Package ‘JADE’ - R

1 Package jade March 25, 2020 TypePackageTitleBlind Source Separation Methods Based on Joint Diagonalizationand Some BSS Performance , graphicsSuggestsICS, ICSNPD escriptionCardoso's jade algorithm as well as his functions for joint diagonaliza-tion are ported to R. Also several other blind source separation (BSS) meth-ods, like AMUSE and SOBI, and some criteria for performance evaluation of BSS algo-rithms, are given. The Package is described in Miettinen, Nordhausen and Taski-nen (2017) < >.LicenseGPL (>= 2)LazyDatatrueNeedsCompilationyesAuthorK laus Nordhausen [aut, cre] (< >),Jean-Francois Cardoso [aut],Jari Miettinen [aut] (< >),Hannu Oja [aut],Esa Ollila [aut],Sara Taskinen [aut] (< >)MaintainerKlaus 11:30:02 UTCR topics documented: jade - Package ..3 AMUSE ..612 jade -packagecjd ..8 ComonGAP ..9 CPPdata .. 10djd .. 11FG .. 13 FOBI .. 14 jade .

2 15k_JADE .. 17MD .. 19multscatter .. 28rjd .. 28 SIR .. 30 SOBI .. 31 Index34 jade -packageBlind Source Separation Methods Based on Joint Diagonalization andSome BSS Performance CriteriaDescriptionCardoso s jade algorithm as well as his functions for joint diagonalization are ported to R. Alsoseveral other blind source separation (BSS) methods, like AMUSE and SOBI, and some criteriafor performance evaluation of BSS algorithms, are given. The Package is described in Miettinen,Nordhausen and Taskinen (2017) doi: :JADEType: :2019-08-19 License:GPL (>= 2)Author(s)Klaus Nordhausen, Jean-Francois Cardoso, Jari Miettinen, Hannu Oja, Esa Ollila, Sara : Klaus Nordhausen J., Nordhausen, K. and Taskinen, S. (2017), Blind Source Separation Based on JointDiagonalization in R: The packages jade and BSSasymp,Journal of Statistical Software,76, 1 31, < >.

3 ErrorDescriptionComputes the Amari Error to evaluate the performance of an ICA ( , A, standardize = F) estimated square unmixing matrix true square mixing matrix value if A and need to be standardized. Default is Amari Error can be used in simulation studies to evaluate the performance of an ICA Amari error is permutation invariant but not scale invariant. Therefore if different algorithmsshould be compared the matrices should be scaled in the same way. Ifstandardizeis TRUE, thiswill be done by the function by standardizing and the inverse of A in such a way, thatevery row has length 1, the largest absolute value of the row has a positive sign and the rows areordered decreasingly according to their largest that this function assumes the ICA model isX=SA , as is assumed Therefore matrices from those functionshave to be transposed Amari Error is scaled in such a way, that it takes a value between 0 and 1.

4 And 0 correspondsto an optimal value of the Amari (s)Klaus Nordhausen4 AMUSER eferencesAmari, S., Cichocki, A. and Yang, (1996), A new learning algorithm for blind signal separa-tion,Advances in Neural Information Processing Systems,8, 757 , K., Ollila, E. and Oja, H. (2011), On the Performance Indices of ICA and Blind SourceSeparation. In the Proceedings of2011 IEEE 12th International Workshop on Signal ProcessingAdvances in Wireless Communications (SPAWC 2011), 486 AlsoComonGAP,SIRE xamplesS <- cbind(rt(1000, 4), rnorm(1000), runif(1000))A <- matrix(rnorm(9), ncol = 3)X <- S %*% t(A) <- jade (X, 3)$ ( , A) ( , A, TRUE)AMUSEAMUSE Method for Blind Source SeparationDescriptionAMUSE method for the second order blind source separation problem. The function estimates theunmixing matrix in a second order stationary source separation model by jointly diagonalizing thecovariance matrix and an autocovariance matrix at lag (x.)

5 ## Default S3 method:AMUSE(x, k = 1, ..)## S3 method for class'ts'AMUSE(x, ..)Argumentsxa numeric matrix or a multivariate time series object of classts. Missing valuesare not lag for the autocovariance matrix, must be larger than 0. Default is arguments to be passed to or from lagkhas a huge effect on the performance and it should be chosen so that the eigenvalues ofautocovariance matrix are distinct. The function assumes always as many sources as there are list with class bss containing the following components:Westimated unmixing of autocovariance of the autocovariance matrix sources as time series objected standardized to have mean 0 and (s)Klaus NordhausenReferencesTong, L., Soon, , Huang, and Liu, R. (1990), AMUSE: a new blind identification algorithm,in Proceedings of IEEE International Symposium onCircuits and Systems 1990, 1784 , J.

6 , Nordhausen, K., Oja, H. and Taskinen, S. (2012), Statistical properties of a blindsource separation estimator for stationary time series,Statistics & Probability Letters, 82, 1865 , J., Nordhausen, K. and Taskinen, S. (2017), Blind Source Separation Based on JointDiagonalization in R: The packages jade and BSSasymp,Journal of Statistical Software,76, 1 31, < >.See AlsotsExamples# creating some toy dataA<- matrix(rnorm(9),3,3)s1 <- (list(ar=c( , )),1000)s2 <- (list(ma=c( , )),1000)s3 <- (list(ar=c( , )),1000)S <- cbind(s1,s2,s3)X <- S %*% t(A)res1<-AMUSE(X)res1coef(res1)plot(res 1) # compare to (S) (coef(res1),A)# input of a time seriesX2<- ts(X, start=c(1961, 1), frequency=12)plot(X2)res2<-AMUSE(X2, k=2)plot(res2) to Extract Estimated Sources from an Object of Class bssDescriptionExtracts the sources estimated by an bss (object)Argumentsobjectobject of class bssAuthor(s)Klaus NordhausenExamplesA<- matrix(rnorm(9),3,3)s1 <- (list(ar=c( , )),1000)s2 <- (list(ma=c( , )),1000)s3 <- (list(ar=c( , )),1000)S <- cbind(s1,s2,s3)X <- S %*% t(A)res1<-AMUSE(X)head( (res1))colMeans( (res1))cov( (res1))

