SimulateComponents {fake}

Data simulation for sparse Principal Component Analysis

Description

Simulates data with with independent groups of variables.

Usage

SimulateComponents(
  n = 100,
  pk = c(10, 10),
  adjacency = NULL,
  nu_within = 1,
  v_within = c(0.5, 1),
  v_sign = -1,
  continuous = TRUE,
  pd_strategy = "min_eigenvalue",
  ev_xx = 0.1,
  scale_ev = TRUE,
  u_list = c(1e-10, 1),
  tol = .Machine$double.eps^0.25,
  scale = TRUE,
  output_matrices = FALSE
)

Arguments

n

number of observations in the simulated dataset.

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.

adjacency

optional binary and symmetric adjacency matrix encoding the conditional graph structure between observations. The clusters encoded in this argument must be in line with those indicated in pk. Edges in off-diagonal blocks are not allowed to ensure that the simulated orthogonal components are sparse. Corresponding entries in the precision matrix will be set to zero.

nu_within

probability of having an edge between two nodes belonging to the same group, as defined in pk. If length(pk)=1, this is the expected density of the graph. If implementation=HugeAdjacency, this argument is only used for topology="random" or topology="cluster" (see argument prob in huge.generator). Only used if nu_mat is not provided.

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_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_ev

logical indicating if the proportion of explained variance by PC1 should be computed from the correlation (scale_ev=TRUE) or covariance (scale_ev=FALSE) matrix. If scale_ev=TRUE, the correlation matrix is used as parameter of the multivariate normal distribution.

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.

scale

logical indicating if the true mean is zero and true variance is one for all simulated variables. The observed mean and variance may be slightly off by chance.

output_matrices

logical indicating if the true precision and (partial) correlation matrices should be included in the output.

Details

The data is simulated from a centered multivariate Normal distribution with a block-diagonal covariance matrix. Independence between variables from the different blocks ensures that sparse orthogonal components can be generated.

The block-diagonal partial correlation matrix is obtained using a graph structure encoding the conditional independence between variables. The orthogonal latent variables are obtained from eigendecomposition of the true correlation matrix. The sparse eigenvectors contain the weights of the linear combination of variables to construct the latent variable (loadings coefficients). The proportion of explained variance by each of the latent variable is computed from eigenvalues.

As latent variables are defined from the true correlation matrix, the number of sparse orthogonal components is not limited by the number of observations and is equal to sum(pk).

Value

A list with:

data

simulated data with n observation and sum(pk) variables.

loadings

loadings coefficients of the orthogonal latent variables (principal components).

theta

support of the loadings coefficients.

ev

proportion of explained variance by each of the orthogonal latent variables.

adjacency

adjacency matrix of the simulated graph.

omega

simulated (true) precision matrix. Only returned if output_matrices=TRUE.

phi

simulated (true) partial correlation matrix. Only returned if output_matrices=TRUE.

C

simulated (true) correlation matrix. Only returned if output_matrices=TRUE.

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

MakePositiveDefinite

Other simulation functions: SimulateAdjacency(), SimulateClustering(), SimulateCorrelation(), SimulateGraphical(), SimulateRegression(), SimulateStructural()

Examples

# Simulation of 3 components with high e.v.
set.seed(1)
simul <- SimulateComponents(pk = c(5, 3, 4), ev_xx = 0.4)
print(simul)
plot(simul)
plot(cumsum(simul$ev), ylim = c(0, 1), las = 1)

# Simulation of 3 components with moderate e.v.
set.seed(1)
simul <- SimulateComponents(pk = c(5, 3, 4), ev_xx = 0.25)
print(simul)
plot(simul)
plot(cumsum(simul$ev), ylim = c(0, 1), las = 1)

# Simulation of multiple components with low e.v.
pk <- sample(3:10, size = 5, replace = TRUE)
simul <- SimulateComponents(
  pk = pk,
  nu_within = 0.3, v_within = c(0.8, 0.5), v_sign = -1, ev_xx = 0.1
)
plot(simul)
plot(cumsum(simul$ev), ylim = c(0, 1), las = 1)