tucker: Tucker Factor Analysis
In multiway: Component Models for Multi-Way Data

Description Usage Arguments Details Value Warnings Note Author(s) References Examples

Fits Ledyard R. Tucker's factor analysis model to 3-way or 4-way data arrays. Parameters are estimated via alternating least squares.

tucker(X, nfac, nstart = 10, Afixed = NULL,
       Bfixed = NULL, Cfixed = NULL, Dfixed = NULL,
       Bstart = NULL, Cstart = NULL, Dstart = NULL,
       maxit = 500, ctol = 1e-4, parallel = FALSE, cl = NULL, 
       output = c("best", "all"), verbose = TRUE)

`X`	Three-way data array with `dim=c(I,J,K)` or four-way data array with `dim=c(I,J,K,L)`. Missing data are allowed (see Note).
`nfac`	Number of factors in each mode.
`nstart`	Number of random starts.
`Afixed`	Fixed Mode A weights. Only used to fit model with fixed weights in Mode A.
`Bfixed`	Fixed Mode B weights. Only used to fit model with fixed weights in Mode B.
`Cfixed`	Fixed Mode C weights. Only used to fit model with fixed weights in Mode C.
`Dfixed`	Fixed Mode D weights. Only used to fit model with fixed weights in Mode D.
`Bstart`	Starting Mode B weights for ALS algorithm. Default uses random weights.
`Cstart`	Starting Mode C weights for ALS algorithm. Default uses random weights.
`Dstart`	Starting Mode D weights for ALS algorithm. Default uses random weights.
`maxit`	Maximum number of iterations.
`ctol`	Convergence tolerance.
`parallel`	Logical indicating if `parLapply` should be used. See Examples.
`cl`	Cluster created by `makeCluster`. Only used when `parallel=TRUE`.
`output`	Output the best solution (default) or output all `nstart` solutions.
`verbose`	If `TRUE`, fitting progress is printed via `txtProgressBar`. Ignored if `parallel=TRUE`.

Given a 3-way array X = array(x,dim=c(I,J,K)), the 3-way Tucker model can be written as

X[i,j,k] = sum sum sum A[i,p]*B[j,q]*C[k,r]*G[p,q,r] + E[i,j,k]

where A = matrix(a,I,P) are the Mode A (first mode) weights, B = matrix(b,J,Q) are the Mode B (second mode) weights, C = matrix(c,K,R) are the Mode C (third mode) weights, G = array(g,dim=c(P,Q,R)) is the 3-way core array, and E = array(e,dim=c(I,J,K)) is the 3-way residual array. The summations are for p = seq(1,P), q = seq(1,Q), and r = seq(1,R).

Given a 4-way array X = array(x,dim=c(I,J,K,L)), the 4-way Tucker model can be written as

X[i,j,k,l] = sum sum sum sum A[i,p]*B[j,q]*C[k,r]*D[l,s]*G[p,q,r,s] + E[i,j,k,l]

where D = matrix(d,L,S) are the Mode D (fourth mode) weights, G = array(g,dim=c(P,Q,R,S)) is the 4-way residual array, E = array(e,dim=c(I,J,K,L)) is the 4-way residual array, and the other terms can be interprered as previously described.

Weight matrices are estimated using an alternating least squares algorithm.

If output="best", returns an object of class "tucker" with the following elements:

`A`	Mode A weight matrix.
`B`	Mode B weight matrix.
`C`	Mode C weight matrix.
`D`	Mode D weight matrix.
`G`	Core array.
`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.

Otherwise returns a list of length nstart where each element is an object of class "tucker".

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

Input matrices in Afixed, Bfixed, Cfixed, Dfixed, Bstart, Cstart, and Dstart must be columnwise orthonormal.

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, and cflag=1 if maximum iteration limit was reached before convergence.

Missing data should be specified as NA values in the input X. The missing data are randomly initialized and then iteratively imputed as a part of the ALS algorithm.

