In multivariate analysis, we routinely observe outcomes which are either continuous, binary, count or mixed types. Often outcomes are interrelated and predictors correlated. In a large dimension, many predictors are irrelevant. Generalized co-sparse factor regression assumes that the outcomes are from the exponential family and models the underlying dependency through a low-rank and sparse coefficient matrix. We express the required coefficient matrix as the sum of unit-rank components with co-sparse singular vectors.
We propose a greedy procedure which sequentially recovers the sparse unit-rank components of the coefficient matrix, and refers it as generalized sequential extraction via constrained unit-rank estimation (GOFAR(S)). In order to be computationally efficient, we also propose an exclusive extraction procedure where the sparse unit-rank components are estimated in parallel and refer it as generalized exclusive extraction via constrained unit-rank estimation (GOFAR(P)).
Here we present examples of using R functions corresponding to the two estimation approach, i.e., GOFAR(S) and GOFAR(P).
rm(list = ls())
# load library
library(gofar)
#
## Model specification:
SD <- 123
set.seed(SD)
n <- 200
p <- 100
pz <- 0
# Model I in the paper
# n <- 200; p <- 300; pz <- 0 ; # Model II in the paper
# q1 <- 0; q2 <- 30; q3 <- 0 # Similar response cases
q1 <- 15
q2 <- 15
q3 <- 0 # mixed response cases
nrank <- 3 # true rank
rank.est <- 4 # estimated rank
nlam <- 40 # number of tuning parameter
s <- 1 # multiplying factor to singular value
snr <- 0.25 # SNR for variance Gaussian error
#
q <- q1 + q2 + q3
respFamily <- c("gaussian", "binomial", "poisson")
family <- list(gaussian(), binomial(), poisson())
familygroup <- c(rep(1, q1), rep(2, q2), rep(3, q3))
cfamily <- unique(familygroup)
nfamily <- length(cfamily)
#
control <- gofar_control()
#
#
## Generate data
D <- rep(0, nrank)
V <- matrix(0, ncol = nrank, nrow = q)
U <- matrix(0, ncol = nrank, nrow = p)
#
U[, 1] <- c(sample(c(1, -1), 8, replace = TRUE), rep(0, p - 8))
U[, 2] <- c(rep(0, 5), sample(c(1, -1), 9, replace = TRUE), rep(0, p - 14))
U[, 3] <- c(rep(0, 11), sample(c(1, -1), 9, replace = TRUE), rep(0, p - 20))
#
if (nfamily == 1) {
# for similar type response type setting
V[, 1] <- c(rep(0, 8), sample(c(1, -1), 8, replace =
TRUE) * runif(8, 0.3, 1), rep(0, q - 16))
V[, 2] <- c(rep(0, 20), sample(c(1, -1), 8, replace =
TRUE) * runif(8, 0.3, 1), rep(0, q - 28))
V[, 3] <- c(
sample(c(1, -1), 5, replace = TRUE) * runif(5, 0.3, 1), rep(0, 23),
sample(c(1, -1), 2, replace = TRUE) * runif(2, 0.3, 1), rep(0, q - 30)
)
} else {
# for mixed type response setting
# V is generated such that joint learning can be emphasised
V1 <- matrix(0, ncol = nrank, nrow = q / 2)
V1[, 1] <- c(sample(c(1, -1), 5, replace = TRUE), rep(0, q / 2 - 5))
V1[, 2] <- c(rep(0, 3), V1[4, 1], -1 * V1[5, 1],
sample(c(1, -1), 3, replace = TRUE), rep(0, q / 2 - 8))
V1[, 3] <- c(V1[1, 1], -1 * V1[2, 1], rep(0, 4),
