Table of SPIs
In the table below, we provide a high-level overview of the pyspi library of statistics for pairwise interactions (SPIs).
For further details about the specific implementation of each SPI, refer to the source code on our GitHub repository. We also provide a detailed list of many of the statistics in this toolkit in the Supplementary Material of the original pyspi paper.
Categories
Modifiers
Many of the algorithms we implement include a number of free parameters that we set either using optimisation procedures or fix to a small number of sensible predefined settings. The combination of both the parameter configuration and the different summary statistics gives each SPI in the library a unique identifier (as a string).
As an example, consider the SPI identifier xcorr_mean_sig-True. Here, xcorr refers to the method of cross-correlation between x and y, which itself does not provide a single statistic, but rather a correlogram. However, the two additional modifiers that are separated by underscores in the identifier, mean and sig-True, collectively return a scalar value. The first modifier, mean indicates that we are taking the average across lags of the cross-correlation function. The second modifier, sig-True indicates that we will only take the mean over statistically significant lags. By using different parameters and modifiers of distinct methodologies, we obtain hundreds of unique SPIs.
In the tables below, we make the distinction between the "base identifier" (xcorr in the above example) and the estimator(s) (e.g., xcorr_mean_sig-True) which include the combination of base identifier and any additional modifiers.
Basic Statistics (42 SPIs)
Basic statistics include SPIs that are foundational to statistical analysis, often because they are widely applicable, relatively straightforward to understand and compute, and can form the basis for more complex methods. For further details about the specific implementation of each basic SPI, refer to the source code on our GitHub repository.
| SPI name | Base Identifier | Estimator(s) |
|---|---|---|
Covariance |
| |
Precision |
| |
Spearman's rank-correlation coefficient |
| |
Kendall's rank-correlation coefficient |
| |
Cross correlation |
| |
Squared cross correlation |
| |
Squared covariance |
|
|
Squared precision |
|
|
Distance Similarity (42 SPIs)
Distance-based similarity measures aim to establish statistical similarity or independence based on the pairwise distance between bivariate observations. For further details about the specific implementation of each distance similarity SPI, refer to the source code on our GitHub repository.
| SPI Name | Base Identifier | Estimator(s) |
|---|---|---|
Pairwise distance |
|
|
Distance correlation |
|
|
Cross distance correlation |
|
|
Multiscale graph correlation |
|
|
Cross multiscale graph correlation |
|
|
Hilbert-Schmidt Independence Criterion |
|
|
Heller-Heller-Gorfine Independence Criterion |
|
|
Dynamic time warping |
|
|
Longest common subsequence |
|
|
Soft dynamic time warping |
|
|
Barycenter |
|
|
Squared Barycenter |
|
|
Gromov-Wasserstein distance |
|
|
Causal Inference (13 SPIs)
Causal inference-based statistics aim to establish directed independence from bivariate observations, typically making assumptions about the underlying model. For further details about the specific implementation of each causal inference SPI, refer to the source code on our GitHub repository.
| SPI Name | Base Identifier | Estimator(s) |
|---|---|---|
Additive noise model |
|
|
Information-geometric conditional independence |
|
|
Conditional distribution similarity fit |
|
|
Regression error-based causal inference |
|
|
Convergent cross-mapping |
|
|
Information Theory (43 SPIs)
Information theoretic based similarity measures which are either intended to operate on serially independent observations or bivariate time series. For further details about the specific implementation of each information theoretic SPI, refer to the source code on our GitHub repository.
| SPI Name | Base Identifier | Estimator(s) |
|---|---|---|
Joint entropy |
|
|
Conditional entropy |
|
|
Mutual information |
|
|
Time-lagged mutual information |
|
|
Transfer entropy |
|
|
Granger causality |
|
|
Causally conditioned entropy |
|
|
Directed information |
|
|
Stochastic interaction |
|
|
Integrated information |
|
|
Cross-Map Entropy |
|
|
Spectral (120 SPIs)
Spectral SPIs are computed in the frequency or time-frequency domain, using either Fourier or wavelet transformations to derive spectral matrices. For further details about the specific implementation of each spectral SPI, refer to the source code on our GitHub repository.
| SPI Name | Base Identifier | Estimator(s) |
|---|---|---|
Coherence magnitude |
|
|
Coherence phase |
|
|
Group delay |
|
|
Phase slope index |
|
|
Imaginary coherence |
|
|
Phase locking value |
|
|
Pairwise phase consistency |
|
|
Phase lag index |
|
|
Weighted phase lag index |
|
|
Debiased squared phase lag index |
|
|
Debiased squared weighted phase lag index |
|
|
Directed transfer function |
|
|
Direct directed transfer function |
|
|
Directed coherence |
|
|
Partial directed coherence |
|
|
Generalised partial directed coherence |
|
|
Spectral granger causality |
|
|
Miscellaneous (24 SPIs)
A small number of methods which do not fit into any of the previous categories. For further details about the specific implementation of each miscellaneous SPI, refer to the source code on our GitHub repository.
| SPI Name | Base Identifier | Estimator(s) |
|---|---|---|
Linear model fit |
|
|
Gaussian process model fit |
|
|
Cointegration |
|
|
Power envelope correlation |
|
|
Short Names
For SPIs in the sonnet subset, we also include a unique `short name' for easier reference. Click the short name for a brief summary of what the SPI calculates.
| Short Name | SPI | Estimator |
|---|---|---|
Power envelope correlation |
| |
Cointegration |
| |
Phase Lag Index |
| |
Phase Slope Index |
| |
Phase Slope Index |
| |
Coherence Magnitude |
| |
Integrated Information |
| |
Directed Information |
| |
Additive Noise Model |
| |
Transfer Entropy |
| |
Granger Causality |
| |
Dynamic Time Warping |
| |
Covariance |
| |
Dynamic Time Warping |
|
Last updated
