# Estimation of Dynamic Bivariate Correlation Using a Weighted Graph Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Method # 1—Sliding-Window (SW) Technique

#### 2.2. Method # 2—Dynamic Conditional Correlation

- 1st step:
- Estimate ${\sigma}_{i,t}^{2}$ using GARCH(1,1) ${\sigma}_{i,t}^{2}={\omega}_{i}+{\alpha}_{i}{y}_{i,t}^{2}+{\beta}_{i}{\sigma}_{i,t-1}^{2}$
- DE-GARCH data: ${\mathbf{r}}_{t}={\mathbf{D}}_{t}^{-1}{\mathbf{y}}_{t},\phantom{\rule{0.277778em}{0ex}}\mathrm{where}\phantom{\rule{0.277778em}{0ex}}{\mathbf{D}}_{t}=\mathrm{diag}\{{\sigma}_{1,t},{\sigma}_{2,t}\}.$

- 2nd step:
- Estimate quasi-correlation matrix ${\mathbf{Q}}_{t}=(1-a-b)\widehat{\mathbf{S}}+a{\mathbf{r}}_{t-1}{\mathbf{r}}_{t-1}^{\prime}+b{\mathbf{Q}}_{t-1}$, where $\widehat{\mathbf{S}}=\frac{1}{T}{\sum}_{i=1}^{T}{\mathbf{r}}_{t-1}{\mathbf{r}}_{t-1}^{\prime}$

- 3rd step:
- Rescale to get a proper correlation matrix: ${\mathbf{R}}_{t}=\mathrm{diag}{\left\{{\mathbf{Q}}_{t}\right\}}^{-1/2}{\mathbf{Q}}_{t}\phantom{\rule{0.166667em}{0ex}}\mathrm{diag}{\left\{{\mathbf{Q}}_{t}\right\}}^{-1/2}.$

#### 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

^{th}time point, the values were mostly above 0.5. These results again confirmed our visual observation of the original time series. Finally, as expected, the WGA estimates of dynamic correlations between hippocampal and striatal LFPs were near zero for most of the time points. Thus, the WGA method led to meaningful results.

## 4. Quantifying Uncertainty and Testing Hypothesis

^{th}and 97.5

^{th}percentiles among all bootstrap based estimates. Our bootstrap sampling method is based on adapting slightly the ‘multivariate linear process bootstrap’ (MLPB) method developed and presented in [27]. The core idea behind our adaptation is same as in ‘DCBootCB’ algorithm presented in [28]. The hypothesis testing framework is based on adapting the statistical tests proposed in [29]. Details of all the approaches are given below.

#### 4.1. Confidence Intervals

^{th}element being $C(i-j)$. Here

^{th}element is ${\Gamma}_{ij}^{\mathrm{JP}}=\kappa (i-j)C(i-j),\phantom{\rule{0.277778em}{0ex}}i,j=1,\dots ,T.$ Here $\kappa $ is a tapering function that leaves the diagonal elements unchanged and downweights the elements farther away from the diagonal. The simplest example (the one used in the simulation study in [27], and also the one used in this paper) is the trapezoid function which leaves the diagonal, super- and supra-diagonal elements unchanged and sets all other elements to zero. This estimation procedure involves a few more technical (but easy to understand) details, mainly to ensure that ${\widehat{\Gamma}}^{\mathrm{JP}}$ is positive definite. We skip these details; a reader interested in these details are referred to [28] or [27]. The steps in MLPB algorithm may be summarized as follows.

