Sparse Covariance Selection by SPACE with EM

Description

A function to estimate partial correlations using SPACE method with EM, missing data is allowed

Usage

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space.miss(Y.m,lam1, lam2=0, sig=NULL, weight=NULL,iter=2, emIter=5,n_iter=1000,t0=0,r=0)

Arguments

Y.m

numeric matrix. Columns are for variables and rows are for samples. 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.

emIter

integer. the maximum number of EM iteration allowed

n_iter

integer. the maximum number of interations in JSRM.

t0

integer. 1<=t0<=n. the time point at which to perform local smoothing estimation.

r

positive value. local smoothing parameter. If r=0, then no smoothing is incorporated.

Details

space.miss Based on the work of J. Peng, et al(2007), this function allows computing with missing data by implementing EM algorithm.

\frac{1}{2}∑_{t=1}^n K(r*|t-t_0|) [∑_{i=1}^p w_i(y_{it}-∑_{j\neq i}√{\frac{σ_{jj}}{σ_{ii}}}ρ_{ij}y_{jt})^2 ] +λ∑_{i<j} |ρ_ij|

where K(\cdot) is the smoothing kernal, K(x)=\exp(-x).

Value

a list of following comopnents.

Y.imputed

The sample matrix Y with missing data imputed by EM

ParCor

the estimated partial correlation matrix.

sig.fit

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

bic

BIC for the current estimate.

Author(s)

Shiyuan He

References

He, S.Y., Wang, X., and Yuan, W.(2012), Discovering Co-movement Structure of Chinese Stock Market by SPACE method with EM.

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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data(finStocksCn)  ##data of finance sector of Chinese Stock Market, from January 4th, 2011 to July 26th, 2011
finStocksCn$names ##stock names
y.m<-scale(finStocksCn$returns)
n=nrow(y.m)
p=ncol(y.m)
alpha=0.1
l1=1/sqrt(n)*qnorm(1-alpha/(2*p^2))
res=space.miss(Y.m=y.m,lam1=l1*25,emIter=25,iter=2,weight=2)
res$bic  ##bic returned