Package 'SILFS'

Selecting Tuning Parameter for Factor Adjusted-Pairwise Fusion Penalty (FA-PFP) Method via corresponding BIC

Description

This function is to select tuning parameters simultaneously for FA-PFP method via minimizing the BIC.

Usage

BIC_PFP(
  Y,
  Fhat,
  Uhat,
  alpha_init,
  lasso_start,
  lasso_stop,
  lam_start,
  lam_stop,
  grid_1,
  grid_2,
  epsilon
)

Arguments

Y	The response vector of length $n$.
Fhat	The estimated common factors matrix of size $n \times r$.
Uhat	The estimated idiosyncratic factors matrix of size $n \times p$.
alpha_init	The initialization of intercept parameter.
lasso_start	The user-supplied start search value of the tuning parameters for LASSO.
lasso_stop	The user-supplied stop search value of the tuning parameters for LASSO.
lam_start	The user-supplied start search value of the tuning parameters for Pairwise Fusion Penalty.
lam_stop	The user-supplied stop search value of the tuning parameters for Pairwise Fusion Penalty.
grid_1	The user-supplied number of search grid points corresponding to the LASSO tuning parameter.
grid_2	The user-supplied number of search grid points corresponding to the tuning parameter for Pairwise Fusion Penalty.
epsilon	The user-supplied stopping tolerance.

Value

A list with the following components:

lasso	The tuning parameter of the LASSO penalty selected using BIC.
lambda	The tuning parameter of the Pairwise Concave Fusion Penalty selected using BIC.

Author(s)

Yong He, Liu Dong, Fuxin Wang, Mingjuan Zhang, Wenxin Zhou.

Examples

n <- 50
p <- 50
r <- 3
lasso_start <- sqrt(log(p)/n)*0.1
lasso_stop <- sqrt(log(p)/n)
lam_start <- 0.3
lam_stop <- 1
grid_1 <- 5
grid_2 <- 5
alpha <- sample(c(-3,3),n,replace=TRUE,prob=c(1/2,1/2))
beta <- c(rep(1,2),rep(0,48))
B <- matrix((rnorm(p*r,1,1)),p,r)
F_1 <- matrix((rnorm(n*r,0,1)),n,r)
U <- matrix(rnorm(p*n,0,0.1),n,p)
X <- F_1%*%t(B)+U
Y <- alpha + X%*%beta + rnorm(n,0,0.5)
alpha_init <- INIT(Y,F_1,0.1)

BIC_PFP(Y,F_1,U,alpha_init,lasso_start,lasso_stop,lam_start,lam_stop,grid_1,grid_2,0.3)

---

Selecting Tuning Parameter for SILFS Method via corresponding BIC

Description

This function is to select tuning parameters simultaneously for SILFS method via minimizing the BIC.

Usage

BIC_SILFS(
  Y,
  Fhat,
  Uhat,
  K,
  alpha_init,
  lasso_start,
  lasso_stop,
  CAR_start,
  CAR_stop,
  grid_1,
  grid_2,
  epsilon
)

Arguments

Y	The response vector of length $n$.
Fhat	The estimated common factors matrix of size $n \times r$.
Uhat	The estimated idiosyncratic factors matrix of size $n \times p$.
K	The estimated subgroup number.
alpha_init	The initialization of intercept parameter.
lasso_start	The user-supplied start search value of the tuning parameters for LASSO.
lasso_stop	The user-supplied stop search value of the tuning parameters for LASSO.
CAR_start	The user-supplied start search value of the tuning parameters for Center-Augmented Regularization.
CAR_stop	The user-supplied stop search value of the tuning parameters for Center-Augmented Regularization.
grid_1	The user-supplied number of search grid points corresponding to the LASSO tuning parameter.
grid_2	The user-supplied number of search grid points corresponding to the tuning parameter for Center-Augmented Regularization.
epsilon	The user-supplied stopping tolerance.

Value

A list with the following components:

lasso	The tuning parameter of the LASSO penalty selected using BIC.
CAR	The tuning parameter of the Center Augmented Regularization selected using BIC.

