sparsifyVAR2: Function that determines the support of autoregression...
In ragt2ridges: Ridge Estimation of Vector Auto-Regressive (VAR) Processes

Description Usage Arguments Details Value Author(s) References See Also Examples

Function that determines the null and non-null elements of \mathbf{A}_1 and \mathbf{A}_2, the matrices of lag one and two (respectively) autoregression coefficients.

sparsifyVAR2(A1, A2, SigmaE, threshold=c("absValue", "localFDR", "top"), 
             absValueCut=c(0.25, 0.25), FDRcut=c(0.8, 0.8), top=c(10,10), 
             zerosA1=matrix(nrow=0, ncol=2), 
             zerosA2=matrix(nrow=0, ncol=2), 
             statistics=FALSE, verbose=TRUE)

`A1`	A `matrix` \mathbf{A}_1 of lag one autoregression parameters.
`A2`	A `matrix` \mathbf{A}_2 of lag two autoregression parameters.
`SigmaE`	Covariance `matrix` of the errors (innovations).
`threshold`	A `character` signifying type of sparsification of both \mathbf{A}_1 and \mathbf{A}_2 by thresholding. Must be one of: "absValue", "localFDR", or "top".
`absValueCut`	A `numeric` (of length two) giving the cut-offs for element selection based on absolute value thresholding. The elements are the cut-offs applied to \mathbf{A}_1 and \mathbf{A}_2, respectively.
`FDRcut`	A `numeric` (of length two) giving the cut-offs for element selection based on local false discovery rate (FDR) thresholding. The elements are the cut-offs applied to \mathbf{A}_1 and \mathbf{A}_2, respectively.
`top`	A `numeric` (of length two) specifying the numbers of elements of \mathbf{A}_1 and \mathbf{A}_2, respectively, which are to be selected.
`zerosA1`	A `matrix` with indices of entries of \mathbf{A}_1 that are (prior to sparsification) known to be zero. The matrix comprises two columns, each row corresponding to an entry of \mathbf{A}_1. The first column contains the row indices and the second the column indices.
`zerosA2`	A `matrix` with indices of entries of \mathbf{A}_2 that are (prior to sparsification) known to be zero. The matrix comprises two columns, each row corresponding to an entry of \mathbf{A}_2. The first column contains the row indices and the second the column indices.
`statistics`	A `logical` indicator: should test statistics be returned. This only applies when `threshold = "localFDR"`
`verbose`	A `logical` indicator: should intermediate output be printed on the screen?

When threshold = "localFDR" the function, following Lutkepohl (2005), divides the elements of (possibly regularized) input matrix \mathbf{A}_1 (or \mathbf{A}_2) of lag one (or two) autoregression coefficients by (an approximation of) their standard errors. Subsequently, the support of the matrix \mathbf{A}_1 (or \mathbf{A}_2) is determined by usage of the local FDR. In that case a mixture model is fitted to the nonredundant (standardized) elements of \mathbf{A}_1 (or \mathbf{A}_2) by fdrtool. The decision to retain elements is then based on the argument FDRcut. Elements with a posterior probability >=q FDRcut (equalling 1 - local FDR) are retained. See Strimmer (2008) for sparsifyfurther details. Alternatively, the support of \mathbf{A}_1 (or \mathbf{A}_2) is determined by simple thresholding on the absolute values of matrix entries (threshold = "absValue"). A third option (threshold = "top") is to retain a prespecified number of matrix entries based on absolute values of the elements of \mathbf{A}_1 (or \mathbf{A}_2). For example, one could wish to retain those entries representing the ten strongest cross-temporal coefficients.

The argument absValueCut is only used when threshold = "absValue". The argument FDRcut is only used when threshold = "localFDR". The argument top is only used when threshold = "top".

When prior to the sparsification knowledge on the support of \mathbf{A}_1 (or \mathbf{A}_2) is specified through the option zerosA1 (or zerosA2), the corresponding elements of \mathbf{A}_1 (or \mathbf{A}_2) are then not taken along in the local FDR procedure.

A list-object with slots:

`zerosA1`	`Matrix` with indices of entries of \mathbf{A}_1 that are identified to be null. It includes the elements of \mathbf{A}_1 assumed to be zero prior to the sparsification as specified through the `zerosA1` option.
`nonzerosA1`	`Matrix` with indices of entries of \mathbf{A}_1 that are identified to be non-null.
`statisticsA1`	`Matrix` with test statistics employed in the local FDR procedure.
`zerosA2`	`Matrix` with indices of entries of \mathbf{A}_2 that are identified to be null. It includes the elements of \mathbf{A}_2 assumed to be zero prior to the sparsification as specified through the `zerosA2` option.
`nonzerosA2`	`Matrix` with indices of entries of \mathbf{A}_2 that are identified to be non-null.
`statisticsA2`	`Matrix` with test statistics employed in the local FDR procedure.

The matrices zerosA1, nonzerosA1, zerosA2 and nonzerosA2 comprise two columns, each row corresponding to an entry of \mathbf{A}_1 and \mathbf{A}_2, respectively. The first column contains the row indices and the second the column indices.

Wessel N. van Wieringen <w.vanwieringen@vumc.nl>, Carel F.W. Peeters.

Lutkepohl, H. (2005), New Introduction to Multiple Time Series Analysis. Springer, Berlin.

Miok, V., Wilting, S.M., Van Wieringen, W.N. (2019), “Ridge estimation of network models from time-course omics data”, Biometrical Journal, 61(2), 391-405.

Strimmer, K. (2008), “fdrtool: a versatile R package for estimating local and tail area-based false discovery rates”, Bioinformatics 24(12): 1461-1462.

Van Wieringen, W.N., Peeters, C.F.W. (2016), “Ridge Estimation of Inverse Covariance Matrices from High-Dimensional Data”, Computational Statistics and Data Analysis, 103, 284-303.

ridgeVAR2, sparsify, sparsifyVAR1.

# set dimensions (p=covariates, n=individuals, T=time points)
p <- 3; n <- 12; T <- 10

# set model parameters
SigmaE <- diag(p)/4
A1 <- -createA(p, "clique", nCliques=1, nonzeroA=0.1)
A2 <- t(createA(p, "chain", nBands=1, nonzeroA=0.1))

# generate data
Y <- dataVAR2(n, T, A1, A2, SigmaE)

# fit VAR(1) model
VAR2hat <- ridgeVAR2(Y, 1, 1, 1)

# obtain support of adjacancy matrix
A1nullornot <- matrix(0, p, p)
A1nullornot[sparsifyVAR2(VAR2hat$A1, VAR2hat$A2, solve(VAR1hat$P), 
                         threshold="top", top=c(3,3))$nonzerosA1] <- 1

## plot non-null structure of A1 
edgeHeat(A1nullornot)