Connectivity Metric Ranges

Direction convention

For xarray results returned by multitaper_connectivity and fourier_connectivity, result.sel(source="a", target="b") is the influence from a to b (or, for phase measures, how much a leads b) for every measure.

The lower-level Connectivity methods return plain arrays whose last two axes are signals, in one of two orders (the Low-level orientation column below, and MeasureInfo.array_orientation from list_measures()):

• j -> i ("target_source"): the Granger and directed-transfer-function families. result[..., i, j] is the influence of signal j on signal i.

• i relative to j ("source_target"): directed_phase_lag_index, phase_slope_index, delay, and group_delay. Positive result[..., i, j] (above 0.5 for the directed phase lag index) means signal i leads j.

The signed, non-directional phase measures (coherence_phase, imaginary_coherency, phase_lag_index, weighted_phase_lag_index) follow the same sign rule: positive [..., i, j] means i leads j.

Measures

Generated from list_measures(). For complex measures the range bounds the magnitude. “Observations” are trials × tapers.

Measure	Range	Units	Low-level orientation	Interpretation
coherence_magnitude	[0, 1]	1		Linear coupling at each frequency: 0 is none, 1 is a perfectly consistent amplitude and phase relationship. Biased upward when trials x tapers is small.
coherence_phase	[-π, π]	rad		Mean phase difference in radians. Positive (source=a, target=b) means a leads b.
debiased_squared_phase_lag_index	[-1, 1]	1		Bias-corrected squared phase lag index. Negative values are finite-sample noise around zero, not negative coupling. Lower bound is -1 / (n_observations - 1).
debiased_squared_weighted_phase_lag_index	[-1, 1]	1		Bias-corrected squared weighted phase lag index. Negative values are finite-sample noise around zero, not negative coupling.
imaginary_coherence	[0, 1]	1		Magnitude of the imaginary part of coherency; blind to zero-lag coupling such as volume conduction.
pairwise_phase_consistency	[-1, 1]	1		Bias-free estimate of the squared phase-locking value. Negative values are finite-sample noise around zero, not negative coupling. Lower bound is -1 / (n_observations - 1).
pairwise_spectral_granger_prediction	[0, ∞)	1	[..., i, j] is j -> i	Nonparametric spectral Granger causality from source to target: 0 is no directed influence; larger values mean more of the target’s power is predicted by the source’s past. Not conditioned on other signals.
phase_lag_index	[-1, 1]	1		Signed asymmetry of the phase-difference distribution; blind to zero-lag coupling. Take the absolute value for the unsigned index. Positive (source=a, target=b) means a leads b.
phase_locking_value	[0, 1]	1		Consistency of the phase difference across trials and tapers, ignoring amplitude: 0 is random, 1 is constant. Biased upward with few observations.
power	[0, ∞)	(input units)^2/Hz		One-sided power spectral density of each signal.
weighted_phase_lag_index	[-1, 1]	1		Phase lag index weighted by the magnitude of the imaginary cross-spectrum; less sensitive to noise than the unweighted index. Positive (source=a, target=b) means a leads b.
coherency	|z| in [0, 1]	1		Complex coherency: its squared magnitude is coherence_magnitude and its angle is coherence_phase.
cross_spectral_density	|z| in [0, ∞)	(input units)^2/Hz		Complex, Hermitian cross-spectrum; unnormalized, so it scales with signal power. Use coherency for a normalized version.
imaginary_coherency	[-1, 1]	1		Signed imaginary part of coherency; blind to zero-lag coupling. Positive (source=a, target=b) means a leads b.
partial_coherence	[0, 1]	1		Magnitude-squared coherence after removing the linear influence of every other signal; near 0 for pairs coupled only through other signals.
corrected_imaginary_phase_locking_value	[0, 1]	1		Phase locking with zero- and pi-lag contributions removed; insensitive to volume conduction.
directed_phase_lag_index	[0, 1]	1	[..., i, j] is i relative to j	Above 0.5, the source phase-leads the target; below 0.5 it lags; 0.5 is no preferred direction.