- Step 1:
- Let ${Y}_{vec}$ denote the centered ${X}_{vec}$ where the centering for elements in ${[{X}_{1}^{\left(1\right)},\dots ,{X}_{T}^{\left(1\right)}]}^{\prime}$ and in ${[{X}_{2}^{\left(1\right)},\dots ,{X}_{T}^{\left(2\right)}]}^{\prime}$ are done separately.
- Step 2:
- Compute ${W}_{vec}={L}^{-1}{Y}_{vec}$ where L denotes the lower left matrix in Cholesky decomposition ($L{L}^{\prime}$) of ${\widehat{\Gamma}}^{\mathrm{JP}}$.
- Step 3:
- Let ${Z}_{vec}$ be the standardized version of ${W}_{vec}$ obtained by subtracting the mean and dividing by the standard deviation (SD). Here mean and SD are computed from all elements in ${W}_{vec}$.
- Step 4:
- Generate ${Z}_{vec}^{boot}$ by performing i.i.d resampling from ${Z}_{vec}$.
- Step 5:
- Compute ${Y}_{vec}^{boot}=L{Z}_{vec}^{boot}$. Split ${Y}_{vec}^{boot}$ into two separate vectors ${Y}^{\left(i\right),boot},\phantom{\rule{0.277778em}{0ex}}i=1,2$ with ${Y}^{\left(1\right),boot}$ containing the first T elements and ${Y}^{\left(2\right),boot}$ containing the last T elements of ${Y}_{vec}^{boot}$.
- Step 6:
- Add mean of ${X}_{1}^{\left(1\right)},\dots ,{X}_{T}^{\left(1\right)}$ to each element of ${Y}^{\left(1\right),boot}$ to obtain ${X}^{\left(1\right),boot}$. Similarly, add mean of ${X}_{1}^{\left(2\right)},\dots ,{X}_{T}^{\left(2\right)}$ to each element of ${Y}^{\left(2\right),boot}$ to obtain ${X}^{\left(2\right),boot}$. The pair of vectors ${X}^{\left(1\right),boot}$ and ${X}^{\left(2\right),boot}$ are the bootstrap sample pair outputted.

- Step 1:
- Divide the data matrix X into K adjacent blocks, where each block is a $2\times (T/K)$ 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 2.5
^{th}and 97.5^{th}percentiles among the bootstrap samples to obtain the bootstrap based 95% confidence interval.

#### 4.2. Hypothesis Testing

^{th}percentile value and adding 97.5

^{th}percentile to the corresponding test statistic; here, the percentiles are obtained from among the test statistic values calculated from bootstrap samples. Null hypothesis may be rejected if the 95% confidence interval does not contain zero.

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Figures and Tables

**Figure A1.**The green curves (DCC estimates) in the bottom panel of Figure 1, with y-axis zoomed in, to see that the curves are not really straight lines.

Maximum Value of Correlations Across Time | |||
---|---|---|---|

SW | WGA | DCC | |

Amygdala vs. Caudate Head | $0.823$ | $0.507$ | $1.000$ |

Amygdala vs. Frontal Medial Cortex | $0.882$ | $0.666$ | $1.000$ |

Amygdala vs. Heschel’s Gyrus | $0.657$ | $0.504$ | $1.000$ |

Amygdala vs. Insula | $0.767$ | $0.584$ | $1.000$ |

Amygdala vs. Subcallosal Cortex | $0.830$ | $0.692$ | $1.000$ |

Caudate Head vs. Frontal Medial Cortex | $0.795$ | $0.701$ | $1.000$ |

Caudate Head vs. Heschel’s Gyrus | $0.606$ | $0.492$ | $1.000$ |

Caudate Head vs. Insula | $0.736$ | $0.602$ | $1.000$ |

Caudate Head vs Subcallosal Cortex | $0.800$ | $0.624$ | $1.000$ |

Frontal Medial Cortex vs. Heschel’s Gyrus | $0.757$ | $0.629$ | $1.000$ |

Frontal Medial Cortex vs. Insula | $0.789$ | $0.684$ | $1.000$ |

Frontal Medial Cortex vs. Subcallosal Cortex | $0.886$ | $0.831$ | $1.000$ |

Heschel’s Gyrus vs. Insula | $0.892$ | $0.851$ | $1.000$ |

Heschel’s Gyrus vs. Subcallosal Cortex | $0.817$ | $0.703$ | $1.000$ |

Insula versus Subcallosal Cortex | $0.693$ | $0.599$ | $1.000$ |

**Figure A3.**LFP time series related to two consecutive time stamps (1024 data points sampled at 1 KHz, approximately 1 sec) from three tetrodes and one electrode. From each of these time series, segments between time points 200 and 400, and between 415 and 600 were cut out and plotted in left and right panels, respectively, of Figure 9. These segments were analyzed for illustrating the dynamic correlation estimation methods.

**Figure A4.**Three plots selected from the pairwise plots in Figure 10. LFP pairs are shown in the left column, the resulting dynamic correlations computed by using the WGA method are shown in the right column. In all cases, the maximum correlation occurs at the time point corresponding to the stimulus artifact. Overall, the highest correlations resulted for the tetrodes 2 and 3 LFPs which are the most similar, while the dynamic correlation between the very different LFPs of tetrode 2 and electrode 4 were dominated by low values.

**Figure A5.**Three plots selected from the pairwise plots in Figure 11. LFP pairs are shown in the left column, the resulting dynamic correlations computed by using the WGA method are shown in the right column. The hippocampal LFPs become more similar after the 50th time point and the dynamic correlations in the top two panels reflect adequately this change. The LFP’s recorded in hippocampus and medial dorsal striatum are very different, a characteristic reflected in the low values of the corresponding dynamic correlation.

