sca: Simultaneous Component Analysis

In multiway: Component Models for Multi-Way Data

Description

Fits Timmerman and Kiers's four Simultaneous Component Analysis (SCA) models to a 3-way data array or a list of 2-way arrays with the same number of columns.

Usage

sca(X, nfac, nstart = 10, maxit = 500,
    type = c("sca-p", "sca-pf2", "sca-ind", "sca-ecp"),
    rotation = c("none", "varimax", "promax"),
    ctol = 1e-4, parallel = FALSE, cl = NULL, verbose = TRUE)

Arguments

X	List of length K where the k-th element contains the I[k]-by-J data matrix X[[k]]. If I[k]=I[1] for all k, can input 3-way data array with dim=c(I,J,K).
nfac	Number of factors.
nstart	Number of random starts.
maxit	Maximum number of iterations.
type	Type of SCA model to fit.
rotation	Rotation to use for type="sca-p" or type="sca-ecp".
ctol	Convergence tolerance.
parallel	Logical indicating if parLapply should be used. See Examples.
cl	Cluster created by makeCluster. Only used when parallel=TRUE.
verbose	If TRUE, fitting progress is printed via txtProgressBar. Ignored if parallel=TRUE.

Details

Given a list of matrices X[[k]] = matrix(xk,I[k],J) for k = seq(1,K), the SCA model is

X[[k]] = tcrossprod(D[[k]],B) + E[[k]]

where D[[k]] = matrix(dk,I[k],R) are the Mode A (first mode) weights for the k-th level of Mode C (third mode), B = matrix(b,J,R) are the Mode B (second mode) weights, and E[[k]] = matrix(ek,I[k],J) is the residual matrix corresponding to k-th level of Mode C.

There are four different versions of the SCA model: SCA with invariant pattern (SCA-P), SCA with Parafac2 constraints (SCA-PF2), SCA with INDSCAL constraints (SCA-IND), and SCA with equal average crossproducts (SCA-ECP). These four models differ with respect to the assumed crossproduct structure of the D[[k]] weights:

SCA-P:		crossprod(D[[k]])/I[k] = Phi[[k]]
SCA-PF2:		crossprod(D[[k]])/I[k] = diag(C[k,])%*%Phi%*%diag(C[k,])
SCA-IND:		crossprod(D[[k]])/I[k] = diag(C[k,]*C[k,])
SCA-ECP:		crossprod(D[[k]])/I[k] = Phi

where Phi[[k]] is specific to the k-th level of Mode C, Phi is common to all K levels of Mode C, and C = matrix(c,K,R) are the Mode C (third mode) weights. This function estimates the weight matrices D[[k]] and B (and C if applicable) using alternating least squares.

Value

D	List of length K where k-th element contains D[[k]].
B	Mode B weight matrix.
C	Mode C weight matrix.
Phi	Mode A common crossproduct matrix (if type!="sca-p").
SSE	Sum of Squared Errors.
Rsq	R-squared value.
GCV	Generalized Cross-Validation.
edf	Effective degrees of freedom.
iter	Number of iterations.
cflag	Convergence flag.
type	Same as input type.
rotation	Same as input rotation.

Warnings

The ALS algorithm can perform poorly if the number of factors nfac is set too large.

Computational Details

The least squares SCA-P solution can be obtained from the singular value decomposition of the stacked matrix rbind(X[[1]],...,X[[K]]).

The least squares SCA-PF2 solution can be obtained using the uncontrained Parafac2 ALS algorithm (see parafac2).

The least squares SCA-IND solution can be obtained using the Parafac2 ALS algorithm with orthogonality constraints on Mode A.

The least squares SCA-ECP solution can be obtained using the Parafac2 ALS algorithm with orthogonality constraints on Mode A and the Mode C weights fixed at C[k,] = rep(I[k]^0.5,R).

Note

Default use is 10 random strarts (nstart=10) with 500 maximum iterations of the ALS algorithm for each start (maxit=500) using a convergence tolerance of 1e-4 (ctol=1e-4). The algorithm is determined to have converged once the change in R^2 is less than or equal to ctol.

Output cflag gives convergence information: cflag=0 if ALS algorithm converged normally, cflag=1 if maximum iteration limit was reached before convergence, and cflag=2 if ALS algorithm terminated abnormally due to problem with non-negativity constraints.

Author(s)

Nathaniel E. Helwig <[e-mail]>

References

Helwig, N. E. (2013). The special sign indeterminacy of the direct-fitting Parafac2 model: Some implications, cautions, and recommendations, for Simultaneous Component Analysis. Psychometrika, 78, 725-739.

Timmerman, M. E., & Kiers, H. A. L. (2003). Four simultaneous component models for the analysis of multivariate time series from more than one subject to model intraindividual and interindividual differences. Psychometrika, 68, 105-121.