subset_pairwise_spectral_granger_prediction	[0, ∞)	1	[..., i, j] is j -> i	pairwise_spectral_granger_prediction for only the requested pairs; other entries are NaN.
conditional_spectral_granger_prediction	[0, ∞)	1	[..., i, j] is j -> i	Spectral Granger causality from source to target conditioned on every other signal, removing influence relayed through observed signals.
time_reversed_spectral_granger_prediction	[0, ∞)	1	[..., i, j] is j -> i	Pairwise spectral Granger causality of the time-reversed data. Genuine directed influence reverses under time reversal; directionality that does not reverse suggests instantaneous mixing.
directed_transfer_function	[0, 1]	1	[..., i, j] is j -> i	Fraction of the target’s inflow at each frequency that comes from the source, including indirect paths; sums to 1 over sources.
directed_coherence	[0, 1]	1	[..., i, j] is j -> i	Noise-weighted directed transfer function: the fraction of the target’s power attributable to the source; sums to 1 over sources. Assumes uncorrelated innovations.
partial_directed_coherence	[0, 1]	1	[..., i, j] is j -> i	Direct influence from source to target, normalized by the source’s total outflow; sums to 1 over targets.
generalized_partial_directed_coherence	[0, 1]	1	[..., i, j] is j -> i	Partial directed coherence with each signal scaled by its innovation variance, making it insensitive to differences in signal scale.
direct_directed_transfer_function	[0, 1]	1	[..., i, j] is j -> i	Direct (not relayed) influence from source to target. Normalized over all frequencies, so values are small: compare pairs, not against 1.
blockwise_spectral_granger_prediction	[0, ∞)	1	[..., i, j] is j -> i	Spectral Granger causality between groups of signals set by group_labels, from source_group to target_group.
canonical_coherence	[0, 1]	1		Largest coherence between linear combinations of two groups of signals (historical estimator; see canonical_coherency).
maximized_imaginary_coherency	[0, 1]	1		Largest imaginary coherency between linear combinations of two groups; blind to zero-lag coupling.
multivariate_interaction_measure	[0, ∞)	1		Total phase-lagged interaction between two groups (the sum of squared imaginary-coherency components); at most the smaller group’s rank.
canonical_coherency	|z| in [0, 1]	1		Complex canonical coherency per component between two groups, with the spatial filters and patterns that produce it.
maximized_imaginary_coherency_components	[0, 1]	1		maximized_imaginary_coherency resolved into components, with the spatial filters and patterns that produce them.
delay	(-∞, ∞)	s	[..., i, j] is i relative to j	Candidate delays in seconds, one per 2*pi phase ambiguity; the true delay is the candidate that is constant across frequency. Frequencies without significant coherence are NaN. Positive (source=a, target=b) means a leads b.
global_coherence	[0, 1]	1		Fraction of the total cross-spectral power in each component; a large leading component indicates one dominant coherent network.
group_delay	(-∞, ∞)	s	[..., i, j] is i relative to j	Delay in seconds from the slope of phase against frequency over the band; check group_delay_r_value for the quality of the fit. Positive (source=a, target=b) means a leads b.
phase_slope_index	(-∞, ∞)	1	[..., i, j] is i relative to j	Coherence-weighted slope of phase against frequency over the band. Unnormalized, so judge it against a null distribution rather than a fixed threshold. Positive (source=a, target=b) means a leads b.

Notes

• For metrics theoretically bounded in [0, 1], numerical implementations should clamp to the interval after computation to avoid tiny overflows (e.g., 1 ± 1e−12). Unbiased estimators such as PPC and the debiased PLI variants must retain legitimate negative finite-sample values.

• For phase‑based metrics, choose an expectation (sample) axis (trials/tapers/segments) and aggregate only across that axis.

• Document the shapes/dtypes of inputs and outputs in each docstring per NumPy style.