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

Basic Statistics (42 SPIs)

Distance Similarity

(42 SPIs)

Causal Inference (13 SPIs)

Information Theory

(43 SPIs)

Spectral (120 SPIs)

Miscellaneous (25 SPIs)

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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.

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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	cov	cov_EmpiricalCovariance cov_EllipticEnvelope cov_MinCovDet cov_GraphicalLasso cov_GraphicalLassoCV cov_ShrunkCovariance cov_LedoitWolf cov_OAS
Precision	prec	prec_EmpiricalCovarianc prec_EllipticEnvelope prec_MinCovDet prec_GraphicalLasso prec_GraphicalLassoCV prec_LedoitWolf prec_OAS prec_ShrunkCovariance
Spearman's rank-correlation coefficient	spearmanr	spearmanr spearmanr-sq
Kendall's rank-correlation coefficient	kendalltau	kendalltau kendalltau-sq
Cross correlation	xcorr	xcorr_max_sig-True xcorr_mean_sig-True xcorr_mean_sig-False
Squared cross correlation	xcorr-sq	xcorr-sq_max_sig-True xcorr-sq_mean_sig-True xcorr-sq_mean_sig-False
Squared covariance	cov-sq	cov-sq_EmpiricalCovariance cov-sq_EllipticEnvelope cov-sq_GraphicalLasso cov-sq_GraphicalLassoCV cov-sq_LedoitWolf cov-sq_MinCovDet cov-sq_OAS cov-sq_ShrunkCovariance
Squared precision	prec-sq	prec-sq_EmpiricalCovariance prec-sq_EllipticEnvelope prec-sq_GraphicalLasso prec-sq_GraphicalLassoCV prec-sq_LedoitWolf prec-sq_MinCovDet prec-sq_OAS prec-sq_ShrunkCovariance

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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	pdist	pdist_euclidean pdist_cityblock pdist_cosine pdist_chebyshev pdist_canberra pdist_braycurtis
Distance correlation	dcorr	dcorr dcorr_biased
Cross distance correlation	dcorrx	dcorrx_maxlag-1 dcorrx_maxlag-10
Multiscale graph correlation	mgc	mgc
Cross multiscale graph correlation	mgcx	mgcx_maxlag-1 mgcx_maxlag-10
Hilbert-Schmidt Independence Criterion	hsic	hsic hsic_biased
Heller-Heller-Gorfine Independence Criterion	hhg	hhg
Dynamic time warping	dtw	dtw dtw_constraint-itakura dtw_constraint-sakoe_chiba
Longest common subsequence	lcss	lcss lcss_constraint-itakura lcss_constraint-sakoe_chiba
Soft dynamic time warping	softdtw	softdtw softdtw_constraint-itakura softdtw_constraint-sakoe_chiba
Barycenter	bary	bary_euclidean_mean bary_euclidean_max bary_dtw_mean bary_dtw_max bary_softdtw_mean bary_softdtw_max bary_sgddtw_mean bary_sgddtw_max
Squared Barycenter	bary-sq	bary-sq_euclidean_mean bary-sq_euclidean_max bary-sq_dtw_mean bary-sq_dtw_max bary-sq_sgddtw_mean bary-sq_sgddtw_max bary-sq_softdtw_mean bary-sq_softdtw_max
Gromov-Wasserstein distance	gwtau	gwtau

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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	anm	anm
Information-geometric conditional independence	igci	igci
Conditional distribution similarity fit	cds	cds
Regression error-based causal inference	reci	reci
Convergent cross-mapping	ccm	ccm_E-1_mean ccm_E-1_max ccm_E-1_diff ccm_E-10_mean ccm_E-10_max ccm_E-10-diff ccm_E-None_mean ccm_E-None_max ccm_E-None_diff