Examples

##########   sca-p   ##########

# create random data list with SCA-P structure
set.seed(3)
mydim <- c(NA,10,20)
nf <- 2
nk <- rep(c(50,100,200), length.out = mydim[3])
Dmat <- matrix(rnorm(sum(nk)*nf),sum(nk),nf)
Bmat <- matrix(runif(mydim[2]*nf),mydim[2],nf)
Dmats <- vector("list",mydim[3])
Xmat <- Emat <- vector("list",mydim[3])
dfc <- 0
for(k in 1:mydim[3]){
  dinds <- 1:nk[k] + dfc
  Dmats[[k]] <- Dmat[dinds,]
  dfc <- dfc + nk[k]
  Xmat[[k]] <- tcrossprod(Dmats[[k]],Bmat)
  Emat[[k]] <- matrix(rnorm(nk[k]*mydim[2]),nk[k],mydim[2])
}
rm(Dmat)
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- mapply("+",Xmat,Emat)

# fit SCA-P model (no rotation)
scamod <- sca(X,nfac=nf,nstart=1)
scamod

# check solution
crossprod(scamod$D[[1]] %*% diag(scamod$C[1,]^-1) ) / nk[1]
crossprod(scamod$D[[5]] %*% diag(scamod$C[5,]^-1) ) / nk[5]
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])

# reorder and resign factors
scamod$B[1:4,]
scamod <- reorder(scamod, 2:1)
scamod$B[1:4,]
scamod <- resign(scamod, mode="B", newsign=c(1,-1))
scamod$B[1:4,]
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])

# rescale factors
colSums(scamod$B^2)
colSums(scamod$C^2)
scamod <- rescale(scamod, mode="C")
colSums(scamod$B^2)
colSums(scamod$C^2)
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])


##########   sca-pf2   ##########

# create random data list with SCA-PF2 (Parafac2) structure
set.seed(3)
mydim <- c(NA,10,20)
nf <- 2
nk <- rep(c(50,100,200), length.out = mydim[3])
Gmat <- 10*matrix(rnorm(nf^2),nf,nf)
Bmat <- matrix(runif(mydim[2]*nf),mydim[2],nf)
Cmat <- matrix(runif(mydim[3]*nf),mydim[3],nf)
Xmat <- Emat <- Fmat <- vector("list",mydim[3])
for(k in 1:mydim[3]){
  Fmat[[k]] <- svd(matrix(rnorm(nk[k]*nf),nk[k],nf),nv=0)$u
  Xmat[[k]] <- tcrossprod(Fmat[[k]]%*%Gmat%*%diag(Cmat[k,]),Bmat)
  Emat[[k]] <- matrix(rnorm(nk[k]*mydim[2]),nk[k],mydim[2])
}
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- mapply("+",Xmat,Emat)

# fit SCA-PF2 model
scamod <- sca(X,nfac=nf,nstart=1,type="sca-pf2")
scamod

# check solution
scamod$Phi
crossprod(scamod$D[[1]] %*% diag(scamod$C[1,]^-1) ) / nk[1]
crossprod(scamod$D[[5]] %*% diag(scamod$C[5,]^-1) ) / nk[5]
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])

# reorder and resign factors
scamod$B[1:4,]
scamod <- reorder(scamod, 2:1)
scamod$B[1:4,]
scamod <- resign(scamod, mode="B", newsign=c(1,-1))
scamod$B[1:4,]
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])

# rescale factors
colSums(scamod$B^2)
colSums(scamod$C^2)
scamod <- rescale(scamod, mode="C")
colSums(scamod$B^2)
colSums(scamod$C^2)
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])


##########   sca-ind   ##########

# create random data list with SCA-IND structure
set.seed(3)
mydim <- c(NA,10,20)
nf <- 2
nk <- rep(c(50,100,200), length.out = mydim[3])
Gmat <- diag(nf)  # SCA-IND is Parafac2 with Gmat=identity
Bmat <- matrix(runif(mydim[2]*nf),mydim[2],nf)
Cmat <- 10*matrix(runif(mydim[3]*nf),mydim[3],nf)
Xmat <- Emat <- Fmat <- vector("list",mydim[3])
for(k in 1:mydim[3]){
  Fmat[[k]] <- svd(matrix(rnorm(nk[k]*nf),nk[k],nf),nv=0)$u
  Xmat[[k]] <- tcrossprod(Fmat[[k]]%*%Gmat%*%diag(Cmat[k,]),Bmat)
  Emat[[k]] <- matrix(rnorm(nk[k]*mydim[2]),nk[k],mydim[2])
}
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- mapply("+",Xmat,Emat)

# fit SCA-IND model
scamod <- sca(X,nfac=nf,nstart=1,type="sca-ind")
scamod

# check solution
scamod$Phi
crossprod(scamod$D[[1]] %*% diag(scamod$C[1,]^-1) ) / nk[1]
crossprod(scamod$D[[5]] %*% diag(scamod$C[5,]^-1) ) / nk[5]
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])

