performance: Performance Measures for Precision Matrix Estimation

In grasps: Groupwise Regularized Adaptive Sparse Precision Solution

Performance Measures for Precision Matrix Estimation

Description

Compute a collection of loss-based and structure-based measures to evaluate the performance of an estimated precision matrix.

Usage

performance(hatOmega, Omega)

Arguments

hatOmega	A numeric d \times d matrix giving the estimated precision matrix.
Omega	A numeric d \times d matrix giving the reference (typically true) precision matrix.

Details

Let \Omega_{d \times d} and \hat{\Omega}_{d \times d} be the reference (true) and estimated precision matrices, respectively, with \Sigma = \Omega^{-1} being the corresponding covariance matrix. Edges are defined by nonzero off-diagonal entries in the upper triangle of the precision matrices.

Sparsity is treated as a structural summary, while the remaining measures are grouped into loss-based measures, raw confusion-matrix counts, and classification-based (structure-recovery) measures.

"sparsity": Sparsity is computed as the proportion of zero entries among the off-diagonal elements in the upper triangle of \hat{\Omega}.

Loss-based measures:

• "Frobenius": Frobenius (Hilbert-Schmidt) norm loss = \Vert \Omega - \hat{\Omega} \Vert_F.

• "KL": Kullback-Leibler divergence = \mathrm{tr}(\Sigma \hat{\Omega}) - \log\det(\Sigma \hat{\Omega}) - d.

• "quadratic": Quadratic norm loss = \Vert \Sigma \hat{\Omega} - I_d \Vert_F^2.

• "spectral": Spectral (operator) norm loss = \Vert \Omega - \hat{\Omega} \Vert_{2,2} = e_1, where e_1^2 is the largest eigenvalue of (\Omega - \hat{\Omega})^2.

Confusion-matrix counts:

• "TP": True positive = number of edges in both \Omega and \hat{\Omega}.

• "TN": True negative = number of edges in neither \Omega nor \hat{\Omega}.

• "FP": False positive = number of edges in \hat{\Omega} but not in \Omega.

• "FN": False negative = number of edges in \Omega but not in \hat{\Omega}.

Classification-based (structure-recovery) measures:

• "TPR": True positive rate (TPR), recall, sensitivity = \mathrm{TP} / (\mathrm{TP} + \mathrm{FN}).

• "FPR": False positive rate (FPR) = \mathrm{FP} / (\mathrm{FP} + \mathrm{TN}).

• "F1": F_1 score = 2\,\mathrm{TP} / (2\,\mathrm{TP} + \mathrm{FN} + \mathrm{FP})

• "MCC": Matthews correlation coefficient (MCC) = (\mathrm{TP}\times\mathrm{TN} - \mathrm{FP}\times\mathrm{FN}) / \sqrt{(\mathrm{TP}+\mathrm{FP})(\mathrm{TP}+\mathrm{FN}) (\mathrm{TN}+\mathrm{FP})(\mathrm{TN}+\mathrm{FN})}

The following table summarizes the confusion matrix and related rates:

	Predicted Positive	Predicted Negative
Real Positive (P)	True positive (TP)	False negative (FN)	True positive rate (TPR), recall, sensitivity = TP / P = 1 - FNR	False negative rate (FNR) = FN / P = 1 - TPR
Real Negative (N)	False positive (FP)	True negative (TN)	False positive rate (FPR) = FP / N = 1 - TNR	True negative rate (TNR), specificity = TN / N = 1 - FPR
	Positive predictive value (PPV), precision = TP / (TP + FP) = 1 - FDR	False omission rate (FOR) = FN / (TN + FN) = 1 - NPV
	False discovery rate (FDR) = FP / (TP + FP) = 1 - PPV	Negative predictive value (NPV) = TN / (TN + FN) = 1 - FOR

Value

A data frame of S3 class "performance", with one row per performance metric and two columns:

measure

The name of each performance metric. The reported metrics include: sparsity, Frobenius norm loss, Kullback-Leibler divergence, quadratic norm loss, spectral norm loss, true positive, true negative, false positive, false negative, true positive rate, false positive rate, F1 score, and Matthews correlation coefficient.

value

The corresponding numeric value.

Examples

library(grasps)

## reproducibility for everything
set.seed(1234)

## block-structured precision matrix based on SBM
sim <- gen_prec_sbm(d = 30, K = 3,
                    within.prob = 0.25, between.prob = 0.05,
                    weight.dists = list("gamma", "unif"),
                    weight.paras = list(c(shape = 20, rate = 10),
                                        c(min = 0, max = 5)),
                    cond.target = 100)
## visualization
plot(sim)

## n-by-d data matrix
library(MASS)
X <- mvrnorm(n = 20, mu = rep(0, 30), Sigma = sim$Sigma)

## adapt, BIC
res <- grasps(X = X, membership = sim$membership, penalty = "adapt", crit = "BIC")

## visualization
plot(res)

## performance
performance(hatOmega = res$hatOmega, Omega = sim$Omega)

grasps documentation built on Nov. 28, 2025, 1:06 a.m.