# An Effective Adaptive Combination Strategy for Distributed Learning Network
## 5 Simulation Results
We now consider target tracking model ([17], Equation (52)), namely  
\[\mathbf{w}_{o}(t) =\mathbf{w}_{o}+\mathbf{\theta}(t) \tag{29}\] \[\mathbf{\theta}(t) =0.99\mathbf{\theta}(t-1)+\mathbf{\zeta}(t) \tag{30}\]  
where \(\mathbf{\zeta}(t)\) is a sequence of independent identically distributed perturbations with zero mean and covariance matrix \(\Xi\), independent of the input regressors and measurement noise at every iteration. We herein consider \(\mathbf{\zeta}(t)\) is Gaussian white with \(\Xi=\sigma_{\xi}^{2}\mathbf{I}_{M}\) and \(\sigma_{\xi}^{2}=1\times 10^{-7}\), i.e., the unknown weight vector is varying slowly. The other simulation conditions are same as Example 1. As illustrated in Figure 4, similar to Example 1, the proposed AC strategy outperforms the uniform combination rule, the classic AC and LS-based AC in terms of superior steady-state performance, and outperforms the optimal AC in terms of accelerated convergence rate, under tracking scenarios. We could also observe that the forgetting factor can further enhance the performance of the proposed AC in the tracking scenario. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Noise \(\mathbf{\tau}\)** & **Combiners** & **Uniform** & **Classic AC** & \(\mathbf{\lambda=0}\) & \(\mathbf{\lambda=0.95}\) \\ \hline \(\tau=0.5\) & \(-49.0\) & \(-50.2\) & \(-51.2\) & \(-53.0\) \\ \(\tau=1\) & \(-46.0\) & \(-48.0\) & \(-49.1\) & \(-50.0\) \\ \(\tau=1.5\) & \(-44.3\) & \(-46.6\) & \(-47.7\) & \(-48.3\) \\ \(\tau=2\) & \(-43.3\) & \(-45.9\) & \(-46.9\) & \(-47.1\) \\ \(\tau=2.5\) & \(-42.1\) & \(-45.1\) & \(-46.1\) & \(-46.1\) \\ \(\tau=3\) & \(-41.2\) & \(-44.3\) & \(-45.2\) & \(-45.3\) \\ \(\tau=3.5\) & \(-40.6\) & \(-43.7\) & \(-44.4\) & \(-44.4\) \\ \(\tau=4\) & \(-40.2\) & \(-43.5\) & \(-44.1\) & \(-44.0\) \\ \(\tau=4.5\) & \(-39.6\) & \(-42.9\) & \(-43.5\) & \(-43.3\) \\ \(\tau=5\) & \(-39.2\) & \(-42.7\) & \(-43.3\) & \(-43.2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The steady-state MSD with respect to the variation of the noise power. Figure 3: The MSD learning curves under different noise levels. **Example 4**. : _We now consider the impact of the factors \(\lambda\) and \(\gamma\) in the performance of the proposed AC strategy. Without loss of generality, we consider the step sizes \(\mu_{k}=0.02\) for each node \(k\). The simulation result is shown in Figure 5._  
It can be seen from Figure 5 that with specific \(\gamma\), the larger \(\lambda\) brings the higher performance gain for the proposed AC. We could also observe that choosing \(\gamma=-2\) brings better steady-state performance than the other two choices, especially for small forgetting factor \(\lambda\). In particular, as compared to Example 1, for the two scenarios \(\gamma=0,\lambda=0\) and \(\gamma=0,\lambda=0.95\), we find that larger step sizes in this example lead to accelerated convergence rate at the cost of degraded steady-state performance. Figure 4: The MSD learning curves under tracking scenarios. Figure 5: The MSD learning curves  
**Example 5**. : _We consider the sparse network with \(N=15\) nodes with its topology same as ([12], Figure 8), depicted in Figure 6a. The measurement noise is real Gaussian white noise with its variance \(\sigma_{v,k}^{2}\) at each node presented in Figure 6c. We consider the heterogenous network with the step sizes \(\mu_{k}=0.004\) for orange shaded nodes and \(\mu_{n}=0.02\) for the rest. Each regressor \(\mathbf{u}_{k}(t)\) at each node \(k\) is real Gaussian white noise sequence with their covariance matrices \(\mathbf{R}_{u,k}=\sigma_{u,k}^{2}\mathbf{I}_{M}\) with \(\sigma_{u,k}^{2}\) presented in Figure 6b for all \(k\)._  
As illustrated in Figure 7, for the sparse network, the proposed AC scheme outperforms the classic AC and LS-based AC in terms of superior steady-state performance while keeping rather approximate convergence rate. It also outperforms the optimal AC scheme in terms of the accelerated convergence rate.