# Data Fusion of Multivariate Time Series: Application to Noisy 12-Lead ECG Signals

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## Abstract

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## 1. Introduction

## 2. The Local Weighted Linear Prediction Algorithm

_{k}is the current state point of the chaotic system, and the future state X

_{k+}

_{1}of the system needs to be predicted. In LWLPA, the neighboring states $\{{X}_{ki},i=1,2,\cdots ,n\}$ of the current state point X

_{k}need to be chosen from the reconstructed trajectories. With the neighborhood X

_{ki}and the linear prediction model ${X}_{k+1}=ae+b{X}_{k}$, where $e={[1,\cdots ,1]}_{m}^{T}$, the future state X

_{k+}

_{1}can be approximately estimated.

## 3. The Novel Data Fusion Algorithm

#### 3.1. Basic Idea of Novel Data Fusion Algorithm

_{1}and L

_{2}, shown in Figure 1. In this example, suppose that the trajectory L

_{F}is the fused result of the trajectories L

_{1}and L

_{2}. Furthermore, the state point X

_{F}on the trajectory L

_{F}should satisfy the linear prediction model ${X}_{F}(p+1)=ae+b{X}_{F}(p)$, where $e={[1,\cdots ,1]}_{m}^{T}$. Here, how to obtain the parameters a and b in the linear prediction model is a critical problem.

_{1}(p) and X

_{2}(p) can be regarded as the neighboring vectors of the current state X

_{F}(p). With the two vectors being employed, we can calculate the parameters a and b by (5). The equation is as follows:

_{F}will be employed for the original signal quality assessment. It implies that to some extent, the characteristic information of original signal ought to be fused in the trajectory L

_{F}. Here, how to effectively inherit the characteristic information by fused result is a key for NDFA.

_{1}(p − 1) on the trajectory L

_{1}and the time-domain characteristic should be well inherited in the fused trajectory L

_{F}. Thus, based on Equation (1), a greater value for weighted coefficient ${\omega}_{1}$ of the state X

_{1}(p − 1) should be chosen, which will further enhance the impact on the final result. Here, the Euclidean distance of two neighboring points and the angle between two neighboring vectors on evolutionary trajectory L are used to estimate the weighted coefficients. In Figure 1, X

_{1}(p) and X

_{2}(p) are the p-step state points on the evolutionary trajectories L

_{1}and L

_{2}, respectively. The vector can be easily calculated through two adjacent state points, e.g., for the evolutionary trajectories L

_{1}, ${\stackrel{\rightharpoonup}{V}}_{p}^{1}={X}_{1}(p)-{X}_{1}(p-1)$, and then the angle ${\theta}_{p}^{1}$ can be obtained via the two neighboring vectors ${\stackrel{\rightharpoonup}{V}}_{p}^{1}$ and ${\stackrel{\rightharpoonup}{V}}_{p-1}^{1}$. The modulus of vector ${\stackrel{\rightharpoonup}{V}}_{p}^{1}$ is the Euclidean distance between X

_{1}(p) and X

_{1}(p − 1). For the lth evolutionary trajectory L

_{l}, the modulus of vector ${\stackrel{\rightharpoonup}{V}}^{l}$ and the angle ${\theta}^{l}$ can well reflect the evolutionary trend of the trajectory. In Figure 1, for the trajectory L

_{1}, the values of the modulus of ${\stackrel{\rightharpoonup}{V}}_{p}^{1}$, ${\stackrel{\rightharpoonup}{V}}_{p-1}^{1}$ and the vector angles ${\theta}_{p}^{1}$, ${\theta}_{p-1}^{1}$ are greater than the values of the trajectory L

_{2}at the same step. Characteristics of evolutionary trajectory can be described objectively by them. According to the idea, the change of evolutionary trajectory is positively related to the values of the two parameters. Based on the relationship, the weighted coefficient of the data point can be approximately estimated.

#### 3.2. Fuzzy Inference System Design for NDFA

_{d}and $\mathrm{FI}{\mathrm{S}}_{\alpha}$, will be devised, which can be used to estimate the weighted coefficients of LWLPA by the modulus of vector and the angle, respectively.

