# A Greedy Pursuit Hierarchical Iteration Algorithm
## 4 Simulation Experiments  
**Experiment 1**. Consider a multi-input system as follows:  
\[\mathcal{A}(t)y(d) =\sum_{i=1}^{5}t^{\tau_{i}}\mathcal{B}_{i}(t)u_{i}(d)+\frac{ \mathcal{F}(t)}{\mathcal{C}(t)}e(d),\] \[\mathcal{A}(t) =1+a_{1}t^{-1}+a_{2}t^{-2}=1-0.80t^{-1}+0.60t^{-2},\] \[\mathcal{B}_{1}(t) =b_{11}t^{-1}+b_{12}t^{-2}=2.00t^{-1}-1.20t^{-2},\] \[\mathcal{B}_{2}(t) =b_{21}t^{-1}+b_{22}t^{-2}=-1.80t^{-1}-0.90t^{-2},\] \[\mathcal{B}_{3}(t) =b_{31}t^{-1}+b_{32}t^{-2}=1.00t^{-1}+0.50t^{-2},\] \[\mathcal{C}(t) =1+c_{1}t^{-1}=1+0.80t^{-1},\] \[\mathcal{F}(t) =1+f_{1}t^{-1}=1-0.40t^{-1}.\] \[\tau_{1} =6,\tau_{2}=15,\tau_{3}=20.\]  
Let \(l=30\); the true value of the parameter vector to be estimated is  
\[\boldsymbol{\theta}=[-0.80,0.60,\boldsymbol{0}_{6},2.00,-1.20,\boldsymbol{0}_ {37},-1.80,-0.90,\boldsymbol{0}_{33},1.00,0.50,\boldsymbol{0}_{8},0.80,-0.40]^ {\intercal}\in\mathbb{R}^{N}.\]  
The \(\{u_{i}(d)\}\) and \(\{y(d)\}\) are measurable. The \(e(d)\thicksim\mathcal{N}(0,\sigma^{2})\) is noise, and the noise variances are taken to be \(\sigma^{2}=0.50^{2}\) and \(\sigma^{2}=1.00^{2}\), respectively. Take the sampled data length \(L=1000\). The dimension of \(\boldsymbol{\theta}\) is \(N:=n_{a}+lr+n_{c}+n_{f}=94\), where the sparsity is \(K:=n_{a}+\frac{5}{i=1}\,n_{bi}+n_{c}+n_{f}=10\). Use the first 200 sampled data to estimate this experiment by applying the GPHI algorithm, and the data from \(d=200\) to \(d=500\) are taken to verify the effectiveness of the obtained results. Define parameter estimation error as \(\delta:=\|\hat{\boldsymbol{b}}_{k}-\boldsymbol{\theta}\|/\boldsymbol{\theta} \times 100\%\). Noise variance is a commonly utilized metric for quantifying noise in measurement data, with higher values indicating reduced accuracy and reliability of the measurements. The impact of noise variance on parameter estimation accuracy cannot be overlooked. To evaluate the robustness of the GPHI algorithm  
Figure 7: The true outputs, estimated outputs, and their bias of Experiment 1 with \(\sigma^{2}=0.50^{2}\). The dimension of sparse parameter vector is \(N:=n_{a}+lr+n_{c}+n_{f}=254\), where the number of key non-zero parameters is \(K:=n_{a}+\frac{5}{i=1}n_{bi}+n_{c}+n_{f}=14\). In this simulation, take the length of measurable input and output data \(L=1000\). The data from \(d=1\) to \(d=250\) are used to realize the parameters and time-delay estimation. The remaining data from \(d=250\) to \(d=500\) are used to contrast the bias between the true outputs and estimated outputs. The other constraints are similar with those in Experiment 1. To show the advantages of the GPHI algorithm, the errors \(\delta\) of the LSI and GPHI algorithms versus the iteration times \(k\) are depicted in Figure 9, where sampled data length \(L\) is 250 and noise variance is \(\  
\[\mathcal{A}(t) =1+a_{1}t^{-1}+a_{2}t^{-2}=1-0.80t^{-1}+0.60t^{-2},\]0 The estimation errors \(\delta\) versus \(k\) of Experiment 2 with \(L=250\) and \((\sigma^{2}=0.50^{2},\sigma^{2}=1.00^{2})\). \[\mathcal{A}(t) =1+a_{1}t^{-1}+a_{2}t^{-2}=1-0.80t^{-1}+0.60t^{-2},\]1 The true outputs, estimated outputs, and their bias of Experiment 2 with \(\sigma^{2}=0.50^{2}\). According to Table 6 and Equation (31), the time delays of Experiment 2 can be estimated:  
\[ \begin{split}\hat{\tau_{1}}&=n_{1}=9,\\ \hat{\tau_{2}}&=n_{2}-(l-\hat{\tau_{1}}-n_{b2})=62-(50-9- 2)=23,\\ \hat{\tau_{3}}&=n_{3}-(l-\hat{\tau_{2}}-n_{b3})=40-(50- 23-2)=15,\\ \hat{\tau_{4}}&=n_{4}-(l-\hat{\tau_{3}}-n_{b4})=63-(50- 15-2)=30,\\ \hat{\tau_{5}}&=n_{5}-(l-\hat{\tau_{4}}-n_{b5})=35-(50- 30-2)=17.\end{split}  \tag{33}\]  
Some results can be obtained from Figures 2-16 and Tables 1-6. * With the increase in noise variance, the error of estimated parameter vector becomes bigger, but the estimation accuracy is still higher. This shows that the GPHI algorithm has certain robustness--see Figures 2 and 9 and Tables 1 and 4.