# Data Fusion of Multivariate Time Series: Application to Noisy 12-Lead ECG Signals
## 2 The Local Weighted Linear Prediction Algorithm  
In the course of signal quality assessment of 12-lead ECG signals, compressing the pending ECG data is effective for further improving the efficiency of the assessment algorithms. Since cardiac signals reveal the possibility of deterministic chaos, here LWLPA--an important prediction method of chaotic time series--was used to fuse 12-lead ECG signals. In this section, as a preliminary, we briefly review the algorithm, which is closely related to NDFA.  
For the chaotic time series \(\{x(i),i=1,2,\cdots,N\}\), the dynamics characteristics of the chaotic time series can be well reflected in \(m\)-dimensional space through vectors \(X_{t}=(x(t),x(t+\tau),\cdots,x(t+(m-1)\tau))\in R^{m}\), \(t=1,2,\cdots,N-(m-1)\tau\), where \(m\) is the embedding dimension and \(\tau\) the delay time. [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON] [PERSON].  
From the point of LWLPA, in Figure 1, the vectors \(X_{1}(p)\) and \(X_{2}(p)\) can be regarded as the neighboring vectors of the current state \(X_{\text{F}}(p)\). With the two vectors being employed, we can calculate the parameters \(a\) and \(b\) by (5). The equation is as follows:  
\[g(a,b)=\min_{a,b\in R}\Bigl{\{}\sum_{l=1}^{2}\omega_{l}[X_{l}(p+1)-ae-bX_{l}(p )]^{2}\Bigr{\}}, \tag{1}\]  
where \(\omega_{l}\) is the weighted coefficient, which reflects the degree of impact from the state point to the fusion result. Then the fused state can be calculated as:  
\[X_{F}(p+1)=ae+bX_{F}(p) \tag{2}\]  
The fused trajectory \(L_{\text{F}}\) 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 \(L_{\text{F}}\). Here, how to effectively inherit the characteristic information by fused result is a key for NDFA.  
In Figure 1, there is an evident amplitude of the point \(X_{1}(p-1)\) on the trajectory \(L_{1}\) and the time-domain characteristic should be well inherited in the fused trajectory \(L_{\text{F}}\). 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 \(L_{1}\), \(\overset{\rightharpoonup 1}{V}_{p}=X_{1}(p)-X_{1}(p-1)\), and then the angle \(\theta_{p}^{1}\) can be obtained via the two neighboring vectors \(\overset{\rightharpoonup 1}{V}_{p}\) and \(\overset{\rightharpoonup 1}{V}_{p-1}\). The modulus of vector \(\overset{\rightharpoonup 1}{V}_{p}\) is the Euclidean distance between \(X_{1}(p)\) and \(X_{1}(p-1)\). For the \(l\)th evolutionary trajectory \(L_{l}\), the modulus of vector \(\overset{\rightharpoonup l}{V}\) 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 \(\overset{\rightharpoonup 1}{V}_{p}\), \(\overset{\rightharpoonup 1}{V}_{p-1}\) 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.  
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).