Module granger

Granger causality testing for embedding trajectories.

Tests whether entity A’s movements in embedding space precede entity B’s. Uses a VAR(L) model on dimensionality-reduced trajectories.

Algorithm

1. Align trajectories to a common time grid (linear interpolation)
2. For each dimension d:
   • Fit restricted model: B_d(t) = Σ β_l · B_d(t-l) + ε
   • Fit unrestricted model: B_d(t) = Σ β_l · B_d(t-l) + Σ γ_l · A_d(t-l) + ε
   • F-test: does the unrestricted model significantly improve?
3. Combine per-dimension p-values via Fisher’s method

References

• Granger, C.W.J. (1969). Investigating causal relations. Econometrica, 37(3).
• Fisher, R.A. (1925). Statistical methods for research workers.

Structs

GrangerResult

Result of a Granger causality test.

Enums

GrangerDirection

Direction of Granger causality.

Functions

align_trajectories 🔒

Align two trajectories to a common time grid via linear interpolation.

cholesky_solve 🔒

Solve Ax = b where A is symmetric positive definite via Cholesky.

erfc 🔒

Complementary error function approximation.

f_statistic 🔒

Compute F-statistic from restricted and unrestricted RSS.

f_to_p 🔒

Approximate p-value from F-statistic using the F-distribution.

fisher_combine 🔒

Fisher’s method: combine independent p-values.

fit_and_rss 🔒

Fit a simple autoregressive model and return residual sum of squares.

granger_causality

Test Granger causality between two embedding trajectories.

interpolate_at 🔒

Linear interpolation of a trajectory at a specific timestamp.

ln_beta 🔒

Log of the Beta function: ln(B(a,b)) = ln(Gamma(a)) + ln(Gamma(b)) - ln(Gamma(a+b))

ln_gamma 🔒

Stirling’s approximation for ln(Gamma(x)) for x > 0.

ols_granger_single_dim 🔒

Fit OLS for a single dimension and compute residual sum of squares.

regularized_incomplete_beta 🔒

Regularized incomplete beta function via continued fraction (Lentz’s method).

solve_ols 🔒

Solve OLS via normal equations with regularization.

test_direction 🔒

Test one direction: does cause Granger-cause effect?