_{d}is applied to estimate the evolutionary trend of the reconstruction trajectory by the modulus D of the vector and the change rate D

_{r}of the modulus of adjacent vectors. Thus, in FIS

_{d}, there are two input variables D, D

_{r}and one output variable O

_{d}.

_{r}can be calculated as:

_{r}(p) are the Euclidean distance and the change rate of X(p) at p-step, respectively.

_{d}, the universe of the three variables D, D

_{r}and O

_{d}are set within the interval [0,1] uniformly. The universe of the variables D, D

_{r}and O

_{d}are divided into several fuzzy sets and the numbers of the fuzzy sets are 3, 3 and 5, respectively, shown in Figure 2a–c. According to the aforementioned relationship, the inference rules of FIS

_{d}can be designed properly, which are summarized in Table 1. Based on FIS

_{d}, the evolutionary trend of trajectory is estimated quantitatively and the parameter ${\omega}_{d}$ can be calculated as:

_{d}, ${y}_{d}(q)$ is the output of the qth rule and ${\beta}_{d}(q)$ the rule activation for the qth rule.

_{l}, the weighted coefficient ${\omega}_{l}$ in the Equation (1) can be computed as:

_{n}is the number of the phase trajectories and the parameter $\gamma $ is set to 1 [26]. Here, the minimum value ${\omega}_{\mathrm{min}}(p)=\mathrm{min}\{{\tilde{\omega}}_{s}(p),s=1,2,\cdots ,{L}_{n}\}$ needs to be selected.

#### 3.3. NDFA Algorithm

_{max}, delay time ${\tau}_{\mathrm{min}}$, initial condition X

_{F}(0), and for each lead of 12-lead ECG signals, construct the vector X

_{l}as:

_{l}is the evolutionary trajectory of the lth lead ECG signal on reconstructed trajectory. X

_{F}(0) is chosen as centroid of all the neighbor neighboring vectors ${X}_{s}\left(0\right)=(x(0),x(0+{\tau}_{\mathrm{min}}),\cdots ,x(0+({m}_{\mathrm{max}}-1){\tau}_{\mathrm{min}}))$, which is the point of the reconstruction trajectory of the sth lead ECG signal at p = 0.

_{l}, calculate the parameters D(p), D

_{r}(p), $\alpha (p)$ and ${\alpha}_{r}(p)$ by Equations (3), (4), (6), and (7), respectively. With the two FISs, ${\omega}_{d}(p)$ and ${\omega}_{\alpha}(p)$ being properly estimated, the weighted coefficient ${\omega}_{l}(p)$ of the state X

_{l}(p) at the p-step can be computed by Equation (10).

_{F}(p + 1) is calculated as:

## 4. Application of NDFA in 12-Lead ECG Signals

_{s}and delay time ${\tau}_{s}$.

#### 4.1. Synthetic Signals Experiments

#### 4.1.1. Ideal Synthetic Signals Experiments

_{x}, V

_{y}and V

_{z}, respectively. Figure 4d represents the fused signal of VCG signals s. From the morphology of reconstructed trajectory perspective, it is clear that there are needle-like features (Feature 1) on the three trajectories. Meanwhile, the longer closed trajectory (Feature 3) and disordered feature of the closed trajectory within a small space (Feature 2) are shown distinctly in Figure 4a–c. In Figure 4a, the local trajectories of Features 2 and 3 essentially reflect the P wave and QRS complexes in ECG signal, respectively. Evidently, the three features of original VCG signals are well described by the fused trajectory s.

#### 4.1.2. Noise Contaminated Synthetic Signals Experiments

_{x}of VCG signals is randomly chosen, which is contaminated by the noise. The parameters of the SNR levels are summarized in Table 3 [11].

_{x}is polluted by BW and the magnitudes of SNR are 12 dB, 6 dB, 0 dB, and −6 dB, respectively. In Table 4, correlation coefficient $Co{r}_{\overline{x}}$ is the mean value of Cor(Vx,Vy) and Cor(Vx,Vz). The parameter $Co{r}_{\overline{s}}$ is the mean value of Cor(s,Vx), Cor(s,Vy), and Cor(s,Vz). The values of parameters $Co{r}_{\overline{x}}$ and $Co{r}_{\overline{s}}$ reflect the degree of correlations between the original and fused signals. From Table 4, we can find that, under the different SNR levels, the correlation coefficient $Co{r}_{\overline{s}}$ is greater than $Co{r}_{\overline{x}}$ consistently, thereby designating the effectiveness of NDFA.

