Miscellaneous

Miscellaneous SPIs Overview

A small number of methods do not fit squarely into any category listed above, and so we place them in a ‘miscellaneous’ category. Here, we outline the use of linear and nonlinear model fits, for which we use scikit-learn, cointegration, for which we use statsmodels, and envelope correlation, for which we use MNE.

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Cointegration

Keywords: undirected, linear, unsigned, bivariate, time-dependent.

Base Identifier: coint

Key References: [1], [2]

If two time series are individually integrated but some linear combination of them has a lower order of integration, then the series are said to be ‘cointegrated’. We implement statistics for quantifying the cointegration of bivariate time series from two tests included in v0.12.0 of statsmodels:

• Augmented Engle-Granger (AEG, modifier aeg) two-step test. For the AEG test, we use a lag that is inferred via either AIC (modifier aic) or BIC (bic), with a maximum lag of 10 and obtain a t-statistic (tstat) of the unit-root test on the residuals. The time series is first detrended by assuming either a constant (c) or a constant and linear trend (ct).

• Johansen test (johansen). For the Johansen test, we output both the maximum eigenvalue (modifier max_eig_stat) and the trace (trace_stat) of the vector error correction model. Similar to the AEG test, we also assume a constant (order-0) or constant and linear trend (order-1) and fixed autoregressive lags of 1 (ardiff-1) and 10 (ardiff-10).

Cointegration Estimators

• 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 (cointegration)

• coint_aeg_tstat_trend-ct_autolag-bic_maxlag-10

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Gaussian Process Model Fit

Keywords: directed, nonlinear, unsigned, bivariate, contemporaneous.

Base Identifier: gpfit

Key References: [1]

Similar to the linear model fits, we also use Gaussian process model fits as a nonparametric measure of influence of x on y. Here, we use a combination of kernels (with parameters chosen from default settings) from the scikit-learn package and compute the MSE of their fit. The fits are computed for the dot-product kernel with inhomogenity parameter σ_0 = 1 (DotProduct) and the radial basis function (RBF) kernel with length scale l = 1 (RBF). Each of these kernels are separately combined with the constant kernel (with a constant of 1.0) and the white kernel (with a noise level of 1).

Gaussian Process Model Fit

• gpfit_DotProduct

• gpfit_RBF

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Linear Model Fit

Keywords: directed, linear, unsigned, bivariate, contemporaneous.

Base Identifier: lmfit

Linear regression is commonly used for establishing independence through model fits (e.g., see additive noise models). As such, we use a number of linear models and record the mean squared error (MSE) of a regression of y on x.

The following models (with the default parameters) from scikit-learn are included: stochastic gradient descent regression with a squared loss (ordinary least squares fit) function (denoted by modifier SGDRegressor); Ridge regression, which uses l2-norm regularization (Ride); the Elastic-Net model, which uses both l1 and l2-norm regularization (ElasticNet); and the Bayesian Ridge regressor uses a gamma distribution prior (with λ1 = λ2 = 10^(-6)}) for the l2-norm regularizor in Ridge regression (BayesianRidge).

Linear Model Fit Estimators

• lmfit_SGDRegressor

• lmfit_Ridge

• lmfit_ElasticNet

• lmfit_BayesianRidge

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Power Envelope Correlation

Keywords: undirected, linear, unsigned, bivariate, time-dependent.

Base Identifier: pec

Key References: [1], [2]

The envelope correlation is the correlation between the two amplitude envelopes of x and y. Power envelope correlation is computed using MNE, where we use six combinations of parameters, including whether the method was orthogonalised (modifier orth), the envelopes were squared and logs were taken prior to correlation (log), or the absolute of the correlation coefficient was used (abs).

Power Envelope Correlation Estimators

• pec

• pec_orth

• pec_log

• pec_orth_log

• pec_orth_abs

• pec_orth_log_abs

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