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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	je	je_gaussian je_kozachenko je_kernel_W-0.5
Conditional entropy	ce	ce_gaussian ce_kozachenko ce_kernel_W-0.5
Mutual information	mi	mi_gaussian mi_kraskov_NN-4 mi_kraskov_NN-4_DCE mi_kernel_W-0.25
Time-lagged mutual information	tlmi	tlmi_gaussian tlmi_kraskov_NN-4 tlmi_kraskov_NN-4_DCE tlmi_kernel_W-0.25
Transfer entropy	te	te_kraskov_NN-4_k-max-10_tau-max-4 te_kraskov_NN-4_DCE_k-max-10_tau-max-4 te_kraskov_NN-4_DCE_k-1_kt-1_l-1_lt-1 te_kraskov_NN-4_DCE_k-2_kt-1_l-1_lt-1 te_kraskov_NN-4_k-1_kt-1_l-1_lt-1 te_kernel_W-0.25_k-1 te_symbolic_k-1_kt-1_l-1_lt-1 te_symbolic_k-10_kt-1_l-1_lt-1
Granger causality	gc	gc_gaussian_k-max-10_tau-max-2 gc_gaussian_k-1_kt-1_l-1_lt-1
Causally conditioned entropy	cce	cce_gaussian cce_kozachenko cce_kernel_W-0.5
Directed information	di	di_gaussian di_kozachenko di_kernel_W-0.5
Stochastic interaction	si	si_gaussian_k-1 si_kozachenko_k-1 si_kernel_W-0.5_k-1
Integrated information	phi	phi_star_t-1_norm-0 phi_star_t-1_norm-1 phi_Geo_t-1_norm-0 phi_Geo_t-1_norm-1
Cross-Map Entropy	xme	xme_kozachenko_k1 xme_kozachenko_k10 xme_gaussian_k1 xme_gaussian_k10 xme_kernel_k1 xme_kernel_k10