_{x}signal is polluted by the noise of EM and MA with different magnitudes of SNR and the correlations coefficients are calculated in Table 5 and Table 6. In the two tables, the relations between noisy signals and fused results are highly consistent with the relation reflected in Table 4.

#### 4.2. Realistic Signals Experiments

_{inv}is given by

#### 4.3. Performance Comparison of Data Fusion Algorithms

_{DTW}, and the smaller value of D

_{DTW}implies that the morphology features of two signals were somewhat similar. Figure 8a shows the similarity between the fused signals by four algorithms and the lead MLII. The means and variances of the parameter D

_{DTW}derived by NDFA, MDF, algorithm 1, and algorithm 2 were 0.0616, 0.0011; 0.8474, 0.0110; 0.2257, 0.0645; 0.2256, and 0.0644, respectively. Analogously, for fused signals and the lead V5, the means and variances of the parameter D

_{DTW}based on NDFA and MDF were 0.0921, 0.0017; 0.8768, 0.0102; 0.1959, 0.0426; 0.1958, and 0.0425 in Figure 8b. As can be seen, via NDFA, the means and the variances of the parameter D

_{DTW}were smaller, compared with the experiment results under the others. Figure 8 demonstrates that the characteristics of two leads ECG signals, MLII and V5, can be well reserved on the fused signal yielded from NDFA, and reflects that the performance of our algorithm exceeds MDF, algorithms 1 and 2.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The reconstructed phase trajectory L

_{1}and L

_{2}, and the fused trajectory L

_{F}to illustrate the basic idea of the novel data fusion algorithm (NDFA).

**Figure 2.**Partitioning results of variables with triangular membership function. (

**a**–

**f**) are partitioning results of D, D

_{r}, O

_{d}, α, α

_{r}and ${O}_{\alpha}$ respectively.

**Figure 4.**Reconstructed phase trajectories of synthetic ECG signals. (

**a**–

**c**) are reconstructed phase trajectories of V

_{x}, V

_{y}and V

_{z}, respectively. (

**d**) Is reconstructed phase trajectory of final fused result via NDFA.

**Figure 6.**Three-lead VCG signals by realistic 12-lead ECG signals transformation and reconstructed trajectories of fused results. (

**a**–

**d**) are results of the signal of No.1027085, No.1075113, No.1027085, and No.1075113, respectively.

**Figure 7.**Reconstructed phase trajectories of ECG signals (No.106). (

**a**) and (

**b**) are reconstructed phase trajectories of lead MLII and lead V5, respectively. (

**c**) is reconstructed phase trajectory of final fused result via NDFA.

**Figure 8.**Performance comparison of NDFA, MDF, Algorithms 1 and 2 by DTW. (

**a**) Is the comparison for the fused signals by four algorithms and the lead MLII. (

**b**) Is the comparison for the fused signals by four algorithms and the lead V5.

**Table 1.**Inference rules of fuzzy inference system (FIS

_{d}) on the evolutionary trend of reconstruction trajectory.

D | ||||

S_{D} | M_{D} | B_{D} | ||

${\mathit{S}}_{{\mathit{D}}_{\mathit{r}}}$ | ${\mathit{S}}_{{\mathit{O}}_{\mathit{d}}}$ | $\mathit{S}{\mathit{R}}_{{\mathit{O}}_{\mathit{d}}}$ | ${\mathit{M}}_{{\mathit{O}}_{\mathit{d}}}$ | |

D_{r} | ${\mathit{M}}_{{\mathit{D}}_{\mathit{r}}}$ | $\mathit{S}{\mathit{R}}_{{\mathit{O}}_{\mathit{d}}}$ | ${\mathit{M}}_{{\mathit{O}}_{\mathit{d}}}$ | $\mathit{B}{\mathit{R}}_{{\mathit{O}}_{\mathit{d}}}$ |

${\mathit{B}}_{{\mathit{D}}_{\mathit{r}}}$ | ${\mathit{M}}_{{\mathit{O}}_{\mathit{d}}}$ | $\mathit{B}{\mathit{R}}_{{\mathit{O}}_{\mathit{d}}}$ | ${\mathit{B}}_{{\mathit{O}}_{\mathit{d}}}$ |

**Table 2.**Inference rules of fuzzy inference system ($\mathrm{FI}{\mathrm{S}}_{\alpha}$) on the angle of adjacent vectors.

