Author Contributions
All authors have contributed substantially to this work: Conceptualization, Z.J. (Ziyu Jia) and J.W.; Data curation, Z.J. (Ziyu Jia) and Y.M.; Formal analysis, Z.J. (Ziyu Jia), Z.J. (Zehui Jiao), Y.M. and J.W.; Funding acquisition, Y.L. and J.W.; Investigation, Z.J. (Ziyu Jia) and J.W.; Methodology, Z.J. (Ziyu Jia) and J.W.; Project administration, Y.L. and J.W.; Resources, Y.L. and J.W.; Software, Z.J. (Ziyu Jia); Supervision, J.W.; Validation, Z.J. (Ziyu Jia) and J.W.; Visualization, Z.J. (Ziyu Jia) and Z.J. (Zehui Jiao); Writing—original draft, Z.J. (Ziyu Jia); Writing—review and editing, Y.L., Z.J. (Zehui Jiao), Y.M. and J.W.
Figure 1.
The flowchart of the low-dimensional approximation of CMI and the mixed search strategy(LM)-partial conditional mutual information from mixed embedding (PMIME) method.
Figure 1.
The flowchart of the low-dimensional approximation of CMI and the mixed search strategy(LM)-partial conditional mutual information from mixed embedding (PMIME) method.
Figure 2.
Matrix representation of causality for the linear vector autoregressive (VAR) process. Retrieved by (a) traditional PMIME method, (b) mixed search strategy (M)-PMIME method, (c) and LM-PMIME method with k-nearest neighbors (k-NNs) estimator. The length of the time series is 512. is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the linear VAR process. The direction of causal influence is from row to column in the matrix. The true causal connections in this linear VAR process are at the matrix elements (1, 2), (1, 4), (2, 4), (4, 5), (5, 1), (5, 2) and (5, 3).
Figure 2.
Matrix representation of causality for the linear vector autoregressive (VAR) process. Retrieved by (a) traditional PMIME method, (b) mixed search strategy (M)-PMIME method, (c) and LM-PMIME method with k-nearest neighbors (k-NNs) estimator. The length of the time series is 512. is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the linear VAR process. The direction of causal influence is from row to column in the matrix. The true causal connections in this linear VAR process are at the matrix elements (1, 2), (1, 4), (2, 4), (4, 5), (5, 1), (5, 2) and (5, 3).
Figure 3.
Matrix representation of causality for . Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The time series length is 512. is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of . The true causal connections in are at the matrix elements (1,2), (1,3), (2,3).
Figure 3.
Matrix representation of causality for . Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The time series length is 512. is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of . The true causal connections in are at the matrix elements (1,2), (1,3), (2,3).
Figure 4.
Matrix representation of causality for variables of the coupled Henon maps (). Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The time series length is 1024. is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the coupled Henon maps. The true causal connections in the coupled Henon maps are at the matrix elements (), where .
Figure 4.
Matrix representation of causality for variables of the coupled Henon maps (). Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The time series length is 1024. is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the coupled Henon maps. The true causal connections in the coupled Henon maps are at the matrix elements (), where .
Figure 5.
Matrix representation of causality for variables of the coupled Henon maps (). Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The time series length is 1024. is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the coupled Henon maps. The true causal connections in the coupled Henon maps are at the matrix elements (), where .
Figure 5.
Matrix representation of causality for variables of the coupled Henon maps (). Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The time series length is 1024. is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the coupled Henon maps. The true causal connections in the coupled Henon maps are at the matrix elements (), where .
Figure 6.
Matrix representation of causality for the three coupled Lorenz oscillators. Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The length of the time series is 512 with coupling strength . is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the three coupled Lorenz oscillators. The true causal connections in the three coupled Lorenz oscillators are at the matrix elements (), where .
Figure 6.
Matrix representation of causality for the three coupled Lorenz oscillators. Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The length of the time series is 512 with coupling strength . is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the three coupled Lorenz oscillators. The true causal connections in the three coupled Lorenz oscillators are at the matrix elements (), where .
Figure 7.
Matrix representation of causality for the three coupled Lorenz oscillators. Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The length of the time series is 512 with coupling strength . is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the three coupled Lorenz oscillators. The direction of causal influence is from row to column in the matrix. The true causal connections in the three coupled Lorenz oscillators are at the matrix elements (), where .
Figure 7.