Nathaniel E. Helwig <helwig@umn.edu>

Kroonenberg, P. M., & de Leeuw, J. (1980). Principal component analysis of three-mode data by means of alternating least squares algorithms. Psychometrika, 45, 69-97.

Tucker, L. R. (1966). Some mathematical notes on three-mode factor analysis. Psychometrika, 31, 279-311.

##########   3-way example   ##########

####****####   TUCKER3   ####****####

# create random data array with Tucker3 structure
set.seed(3)
mydim <- c(50,20,5)
nf <- c(3,2,3)
Amat <- matrix(rnorm(mydim[1]*nf[1]), mydim[1], nf[1])
Amat <- svd(Amat, nu = nf[1], nv = 0)$u
Bmat <- matrix(rnorm(mydim[2]*nf[2]), mydim[2], nf[2])
Bmat <- svd(Bmat, nu = nf[2], nv = 0)$u
Cmat <- matrix(rnorm(mydim[3]*nf[3]), mydim[3], nf[3])
Cmat <- svd(Cmat, nu = nf[3], nv = 0)$u
Gmat <- matrix(rnorm(prod(nf)), nf[1], prod(nf[2:3]))
Xmat <- tcrossprod(Amat %*% Gmat, kronecker(Cmat, Bmat))
Xmat <- array(Xmat, dim = mydim)
Emat <- array(rnorm(prod(mydim)), dim = mydim)
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- Xmat + Emat

# fit Tucker3 model
tuck <- tucker(X, nfac = nf, nstart = 1)
tuck

# check solution
Xhat <- fitted(tuck)
sum((Xmat-Xhat)^2) / prod(mydim)

# reorder mode="A"
tuck$A[1:4,]
tuck$G
tuck <- reorder(tuck, neworder = c(3,1,2), mode = "A")
tuck$A[1:4,]
tuck$G
Xhat <- fitted(tuck)
sum((Xmat-Xhat)^2)/prod(mydim)

# reorder mode="B"
tuck$B[1:4,]
tuck$G
tuck <- reorder(tuck, neworder=2:1, mode="B")
tuck$B[1:4,]
tuck$G
Xhat <- fitted(tuck)
sum((Xmat-Xhat)^2)/prod(mydim)

# resign mode="C"
tuck$C[1:4,]
tuck <- resign(tuck, mode="C")
tuck$C[1:4,]
Xhat <- fitted(tuck)
sum((Xmat-Xhat)^2)/prod(mydim)


####****####   TUCKER2   ####****####

# create random data array with Tucker2 structure
set.seed(3)
mydim <- c(50, 20, 5)
nf <- c(3, 2, mydim[3])
Amat <- matrix(rnorm(mydim[1]*nf[1]), mydim[1], nf[1])
Amat <- svd(Amat, nu = nf[1], nv = 0)$u
Bmat <- matrix(rnorm(mydim[2]*nf[2]), mydim[2], nf[2])
Bmat <- svd(Bmat, nu = nf[2], nv = 0)$u
Cmat <- diag(nf[3])
Gmat <- matrix(rnorm(prod(nf)), nf[1], prod(nf[2:3]))
Xmat <- tcrossprod(Amat %*% Gmat, kronecker(Cmat, Bmat))
Xmat <- array(Xmat, dim = mydim)
Emat <- array(rnorm(prod(mydim)), dim = mydim)
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- Xmat + Emat

# fit Tucker2 model
tuck <- tucker(X, nfac = nf, nstart = 1, Cfixed = diag(nf[3]))
tuck

# check solution
Xhat <- fitted(tuck)
sum((Xmat-Xhat)^2) / prod(mydim)


####****####   TUCKER1   ####****####

# create random data array with Tucker1 structure
set.seed(3)
mydim <- c(50, 20, 5)
nf <- c(3, mydim[2:3])
Amat <- matrix(rnorm(mydim[1]*nf[1]), mydim[1], nf[1])
Amat <- svd(Amat, nu = nf[1], nv = 0)$u
Bmat <- diag(nf[2])
Cmat <- diag(nf[3])
Gmat <- matrix(rnorm(prod(nf)), nf[1], prod(nf[2:3]))
Xmat <- tcrossprod(Amat %*% Gmat, kronecker(Cmat, Bmat))
Xmat <- array(Xmat, dim = mydim)
Emat <- array(rnorm(prod(mydim)), dim = mydim)
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- Xmat + Emat