# reorder and resign factors
scamod$B[1:4,]
scamod <- reorder(scamod, 2:1)
scamod$B[1:4,]
scamod <- resign(scamod, mode="B", newsign=c(1,-1))
scamod$B[1:4,]
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])

# rescale factors
colSums(scamod$B^2)
colSums(scamod$C^2)
scamod <- rescale(scamod, mode="C")
colSums(scamod$B^2)
colSums(scamod$C^2)
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])


##########   sca-ecp   ##########

# create random data list with SCA-ECP structure
set.seed(3)
mydim <- c(NA,10,20)
nf <- 2
nk <- rep(c(50,100,200), length.out = mydim[3])
Gmat <- diag(nf)
Bmat <- matrix(runif(mydim[2]*nf),mydim[2],nf)
Cmat <- matrix(sqrt(nk),mydim[3],nf)
Xmat <- Emat <- Fmat <- vector("list",mydim[3])
for(k in 1:mydim[3]){
  Fmat[[k]] <- svd(matrix(rnorm(nk[k]*nf),nk[k],nf),nv=0)$u
  Xmat[[k]] <- tcrossprod(Fmat[[k]]%*%Gmat%*%diag(Cmat[k,]),Bmat)
  Emat[[k]] <- matrix(rnorm(nk[k]*mydim[2]),nk[k],mydim[2])
}
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- mapply("+",Xmat,Emat)

# fit SCA-ECP model
scamod <- sca(X,nfac=nf,nstart=1,type="sca-ecp")
scamod

# check solution
scamod$Phi
crossprod(scamod$D[[1]] %*% diag(scamod$C[1,]^-1) ) / nk[1]
crossprod(scamod$D[[5]] %*% diag(scamod$C[5,]^-1) ) / nk[5]
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])

# reorder and resign factors
scamod$B[1:4,]
scamod <- reorder(scamod, 2:1)
scamod$B[1:4,]
scamod <- resign(scamod, mode="B", newsign=c(-1,1))
scamod$B[1:4,]
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])

# rescale factors
colSums(scamod$B^2)
colSums(scamod$C^2)
scamod <- rescale(scamod, mode="B")
colSums(scamod$B^2)
colSums(scamod$C^2)
Xhat <- fitted(scamod)
sse <- sumsq(mapply("-",Xmat,Xhat))
sse/(sum(nk)*mydim[2])


## Not run: 

##########   parallel computation   ##########

# create random data list with SCA-IND structure
set.seed(3)
mydim <- c(NA,10,20)
nf <- 2
nk <- rep(c(50,100,200), length.out = mydim[3])
Gmat <- diag(nf)  # SCA-IND is Parafac2 with Gmat=identity
Bmat <- matrix(runif(mydim[2]*nf),mydim[2],nf)
Cmat <- 10*matrix(runif(mydim[3]*nf),mydim[3],nf)
Xmat <- Emat <- Fmat <- vector("list",mydim[3])
for(k in 1:mydim[3]){
  Fmat[[k]] <- svd(matrix(rnorm(nk[k]*nf),nk[k],nf),nv=0)$u
  Xmat[[k]] <- tcrossprod(Fmat[[k]]%*%Gmat%*%diag(Cmat[k,]),Bmat)
  Emat[[k]] <- matrix(rnorm(nk[k]*mydim[2]),nk[k],mydim[2])
}
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- mapply("+",Xmat,Emat)

# fit SCA-PF2 model (10 random starts -- sequential computation)
set.seed(1)
system.time({scamod <- sca(X,nfac=nf,type="sca-pf2")})
scamod

# fit SCA-PF2 model (10 random starts -- parallel computation)
cl <- makeCluster(detectCores())
ce <- clusterEvalQ(cl,library(multiway))
clusterSetRNGStream(cl, 1)
system.time({scamod <- sca(X,nfac=nf,type="sca-pf2",parallel=TRUE,cl=cl)})
scamod
stopCluster(cl)

# fit SCA-IND model (10 random starts -- sequential computation)
set.seed(1)
system.time({scamod <- sca(X,nfac=nf,type="sca-ind")})
scamod

# fit SCA-IND model (10 random starts -- parallel computation)
cl <- makeCluster(detectCores())
ce <- clusterEvalQ(cl,library(multiway))
clusterSetRNGStream(cl, 1)
system.time({scamod <- sca(X,nfac=nf,type="sca-ind",parallel=TRUE,cl=cl)})
scamod
stopCluster(cl)

# fit SCA-ECP model (10 random starts -- sequential computation)
set.seed(1)
system.time({scamod <- sca(X,nfac=nf,type="sca-ecp")})
scamod

# fit SCA-ECP model (10 random starts -- parallel computation)
cl <- makeCluster(detectCores())
ce <- clusterEvalQ(cl,library(multiway))
clusterSetRNGStream(cl, 1)
system.time({scamod <- sca(X,nfac=nf,type="sca-ecp",parallel=TRUE,cl=cl)})
scamod
stopCluster(cl)

## End(Not run)

multiway documentation built on May 2, 2019, 6:47 a.m.