7 Cjd7cjdJoint Diagonalization of Complex MatricesDescriptionThis is anRversion of Cardoso s joint_diag matlab function for joint diagonalization of k complex-valued square (X, eps = 1e-06, maxiter = 100)ArgumentsXA matrix of k stacked pxp complex matrices with dimension c(kp,p) or an arraywith dimension c(p,p,k).epsConvergence number of orthogonal stacked matrix with the diagonal matrices or an array with the diagonal ma-trices. The form of the output depends on the form of the (s)Jean-Francois Cardoso. Ported toRby Klaus , and Souloumiac, A., (1996), Jacobi angles for simultaneous diagonalization,SIAMJ. Mat. Anal. Appl.,17, 161 Alsorjd, <- diag(complex(real=runif(3,0,2), imaginary=runif(3)))D2 <- diag(complex(real=runif(3,0,2), imaginary=runif(3)))D3 <- diag(complex(real=runif(3,0,2), imaginary=runif(3)))D4 <- diag(complex(real=runif(3,0,2), imaginary=runif(3)))Z <- matrix(runif(9), ncol = 3)V <- eigen(Z %*% t(Z))$ <- t(V)%*%D1%*%VM2 <- t(V)%*%D2%*%VM3 <- t(V)%*%D3%*%VM4 <- t(V)%*%D4%*%VMS <- rbind(M1,M2,M3,M4)Ms <- array(0,dim=c(3,3,4))Ms[,,1]<-M1Ms[,,3]< -M3Ms[,,2]<-M2Ms[,,4]< <- cjd(Ms) <- cjd(MS)Re( $V)Vround(V%*%Re( $V),2)round(V%*%Re( $V),2) of a bss ObjectDescriptionExtracts the estimated unmixing matrix from an object of class ## S3 method for class'bss'coef(object.)

8 Argumentsobjectobject of class arguments to be passed to or from (s)Klaus NordhausenExamplesA<- matrix(rnorm(9),3,3)s1 <- (list(ar=c( , )),1000)s2 <- (list(ma=c( , )),1000)s3 <- (list(ar=c( , )),1000)S <- cbind(s1,s2,s3)X <- S %*% t(A)res1<-AMUSE(X)coef(res1)coef(res1) %*% A # should be a matrix with one dominant element in each row and columnComonGAP9 ComonGAPC omon s GapDescriptionComon s GAP criterion to evaluate the performance of an ICA (A, )ArgumentsAThe true square mixing estimated square mixing s GAP criterion is permutation and scale invariant. It can take every positive value and 0corresponds to an optimal separation. IfAis however nearly singular the values of the criterion canbe that this function assumes the ICA model isX=SA , as is assumed Therefore matrices from those functionshave to be transposed value of the Comon s (s)Klaus NordhausenReferencesComon, P.

9 , (1994), Independent Component Analysis, A new concept?,Signal Processing,36,287 ,SIR10 CPPdataExamplesS <- cbind(rt(1000, 4), rnorm(1000), runif(1000))A <- matrix(rnorm(9), ncol = 3)X <- S %*% t(A) <- jade (X, 3)$AComonGAP(A, )CPPdataCocktail Party Problem DataDescriptionThis data set is a toy example for the so called cocktail party problem. In this case three sounds aremixed together with one noise source using four ("CPPdata")FormatA data frame with 50000 observations on the following 4 mixture recorded by the first mixture recorded by the second mixture recorded by the third mixture recorded by the fourth three original source files were kindly provided by Ella Bingham and are also available on-line at the following locations: , that the original sound files are included in the Package s subfolder datafiles. In the examplesection we illustrate how the CPPdata was created.

10 An example analysis of the data is given inMiettinen et al. (2017). , J., Nordhausen, K. and Taskinen, S. (2017), Blind Source Separation Based on JointDiagonalization in R: The packages jade and BSSasymp,Journal of Statistical Software,76, 1 31, < >.djd11 Examples## Not run:# the data was created as follows:library("tuneR")S1 <- readWave( (" ", Package = " jade "))S2 <- readWave( (" ", Package = " jade "))S3 <- readWave( (" ", Package = " jade ")) (321)NOISE <- noise("white", duration = 50000)S <- cbind(S1@left, S2@left, S3@left, NOISE@left)S <- scale(S, center = FALSE, scale = apply(S, 2, sd))St <- ts(S, start = 0, frequency = 8000)p <- 4A <- matrix(runif(p^2, 0, 1), p, p)AX <- tcrossprod(St, A)Xt <- (X)colnames(X) <- c("Mic1", "Mic2", "Mic3", "Mic4")CPPdata <- (X)## End(Not run)djdFunction for Joint Diagonalization of k Square Matrices in a DeflationBased MannerDescriptionThis function jointly diagonalizes k real-valued square matrices by searching an orthogonal matrixin a deflation based (X, G = "max", r = 2, eps = 1e-06, maxiter = 500)ArgumentsXan array containing the k p times p real valued matrices of dimension c(p, p, k).