基于多維獨(dú)立成分分析的數(shù)值仿真與分析

1、基于多維獨(dú)立成分分析的數(shù)值仿真與分析摘要:通過(guò)引入一個(gè)用于評(píng)價(jià)多維獨(dú)立成分分析(MICA)算法性能的指標(biāo)，進(jìn)行數(shù)值仿真來(lái)研究其分離性。將多維Amari分離誤差作為度量多維獨(dú)立成分分析算法性能的一個(gè)重要指標(biāo)，在比較分析研究vkMICA、cfMICA、MSOBI、SJADE等四個(gè)算法的分離性能的基礎(chǔ)上，使用隨機(jī)分布生成的字母信號(hào)進(jìn)行仿真與測(cè)試，直觀地顯示了MICA模型的分離效果和不確定性。研究結(jié)果顯示，MICA是一種非常有效的進(jìn)行多維源信號(hào)分析的方法。 關(guān)鍵詞:多維獨(dú)立成分分析；多維Amari； 數(shù)值仿真；信號(hào)測(cè)試 中圖分類號(hào): TP301.6；O242.1 文獻(xiàn)標(biāo)志碼:A ? Numerical

2、 simulation and analysis based on ?multidimensional independent component analysis ? XIE Yong.hong?1?*?, ZHANG Guo.wei?2 ?(1.Department of Computer Science, Harbin Finance University, Harbin Heilongjiang 150030, China ;? 2.School of Electronics and Information Engineering, Xi?an Jiaotong University,

3、 Xi?an Shaanxi 710049, China Abstract: By introducing a indicator to evaluate performance of Multidimensional Independent Component Analysis (MICA) algorithm, the separation was studied by numerical simulation. Using multidimensional Amari separation error as an important indicator of a measurement

4、of multidimensional independent component analysis algorithm performance. In the comparative analysis of four algorithm named vkMICA, cfMICA, MSOBI, SJADE in the separation performance, a random distribution of letters signal was used for simulation and testing, and get a visual representation of MI

5、CA model of separation and uncertainty. The results show that MICA is a very effective method for multidimensional source signal analysis. By introducing an indicator to evaluate performance of Multidimensional Independent Component Analysis (MICA) algorithm, the separation was studied by numerical

6、simulation. The multidimensional Amari separation error was used as an important indicator of the measurement of MICA algorithm performance. In the comparative separation performance analysis of four algorithms named vkMICA, cfMICA, MSOBI, SJADE, a random distribution of letters signal was used for

7、simulation and testing, and a visual representation of MICA model of separation and uncertainty was got. The results show that MICA is a very effective method for multidimensional source signal analysis.?Key words: Multidimensional Independent Component Analysis (MICA); multidimensional Amari; numer

8、ical simulation; signal testing ? 0 引言? 法國(guó)學(xué)者Cardoso?1?于1998年首先給出了標(biāo)準(zhǔn)的多維獨(dú)立成分分析(Multidimensional Independent Component Analysis, MICA)定義，提出了加法模型并通過(guò)幾何參數(shù)化方法對(duì)MICA的算法進(jìn)行分解。MICA算法估計(jì)的系統(tǒng)框架如圖1所示。? 在圖1中，?A是混合矩陣，V是白化矩陣，U是局部的正交分離矩陣，B是利用算法最終確定的全局解混矩陣，它實(shí)際上是對(duì)A?1?的估計(jì)，而得到的y?是對(duì)源的估計(jì)。所以，MICA估計(jì)算法的中心任務(wù)就是確定分離矩陣?B，使得能夠?qū)υ碨【圖上是

9、小寫(xiě)？】或混合矩陣A?實(shí)現(xiàn)有效地估計(jì)。對(duì)于MICA估計(jì)算法，可分為批處理(離線)和在線算法，把批處理算法結(jié)合獨(dú)立成分分析（Independent Component Analysis，ICA）算法，并將其推廣到多維的情況，從而討論四種MICA算法的分離性能，即基于向量峭度的不動(dòng)點(diǎn)算法?2?(vector kurtosis based MICA, vkMICA)，基于第二特征函數(shù)Hessian矩陣聯(lián)合塊對(duì)角化的算法?3?(characteristic function based MICA, cfMICA)， 基于特征矩陣聯(lián)合塊對(duì)角化的算法?4?(Subspace JADE, SJADE)和基于

10、時(shí)延協(xié)方差矩陣聯(lián)合塊對(duì)角化的算法?5?(Multidimensional Second Order Blind Identification, MSOBI)。通過(guò)引入一個(gè)用于評(píng)價(jià)MICA算法性能的指標(biāo)，進(jìn)行數(shù)值仿真來(lái)研究其分離性。? 2.2 處理點(diǎn)數(shù)?K?的選取對(duì)算法的影響? 以cfMICA算法為例，使用式(7)所示的混合矩陣得到觀測(cè)信號(hào)?X(t)=As(t)，取處理點(diǎn)數(shù)K分別為K=10,20,100，對(duì)每個(gè)K分別運(yùn)行50次，得到混合矩陣的估計(jì)A?U，求取每個(gè)K?對(duì)應(yīng)的平均多維Amari分離誤差?E?(2)?(A?U?-1?A)，得到的結(jié)果如圖5所示。? 由圖5可知，對(duì)于不同的處理點(diǎn)數(shù)K，算法

11、的分離誤差E?(2)?(A?U?-1?A)約為0.05?9，且波動(dòng)范圍不大，這說(shuō)明在處理點(diǎn)選取為-0.025,0.025的均勻分布時(shí)，處理點(diǎn)的個(gè)數(shù)對(duì)于算法的性能并無(wú)明顯的影響，算法是比較穩(wěn)健的。但當(dāng)K=60?時(shí)，分離誤差是最小的，故在比較各算法的性能時(shí)，將選取此值。? 2.3 時(shí)延個(gè)數(shù)?L?的選取對(duì)算法的影響? 以MSOBI算法為例，仍然使用式(7)所示的混合矩陣得到觀測(cè)信號(hào)?X(t)=As(t)，取時(shí)延個(gè)數(shù)分別為L(zhǎng)=10,20,100，對(duì)每個(gè)L分別運(yùn)行50次，得到混合矩陣的估計(jì)A?U，求取每個(gè)L?對(duì)應(yīng)的平均多維Amari分離誤差?E?(2)?(A?U?-1?A)，?得到的結(jié)果如圖6所示。?

12、4 結(jié)語(yǔ)? 本文檢測(cè)了MICA的四個(gè)算法的性能指標(biāo)，主要分析了處理點(diǎn)的個(gè)數(shù)的選取、不同時(shí)延個(gè)數(shù)的選取對(duì)算法的影響；在無(wú)噪聲和有噪聲條件下，基于聯(lián)合塊對(duì)角化(JBD)的CFMICA、SJADE、MSOBI三個(gè)算法都表現(xiàn)了一定的抗噪聲性能。但當(dāng)高能量的噪聲下，需要在算法處理前加上降噪處理環(huán)節(jié)。仿真與測(cè)試分析表明，四個(gè)算法均能夠在一定程度上完成對(duì)源信號(hào)的分離，在此基礎(chǔ)上，再進(jìn)行聯(lián)合塊對(duì)角化的協(xié)方差矩陣相應(yīng)增加，從而更好地反映了信號(hào)空間的“平均特征”，得到更好的分離效果。MICA的算法可以非常有效地進(jìn)行多維源信號(hào)分析，也是后期研究和實(shí)際應(yīng)用的內(nèi)容。 ? 圖片 圖12 以SJADE算法為例分析得到的對(duì)3

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