glamlassoRR {glamlasso} | R Documentation |
Efficient design matrix free procedure for fitting large scale penalized reduced rank
regressions in a 3-dimensional generalized linear array model. To obtain a factorization of the parameter array,
the glamlassoRR
function performes a block relaxation scheme within the gdpg algorithm, see Lund et al., 2017.
glamlassoRR(X, Y, Z = NULL, family = "gaussian", penalty = "lasso", intercept = FALSE, weights = NULL, thetainit = NULL, alphainit = NULL, nlambda = 100, lambdaminratio = 1e-04, lambda = NULL, penaltyfactor = NULL, penaltyfactoralpha = NULL, reltolinner = 1e-07, reltolouter = 1e-04, reltolalt = 1e-04, maxiter = 15000, steps = 1, maxiterinner = 3000, maxiterouter = 25, maxalt = 10, btinnermax = 100, btoutermax = 100, iwls = "exact", nu = 1)
X |
A list containing the 3 tensor components of the tensor design matrix. These are matrices of sizes n_i \times p_i. |
Y |
The response values, an array of size n_1 \times n_2\times n_3. For option
|
Z |
The non tensor structrured part of the design matrix. A matrix of size n_1 n_2 n_3\times q.
Is set to |
family |
A string specifying the model family (essentially the response distribution). Possible values
are |
penalty |
A string specifying the penalty. Possible values are |
intercept |
Logical variable indicating if the model includes an intercept. When |
weights |
Observation weights, an array of size n_1 \times \cdots \times n_d. For option
|
thetainit |
A list (length 2) containing the initial parameter values for each of the parameter factors. Default is NULL in which case all parameters are initialized at 0.01. |
alphainit |
A q\times 1 vector containing the initial parameter values for the non-tensor parameter. Default is NULL in which case all parameters are initialized at 0. |
nlambda |
The number of |
lambdaminratio |
The smallest value for |
lambda |
The sequence of penalty parameters for the regularization path. |
penaltyfactor |
A list of length two containing an array of size p_1 \times p_2 and a p_3 \times 1 vector.
Multiplied with each element in |
penaltyfactoralpha |
A q \times 1 vector multiplied with each element in |
reltolinner |
The convergence tolerance for the inner loop |
reltolouter |
The convergence tolerance for the outer loop. |
reltolalt |
The convergence tolerance for the alternation loop over the two parameter blocks. |
maxiter |
The maximum number of inner iterations allowed for each |
steps |
The number of steps used in the multi-step adaptive lasso algorithm for non-convex penalties. Automatically set to 1 when |
maxiterinner |
The maximum number of inner iterations allowed for each outer iteration. |
maxiterouter |
The maximum number of outer iterations allowed for each lambda. |
maxalt |
The maximum number of alternations over parameter blocks. |
btinnermax |
Maximum number of backtracking steps allowed in each inner iteration. Default is |
btoutermax |
Maximum number of backtracking steps allowed in each outer iteration. Default is |
iwls |
A string indicating whether to use the exact iwls weight matrix or use a tensor structured approximation to it. |
nu |
A number between 0 and 1 that controls the step size δ in the proximal algorithm (inner loop) by
scaling the upper bound \hat{L}_h on the Lipschitz constant L_h (see Lund et al., 2017).
For |
Given the setting from glamlasso
we place a reduced rank
restriction on the p_1\times p_2\times p _3 parameter array Θ given by
Θ=(Θ_{i,j,k})_{i,j,k} = (γ_{k}β_{i,j})_{i,j,k}, \ \ \ γ_k,β_{i,j}\in \mathcal{R}.
The glamlassoRR
function solves the PMLE problem by combining a block relaxation scheme with the gdpg algorithm. This scheme alternates between optimizing over the first
parameter block β=(β_{i,j})_{i,j} and the second block γ=(γ_k)_k while fixing the second resp. first block. We note that the
individual parameter blocks are only identified up to a multiplicative constant.
Adam Lund
Maintainer: Adam Lund, adam.lund@math.ku.dk
Lund, A. and N. R. Hansen (2017). Sparse Network Estimation for Dynamical Spatio-temporal Array Models. ArXiv.