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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	cohmag	cohmag_multitaper_mean_fs-1_fmin-0_fmax-0-5 cohmag_multitaper_mean_fs-1_fmin-0_fmax-0-25 cohmag_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 cohmag_multitaper_max_fs-1_fmin-0_fmax-0-5 cohmag_multitaper_max_fs-1_fmin-0_fmax-0-25 cohmag_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Coherence phase	phase	phase_multitaper_mean_fs-1_fmin-0_fmax-0-5 phase_multitaper_mean_fs-1_fmin-0_fmax-0-25 phase_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 phase_multitaper_max_fs-1_fmin-0_fmax-0-5 phase_multitaper_max_fs-1_fmin-0_fmax-0-25 phase_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Group delay	gd	gd_multitaper_delay_fs-1_fmin-0_fmax-0-5 gd_multitaper_delay_fs-1_fmin-0_fmax-0-25 gd_multitaper_delay_fs-1_fmin-0-25_fmax-0-5
Phase slope index	psi	psi_multitaper_mean_fs-1_fmin-0_fmax-0-5 psi_multitaper_mean_fs-1_fmin-0_fmax-0-25 psi_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 psi_wavelet_mean_fs-1_fmin-0_fmax-0-5_mean psi_wavelet_mean_fs-1_fmin-0_fmax-0-25_mean psi_wavelet_mean_fs-1_fmin-0-25_fmax-0-5_mean psi_wavelet_max_fs-1_fmin-0_fmax-0-5_max psi_wavelet_max_fs-1_fmin-0_fmax-0-25_max psi_wavelet_max_fs-1_fmin-0-25_fmax-0-5_max
Imaginary coherence	icoh	icoh_multitaper_mean_fs-1_fmin-0_fmax-0-5 icoh_multitaper_mean_fs-1_fmin-0_fmax-0-25 icoh_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 icoh_multitaper_max_fs-1_fmin-0_fmax-0-5 icoh_multitaper_max_fs-1_fmin-0_fmax-0-25 icoh_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Phase locking value	plv	plv_multitaper_mean_fs-1_fmin-0_fmax-0-5 plv_multitaper_mean_fs-1_fmin-0_fmax-0-25 plv_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 plv_multitaper_max_fs-1_fmin-0_fmax-0-5 plv_multitaper_max_fs-1_fmin-0_fmax-0-25 plv_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Pairwise phase consistency	ppc	ppc_multitaper_mean_fs-1_fmin-0_fmax-0-5 ppc_multitaper_mean_fs-1_fmin-0_fmax-0-25 ppc_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 ppc_multitaper_max_fs-1_fmin-0_fmax-0-5 ppc_multitaper_max_fs-1_fmin-0_fmax-0-25 ppc_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Phase lag index	pli	pli_multitaper_mean_fs-1_fmin-0_fmax-0-5 pli_multitaper_mean_fs-1_fmin-0_fmax-0-25 pli_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 pli_multitaper_max_fs-1_fmin-0_fmax-0-5 pli_multitaper_max_fs-1_fmin-0_fmax-0-25 pli_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Weighted phase lag index	wpli	wpli_multitaper_mean_fs-1_fmin-0_fmax-0-5 wpli_multitaper_mean_fs-1_fmin-0_fmax-0-25 wpli_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 wpli_multitaper_max_fs-1_fmin-0_fmax-0-5 wpli_multitaper_max_fs-1_fmin-0_fmax-0-25 wpli_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Debiased squared phase lag index	dspli	dspli_multitaper_mean_fs-1_fmin-0_fmax-0-5 dspli_multitaper_mean_fs-1_fmin-0_fmax-0-25 dspli_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 dspli_multitaper_max_fs-1_fmin-0_fmax-0-5 dspli_multitaper_max_fs-1_fmin-0_fmax-0-25 dspli_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Debiased squared weighted phase lag index	dswpli	dswpli_multitaper_mean_fs-1_fmin-0_fmax-0-5 dswpli_multitaper_mean_fs-1_fmin-0_fmax-0-25 dswpli_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 dswpli_multitaper_max_fs-1_fmin-0_fmax-0-5 dswpli_multitaper_max_fs-1_fmin-0_fmax-0-25 dswpli_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Directed transfer function	dtf	dtf_multitaper_mean_fs-1_fmin-0_fmax-0-5 dtf_multitaper_mean_fs-1_fmin-0_fmax-0-25 dtf_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 dtf_multitaper_max_fs-1_fmin-0_fmax-0-5 dtf_multitaper_max_fs-1_fmin-0_fmax-0-25 dtf_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Direct directed transfer function	ddtf	ddtf_multitaper_mean_fs-1_fmin-0_fmax-0-5 ddtf_multitaper_mean_fs-1_fmin-0_fmax-0-25 ddtf_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 ddtf_multitaper_max_fs-1_fmin-0_fmax-0-5 ddtf_multitaper_max_fs-1_fmin-0_fmax-0-25 ddtf_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Directed coherence	dcoh	dcoh_multitaper_mean_fs-1_fmin-0_fmax-0-5 dcoh_multitaper_mean_fs-1_fmin-0_fmax-0-25 dcoh_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 dcoh_multitaper_max_fs-1_fmin-0_fmax-0-5 dcoh_multitaper_max_fs-1_fmin-0_fmax-0-25 dcoh_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Partial directed coherence	pdcoh	pdcoh_multitaper_mean_fs-1_fmin-0_fmax-0-5 pdcoh_multitaper_mean_fs-1_fmin-0_fmax-0-25 pdcoh_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 pdcoh_multitaper_max_fs-1_fmin-0_fmax-0-5 pdcoh_multitaper_max_fs-1_fmin-0_fmax-0-25 pdcoh_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Generalised partial directed coherence	gpdcoh	gpdcoh_multitaper_mean_fs-1_fmin-0_fmax-0-5 gpdcoh_multitaper_mean_fs-1_fmin-0_fmax-0-25 gpdcoh_multitaper_mean_fs-1_fmin-0-25_fmax-0-5 gpdcoh_multitaper_max_fs-1_fmin-0_fmax-0-5 gpdcoh_multitaper_max_fs-1_fmin-0_fmax-0-25 gpdcoh_multitaper_max_fs-1_fmin-0-25_fmax-0-5
Spectral granger causality	sgc	sgc_nonparametric_mean_fs-1_fmin-0_fmax-0-5 sgc_nonparametric_mean_fs-1_fmin-0_fmax-0-25 sgc_nonparametric_mean_fs-1_fmin-0-25_fmax-0-5 sgc_nonparametric_max_fs-1_fmin-0_fmax-0-5 sgc_nonparametric_max_fs-1_fmin-0_fmax-0-25 sgc_nonparametric_max_fs-1_fmin-0-25_fmax-0-5 sgc_parametric_mean_fs-1_fmin-0_fmax-0-5_order-None sgc_parametric_mean_fs-1_fmin-0_fmax-0-25_order-None sgc_parametric_mean_fs-1_fmin-0-25_fmax-0-5_order-None sgc_parametric_mean_fs-1_fmin-0_fmax-0-5_order-1 sgc_parametric_mean_fs-1_fmin-0_fmax-0-25_order-1 sgc_parametric_mean_fs-1_fmin-0-25_fmax-0-5_order-1 sgc_parametric_mean_fs-1_fmin-0_fmax-0-5_order-20 sgc_parametric_mean_fs-1_fmin-0_fmax-0-25_order-20 sgc_parametric_mean_fs-1_fmin-0-25_fmax-0-5_order-20 sgc_parametric_max_fs-1_fmin-0_fmax-0-5_order-None sgc_parametric_max_fs-1_fmin-0_fmax-0-25_order-None sgc_parametric_max_fs-1_fmin-0-25_fmax-0-5_order-None sgc_parametric_max_fs-1_fmin-0_fmax-0-5_order-1 sgc_parametric_max_fs-1_fmin-0_fmax-0-25_order-1 sgc_parametric_max_fs-1_fmin-0-25_fmax-0-5_order-1 sgc_parametric_max_fs-1_fmin-0_fmax-0-5_order-20 sgc_parametric_max_fs-1_fmin-0_fmax-0-25_order-20 sgc_parametric_max_fs-1_fmin-0-25_fmax-0-5_order-20

