中''&omega;''是計算讓特定範數（例如logdet或[[跡]]）最小化。若是較高[[維度]]問題需要求解[[最佳化問題]]，不過在較低維度下有[[解析解]]<ref>{{cite conference | first1 = Marc | last1 = Reinhardt | first2 = Benjamin | last2 = Noack | first3 = Uwe D. | last3 = Hanebeck | year = 2012 | url = http://isas.uka.de/Publikationen/Fusion12_Reinhardt-FastCI.pdf | title = Closed-form Optimization of Covariance Intersection for Low-dimensional Matrices | conference = International Conference on Information Fusion 2012 | access-date = 2018-12-02 | archive-date = 2016-03-03 | archive-url = https://web.archive.org/web/20160303230006/http://isas.uka.de/Publikationen/Fusion12_Reinhardt-FastCI.pdf }}</ref>。協方差交叉可以用來取代傳統的卡尔曼更新方程，確定所得的估測值是保守的，不論二個估測值之間的相关如何，而協方差會依選定的範而出現嚴格的未遞增<!--。The use of a fixed measure is necessary for rigor to ensure that a sequence of updates does not cause the filtered [[协方差]] to increase.--><ref name="uthesis"/><ref name="infofuse">{{cite journal |publisher=Elsevier |first=Jeffrey |last=Uhlmann |title=Covariance Consistency Methods for Fault-Tolerant Distributed Data Fusion |volume=4 | pages=201–215 | year=2003}}</ref>。

== 優點 ==
根據最近的研究論文<ref name=":0">Wangyan Li, Zidong Wang, Guoliang Wei, Lifeng Ma, Jun Hu, and Derui Ding. "A Survey on Multi-Sensor Fusion and Consensus Filtering for Sensor Networks." ''Discrete Dynamics in Nature and Society'', vol. 2015, Article ID 683701, 12 pages, 2015. [http://www.hindawi.com/journals/ddns/2015/683701/] {{Wayback|url=http://www.hindawi.com/journals/ddns/2015/683701/ |date=20211108015234 }}</ref>及<ref>{{Cite journal|title = Sequential covariance intersection fusion Kalman filter|url = http://www.sciencedirect.com/science/article/pii/S0020025511006232|journal = Information Sciences|date = 2012-04-15|pages = 293–309|volume = 189|doi = 10.1016/j.ins.2011.11.038|first = Zili|last = Deng|first2 = Peng|last2 = Zhang|first3 = Wenjuan|last3 = Qi|first4 = Jinfang|last4 = Liu|first5 = Yuan|last5 = Gao|access-date = 2018-12-02|archive-date = 2020-08-14|archive-url = https://web.archive.org/web/20200814224148/https://www.sciencedirect.com/science/article/pii/S0020025511006232}}</ref>，協方差交叉有以下的優點：
# 避免識別以及計算交叉協方差
# 可以獲得一致的融合估測值，也可以得到無發散的濾波器
# 融合估測值的準確性比其他方式要好
# 對實際的估測誤差變異有常見的上界，且對未知的相关性具有強健性。

== 發展 ==
=== 前協方差交叉 ===
一般認為在許多[[感測器整合]]問題中，都存在著未知相關性的情形。忽略未知相關性的後果可能會讓性能惡化甚至發散。因此這類問題在幾十年來吸引了研究者的關注。不過因為未知相關性融合問題複雜、未知的特性，要找到一個令人滿意的架構並不容易。若直接省略相關性，即為樸素融合（Naive fusion）<ref>{{Cite journal|title = Analytical and Computational Evaluation of Scalable Distributed Fusion Algorithms|url = http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5595611|journal = IEEE Transactions on Aerospace and Electronic Systems|date = 2010-10-01|issn = 0018-9251|pages = 2022–2034|volume = 46|issue = 4|doi = 10.1109/TAES.2010.5595611|first = K.C.|last = Chang|first2 = Chee-Yee|last2 = Chong|first3 = S.|last3 = Mori}}</ref>，會讓濾波器發散。了為補償這類的發散，正規的次最佳化作法是人為的增加系統雜訊，不過這種[[启发法]]需要大量的專業知識，而且會破壞卡尔曼滤波的完整性<ref>{{Cite journal|title = Information fusion based on fast covariance intersection filtering|url = http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=1020907|journal = Proceedings of the Fifth International Conference on Information Fusion, 2002|date = 2002-07-01|pages = 901–904 vol.2|volume = 2|doi = 10.1109/ICIF.2002.1020907|first = W.|last = Niehsen|isbn = 0-9721844-1-4}}</ref>。

==參考資料==
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