Examples

n <- 50
p <- 50
r <- 3
K <- 2
lasso_start <- sqrt(log(p)/n)*0.01
lasso_stop <- sqrt(log(p)/n)*10^(0.5)
CAR_start <- 0.001
CAR_stop <- 0.1
grid_1 <- 5
grid_2 <- 5
alpha <- sample(c(-3,3),n,replace=TRUE,prob=c(1/2,1/2))
beta <- c(rep(1,2),rep(0,48))
B <- matrix((rnorm(p*r,1,1)),p,r)
F_1 <- matrix((rnorm(n*r,0,1)),n,r)
U <- matrix(rnorm(p*n,0,0.1),n,p)
X <- F_1%*%t(B)+U
Y <- alpha + X%*%beta + rnorm(n,0,0.5)
alpha_init <- INIT(Y,F_1,0.1)

BIC_SILFS(Y,F_1,U,K,alpha_init,lasso_start,lasso_stop,CAR_start,CAR_stop,grid_1,grid_2,0.3)

---

SILFS-Based Subgroup Identification and Variable Selection Optimized by DC-ADMM under the L1 Distance

Description

This function employs SILFS method and uses the corresponding Difference of Convex functions-Alternating Direction Method of Multipliers (DC-ADMM) algorithm for optimization to identify subgroup structures and conduct variable selection under the L1 Distance.

Usage

DCADMM_iter_l1(
  Y,
  F_hat,
  U_hat,
  r_1,
  r_2,
  r_3,
  lambda_1,
  lambda_2,
  K,
  alpha_init,
  epsilon_1,
  epsilon_2
)

Arguments

Y	The response vector of length $n$.
F_hat	The estimated factor matrix of size $n \times r$.
U_hat	The estimated idiosyncratic factors matrix of size $n \times p$.
r_1	The Lagrangian augmentation parameter for constraints of intercepts.
r_2	The Lagrangian augmentation parameter for constraints of group centers.
r_3	The Lagrangian augmentation parameter for constraints of coefficients.
lambda_1	The tuning parameter for Center-Augmented Regularization.
lambda_2	The tuning parameter for LASSO.
K	The estimated group number.
alpha_init	The initialization of intercept parameter.
epsilon_1	The user-supplied stopping error for outer loop.
epsilon_2	The user-supplied stopping error for inner loop.

Value

A list with the following components:

alpha_curr	The estimated intercept parameter vector of length $n$.
gamma_curr	The estimated vector of subgroup centers of length $K$.
theta_curr	The estimated regression coefficient vector, matched with common factor terms, with a dimension of $r$.
beta_curr	The estimated regression coefficients matched with idiosyncratic factors, with a dimension of $p$.

Author(s)

Yong He, Liu Dong, Fuxin Wang, Mingjuan Zhang, Wenxin Zhou.

References

He, Y., Liu, D., Wang, F., Zhang, M., Zhou, W., 2024. High-Dimensional Subgroup Identification under Latent Factor Structures.

Examples

n <- 50
p <- 50
r <- 3
K <- 2
alpha <- sample(c(-3,3),n,replace=TRUE,prob=c(1/2,1/2))
beta <- c(rep(1,2),rep(0,48))
B <- matrix((rnorm(p*r,1,1)),p,r)
F_1 <- matrix((rnorm(n*r,0,1)),n,r)
U <- matrix(rnorm(p*n,0,0.1),n,p)
X <- F_1%*%t(B)+U
Y <- alpha + X%*%beta + rnorm(n,0,0.5)
alpha_init <- INIT(Y,F_1,0.1)
DCADMM_iter_l1(Y,F_1,U,0.5,0.5,0.5,0.01,0.05,K,alpha_init,1,0.3)

---

SILFS-Based Subgroup Identification and Variable Selection Optimized by DC-ADMM under the L2 Distance

Description

This function employs SILFS method and uses the corresponding Difference of Convex functions-Alternating Direction Method of Multipliers (DC-ADMM) algorithm for optimization to identify subgroup structures and conduct variable selection under the L2 Distance.

Usage

DCADMM_iter_l2(
  Y,
  F_hat,
  U_hat,
  r_1,
  r_2,
  r_3,
  lambda_1,
  lambda_2,
  K,
  alpha_init,
  epsilon_1,
  epsilon_2
)

Arguments

Y	The response vector of length $n$.
F_hat	The estimated factor matrix of size $n \times r$.
U_hat	The estimated idiosyncratic factors matrix of size $n \times p$.
r_1	The Lagrangian augmentation parameter for constraints of intercepts.
r_2	The Lagrangian augmentation parameter for constraints of group centers.
r_3	The Lagrangian augmentation parameter for constraints of coefficients.
lambda_1	The tuning parameter for Center-Augmented Regularization.
lambda_2	The tuning parameter for LASSO.
K	The estimated group number.
alpha_init	The initialization of intercept parameter.
epsilon_1	The user-supplied stopping error for outer loop.
epsilon_2	The user-supplied stopping error for inner loop.