V1[7, 2], -1 * V1[8, 2], sample(c(1, -1), 2, replace = TRUE),
rep(0, q / 2 - 10))
#
V2 <- matrix(0, ncol = nrank, nrow = q / 2)
V2[, 1] <- c(sample(c(1, -1), 5, replace = TRUE), rep(0, q / 2 - 5))
V2[, 2] <- c(rep(0, 3), V2[4, 1], -1 * V2[5, 1],
sample(c(1, -1), 3, replace = TRUE), rep(0, q / 2 - 8))
V2[, 3] <- c(V2[1, 1], -1 * V2[2, 1], rep(0, 4),
V2[7, 2], -1 * V2[8, 2],
sample(c(1, -1), 2, replace = TRUE), rep(0, q / 2 - 10))
#
V <- rbind(V1, V2)
}
U[, 1:3] <- apply(U[, 1:3], 2, function(x) x / sqrt(sum(x^2)))
V[, 1:3] <- apply(V[, 1:3], 2, function(x) x / sqrt(sum(x^2)))
#
D <- s * c(4, 6, 5) # signal strength varries as per the value of s
or <- order(D, decreasing = T)
U <- U[, or]
V <- V[, or]
D <- D[or]
C <- U %*% (D * t(V)) # simulated coefficient matrix
intercept <- rep(0.5, q) # specifying intercept to the model:
C0 <- rbind(intercept, C)
#
Xsigma <- 0.5^abs(outer(1:p, 1:p, FUN = "-"))
# Simulated data
sim.sample <- gofar_sim(U, D, V, n, Xsigma, C0, familygroup, snr)
# Dispersion parameter
pHI <- c(rep(sim.sample$sigmaG, q1), rep(1, q2), rep(1, q3))
X <- sim.sample$X[1:n, ]
Y <- sim.sample$Y[1:n, ]
# Test data
sim.sample <- gofar_sim(U, D, V, 1000, Xsigma, C0, familygroup, snr)
Xt <- sim.sample$X
Yt <- sim.sample$Y
#
crossprod(X %*% U)
apply(X, 2, norm, "2")
X0 <- cbind(1, X)
#
# Simulate data with 20% missing entries
miss <- 0.20 # Proportion of entries missing
t.ind <- sample.int(n * q, size = miss * n * q)
y <- as.vector(Y)
y[t.ind] <- NA
Ym <- matrix(y, n, q)
naind <- (!is.na(Ym)) + 0 # matrix(1,n,q)
misind <- any(naind == 0) + 0
#
An R user will need the following package to connect to the database: a) “DBI”, for database interface; b) “odbc” for connecting to the database using DBI interface.
In addition, user may require some additional R package for downloading, processing and visualizing the data. Run the following commands to install some of the essential packages.
# Model fit:
control$epsilon <- 1e-7
control$spU <- 50 / p # estimated percentage of non zero entries in U
control$spV <- 25 / q # estimated percentage of non zero entries in V
control$maxit <- 1000
# Model fitting: GOFAR(S) (full data)
set.seed(SD)
rank.est <- 5
fit.seq <- gofar_s(Y, X, nrank = rank.est, family = family,
nlambda = nlam, familygroup = familygroup,
control = control, nfold = 5)
# Model fitting: GOFAR(S) (missing data)
set.seed(SD)
rank.est <- 5
fit.seq.m <- gofar_s(Ym, X, nrank = rank.est, family = family,
nlambda = nlam, familygroup = familygroup,
control = control, nfold = 5)
# Model fit:
control$epsilon <- 1e-7
control$spU <- 50 / p
control$spV <- 25 / q
control$maxit <- 1000
# Model fitting: GOFAR(P) (full data)
set.seed(SD)
rank.est <- 5
fit.eea <- gofar_p(Y, X, nrank = rank.est, nlambda = nlam,
family = family, familygroup = familygroup,
control = control, nfold = 5)
# Model fitting: GOFAR(P) (missing data)
set.seed(SD)
rank.est <- 5
fit.eea.m <- gofar_p(Ym, X, nrank = rank.est, nlambda = nlam,
family = family, familygroup = familygroup,
control = control, nfold = 5)