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Miscellaneous (25 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	lmfit	lmfit_SGDRegressor lmfit_Ridge lmfit_Lasso lmfit_ElasticNet lmfit_BayesianRidge
Gaussian process model fit	gpfit	gpfit_DotProduct gpfit_RBF
Cointegration	coint	coint_johansen_max_eig_stat_order-0_ardiff-10 coint_johansen_trace_stat_order-0_ardiff-10 coint_johansen_max_eig_stat_order-0_ardiff-1 coint_johansen_trace_stat_order-0_ardiff-1 coint_johansen_max_eig_stat_order-1_ardiff-10 coint_johansen_trace_stat_order-1_ardiff-10 coint_johansen_max_eig_stat_order-1_ardiff-1 coint_johansen_trace_stat_order-1_ardiff-1 coint_aeg_tstat_trend-c_autolag-aic_maxlag-10 coint_aeg_tstat_trend-ct_autolag-aic_maxlag-10 coint_aeg_tstat_trend-ct_autolag-bic_maxlag-10
Power envelope correlation	pec	pec pec_orth pec_log pec_orth_log pec_orth_abs pec_orth_log_abs
Interdependence score	ids	ids

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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	pec
	Cointegration	coint_aeg_tstat_trend-ct_autolag-aic_maxlag-10
	Phase Lag Index	pli_multitaper_max_fs-1_fmin-0_fmax-0-5
	Phase Slope Index	psi_wavelet_mean_fs-1_fmin-0_fmax-0-5_mean
	Phase Slope Index	psi_multitaper_mean_fs-1_fmin-0_fmax-0-5
	Coherence Magnitude	cohmag_multitaper_mean_fs-1_fmin-0_fmax-0-5
	Integrated Information	phi_star_t-1_norm-0
	Directed Information	di_gaussian
	Additive Noise Model	anm
	Transfer Entropy	te_kraskov_NN-4_DCE_k-max-10_tau-max-4
	Granger Causality	sgc_nonparametric_mean_fs-1_fmin-0_fmax-0-5
	Dynamic Time Warping	bary_dtw_mean
	Covariance	cov_EmpiricalCovariance
	Dynamic Time Warping	dtw_constraint-itakura

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