Value

A list with the following components:

alpha_curr	The estimated intercept parameter vector of length $n$.
gamma_curr	The estimated vector of subgroup centers of length $K$.
theta_curr	The estimated regression coefficient vector, matched with common factor terms, with a dimension of $r$.
beta_curr	The estimated regression coefficients matched with idiosyncratic factors, with a dimension of $p$.

Author(s)

Yong He, Liu Dong, Fuxin Wang, Mingjuan Zhang, Wenxin Zhou.

References

He, Y., Liu, D., Wang, F., Zhang, M., Zhou, W., 2024. High-Dimensional Subgroup Identification under Latent Factor Structures.

Examples

n <- 50
p <- 50
r <- 3
K <- 2
alpha <- sample(c(-3,3),n,replace=TRUE,prob=c(1/2,1/2))
beta <- c(rep(1,2),rep(0,48))
B <- matrix((rnorm(p*r,1,1)),p,r)
F_1 <- matrix((rnorm(n*r,0,1)),n,r)
U <- matrix(rnorm(p*n,0,0.1),n,p)
X <- F_1%*%t(B)+U
Y <- alpha + X%*%beta + rnorm(n,0,0.5)
alpha_init <- INIT(Y,F_1,0.1)
DCADMM_iter_l2(Y,F_1,U,0.5,0.5,0.5,0.01,0.05,K,alpha_init,1,0.3)

---

Factor Adjusted-Pairwise Fusion Penalty (FA-PFP) Method for Subgroup Identification and Variable Selection

Description

This function utilizes the FA-PFP method implemented via the Alternating Direction Method of Multipliers (ADMM) algorithm to identify subgroup structures and conduct variable selection.

Usage

FA_PFP(Y, Fhat, Uhat, vartheta, lam, gam, alpha_init, lam_lasso, epsilon)

Arguments

Y	The response vector of length $n$.
Fhat	The estimated common factors matrix of size $n \times r$.
Uhat	The estimated idiosyncratic factors matrix of size $n \times p$.
vartheta	The Lagrangian augmentation parameter for intercepts.
lam	The tuning parameter for Pairwise Fusion Penalty.
gam	The user-supplied parameter for Alternating Direction Method of Multipliers (ADMM) algorithm.
alpha_init	The initialization of intercept parameter.
lam_lasso	The tuning parameter for LASSO.
epsilon	The user-supplied stopping tolerance.

Value

A list with the following components:

alpha_m	The estimated intercept parameter vector of length $n$.
theta_m	The estimated regression coefficient vector, matched with common factor terms, with a dimension of $r$.
beta_m	The estimated regression coefficients matched with idiosyncratic factors, with a dimension of $p$.
eta_m	A numeric matrix storing the pairwise differences of the estimated intercepts, with size of $n \times (n\times(n-1)/2)$.

Author(s)

Yong He, Liu Dong, Fuxin Wang, Mingjuan Zhang, Wenxin Zhou.

References

Ma, S., Huang, J., 2017. A concave pairwise fusion approach to subgroup analysis.

Examples

n <- 50
p <- 50
r <- 3
alpha <- sample(c(-3,3),n,replace=TRUE,prob=c(1/2,1/2))
beta <- c(rep(1,2),rep(0,48))
B <- matrix((rnorm(p*r,1,1)),p,r)
F_1 <- matrix((rnorm(n*r,0,1)),n,r)
U <- matrix(rnorm(p*n,0,0.1),n,p)
X <- F_1%*%t(B)+U
Y <- alpha + X%*%beta + rnorm(n,0,0.5)
alpha_init <- INIT(Y,F_1,0.1)
FA_PFP(Y,F_1,U,1,0.67,3,alpha_init,0.05,0.3)

---

Initialization Function for the Intercept Parameter

Description

This function computes initial values for intercept parameter by solving a ridge regression problem.

Usage

INIT(Y, X, lam_ridge)

Arguments

Y	The response vector of length $n$.
X	The design matrix of size $n\times p$.
lam_ridge	The tuning parameter for ridge regression.

