SimulatePrecision {fake}

Simulation of precision matrix

Description

Simulates a sparse precision matrix from a binary adjacency matrix theta encoding conditional independence in a Gaussian Graphical Model.

Usage

SimulatePrecision(
  pk = NULL,
  theta,
  v_within = c(0.5, 1),
  v_between = c(0, 0.1),
  v_sign = c(-1, 1),
  continuous = TRUE,
  pd_strategy = "diagonally_dominant",
  ev_xx = NULL,
  scale = TRUE,
  u_list = c(1e-10, 1),
  tol = .Machine$double.eps^0.25
)

Arguments

pk

vector of the number of variables per group in the simulated dataset. The number of nodes in the simulated graph is sum(pk). With multiple groups, the simulated (partial) correlation matrix has a block structure, where blocks arise from the integration of the length(pk) groups. This argument is only used if theta is not provided.

theta

binary and symmetric adjacency matrix encoding the conditional independence structure.

v_within

vector defining the (range of) nonzero entries in the diagonal blocks of the precision matrix. These values must be between -1 and 1 if pd_strategy="min_eigenvalue". If continuous=FALSE, v_within is the set of possible precision values. If continuous=TRUE, v_within is the range of possible precision values.

v_between

vector defining the (range of) nonzero entries in the off-diagonal blocks of the precision matrix. This argument is the same as v_within but for off-diagonal blocks. It is only used if length(pk)>1.

v_sign

vector of possible signs for precision matrix entries. Possible inputs are: -1 for positive partial correlations, 1 for negative partial correlations, or c(-1, 1) for both positive and negative partial correlations.

continuous

logical indicating whether to sample precision values from a uniform distribution between the minimum and maximum values in v_within (diagonal blocks) or v_between (off-diagonal blocks) (if continuous=TRUE) or from proposed values in v_within (diagonal blocks) or v_between (off-diagonal blocks) (if continuous=FALSE).

pd_strategy

method to ensure that the generated precision matrix is positive definite (and hence can be a covariance matrix). If pd_strategy="diagonally_dominant", the precision matrix is made diagonally dominant by setting the diagonal entries to the sum of absolute values on the corresponding row and a constant u. If pd_strategy="min_eigenvalue", diagonal entries are set to the sum of the absolute value of the smallest eigenvalue of the precision matrix with zeros on the diagonal and a constant u.

ev_xx

expected proportion of explained variance by the first Principal Component (PC1) of a Principal Component Analysis. This is the largest eigenvalue of the correlation (if scale_ev=TRUE) or covariance (if scale_ev=FALSE) matrix divided by the sum of eigenvalues. If ev_xx=NULL (the default), the constant u is chosen by maximising the contrast of the correlation matrix.

scale

logical indicating if the proportion of explained variance by PC1 should be computed from the correlation (scale=TRUE) or covariance (scale=FALSE) matrix.

u_list

vector with two numeric values defining the range of values to explore for constant u.

tol

accuracy for the search of parameter u as defined in optimise.

Details

Entries that are equal to zero in the adjacency matrix theta are also equal to zero in the generated precision matrix. These zero entries indicate conditional independence between the corresponding pair of variables (see SimulateGraphical).

Argument pk can be specified to create groups of variables and allow for nonzero precision entries to be sampled from different distributions between two variables belonging to the same group or to different groups.

If continuous=FALSE, nonzero off-diagonal entries of the precision matrix are sampled from a discrete uniform distribution taking values in v_within (for entries in the diagonal block) or v_between (for entries in off-diagonal blocks). If continuous=TRUE, nonzero off-diagonal entries are sampled from a continuous uniform distribution taking values in the range given by v_within or v_between.

Diagonal entries of the precision matrix are defined to ensure positive definiteness as described in MakePositiveDefinite.

Value

A list with:

omega

true simulated precision matrix.

u

value of the constant u used to ensure that omega is positive definite.

References

Bodinier B, Filippi S, Nost TH, Chiquet J, Chadeau-Hyam M (2021). “Automated calibration for stability selection in penalised regression and graphical models: a multi-OMICs network application exploring the molecular response to tobacco smoking.” https://arxiv.org/abs/2106.02521.

See Also

SimulateGraphical, MakePositiveDefinite

Examples

# Simulation of an adjacency matrix
theta <- SimulateAdjacency(pk = c(5, 5), nu_within = 0.7)
print(theta)

# Simulation of a precision matrix maximising the contrast
simul <- SimulatePrecision(theta = theta)
print(simul$omega)

# Simulation of a precision matrix with specific ev by PC1
simul <- SimulatePrecision(
  theta = theta,
  pd_strategy = "min_eigenvalue",
  ev_xx = 0.3, scale = TRUE
)
print(simul$omega)