Estimation of Dynamic Bivariate Correlation Using a Weighted Graph Algorithm
Abstract
:1. Introduction
2. Methods
2.1. Method # 1—Sliding-Window (SW) Technique
2.2. Method # 2—Dynamic Conditional Correlation
- 1st step:
- Estimate using GARCH(1,1)
- DE-GARCH data:
- 2nd step:
- Estimate quasi-correlation matrix , where
- 3rd step:
- Rescale to get a proper correlation matrix:
2.3. Method # 3—Weighted Graph Algorithm (WGA)
2.4. Simulation Design
2.5. Real Data Analysis
3. Results
3.1. Simulations Results
3.2. Stroop Task Data Results
4. Quantifying Uncertainty and Testing Hypothesis
4.1. Confidence Intervals
- Step 1:
- Let denote the centered where the centering for elements in and in are done separately.
- Step 2:
- Compute where L denotes the lower left matrix in Cholesky decomposition () of .
- Step 3:
- Let be the standardized version of obtained by subtracting the mean and dividing by the standard deviation (SD). Here mean and SD are computed from all elements in .
- Step 4:
- Generate by performing i.i.d resampling from .
- Step 5:
- Compute . Split into two separate vectors with containing the first T elements and containing the last T elements of .
- Step 6:
- Add mean of to each element of to obtain . Similarly, add mean of to each element of to obtain . The pair of vectors and are the bootstrap sample pair outputted.
- Step 1:
- Divide the data matrix X into K adjacent blocks, where each block is a matrix.
- Step 2:
- For each block run the MLPB algorithm to get a bootstrap sample pair corresponding to that block.
- Step 3:
- Combine the K adjacent blocks of bootstrap sample pairs and apply the WGA algorithm on the combined bootstrap sample pair.
- Step 4:
- Repeat the above steps B times to get B dynamic correlation time series based on the WGA algorithm.
- Step 5:
- At each time point calculate the 2.5th and 97.5th percentiles among the bootstrap samples to obtain the bootstrap based 95% confidence interval.
4.2. Hypothesis Testing
5. Conclusions and Discussion
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Figures and Tables
Maximum Value of Correlations Across Time | |||
---|---|---|---|
SW | WGA | DCC | |
Amygdala vs. Caudate Head | |||
Amygdala vs. Frontal Medial Cortex | |||
Amygdala vs. Heschel’s Gyrus | |||
Amygdala vs. Insula | |||
Amygdala vs. Subcallosal Cortex | |||
Caudate Head vs. Frontal Medial Cortex | |||
Caudate Head vs. Heschel’s Gyrus | |||
Caudate Head vs. Insula | |||
Caudate Head vs Subcallosal Cortex | |||
Frontal Medial Cortex vs. Heschel’s Gyrus | |||
Frontal Medial Cortex vs. Insula | |||
Frontal Medial Cortex vs. Subcallosal Cortex | |||
Heschel’s Gyrus vs. Insula | |||
Heschel’s Gyrus vs. Subcallosal Cortex | |||
Insula versus Subcallosal Cortex |
Appendix B. Visibility Criteria
Schematic Diagram
Appendix C. Theoretical Considerations
Appendix C.1. Heuristic Justification
Appendix C.2. A Theoretical Framework
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Mean of Absolute Value of Correlations Across Time | ||||
---|---|---|---|---|
T = 150 | T = 300 | T = 600 | T = 1000 | |
SW | ||||
WGA | ||||
DCC | ||||
Maximum of Absolute Value of Correlations Across Time | ||||
T = 150 | T = 300 | T = 600 | T = 1000 | |
SW | ||||
WGA | ||||
DCC |
Mean of Absolute Value of Correlations Across Time | ||||
---|---|---|---|---|
T = 150 | T = 300 | T = 600 | T = 1000 | |
SW | ||||
WGA | ||||
DCC | ||||
Maximum of Absolute Value of Correlations Across Time | ||||
T = 150 | T = 300 | T = 600 | T = 1000 | |
SW | ||||
WGA | ||||
DCC |
Mean Value of Correlations Across Time | |||
---|---|---|---|
SW | WGA | DCC | |
Amygdala vs. Caudate Head | |||
Amygdala vs. Frontal Medial Cortex | |||
Amygdala vs. Heschel’s Gyrus | |||
Amygdala vs. Insula | |||
Amygdala vs. Subcallosal Cortex | |||
Caudate Head vs. Frontal Medial Cortex | |||
Caudate Head vs. Heschel’s Gyrus | |||
Caudate Head vs. Insula | |||
Caudate Head vs Subcallosal Cortex | |||
Frontal Medial Cortex vs. Heschel’s Gyrus | |||
Frontal Medial Cortex vs. Insula | |||
Frontal Medial Cortex vs. Subcallosal Cortex | |||
Heschel’s Gyrus vs. Insula | |||
Heschel’s Gyrus vs. Subcallosal Cortex | |||
Insula versus Subcallosal Cortex |
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John, M.; Wu, Y.; Narayan, M.; John, A.; Ikuta, T.; Ferbinteanu, J. Estimation of Dynamic Bivariate Correlation Using a Weighted Graph Algorithm. Entropy 2020, 22, 617. https://doi.org/10.3390/e22060617
John M, Wu Y, Narayan M, John A, Ikuta T, Ferbinteanu J. Estimation of Dynamic Bivariate Correlation Using a Weighted Graph Algorithm. Entropy. 2020; 22(6):617. https://doi.org/10.3390/e22060617
Chicago/Turabian StyleJohn, Majnu, Yihren Wu, Manjari Narayan, Aparna John, Toshikazu Ikuta, and Janina Ferbinteanu. 2020. "Estimation of Dynamic Bivariate Correlation Using a Weighted Graph Algorithm" Entropy 22, no. 6: 617. https://doi.org/10.3390/e22060617