sparsifyVAR1: 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}, the matrix of autoregression coefficients.

sparsifyVAR1(A, SigmaE, threshold=c("absValue", "localFDR", "top"), 
             absValueCut=0.25, FDRcut=0.8, top=10, 
             zerosA=matrix(nrow=0, ncol=2), statistics=FALSE, 
             verbose=TRUE)

`A`	`Matrix` \mathbf{A} of autoregression parameters.
`SigmaE`	Covariance `matrix` of the errors (innovations).
`threshold`	A `character` signifying type of sparsification of \mathbf{A} by thresholding. Must be one of: "absValue", "localFDR", or "top".
`absValueCut`	A `numeric` giving the cut-off for element selection based on absolute value thresholding.
`FDRcut`	A `numeric` giving the cut-off for element selection based on local false discovery rate (FDR) thresholding.
`top`	A `numeric` giving the number of elements of \code{A} which is to be selected, based on absolute value thresholding.
`zerosA`	`Matrix` with indices of entries of \mathbf{A} that are (prior to sparsification) known to be zero. The matrix comprises two columns, each row corresponding to an entry of \mathbf{A}. The first column contains the row indices and the second the column indices.
`statistics`	`Logical` indicator: should test statistics be returned. This only applies when `threshold = "localFDR"`
`verbose`	`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} of autoregression coefficients by (an approximation of) their standard errors. Subsequently, the support of the matrix \mathbf{A} is determined by usage of local FDR. In that case a mixture model is fitted to the nonredundant (standardized) elements of \mathbf{A} 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 further details. Alternatively, the support of \mathbf{A} 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}. 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} is specified through the option zerosA, the corresponding elements of \mathbf{A} are then not taken along in the local FDR procedure.

A list-object with slots:

`zerosA`	`Matrix` with indices of entries of \mathbf{A} that are identified to be null. It includes the elements of \mathbf{A} assumed to be zero prior to the sparsification as specified through the `zerosAknown` option.
`nonzerosA`	`Matrix` with indices of entries of \mathbf{A} that are identified to be non-null.
`statisticsA`	`Matrix` with test statistics employed in the local FDR procedure.

The matrices zerosA and nonzerosA comprises two columns, each row corresponding to an entry of \mathbf{A}. 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. (2017), "Ridge estimation of the VAR(1) model and its time series chain graph from multivariate time-course omics data", Biometrical Journal, 59(1), 172-191.

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.

ridgeVAR1, sparsify.

# set dimensions
p <- 3
n <- 4
T <- 10

# set model parameters
SigmaE <- diag(p)/4
A <- matrix(c(-0.1, -0.3, 0.6, 0.5, -0.4, 0, 0.3, -0.5, -0.2), 
     byrow=TRUE, ncol=3)

# generate data
Y <- dataVAR1(n, T, A, SigmaE)

# fit VAR(1) model
VAR1hat <- ridgeVAR1(Y, 1, 1)

## determine which elements of A are non-null
## Not run:  Anullornot <- matrix(0, p, p) 
## Not run:  Anullornot[sparsifyVAR1(VAR1hat$A, solve(VAR1hat$P), 
threshold="localFDR")$nonzeros] <- 1 
## End(Not run)
## REASON FOR NOT RUN:
## the employed local FDR approximation is only valid for reasonably sized 
## number of elements of A (say) at least p > 10 and, 
## consequently, a vector of 100 regression coefficients.

## plot non-null structure of A 
## Not run:  edgeHeat(Anullornot)