Package 'LassoNet'

LassoNet: package for 3CoSE algorithm.

Description

LassoNet contains functions to estimate a penalized regression model using 3CoSE algorithm described in the paper Weber, Striaukas, Schumacher and Binder (2018). The main function of the package is the function lasso.net.grid, see the example below.

Details

Package:	LassoNet
Type:	Package
Version:	0.8.3
Date:	2019-12-16
License:	Open source

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

References

Weber, M., Striaukas, J., Schumacher, M., Binder, H. "Network-Constrained Covariate Coefficient and Connection Sign Estimation" (2018) <doi:10.2139/ssrn.3211163>

See Also

Rcpp, glmnet

---

Updates $\beta$ coefficients.

Description

This function updates $\beta$ for given penalty parameters.

Usage

beta.update.net(x,y,beta,lambda1,lambda2,M1,n.iter,iscpp,tol)

Arguments

x	input data matrix of size $n \times p$; n - number of observations; p - number of covariates
y	response vector or size $n \times 1$
beta	initial value for $\beta$; default - zero vector of size $n \times 1$
lambda1	lasso penalty parameter
lambda2	network penalty parameter
M1	penalty matrix
n.iter	maximum number of iterations for $\beta$ step; default - 1e5
iscpp	binary choice for using cpp function in coordinate updates; 1 - use C++ (default), 0 - use R
tol	convergence tolerance level; default - 1e-6

Details

Updates the coefficient vector $\beta$ given the data and penalty parameters $\lambda$1 and $\lambda$2. Convergence criterion is defined as $\sum_{i=1}^p |\beta_{i,j} - \beta_{i,j-1}| \leq$ to.

Value

beta	updated $\beta$ vector
convergence	binary variable; 1 - yes
steps	number of steps until convergence

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

References

Weber, M., Striaukas, J., Schumacher, M., Binder, H. "Network-Constrained Covariate Coefficient and Connection Sign Estimation" (2018) <doi:10.2139/ssrn.3211163>

Examples

p<-200
n<-100
beta.0=array(1,c(p,1))
x<-matrix(rnorm(n*p),n,p)
y<-rnorm(n,mean=0,sd=1)
lambda1<-1
lambda2<-1
M1<-diag(p)
updates<-beta.update.net(x, y, beta.0, lambda1, lambda2, M1)

---

C++ subroutine that updates $\beta$ coefficients.

Description

This function updates $\beta$ for given penalty parameters.

Usage

betanew_lasso_cpp(xx, xy, beta, M, y, Lambda1, Lambda2, iter, tol)

Arguments

xx	Bx matrix
xy	By vector
beta	initial value for $\beta$; default - zero vector of size $p \times 1$
M	penalty matrix
y	response vector or size $n \times 1$
Lambda1	lasso penalty parameter
Lambda2	network penalty parameter
iter	maximum number of iterations for $\beta$ step
tol	convergence tolerance level

Details

See beta.update.net

Value

beta	updated $\beta$ vector
steps	number of steps until convergence

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

References

Weber, M., Striaukas, J., Schumacher, M., Binder, H. "Network-Constrained Covariate Coefficient and Connection Sign Estimation" (2018) <doi:10.2139/ssrn.3211163>

Examples

p<-200
n<-100
beta.0=array(1,c(p,1))
x<-matrix(rnorm(n*p),n,p)
y<-rnorm(n,mean=0,sd=1)
lambda1<-1
lambda2<-1
M1<-diag(p)
updates<-beta.update.net(x, y, beta.0, lambda1, lambda2, M1)

---

Fast least squares estimate.

Description

Computes least squares estimate in an efficient way.

Usage

fastols(y, x)

Arguments

y	dependent variable
x	response variable

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

Examples

p<-10
n<-100
x<-matrix(rnorm(n*p),n,p)
beta<-array(5, c(p,1))
y<-x%*%beta + rnorm(n,mean=0,sd=0.1)
fastols(y,x)

---

Computes decomposition elements.

Description

Computes matrices $B^{ij}_X$ and $B^{ij}_y$ to speed up estimation of connection signs. These matrices are stored only for indices that have non zero entries in penalty matrix M.