$\mathit{\alpha}$ | ||||||

$\mathit{N}{\mathit{B}}_{\mathit{\alpha}}$ | $\mathit{N}{\mathit{M}}_{\mathit{\alpha}}$ | ${\mathit{Z}}_{\mathit{\alpha}}$ | $\mathit{P}{\mathit{M}}_{\mathit{\alpha}}$ | $\mathit{P}{\mathit{B}}_{\mathit{\alpha}}$ | ||

${\mathit{S}}_{{\mathit{\alpha}}_{\mathit{r}}}$ | $\mathit{V}{\mathit{B}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | ${\mathit{B}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | $\mathit{B}{\mathit{R}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | $\mathit{M}{\mathit{R}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | ${\mathit{M}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | |

${\mathit{\alpha}}_{\mathit{r}}$ | ${\mathit{M}}_{{\mathit{\alpha}}_{\mathit{r}}}$ | ${\mathit{B}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | $\mathit{B}{\mathit{R}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | $\mathit{M}{\mathit{R}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | ${\mathit{M}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | $\mathit{S}{\mathit{R}}_{{\mathit{o}}_{\mathit{\alpha}}}$ |

${\mathit{B}}_{{\mathit{\alpha}}_{\mathit{r}}}$ | $\mathit{B}{\mathit{R}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | $\mathit{M}{\mathit{R}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | ${\mathit{M}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | $\mathit{S}{\mathit{R}}_{{\mathit{o}}_{\mathit{\alpha}}}$ | ${\mathit{S}}_{{\mathit{o}}_{\mathit{\alpha}}}$ |

**Table 3.**SNR magnitudes for noise, baseline wander (BW), electrode movement (EM) and muscle artifact (MA).

SNR Levels (dB) | ||||
---|---|---|---|---|

BW | 12 | 6 | 0 | −6 |

EM | 6 | 0 | −6 | −12 |

MA | 12 | 6 | 0 | −6 |

SNR (dB) | 12 | 6 | 0 | −6 |
---|---|---|---|---|

$Co{r}_{\overline{x}}$ | 0.3449 | 0.3103 | 0.2340 | 0.1368 |

$Co{r}_{\overline{s}}$ | 0.5369 | 0.5329 | 0.5177 | 0.4566 |

SNR (dB) | 6 | 0 | −6 | −12 |
---|---|---|---|---|

$Co{r}_{\overline{x}}$ | 0.3281 | 0.2598 | 0.1665 | 0.0944 |

$Co{r}_{\overline{s}}$ | 0.5347 | 0.5156 | 0.4728 | 0.4080 |

SNR (dB) | 12 | 6 | 0 | −6 |
---|---|---|---|---|

$Co{r}_{\overline{x}}$ | 0.3498 | 0.3192 | 0.2476 | 0.1539 |

$Co{r}_{\overline{s}}$ | 0.5439 | 0.5221 | 0.4970 | 0.4379 |

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**MDPI and ACS Style**

Diao, C.; Wang, B.; Cai, N.
Data Fusion of Multivariate Time Series: Application to Noisy 12-Lead ECG Signals. *Appl. Sci.* **2019**, *9*, 105.
https://doi.org/10.3390/app9010105

**AMA Style**

Diao C, Wang B, Cai N.
Data Fusion of Multivariate Time Series: Application to Noisy 12-Lead ECG Signals. *Applied Sciences*. 2019; 9(1):105.
https://doi.org/10.3390/app9010105

**Chicago/Turabian Style**

Diao, Chen, Bin Wang, and Ning Cai.
2019. "Data Fusion of Multivariate Time Series: Application to Noisy 12-Lead ECG Signals" *Applied Sciences* 9, no. 1: 105.
https://doi.org/10.3390/app9010105