fusionbase {FusionLearn}

Fusion learning method for continuous responses

Description

fusionbase conducts the group penalization to multiple linear models with a specified penalty value. fusionbase.fit can be used to search the best candidate model based on the pseudo Bayesian information criterion with a sequence of penalty values.

Usage

fusionbase(x, y, lambda, N, p, m, beta=0.1, thresh=0.05, 
           maxiter=30, methods="scad",Complete=TRUE)

fusionbase.fit(x, y, lambda, N, p, m, beta=0.1, thresh=0.05, 
               maxiter=30, methods="scad", Complete=TRUE, depen ="IND", a=1)

Arguments

x

List. Listing matrices of the predictors from different platforms.

y

List. A list of continuous responses vectors from different platforms following the same order as in x.

lambda

Numeric or vector. For fusionbase, lambda is a numeric value for the penalty; for fusionbase.fit, lambda is a vector with a list of penalty values.

N

Numeric or vector. If only one numeric value is provided, equal sample size will be assumed for each data set. If a vector is provided, then the elements are the sample sizes for all the platforms.

p

Numeric. The number of predictors.

m

Numeric. The number of platforms.

beta

Numeric or Matrix. An initial value for the estimated parameters with dimensions nvars x nplatforms. The defaul value is 0.1.

thresh

Numeric. The stopping criteria. The default value is 0.05.

maxiter

Numeric. Maximum number of iterations. The default value is 30.

methods

Character ("lass" or "scad"). lass: LASSO; scad: SCAD.

Complete

Logic input. If Complete == TRUE, the predictors M_1,...,M_p are measured in all platforms. If Compelte == FALSE, in some platforms, not all of the predictors \{M_1,M_2,...,M_p\} are measured. The values of the corresponding estimated coefficients for the missing predictors will be NA.

depen

Character. Input only for function fusionbase.fit. "IND" means the observations across different platforms are independent; "CORR" means the observations are correlated, and the sample sizes should be equal for different platforms.

a

Numeric. Input only for function fusionbase.fit. The free multiplicative constant used in \gamma_n. The default value is 1.

Details

The basic fusion learning function to learn from multiple linear models with continuous responses. More details regarding the model assumptions and the algorithm can be found in FusionLearn.

Value

fusionbase returns a list that has components:

beta

A matrix (nvars x nplatforms) containing estimated coefficients of each linear model. If some data sets do not have the complete set of predictors, the corresponding coefficients are output as NA.

method

Penalty function LASSO or SCAD.

threshold

The numeric value shows the difference in the estimates between the successive updates upon convergence.

iteration

The numeric value shows the number of iterations upon convergence.

fusionbase.fit provides the results in a table:

lambda

The sequence of penalty values.

BIC

The pseudolikelihood Bayesian information criterion evaluated at the sequence of the penalty values.

-2Loglkh

Minus twice the pseudo loglikelihood of the chosen model.

Est_df

The estimated degrees of freedom quantifying the model complexity.

fusionbase.fit also returns a model selection plot showing the results above.

Note

The range of the penalty values should be carefully chosen. For some penalty values, the resulting models may have singular information matrix or the fitting of the glm cannot converge.

Author(s)

Xin Gao, Yuan Zhong, and Raymond J. Carroll

References

Gao, X and Carroll, R. J. (2017) Data integration with high dimensionality. Biometrika, 104, 2, pp. 251-272

Examples

##analysis of the stock index data
#Responses contain indices "VIX","GSPC", and "DJI" 
y <- list(stockindexVIX[,1],stockindexGSPC[,1],stockindexDJI[,1]) 

#Predictors include 46 stocks
x <- list(stockindexVIX[,2:47],stockindexGSPC[,2:47],stockindexDJI[,2:47])  

##Implementing the model selection algorithm based on the psuedolikelihood 
##information criteria  
model <- fusionbase.fit(x,y,seq(0.03,5,length.out = 10),232,46,3,depen="CORR")
lambda <- model[which.min(model[,2]),1]
result <- fusionbase(x,y,lambda,232,46,3)

##Identify the significant predictors for the three indices
id <- which(result$beta[,1]!=0)+1
colnames(stockindexVIX)[id]