## Not run: ##size of example n1 <- 65; n2 <- 26; n3 <- 13; p1 <- 12; p2 <- 6; p3 <- 4 ##marginal design matrices (tensor components) X1 <- matrix(rnorm(n1 * p1), n1, p1) X2 <- matrix(rnorm(n2 * p2), n2, p2) X3 <- matrix(rnorm(n3 * p3), n3, p3) X <- list(X1, X2, X3) Beta12 <- matrix(rnorm(p1 * p2), p1, p2) * matrix(rbinom(p1 * p2, 1, 0.5), p1, p2) Beta3 <- matrix(rnorm(p3) * rbinom(p3, 1, 0.5), p3, 1) Beta <- outer(Beta12, c(Beta3)) Mu <- RH(X3, RH(X2, RH(X1, Beta))) Y <- array(rnorm(n, Mu), dim = c(n1, n2, n3)) system.time(fit <- glamlassoRR(X, Y)) modelno <- length(fit$lambda) par(mfrow = c(1, 3)) plot(c(Beta), type = "h") points(c(Beta)) lines(c(outer(fit$coef12[, modelno], c(fit$coef3[, modelno]))), col = "red", type = "h") plot(c(Beta12), ylim = range(Beta12, fit$coef12[, modelno]), type = "h") points(c(Beta12)) lines(fit$coef12[, modelno], col = "red", type = "h") plot(c(Beta3), ylim = range(Beta3, fit$coef3[, modelno]), type = "h") points(c(Beta3)) lines(fit$coef3[, modelno], col = "red", type = "h") ###with non tensor design component Z q <- 5 alpha <- matrix(rnorm(q)) * rbinom(q, 1, 0.5) Z <- matrix(rnorm(n1 * n2 * n3 * q), n1 * n2 * n3, q) Y <- array(rnorm(n1 * n2 * n3, Mu + array(Z %*% alpha, c(n1, n2, n3))), c(n1, n2, n3)) system.time(fit <- glamlassoRR(X, Y, Z)) modelno <- length(fit$lambda) par(mfrow = c(2, 2)) plot(c(Beta), type = "h") points(c(Beta)) lines(c(outer(fit$coef12[, modelno], c(fit$coef3[, modelno]))), col = "red", type = "h") plot(c(Beta12), ylim = range(Beta12,fit$coef12[, modelno]), type = "h") points(c(Beta12)) lines(fit$coef12[, modelno], col = "red", type = "h") plot(c(Beta3), ylim = range(Beta3, fit$coef3[, modelno]), type = "h") points(c(Beta3)) lines(fit$coef3[, modelno], col = "red", type = "h") plot(c(alpha), ylim = range(alpha, fit$alpha[, modelno]), type = "h") points(c(alpha)) lines(fit$alpha[, modelno], col = "red", type = "h") ################ poisson example Beta12 <- matrix(rnorm(p1 * p2, 0, 0.5), p1, p2) * matrix(rbinom(p1 * p2, 1, 0.1), p1, p2) Beta3 <- matrix(rnorm(p3, 0, 0.5) * rbinom(p3, 1, 0.5), p3, 1) Beta <- outer(Beta12, c(Beta3)) Mu <- RH(X3, RH(X2, RH(X1, Beta))) Y <- array(rpois(n1 * n2 * n3, exp(Mu)), dim = c(n1, n2, n3)) system.time(fit <- glamlassoRR(X, Y, family = "poisson")) modelno <- length(fit$lambda) par(mfrow = c(1, 3)) plot(c(Beta), type = "h") points(c(Beta)) lines(c(outer(fit$coef12[, modelno], c(fit$coef3[, modelno]))), col = "red", type = "h") plot(c(Beta12), ylim = range(Beta12, fit$coef12[, modelno]), type = "h") points(c(Beta12)) lines(fit$coef12[, modelno], col = "red", type = "h") plot(c(Beta3), ylim = range(Beta3, fit$coef3[, modelno]), type = "h") points(c(Beta3)) lines(fit$coef3[, modelno], col = "red", type = "h") ## End(Not run)