Replace direct matrix inverse with Tikhonov-regularized solve for MVAR Fourier

[edeno]

Summary

• Replace direct matrix inverse with Tikhonov-regularized solve in MVAR Fourier coefficient computations
• Add scale-aware regularization parameter: λ = 1e-12 * mean(||H||²)
• Add stress test for near-singular frequency bins

Test plan

• New stress test test_mvar_regularized_inverse_near_singular passes
• All existing MVAR/Granger causality tests continue to pass
• Directed connectivity measures (DTF, etc.) work with near-singular inputs
• No LinAlgError exceptions thrown for highly correlated signal inputs

🤖 Generated with Claude Code

[Copilot]

Pull Request Overview

This PR replaces direct matrix inversion operations in MVAR Fourier coefficient computations with Tikhonov-regularized linear solves to prevent numerical instability when dealing with near-singular frequency bins.

• Replaces inv(H) with solve(H + λI, I) using scale-aware regularization
• Adds comprehensive stress test for near-singular cross-spectral matrices
• Applies regularization to both transfer function estimation and MVAR coefficient computation

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 3 comments.

File	Description
spectral_connectivity/connectivity.py	Implements Tikhonov regularization in _MVAR_Fourier_coefficients property and _estimate_transfer_function function
tests/test_connectivity.py	Adds stress test test_mvar_regularized_inverse_near_singular for near-singular frequency bins

---

_{Tip: Customize your code reviews with copilot-instructions.md. Create the file or learn how to get started.}

[Copilot]

The regularization parameter lam uses a magic number 1e-12. Consider defining this as a named constant (e.g., TIKHONOV_REGULARIZATION_FACTOR = 1e-12) to improve maintainability and make it easier to adjust if needed.

[Copilot]

The regularization parameter uses the same magic number 1e-12 as in the other function. Consider extracting this to a shared constant to ensure consistency and easier maintenance.

[Copilot]

[nitpick] The complex signal generation spans multiple lines with backslash continuation. Consider using parentheses for line continuation instead: base_signal = (np.random.randn(...) + 1j * np.random.randn(...))

Suggested change

	base_signal = np.random.randn(n_time_samples, n_trials, n_tapers, n_fft_samples) + \
	1j * np.random.randn(n_time_samples, n_trials, n_tapers, n_fft_samples)
	base_signal = (
	np.random.randn(n_time_samples, n_trials, n_tapers, n_fft_samples)
	+ 1j * np.random.randn(n_time_samples, n_trials, n_tapers, n_fft_samples)
	)

[coveralls]

coverage: 76.304% (-0.2%) from 76.509%
when pulling 024dbc8 on feat/mvar-regularized-inverse
into 3e31c94 on master.