Package 'jewel'

Construct weights for _jewel_ minimization problem from prior information on vertices degrees.

Description

Function takes a numerical vector of vertices degrees and constructs weights with the rule W_ij = 1 / sqrt(d_i * d_j) and then the whole matrix is normilized by the maximum.

Usage

constructWeights(d, K = NULL)

Arguments

d

either one numerical vector or a list of K numerical vectors of length p with user-provided degrees of vertices for each class. If there is only one vector, we assume degrees are the same for all K classes. In that case parameter K (number of classes) must be provided. Note that for successful _jewel_ estimation true degrees are not necessary: for example, user can provide a vector where known hubs have degree 10 and the rest of the vertices have degree 1.

K

number of classes (i.e. datasets, i.e. desired graphs). By default it is length(d). In length(d) = 1, K must be provided by the user.

Value

W - a list of K numeric matrices of the size p by p

Examples

{
K <- 3
p <- 50
n <- 20
data <- generateData_rewire(K = K, p = p, n = n, ncores = 1, verbose = FALSE)
G_list_true <- data$Graphs
true_degrees <- rowSums(G_list_true[[1]])
cut <- sort(true_degrees, decreasing = TRUE)[ceiling(p * 0.03)]
apriori_hubs <- ifelse(true_degrees >= cut, 10, 1)
W <- constructWeights(apriori_hubs, K = K)
}

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Evaluation of graph estimation method's performance if the true graph is known.

Description

Function compares adjacency matrices of the true and estimated simple graphs and calculates the number of true positives (correctly estimated edges), true negatives (correctly estimated absence of edges), false positives (edges present in the estimator but not in the true graph) and false negatives (failure to identify an edge).

Usage

evaluatePerformance(G, G_hat)

Arguments

G	true graph's adjacency matrix.
G_hat	estimated graph's adjacency matrix. Must have the same dimensions as G.

Value

performance - a numeric vector of length 4 with TP, TN, FP, FN.

Examples

{
K <- 3
p <- 50
n <- 20
data <- generateData_rewire(K = K, p = p, n = n, ncores = 1, verbose = FALSE)
G_common_true <- data$CommonGraph
X <- data$Data
res <- jewel(X, lambda1 = 0.25)
G_common_est <- res$CommonG
evaluatePerformance(G = G_common_true, G_hat = G_common_est)
}

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Generate a set of scale-free graphs and corresponding datasets (using the graphs as their Gaussian graphical models)

Description

Function first generates K scale-free graphs with p vertices. They have the same order and degree distribution and share most of the edges, but some edges may vary (user can control how many). Function then generates corresponding precision and covariance matrices, all of the size p by p (see the paper for the details of the procedure). Then for each l-th element of vector n it generates K data matrices, each of the size n_l by p, i.e., for the same underlying graphs we can generate several sets of K datasets with different sample sizes.

Usage

generateData_rewire(
  K,
  p,
  n,
  power = 1,
  m = 1,
  perc = 0.05,
  int = NULL,
  ncores = NULL,
  makePlot = TRUE,
  verbose = TRUE
)

Arguments

K	number of graphs/data matrices.
p	number of nodes in the true graphs.
n	a numerical vector of the sample sizes for each desired set of K data matrices. Can be a vector of one element if the user wishes to obtain only one dataset of K matrices.
power	a number, power of preferential attachment for the Barabasi-Albert algorithm for the generation of the scale-free graph. Bigger number means more connected hubs. The default value is 1.
m	number of edges to add at each step of Barabasi-Albert algorithm for generation of the scale-free graph. The default value is 1.
perc	a number, tuning parameter for the difference between graphs. Number of trials to perform in the rewiring procedure of the first graph is p * perc. Bigger the number, more different are the graphs.
int	a vector of two numbers, a and b. Entries of precision matrices are sampled from the uniform distribution on the interval [-b, -a] + [a, b]. The default values are a = 0.2, b = 0.8.
ncores	number of cores to use in parallel data generation. If NULL, set to $\#physical cores - 1$.
makePlot	If makePlot = FALSE, plotting of the generated graphs is disabled. The default value is TRUE.
verbose	If verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

Value

The following list is returned

• Graphs – a list of adjacency matrices of the K generated graphs.

• CommomGraph - a matrix, common part (intersection) of the K generated graphs.

• Data - a list of lists, for each sample size of the input vector n one obtains K data matrices, each of the size n_l by p.

• Sigma - a list of K covariance matrices of the size p by p.

Examples

data <- generateData_rewire(K = 3, p = 50, n = 20, ncores = 1, verbose = FALSE)

---

Estimate Gaussian graphical models from multiple datasets

Description

This function estimates Gaussian graphical models (i.e. networks of conditional dependencies, direct connections between variables) given multiple datasets. We assume that datasets contain measurements of the same variables collected under different conditions (different equipment, locations, even sub-types of disease).

Usage

jewel(
  X,
  lambda1,
  lambda2 = NULL,
  Theta = NULL,
  W = NULL,
  tol = 0.01,
  maxIter = 10000,
  stability = FALSE,
  stability_nsubsets = 25,
  stability_frac = 0.8,
  verbose = TRUE
)

Arguments

X	a list of K numeric data matrices of n_k samples and p variables (n_k can be different for each matrix).
lambda1	a number, first regularization parameter (of the common penalty).
lambda2	an optional number, second regularization parameter (of the class-specific penalty). If NULL, set to lambda_2 = lambda_1 * 1.4
Theta	an optional list of K regression coefficient matrices of the size p by p. User-provided initialization can be used for warm-start procedures. If NULL, initialized as all zeros.
W	an optional list of K weights matrices of the size p by p. User-provided initialization can be used when some vertices are believed to be hubs. If NULL, initialized as all ones.
tol	an optional number, convergence threshold controlling the relative error between iterations. The default value is 0.01.
maxIter	an optional number, maximum allowed number of iterations. The default value is 10 000.
stability	if stability = TRUE, stability selection procedure to reduce the number of false positives will be applied. n_k / 2 samples are randomly chosen in each dataset stability_nsubsets times and then __jewel__ method is applied to each subset. In the final estimate, we include only the edges that appear in at least stability_frac proportion of the subsets. By default this procedure is disabled since it increases the running time.
stability_nsubsets	an optional number, how many times to subsample datasets and apply __jewel__ for stability selection procedure. The default value is 25.
stability_frac	an optional number, in what proportion of the stability results on subsampled data an edge has to be present to be included into the final estimate. The default value is 0.8.
verbose	if verbose = FALSE, tracing information printing is disabled. The default value is TRUE.

Value

The following list is returned

• CommonG - an adjacency matrix of the common estimated graph (intersection of K estimated graphs).

• G_list - a list of K adjacency matrices for each estimated graph.

• Theta - a list of K estimated covariance matrices (when stability selection is disabled).

• BIC – a number, value of Bayesian information criterion for resulting graphs (when stability selection is disabled).

Examples

{
K <- 3
p <- 50
n <- 20
data <- generateData_rewire(K = K, p = p, n = n, ncores = 1, verbose = FALSE)
G_list_true <- data$Graphs
X <- data$Data
true_degrees <- rowSums(G_list_true[[1]])
cut <- sort(true_degrees, decreasing = TRUE)[ceiling(p * 0.03)]
apriori_hubs <- ifelse(true_degrees >= cut, 10, 1)
W <- constructWeights(apriori_hubs, K = K)
res <- jewel(X, lambda1 = 0.25, W = W, verbose = FALSE)
}

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