Usage

get.BxBy(x, y, M)

Arguments

x	Input data matrix of size $n \times p$, n - number of observations, p - number of covariates
y	y Response vector or size $n \times 1$
M	penalty matrix

Details

Calculates matrices all for i and j indices that have non zero values in a given penalty matrix.

Value

Bx	array of $B^{ij}_X$ stored matrices. $Bx[,,k]$ are the k-th combination of i and j non zero entry in the penalty matrix M
By	array of $B^{ij}_y$ stored matrices. $By[,k]$ are the k-th combination of i and j non zero entry in the penalty matrix M

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

References

Weber, M., Striaukas, J., Schumacher, M., Binder, H. "Network-Constrained Covariate Coefficient and Connection Sign Estimation" (2018) <doi:10.2139/ssrn.3211163>

Examples

p<-200
n<-100
x<-matrix(rnorm(n*p),n,p)
y<-rnorm(n,mean=0,sd=1)
M<-diag(p)
get.BxBy(x, y, M)

---

Vetorizes connection sign matrix.

Description

Stores a matrix of connection signs to a vector.

Usage

get.signs.M(MAT)

Arguments

MAT	matrix of connection signs that contains -1, 1 or 0

Value

vec.out

vectorized MAT matrix

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

---

Updates the estimates of the connection signs by running mini OLS models.

Description

Updates connection signs $\hat{\xi}$.

Usage

get.xi(Bx,By,beta,xi,M)

Arguments

Bx	Bx element
By	By element
beta	$\hat{\beta}$ estimated value
xi	$\hat{\xi}$ matrix estimated at the previous step
M	penalty matrix

Value

xi	$\hat{\xi}$ matrix

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

References

Weber, M., Striaukas, J., Schumacher, M., Binder, H. "Network-Constrained Covariate Coefficient and Connection Sign Estimation" (2018) <doi:10.2139/ssrn.3211163>

---

Estimates coefficients over the grid values of penalty parameters.

Description

See lasso.net.grid

Usage

lasso.net.fixed(x,y,beta.0,lambda1,lambda2,M1,n.iter,iscpp,tol)

Arguments

x	$n \times p$ input data matrix
y	response vector or size $n \times 1$
beta.0	initial value for $\beta$; default - zero vector of size $n \times 1$
lambda1	lasso penalty coefficient
lambda2	network penalty coefficient
M1	penalty matrix
n.iter	maximum number of iterations for $\beta$ updating; default - 1e5
iscpp	binary choice for using cpp function in coordinate updates; 1 - use C++ (default), 0 - use R.
tol	convergence in $\beta$ tolerance level; default - 1e-6

Details

Function loops through the grid of values of penalty parameters $\lambda$1 and $\lambda$2 until convergence is reached. Warm starts are stored for each iterator. The warm starts are stored once the coordinate updating converges.

Value

beta	Matrix of $\beta$ coefficients. Columns denote different $\lambda$1 coefficients, rows - $\lambda$2 coefficients
mse	Mean squared error value
iterations	matrix with stored number of steps for sign matrix to converge
update.steps	matrix with stored number of steps for $\beta$ updates to converge. (only stores the last values from connection signs iterations)
convergence.in.grid	matrix with stored values for convergence in $\beta$ coefficients. If at least one $\beta$ did not converge in sign matrix iterations, 0 (false) is stored, otherwise 1 (true)

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

References

Weber, M., Striaukas, J., Schumacher, M., Binder, H. "Network-Constrained Covariate Coefficient and Connection Sign Estimation" (2018) <doi:10.2139/ssrn.3211163>

Examples

p=200
n=100
beta.0=array(1,c(p,1))
x=matrix(rnorm(n*p),n,p)
y=rnorm(n,mean=0,sd=1)
lambda1=c(0,1)
lambda2=c(0,1)
M1=diag(p)
lasso.net.fixed(x, y, beta.0, lambda1, lambda2, M1)

---

Estimates coefficients and connection signs over the grid of values of penalty parameters $\lambda$1 and $\lambda$2.