Value

A numeric vector of length $n$, representing the initial estimation for intercept parameter.

Examples

n <- 100
p <- 100
beta <- rep(1,p)
X <- matrix(rnorm(100*100), n, p)
Y <- sample(c(-3,3),n,replace=TRUE,prob=c(1/2,1/2)) +  X%*%beta
lam_ridge <- 0.1
alpha_init <- INIT(Y, X, lam_ridge)

---

Standard Center Augmented Regularization (S-CAR) Method for Subgroup Identification and Variable Selection

Description

This function employs the S-CAR method under L2 distance and uses the Coordinate Descent Algorithm for optimization to identify subgroup structures and execute variable selection.

Usage

SCAR(Y, X, lam_CAR, lam_lasso, alpha_init, K, epsilon)

Arguments

Y	The response vector of length $n$.
X	The design matrix of size $n \times p$.
lam_CAR	The tuning parameter for Center-Augmented Regularization.
lam_lasso	The tuning parameter for lasso.
alpha_init	The initialization of intercept parameter.
K	The estimated group number.
epsilon	The user-supplied stopping tolerance.

Value

A list with the following components:

alpha_m	The estimated intercept parameter vector of length $n$.
gamma	The estimated vector of subgroup centers of length $K$.
beta_m	The estimated regression coefficient vector of dimension $p$.

Author(s)

Yong He, Liu Dong, Fuxin Wang, Mingjuan Zhang, Wenxin Zhou.

Examples

n <- 50
p <- 50
r <- 3
K <- 2
alpha <- sample(c(-3,3),n,replace=TRUE,prob=c(1/2,1/2))
beta <- c(rep(1,2),rep(0,48))
B <- matrix((rnorm(p*r,1,1)),p,r)
F_1 <- matrix((rnorm(n*r,0,1)),n,r)
U <- matrix(rnorm(p*n,0,0.1),n,p)
X <- F_1%*%t(B)+U
Y <- alpha + X%*%beta + rnorm(n,0,0.5)
alpha_init <- INIT(Y,X,0.1)
SCAR(Y,X,0.01,0.05,alpha_init,K,0.3)

---

SILFS-Based Subgroup Identification and Variable Selection Optimized by Coordinate Descent under the L2 Distance

Description

This function employs SILFS method under L2 distance and uses the Coordinate Descent Algorithm for optimization to effectively identify subgroup structures and perform variable selection.

Usage

SILFS(Y, X_aug, r, lam_CAR, lam_lasso, alpha_init, K, epsilon)

Arguments

Y	The response vector of length $n$.
X_aug	The augmented design matrix created by row concatenation of common and idiosyncratic factor matrices, with a size of $n \times (r+p)$.
r	The user supplied number of common factors.
lam_CAR	The tuning parameter for Center-Augmented Regularization.
lam_lasso	The tuning parameter for LASSO.
alpha_init	The initialization of intercept parameter.
K	The user-supplied group number.
epsilon	The user-supplied stopping tolerance.

Value

A vector containing the following components:

alpha_m	The estimated intercept parameter vector of length $n$.
gamma	The estimated vector of subgroup centers of length $K$.
theta_m	The estimated regression coefficient vector, matched with common factor terms, with a dimension of $r$.
beta_m	The estimated regression coefficients matched with idiosyncratic factors, with a dimension of $p$.

Author(s)

Yong He, Liu Dong, Fuxin Wang, Mingjuan Zhang, Wenxin Zhou.

References

He, Y., Liu, D., Wang, F., Zhang, M., Zhou, W., 2024. High-Dimensional Subgroup Identification under Latent Factor Structures.

Examples

n <- 50
p <- 50
r <- 3
K <- 2
alpha <- sample(c(-3,3),n,replace=TRUE,prob=c(1/2,1/2))
beta <- c(rep(1,2),rep(0,48))
B <- matrix((rnorm(p*r,1,1)),p,r)
F_1 <- matrix((rnorm(n*r,0,1)),n,r)
U <- matrix(rnorm(p*n,0,0.1),n,p)
X <- F_1%*%t(B)+U
Y <- alpha + X%*%beta + rnorm(n,0,0.5)
alpha_init <- INIT(Y,F_1,0.1)
SILFS(Y,cbind(F_1,U),3,0.01,0.05,alpha_init,K,0.3)

---