netcoh (version 0.2)

rncreg: Fits a regression model with network cohesion effects.

Description

fits a regression model such that each samples are following a different regression curve such that connected individuals in a network tend to have similar curves. The function currently fits linear, logistic and Cox's regression model.

Usage

rncreg(A,lambda,Y=NULL,X=NULL,dt=NULL,gamma=0.05,
     model=c("linear","logistic","cox"), max.iter=50,tol=1e-4,
     init=NULL, cv=NULL,cv.seed=999,low.dim=NULL,verbose=FALSE)

Arguments

A

An nxn symmetric adjacency matrix for the network.

lambda

Tuning parameter for the cohesion penalty.

Y

An nx1 matrix of response, if the model to fit is linear or logistic. It will not be used if one fits Cox's model.

X

An nxp covariate matrix with each row being the covariates for one individual. If one want to fits a model without using any covariate, it can be empty.

dt

Only used to fit Cox's model. An nx2 data.frame such that the first column is the observed time while the second column is the event indicator which is 1 for truely observed events and 0 for censored events.

gamma

The amount of diagonal regularization added to graph Laplacian.

model

Can only be one of "linear", "logistic" or "cox".

max.iter

The maximum number of newton steps to iterate. Only used for logistic model or Cox's model.

tol

The tolerance level for convergence. Only used for logistic model or Cox's model.

init

The initial point for newton algoritm. It should be an (n+p)x1 matrix that stacks alpha and beta. Only used for logistic model or Cox's model. If not specified, zeros will be used.

cv

Number of folds for cross-validation. If unspecified, then no cross-validation will be done.

cv.seed

Random number generator seed for cross-validation.

low.dim

Only used for linear model. If the probelm is a low dimensional problem such that n>>p, then using low.dim=TRUE is potentially faster.

verbose

If TRUE, the log likelihood in each newton step will be printed. Only used for logistic model and Cox's model.

Value

• An object from class rncReg will be returned.

Details

The function solves $$max L(\alpha, \beta) - \lambda\alpha^TL\alpha.$$ With a proper choice of L function according to the specific model. When the model is linear regression, L is the negative squared error (or gaussian kernel); when the model is logistic regression, L is the binomial log likelihood; when the model is Cox's model, L is the log partial likelihood. gamma is used to regularize the graph Laplacian and is potentially helpful for numerical stability and Cox's model identifiability. Notice that having a positive gamma tends to shrink individual effects to zeros. Thus in linear regression, we suggest first center the data (both predictors and response) before fitting the model. For full details, please check the reference paper.

References

Tianxi Li, Elizaveta Levina and Ji Zhu. (2016) Regression with network cohesion, http://arxiv.org/pdf/1602.01192v1.pdf

See Also

rncReg,predict.rncReg

Examples

set.seed(100)

A <- matrix(0,200,200)
A[1:100,1:100] <- 1
A[101:200,101:200] <- 1
diag(A) <- 0

alpha <- c(rep(1,100),rep(-1,100)) + rnorm(200)*0.5
A <- A[c(1:50,101:150,51:100,151:200),c(1:50,101:150,51:100,151:200)]
alpha <- alpha[c(1:50,101:150,51:100,151:200)]

beta <- rnorm(2)

X <- matrix(rnorm(400),ncol=2)

Y <- X
delta <- Y
delta[Y>0] <- 1
delta[Y<=0] <- 0

A1 <- A[1:100,1:100]
X1 <- X[1:100,]
Y1 <- matrix(Y[1:100],ncol=1)
delta1 <- matrix(delta[1:100],ncol=1)


## If one wants to regularize the Laplacian
## by using gamma > 0 in rncreg,
## we suggest use centered data.
#mean.x <- colMeans(X1)
#mean.y <- mean(Y1)
#Y1 <- Y1-mean.y
#X1 <- t(t(X1)-mean.x)
#Y <- Y-mean.y
#X <- t(t(X)-mean.x)

m <- rncreg(A=A1,X=X1,Y=Y1,model="linear",lambda=10,gamma=0,cv=5)
p <- predict(m,full.A=A,full.X=X)

#m <- rncreg(A=A1,X=X1,Y=Y1,model="logistic",lambda=10,gamma=0.01,cv=5)

#m <- rncreg(A=A1,X=X1,dt=data.frame(y=Y1,delta=delta1),model="cox",lambda=10,gamma=0.01,cv=5)