## Appendix B. Visibility Criteria

**Figure A6.**A time series and the corresponding visibility graph. ${t}_{1},{t}_{2}$ and so forth denote the time points as well as the corresponding nodes in the visibility graph.

#### Schematic Diagram

**Figure A7.**Schematic diagram illustrating the median-filtered weight vectors used in the algorithm presented in this paper.

## Appendix C. Theoretical Considerations

#### Appendix C.1. Heuristic Justification

#### Appendix C.2. A Theoretical Framework

**Lemma**

**A1.**

**Proof.**

**Lemma**

**A2.**

**Proof.**

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**Figure 1.**

**Top panel**: The two time series used for the first illustrative simulated example; data were generated from a bivariate normal distribution.

**Bottom Panel**—Dynamic correlations estimated for the above two time series using 3 different methods—method #1, sliding window (SW) (blue); method #2, dynamic conditional correlation (DCC) (green) (see also Figure A1); and method #3, weighted graph algorithm (WGA) (magenta). The red horizontal line in the center through zero represents the underlying true dynamic correlation for this example.

**Figure 2.**

**Top panel**: The two time series used for the second illustrative simulated example; data were generated from a bivariate Cauchy distribution.

**Bottom Panel**: Dynamic correlations estimated for the above two time series using various methods. Method #1, SW (blue), method #2, DCC (green), and method #3, based on WGA (magenta). The red horizontal line in the center through zero represents the underlying true dynamic correlation for this example.

**Figure 3.**Results for single iteration from simulations design D2. (

**Top row**): Underlying pair of time series from bivariate normal. (

**Bottom row**): Underlying pair of time series from bivariate Cauchy. (

**Left column**): Design 2a (that is, $k=3$). (

**Right column**): Design 2b ($k=4$). The underlying true dynamic correlations are plotted as the black curve. Green (SW), magenta (WGA) and red (DCC) represents the estimates of dynamic correlation.

**Figure 4.**Boxplots of mean squared error of the estimation methods based on 1000 iterations from simulations design D2. The panels correspond to bivariate Normal (

**a**,

**b**), bivariate Cauchy (

**c**,

**d**), design D2a (

**a**,

**c**) and design D2b (

**b**,

**d**), as in Figure 3.

**Figure 5.**Results for single iteration from simulations design D2. (

**Top row**): Underlying pair of time series from bivariate Normal. (

**Bottom row**): Underlying pair of time series from bivariate Cauchy. (

**Left column**): Design 3a (that is, $k=3$). (

**Right column**): Design 3b ($k=4$). The underlying true dynamic correlations are plotted as the black curve. Green (SW), magenta (WGA) and red (DCC) represents the estimates of dynamic correlation.

**Figure 6.**Boxplots of mean squared error of the estimation methods based on 1000 iterations from simulations design D3. The panels correspond to bivariate Normal (

**a**,

**b**), bivariate Cauchy (

**c**,

**d**), design D3a (

**a**,

**c**) and design D3b (

**b**,

**d**), as in Figure 5.

**Figure 8.**Pairwise dynamic correlation estimates for the Stroop task fMRI data. Green (SW), magenta (WGA), red (DCC).

**Figure 9.**Segments of local field potential (LFP) time series from four electrodes implanted in the CA1 field of the hippocampus (tetrodes 1-3) and medial dorsal striatum (electrode 4) in the brain of the same rat.

**Left column**: Segments between time points 200 and 400 from the respective time series plotted in appendix Figure A3.

**Right column**: segments between time points 415 and 600 of time series seen in Figure A3. 50

^{th}time point within these segments is marked by a blue vertical line.

**Figure 10.**Pairwise dynamic correlations for time series plotted in the left column panels in Figure 9. Green (SW), red (DCC) and magenta (WGA).

**Figure 11.**Pairwise dynamic correlations for time series plotted in the right column panels in Figure 9. Blue (SW), magenta (WGA), green (DCC). 50

^{th}time point within these segments is marked by a light blue vertical line.

**Figure 12.**Plots of WGA estimates and 95% confidence intervals for simulated data with D2b design, with bivariate normal (

**a**,

**b**) and bivariate Cauchy (

**c**,

**d**). Upper panels plot WGA estimate (black), 95% CI with ‘BootCI’ algorithm (blue) and with z-transformation (red). Lower panels plot WGA estimate and BootCI 95% CI (both blue), and corresponding estimates for randomly permuted series (green).