Matrix representation of causality for the three coupled Lorenz oscillators. Retrieved by (a) traditional PMIME method, (b) M-PMIME method, (c) and LM-PMIME method with k-NNs estimator. The length of the time series is 512 with coupling strength . is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same (, ). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the three coupled Lorenz oscillators. The direction of causal influence is from row to column in the matrix. The true causal connections in the three coupled Lorenz oscillators are at the matrix elements (), where .
Figure 8.
Results for multivariate electrocorticographic (ECoG) data. Matrices of causalities reflect the pre-seizure state (top) and the seizure state (bottom)) estimated by the PMIME method and the LM-PMIME method. The causal strengths are averaged (the mean values of the coupling measurements over all epochs in the same physiological state). Contacts 1 to 64 belong to an eight-by-eight electrode grid, and contacts 65 to 76 belong to two depth electrodes. The direction of causal influence is from row to column in the matrices. The brighter colors indicate more significant values. The key contact is marked by a rectangular box. The parameter and are set for the different methods.
Figure 8.
Results for multivariate electrocorticographic (ECoG) data. Matrices of causalities reflect the pre-seizure state (top) and the seizure state (bottom)) estimated by the PMIME method and the LM-PMIME method. The causal strengths are averaged (the mean values of the coupling measurements over all epochs in the same physiological state). Contacts 1 to 64 belong to an eight-by-eight electrode grid, and contacts 65 to 76 belong to two depth electrodes. The direction of causal influence is from row to column in the matrices. The brighter colors indicate more significant values. The key contact is marked by a rectangular box. The parameter and are set for the different methods.
Figure 9.
Results for multivariate ECoG data. Matrices reflect the difference of total numbers of significant connections between the seizure state and the pre-seizure state (seizure minus pre-seizure). The numbers are respectively summed from 8 seizure epochs and eight pre-seizure epochs. Contacts 1 to 64 belong to an eight-by-eight electrode grid, and contacts 65 to 76 belong to two depth electrodes. The brighter colors indicate more significant values. The key contact is marked by a rectangular box. The parameter and are set for the different methods.
Figure 9.
Results for multivariate ECoG data. Matrices reflect the difference of total numbers of significant connections between the seizure state and the pre-seizure state (seizure minus pre-seizure). The numbers are respectively summed from 8 seizure epochs and eight pre-seizure epochs. Contacts 1 to 64 belong to an eight-by-eight electrode grid, and contacts 65 to 76 belong to two depth electrodes. The brighter colors indicate more significant values. The key contact is marked by a rectangular box. The parameter and are set for the different methods.
Table 1.
Evaluation indicators are obtained from 100 realizations of linear VAR process with varying time series length for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
Table 1.
Evaluation indicators are obtained from 100 realizations of linear VAR process with varying time series length for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
| Sensitivity | Specificity | F1 Score |
---|
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
Table 2.
Evaluation indicators are obtained from 100 realizations of with varying time series length for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
Table 2.
Evaluation indicators are obtained from 100 realizations of with varying time series length for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
| Sensitivity | Specificity | F1 Score |
---|
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
Table 3.
Evaluation indicators are obtained from 100 realizations of K variables of the Henon maps () for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
Table 3.
Evaluation indicators are obtained from 100 realizations of K variables of the Henon maps () for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
| Sensitivity | Specificity | F1 Score |
---|
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
Table 4.
Evaluation indicators are obtained from 100 realizations of K variables of the Henon maps () for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
Table 4.
Evaluation indicators are obtained from 100 realizations of K variables of the Henon maps () for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
| Sensitivity | Specificity | F1 Score |
---|
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
Table 5.
Evaluation indicators are obtained from 100 realizations of the three coupled Lorenz oscillators () with varying time series length for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
Table 5.
Evaluation indicators are obtained from 100 realizations of the three coupled Lorenz oscillators () with varying time series length for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
| Sensitivity | Specificity | F1 Score |
---|
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
Table 6.
Evaluation indicators are obtained from 100 realizations of the three coupled Lorenz oscillators () with coupling strength C from 1 to 5 for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
Table 6.
Evaluation indicators are obtained from 100 realizations of the three coupled Lorenz oscillators () with coupling strength C from 1 to 5 for the three different methods. and are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter .
| Sensitivity | Specificity | F1 Score |
---|
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |
| | | |
PMIME | | | |
M-PMIME | | | |
LM-PMIME | | | |