sparsifyVARX1: Function that determines the support of (auto)regression...
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 support of \mathbf{A}, the matrix with autoregression coefficients, and \mathbf{B}, the matrix with regression coefficients of the time-varying covariates, of the VARX(1) model.

sparsifyVARX1(X, A, B, SigmaE, threshold=c("absValue", "localFDR", "top"), 
              absValueCut=rep(0.25, 2), FDRcut=rep(0.8, 2), top=rep(10, 2), 
              zerosA=matrix(nrow=0, ncol=2), zerosB=matrix(nrow=0, ncol=2), 
              statistics=FALSE, verbose=TRUE)

`X`	Three-dimensional `array` containing the time-varying covariate data. The first, second and third dimensions correspond to covariates, time and samples, respectively. The data are assumed to be centered covariate-wise.
`A`	A `matrix` \mathbf{A} of autoregression parameters.
`B`	A `matrix` \mathbf{B} of regression parameters of the time-varying covariates stored in the array `X`.
`SigmaE`	Covariance `matrix` of the errors (innovations).
`threshold`	A `character` signifying type of sparsification of \mathbf{A} and \mathbf{B} 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.
`FDRcut`	A `numeric` of length two giving the cut-off (for \mathbf{A} and \mathbf{B}, respectively) for element selection based on local false discovery rate (FDR) thresholding.
`top`	A `numeric` of length two giving the number of elements of \code{A} and \mathbf{B}, respectively, 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.
`zerosB`	A `Matrix` with indices of entries of \mathbf{B} that are (prior to sparsification) known to be zero. The matrix comprises two columns, each row corresponding to an entry of \mathbf{B}. 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`	`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 (approximations 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} and/or \mathbf{B} is specified through the options zerosA and/or zerosB, the corresponding elements of \mathbf{A} and \mathbf{B} 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 for the sparsification of \mathbf{A}.
`zerosB`	`Matrix` with indices of entries of \mathbf{B} that are identified to be null. It includes the elements of \mathbf{B} assumed to be zero prior to the sparsification as specified through the `zerosBknown` option.
`nonzerosB`	`Matrix` with indices of entries of \mathbf{B} that are identified to be non-null.
`statisticsB`	`Matrix` with test statistics employed in the local FDR procedure for the sparsification of \mathbf{B}.

The matrices zerosA, nonzerosA, zerosB and nonzerosB comprise two columns, each row corresponding to an entry of either \mathbf{A} or \mathbf{B}. 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.

ridgeVAR1, sparsify, sparsifyVAR1.

# set dimensions (p=covariates, n=individuals, T=time points)
p <- 3; n <- 4; T <- 10

# set model parameters
SigmaE <- diag(p)/4
Ax     <- createA(p, "chain", nBands=1)

# generate time-varying covariates in accordance with VAR(1) process
X <- dataVAR1(n, T, Ax, SigmaE)

# set model parameters
B <- createA(p, "clique", nCliques=1)
A <- createA(p, "hub", nHubs=1)

# generate time-varying covariates in accordance with VAR(1) process
Y <- dataVARX1(X, A, B, SigmaE, lagX=0)

# fit VARX(1) model
VARX1hat <- ridgeVARX1(Y, X, 1, 1, 1, lagX=0)

## determine which elements of A are non-null
## Not run:  Anullornot <- matrix(0, p, p) 
## Not run:  Anullornot[sparsifyVARX1(X, VARX1hat$A, VARX1hat$B,
        solve(VARX1hat$P), threshold="localFDR")$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)