Mean of Absolute Value of Correlations Across Time | ||||
---|---|---|---|---|

T = 150 | T = 300 | T = 600 | T = 1000 | |

SW | $0.219\left(0.038\right)$ | $0.218\left(0.027\right)$ | $0.218\left(0.019\right)$ | $0.218\left(0.014\right)$ |

WGA | $0.134\left(0.033\right)$ | $0.129\left(0.022\right)$ | $0.127\left(0.015\right)$ | $0.126\left(0.012\right)$ |

DCC | $0.083\left(0.052\right)$ | $0.059\left(0.034\right)$ | $0.042\left(0.025\right)$ | $0.033\left(0.019\right)$ |

Maximum of Absolute Value of Correlations Across Time | ||||

T = 150 | T = 300 | T = 600 | T = 1000 | |

SW | $0.615\left(0.090\right)$ | $0.669\left(0.076\right)$ | $0.716\left(0.062\right)$ | $0.741\left(0.055\right)$ |

WGA | $0.394\left(0.076\right)$ | $0.424\left(0.066\right)$ | $0.456\left(0.059\right)$ | $0.477\left(0.052\right)$ |

DCC | $0.199\left(0.164\right)$ | $0.164\left(0.138\right)$ | $0.131\left(0.111\right)$ | $0.105\left(0.092\right)$ |

Mean of Absolute Value of Correlations Across Time | ||||
---|---|---|---|---|

T = 150 | T = 300 | T = 600 | T = 1000 | |

SW | $0.526\left(0.079\right)$ | $0.529\left(0.057\right)$ | $0.530\left(0.040\right)$ | $0.529\left(0.031\right)$ |

WGA | $0.241\left(0.097\right)$ | $0.220\left(0.070\right)$ | $0.209\left(0.047\right)$ | $0.203\left(0.036\right)$ |

DCC | $0.338\left(0.155\right)$ | $0.252\left(0.129\right)$ | $0.192\left(0.099\right)$ | $0.148\left(0.081\right)$ |

Maximum of Absolute Value of Correlations Across Time | ||||

T = 150 | T = 300 | T = 600 | T = 1000 | |

SW | $0.972\left(0.030\right)$ | $0.987\left(0.010\right)$ | $0.992\left(0.005\right)$ | $0.994\left(0.003\right)$ |

WGA | $0.535\left(0.117\right)$ | $0.552\left(0.100\right)$ | $0.578\left(0.086\right)$ | $0.593\left(0.073\right)$ |

DCC | $0.801\left(0.267\right)$ | $0.728\left(0.312\right)$ | $0.657\left(0.323\right)$ | $0.588\left(0.329\right)$ |

Mean Value of Correlations Across Time | |||
---|---|---|---|

SW | WGA | DCC | |

Amygdala vs. Caudate Head | $0.075$ | $0.031$ | $1.000$ |

Amygdala vs. Frontal Medial Cortex | $0.210$ | $0.173$ | $1.000$ |

Amygdala vs. Heschel’s Gyrus | $0.138$ | $0.117$ | $1.000$ |

Amygdala vs. Insula | $0.022$ | $-0.020$ | $1.000$ |

Amygdala vs. Subcallosal Cortex | $0.417$ | $0.378$ | $1.000$ |

Caudate Head vs. Frontal Medial Cortex | $0.280$ | $0.301$ | $1.000$ |

Caudate Head vs. Heschel’s Gyrus | $0.010$ | $0.039$ | $1.000$ |

Caudate Head vs. Insula | $0.214$ | $0.223$ | $1.000$ |

Caudate Head vs Subcallosal Cortex | $0.174$ | $0.155$ | $1.000$ |

Frontal Medial Cortex vs. Heschel’s Gyrus | $0.219$ | $0.275$ | $1.000$ |

Frontal Medial Cortex vs. Insula | $0.151$ | $0.179$ | $1.000$ |

Frontal Medial Cortex vs. Subcallosal Cortex | $0.440$ | $0.501$ | $1.000$ |

Heschel’s Gyrus vs. Insula | $0.619$ | $0.622$ | $1.000$ |

Heschel’s Gyrus vs. Subcallosal Cortex | $0.274$ | $0.313$ | $1.000$ |

Insula versus Subcallosal Cortex | $0.142$ | $0.142$ | $1.000$ |

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## Share and Cite

**MDPI and ACS Style**

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

**AMA Style**

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 Style**

John, 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