# fit Tucker1 model
tuck <- tucker(X, nfac = nf, nstart = 1,
               Bfixed = diag(nf[2]), Cfixed = diag(nf[3]))
tuck

# check solution
Xhat <- fitted(tuck)
sum((Xmat-Xhat)^2) / prod(mydim)

# closed-form Tucker1 solution via SVD
tsvd <- svd(matrix(X, nrow = mydim[1]), nu = nf[1], nv = nf[1])
Gmat0 <- t(tsvd$v %*% diag(tsvd$d[1:nf[1]]))
Xhat0 <- array(tsvd$u %*% Gmat0, dim = mydim)
sum((Xmat-Xhat0)^2) / prod(mydim)

# get Mode A weights and core array 
tuck0 <- NULL
tuck0$A <- tsvd$u                   # A weights
tuck0$G <- array(Gmat0, dim = nf)   # core array



##########   4-way example   ##########

# create random data array with Tucker structure
set.seed(4)
mydim <- c(30,10,8,10)
nf <- c(2,3,4,3)
Amat <- svd(matrix(rnorm(mydim[1]*nf[1]),mydim[1],nf[1]),nu=nf[1])$u
Bmat <- svd(matrix(rnorm(mydim[2]*nf[2]),mydim[2],nf[2]),nu=nf[2])$u
Cmat <- svd(matrix(rnorm(mydim[3]*nf[3]),mydim[3],nf[3]),nu=nf[3])$u
Dmat <- svd(matrix(rnorm(mydim[4]*nf[4]),mydim[4],nf[4]),nu=nf[4])$u
Gmat <- array(rnorm(prod(nf)),dim=nf)
Xmat <- array(tcrossprod(Amat%*%matrix(Gmat,nf[1],prod(nf[2:4])),
                      kronecker(Dmat,kronecker(Cmat,Bmat))),dim=mydim)
Emat <- array(rnorm(prod(mydim)),dim=mydim)
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- Xmat + Emat

# fit Tucker model
tuck <- tucker(X,nfac=nf,nstart=1)
tuck

# check solution
Xhat <- fitted(tuck)
sum((Xmat-Xhat)^2)/prod(mydim)


## Not run: 

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

# create random data array with Tucker structure
set.seed(3)
mydim <- c(50,20,5)
nf <- c(3,2,3)
Amat <- svd(matrix(rnorm(mydim[1]*nf[1]),mydim[1],nf[1]),nu=nf[1])$u
Bmat <- svd(matrix(rnorm(mydim[2]*nf[2]),mydim[2],nf[2]),nu=nf[2])$u
Cmat <- svd(matrix(rnorm(mydim[3]*nf[3]),mydim[3],nf[3]),nu=nf[3])$u
Gmat <- array(rnorm(prod(nf)),dim=nf)
Xmat <- array(tcrossprod(Amat%*%matrix(Gmat,nf[1],nf[2]*nf[3]),kronecker(Cmat,Bmat)),dim=mydim)
Emat <- array(rnorm(prod(mydim)),dim=mydim)
Emat <- nscale(Emat, 0, ssnew = sumsq(Xmat))   # SNR=1
X <- Xmat + Emat

# fit Tucker model (10 random starts -- sequential computation)
set.seed(1)
system.time({tuck <- tucker(X,nfac=nf)})
tuck$Rsq

# fit Tucker model (10 random starts -- parallel computation)
cl <- makeCluster(detectCores())
ce <- clusterEvalQ(cl,library(multiway))
clusterSetRNGStream(cl, 1)
system.time({tuck <- tucker(X,nfac=nf,parallel=TRUE,cl=cl)})
tuck$Rsq
stopCluster(cl)

## End(Not run)