A function to estimate partial correlations using the Joint Sparse Regression Model

Description

A function to estimate partial correlations using the Joint Sparse Regression Model

Usage

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space.joint(Y.m, lam1, lam2=0, sig=NULL, weight=NULL,iter=2)

Arguments

Y.m

numeric matrix. Columns are for variables and rows are for samples. Missing values are not allowed. It's recommended to first standardize each column to have mean 0 and l_2 norm 1.

lam1

numeric value. This is the l_1 norm penalty parameter. If the columns of Y.m have norm one, then the suggested range of lam1 is: O(n^{3/2}Φ^{-1}(1-α/(2p^2))) for small α such as 0.1.

lam2

numeric value. If not specified, lasso regression is used in the Joint Sparse Regression Model (JSRM). Otherwise, elastic net regression is used in JSRM and lam2 serves as the l_2 norm penalty parameter.

sig

numeric vector. Its length should be the same as the number of columns of Y.m. It is the vector of σ^{ii} (the diagonal of the inverse covariance matrix). If not specified, σ^{ii} will be estimated during the model fitting with initial values rep(1,p). The number of the iteration of the model fitting (iter) will then be at least 2. Note, the scale of sig does not matter.

weight

numeric value or vector. It specifies the weights or the type of weights used for each regression in JSRM. The default value is NULL, which means all regressions will be weighted equally in the joint model. If weight=1, residue variances will be used for weights. If weight=2, the estimated degree of each variable will be used for weights. Otherwise, it should be a positive numeric vector, whose length is equal to the number of columns of Y.m.

iter

integer. It is the total number of interactions in JSRM for estimating σ^{ii} and partial correlations. When sig=NULL and/or weight=NULL or 2, iter should be at least 2.

Details

space.joint uses a computationally efficient approach for selecting non-zero partial correlations under the high-dimension-low-sample-size setting (Peng and et.al., 2007).

Value

A list with two components

ParCor

the estimated partial correlation matrix.

sig.fit

numeric vector of the estimated diagonal σ^{ii}.

Author(s)

J. Peng, P. Wang, Nengfeng Zhou, Ji Zhu

References

J. Peng, P. Wang, N. Zhou, J. Zhu (2007), Partial Correlation Estimation by Joint Sparse Regression Model.

Meinshausen, N., and Buhlmann, P. (2006), High Dimensional Graphs and Variable Selection with the Lasso, Annals of Statistics, 34, 1436-1462.

Examples

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#############################################################################################
############################ (A) The simulated Hub.net example in Peng et. al. (2007).
#############################################################################################
data(spaceSimu)

n=nrow(spaceSimu$Y.data)
p=ncol(spaceSimu$Y.data)
true.adj=abs(spaceSimu$ParCor.true)>1e-6

#################### view the network corresponding to the parcial correlation matrix in the simulation example
########### the following code can run only if the "igraph" is installed in the system.
#library(igraph)
#plot.adj=true.adj
#diag(plot.adj)=0
#temp=graph.adjacency(adjmatrix=plot.adj, mode="undirected")
#temp.degree=apply(plot.adj, 2, sum)
#V(temp)$color=(temp.degree>9)+3
#plot(temp, vertex.size=3, vertex.frame.color="white",layout=layout.fruchterman.reingold, vertex.label=NA, edge.color=grey(0.5))


#################### estimate the parcial correlation matrix with various methods
alpha=1
l1=1/sqrt(n)*qnorm(1-alpha/(2*p^2))
iter=3

########### the values of lam1 were selected to make the results of different methods comparable. 
#### 1. MB method
result1=space.neighbor(spaceSimu$Y.data, lam1=l1*0.7, lam2=0)
fit.adj=abs(result1$ParCor)>1e-6
sum(fit.adj==1)/2                  ##total number of edges detected      
sum(fit.adj[true.adj==1]==1)/2  ##total number of true edges detected        
 
#### 2. Joint method with no weight
result2=space.joint(spaceSimu$Y.data, lam1=l1*n*1.56, lam2=0, iter=iter)
fit.adj=abs(result2$ParCor)>1e-6
sum(fit.adj==1)/2                  ##total number of edges detected      
sum(fit.adj[true.adj==1]==1)/2  ##total number of true edges detected        

#### 3. Joint method with residue variance based weights
result3=space.joint(spaceSimu$Y.data, lam1=l1*n*1.86, lam2=0, weight=1, iter=iter)
fit.adj=abs(result3$ParCor)>1e-6
sum(fit.adj==1)/2                  ##total number of edges detected      
sum(fit.adj[true.adj==1]==1)/2  ##total number of true edges detected        

#### 4. Joint method with degree based weights
result4=space.joint(spaceSimu$Y.data, lam1=l1*n*1.61, lam2=0, weight=2, iter=iter)
fit.adj=abs(result4$ParCor)>1e-6
sum(fit.adj==1)/2                  ##total number of edges detected      
sum(fit.adj[true.adj==1]==1)/2  ##total number of true edges detected