Description

Fits network regressions over the grid of values of penalty parameters $\lambda$1 and $\lambda$2, stores connection signs, number of iterations until convergence and convergence outcome.

Usage

lasso.net.grid(x,y ,beta.0,lambda1,lambda2,M1,m.iter,n.iter,iscpp=TRUE,tol,alt.num)

Arguments

x	$n \times p$ input data matrix
y	response vector or size $n \times 1$
beta.0	initial value for $\beta$. default - zero vector of size $n \times 1$
lambda1	lasso penalty coefficient
lambda2	network penalty coefficient
M1	penalty matrix
m.iter	maximum number of iterations for sign matrix updating; default - 100
n.iter	maximum number of iterations for $\beta$ updating; default - 1e5
iscpp	binary choice for using cpp function in coordinate updates; 1 - use C++ (default), 0 - use R
tol	convergence in $\beta$ tolerance level; default - 1e-6
alt.num	alt.num remaining iterataions are stored; default - 12

Details

Fits network regression for the grid values of $\lambda$1 and $\lambda$2 using warm starts.

Value

beta	matrix of $\beta$ coefficients, columns are for different $\lambda$1 parameters, rows $\lambda$2 parameters
mse	mean squared error value
M	array of connection signs. $M[,,i,j]$ is the connection sign matrix for j-th $\lambda$1 value and i-th $\lambda$2 value
iterations	matrix with stored number of steps for sign matrix to converge
update.steps	matrix with stored number of steps for $\beta$ updates to converge. (only stores the last values from connection signs iterations)
convergence.in.M	matrix with stored values for convergence in sign matrix
convergence.in.grid	matrix with stored values for convergence in $\beta$ coefficients. If at least one $\beta$ did not converge in sign matrix iterations, 0 (false) is stored, otherwise 1 (true)
xi.conv	array with stored connection signs changes in each iteration
beta.alt	array of coefficient vectors in case connection signs alternate

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

References

Weber, M., Striaukas, J., Schumacher, M., Binder, H. "Network-Constrained Covariate Coefficient and Connection Sign Estimation" (2018) <doi:10.2139/ssrn.3211163>

Examples

p=200
n=100
beta.0=array(1,c(p,1))
x=matrix(rnorm(n*p),n,p)
y=rnorm(n,mean=0,sd=1)
lambda1=c(0,1)
lambda2=c(0,1)
M1=diag(p)
lasso.net.grid(x, y, beta.0, lambda1, lambda2, M1)

---

Computes Laplacian matrix.

Description

Computes Laplacian matrix.

Usage

mat.to.laplacian(M1,type)

Arguments

M1	$p \times p$ matrix
type	Laplacian types: 1) "normalized" (default) - normalized Laplacian, 2) "combinatorial" - combinatorial Laplacian

Value

L

Laplacian

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

---

Updates connection sign matrix.

Description

Updates M using relation $(M)_{ij} = -\hat{\xi}_{ij} |(M_1)|_{ij}$.

Usage

matrix.M.update(M, xi)

Arguments

M	penalty matrix
xi	estimated $\hat{\xi}_{ij}$ matrix

Details

Updates M

Value

M

updated M

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

References

Weber, M., Striaukas, J., Schumacher, M., Binder, H. "Network-Constrained Covariate Coefficient and Connection Sign Estimation" (2018) <doi:10.2139/ssrn.3211163>

Examples

p<-100
M<-diag(p)
xi<-matrix(rnorm(p*p), p, p)
matrix.M.update(M,xi)

---

Soft thresholding operator.

Description

Soft thresholding operator.

Usage

soft.thresh(x, kappa)

Arguments

x	$\beta$ coordinate
kappa	$\kappa$ value in general or $\lambda_1$ for covariance updating

Details

Soft thresholding definition: $S(x,\kappa) = sign(x) (|x| - \kappa)_{+}$

Value

x	value after applying soft thresholding operator

Author(s)

Maintainer: Jonas Striaukas <[email protected]>

Examples

kappa<-0.2
x<-0.7
soft.thresh(x, kappa)

---