Module causality 

Functions

conditional_granger_causality

Conditional Granger causality (Geweke 1984). Tests if source Granger-causes target controlling for condition.

directed_transfer_function

Directed transfer function (Kaminski & Blinowska 1991). Returns (d × d × n_freqs) flat Vec.

identity_flat 🔒

pairwise_granger_causality

Pairwise Granger causality (Granger 1969). Returns log-likelihood ratio. Positive = source Granger-causes target.

partial_directed_coherence

Partial directed coherence (Baccala & Sameshima 2001). Returns (d × d × n_freqs) flat Vec.

solve_spd 🔒

Solve the symmetric positive-definite system S X = B via Cholesky factorisation. S (n×n, row-major) is a ridge-regularised normal-equations matrix XᵀX + εI, which is positive-definite for any ε > 0; B is n×m, row-major. Returns X (n×m, row-major).

spectral_granger_causality

Spectral Granger causality (Geweke 1982). Returns (d × d × n_freqs) as flat Vec, row-major in [i][j][f] order.

spectral_matrix 🔒

Build the multivariate-autoregressive spectral matrix A(f) = I − Σ_{k=0}^{order−1} A_{k+1} · e^{−i2πf(k+1)} at normalised frequency f ∈ [0, 0.5), from the row-major VAR coefficient stack beta ((order·d) × d). Block k is transposed on the fly: coeff_block.T[i, j] = beta[k·d + j, i]. A(f) is a general complex, non-Hermitian matrix.

spectral_transfer_inverse 🔒

Invert the MVAR spectral matrix A(f) to the transfer function H(f) = A(f)⁻¹ via LU factorisation, returning None when A(f) is numerically singular (|det A(f)| < 1e-30).

sse_ols 🔒

Sum of squared errors for OLS regression: SSE = ||y - X beta||^2.

var_coefficients 🔒

Fit VAR(order) model. Returns (beta [order*d × d, row-major], sigma [d×d, row-major]).