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)A matrix \(\mathbf{A}_1\) of lag one autoregression parameters.
A matrix \(\mathbf{A}_2\) of lag two autoregression parameters.
Covariance matrix of the errors (innovations).
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".
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.
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.
A numeric (of length two) specifying the numbers of elements of \(\mathbf{A}_1\) and \(\mathbf{A}_2\), respectively, which are to be selected.
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.
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.
A logical indicator: should test statistics be returned. This only applies when
threshold = "localFDR"
A logical indicator: should intermediate output be printed on the screen?
A list-object with slots:
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.
Matrix with indices of entries of \(\mathbf{A}_1\) that are identified to be non-null.
Matrix with test statistics employed in the local FDR procedure.
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.
Matrix with indices of entries of \(\mathbf{A}_2\) that are identified to be non-null.
Matrix with test statistics employed in the local FDR procedure.
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.
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.
# NOT RUN {
# 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)
# }
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