Tucker Factor Analysis
Fits Ledyard R. Tucker's factor analysis model to 3-way or 4-way data arrays. Parameters are estimated via alternating least squares.
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.
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.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".
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.
Nathaniel E. Helwig helwig@umn.edu
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.
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.
########## 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)
Useful links