Improper Complex-Valued Bhattacharyya Distance
Bibliographic record
Abstract
Motivated by application of complex-valued signal processing techniques in statistical pattern recognition, classification, and Gaussian mixture (GM) modeling, this paper derives analytical expressions for computing the Bhattacharyya coefficient/distance (BC/BD) between two improper complex-valued Gaussian distributions. The BC/BD is one of the most widely used statistical measures for evaluating class separability in classification problems, feature extraction in pattern recognition, and for GM reduction (GMR) purposes. The BC provides an upper bound on the Bayes error, which is commonly known as the best criterion to evaluate feature sets. Although the computation of the BC/BD between real-valued signals is a well-known result, it has not yet been extended to the case of improper complex-valued Gaussian densities. This paper addresses this gap. We analyze the role of the pseudocovariance matrix, which characterizes the noncircularity of the signal, and show that it carries critical second-order statistical information for computing the BC/BD. We derive upper and lower bounds on the BD in terms of the eigenvalues of the covariance and pseudocovariance matrices of the underlying densities. The theoretical bounds are then used to introduce the concept of β -dominance in the context of statistical distance measures. The BC is pseudometric, since it fails to satisfy the triangle inequality. Using the Matusita distance (a full-metric variant of the BC), we propose an intuitively pleasing indirect distance measure for comparing two general GMs. Finally, we investigate the application of the proposed BC/BD measures for GMR purposes and develop two BC-based GMR algorithms.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".