# A Greedy Pursuit Hierarchical Iteration Algorithm
## 3 Greedy Pursuit Hierarchical Iterative Parameter Estimation Algorithm
The above steps from Equations (12)-(17) cannot be directly implemented since \(\mathbf{\Phi}\) contains some unknown items according to Equation (2)._  
Replace the true unmeasurable values \(w(d)\) and \(e(d)\) with their last iteration estimate by using the hierarchical principle [29]. Let \(k\) be the external iteration variable. \(\hat{\mathbf{\theta}}_{k},\hat{\mathbf{\theta}}_{k,s},\hat{\mathbf{\theta}}_{k,n}\) are iteration estimates of \(\mathbf{\theta},\mathbf{\theta}_{s},\mathbf{\theta}_{n}\), respectively. By replacing \(d\) with \(d-j\) in Equations (2) and (8), the following results can be obtained:  
\[w(d-j)=\mathbf{\varphi}_{n}^{\mathrm{r}}(d-j)\mathbf{\theta}_{n}+e(d-j), \tag{18}\]  
\[y(d-j)=\mathbf{\varphi}_{s}^{\mathrm{r}}(d-j)\mathbf{\theta}_{s}+w(d-j). \tag{19}\]  
Then, the unknown noise items \(w(d-j),e(d-j)\) can be replaced by their last estimate \(\hat{w}_{k-1}(d-j),\hat{e}_{k-1}(d-j)\) based on Equation (3) and the hierarchical principle,  
\[ \begin{split}\hat{\mathbf{\varphi}}_{k,n}(d)&=[-\hat{w }_{k-1}(d-1),-\hat{w}_{k-1}(d-2),\cdots,-\hat{w}_{k-1}(d-n_{c}),\\ \hat{e}_{k-1}(d-1),\hat{e}_{k-1}(d-2),\cdots,\hat{e}_{k-1}(d-n_{f })]^{\mathrm{r}}\in\mathbb{R}^{n_{c}+n_{f}}.\end{split}  \tag{20}\]  
Replace \(\mathbf{\varphi}_{n}^{\mathrm{r}}(d-j),\mathbf{\theta}_{n},\mathbf{\theta}_{s}\) with their iteration estimates \(\hat{\mathbf{\varphi}}_{k,n}^{\mathrm{r}}(d-j),\hat{\mathbf{\theta}}_{k,n},\hat\mathbf{{\theta}}_{k,s}\) in Equations (18) and (19); the estimation values of \(w(d-j)\) and \(e(d-j)\) at \(k\)-th iteration can be achieved by  
\[\hat{w}_{k}(d-j)=y(d-j)-\mathbf{\varphi}_{k,s}^{\mathrm{r}}(d-j)\hat{\mathbf{\theta}} _{k,s}, \tag{21}\]  
\[\hat{e}_{k}(d-j)=\hat{w}_{k}(d-j)-\hat{\mathbf{\varphi}}_{k,n}^{\mathrm{r}}(d-j) \hat{\mathbf{\theta}}_{k,n}. \tag{22}\]  
According to Equation (20), the estimation of information matrix \(\hat{\mathbf{\Phi}}_{k}\) can be denoted at the \(k\)-th external iteration, where \(\hat{\mathbf{\phi}}_{k}(d)=[\mathbf{\varphi}_{s}^{\mathrm{r}}(d),\hat{\mathbf{\varphi}}_{k,n}^{\mathrm{r}}(d)]^{\mathrm{r}}\). Use \(\hat{\mathbf{\Phi}}_{k}\) to replace \(\mathbf{\Phi}\) in Equation (9). Construct the normalized transformation matrix \(\hat{\mathbf{T}}\) using \(\hat{\mathbf{\Phi}}_{k}\) based on Equation (12), and construct \(\mathbf{\Psi}_{k}\) according to \(\hat{\mathbf{\Phi}}_{k}\) and \(\hat{\mathbf{T}}\). Then, Equations (14) and (15) can be expressed as  
\[\mathbf{r}_{k,q}=\mathbf{Y}-\hat{\mathbf{\Psi}}_{k,\Lambda_{q}}\hat{\mathbf{\vartheta}}_{k, \Lambda_{q}}, \tag{23}\]  
\[\lambda_{k,q}=\arg\max_{j=1,2,\cdots,N}|\langle\mathbf{r}_{k,q-1},\hat{\mathbf{\psi}}_ {j}\rangle|, \tag{24}\]where the double subscripts \(k,q\) denote that the key index at \(q\) inner iteration can be found using the \(k\)th external iteration updated data. Applying the index set to update the normalization sub-matrix, we have  
\[\hat{\boldsymbol{\Psi}}_{k,\Lambda_{q}}=\hat{\boldsymbol{\Psi}}_{k,\Lambda_{q-1 }}\cup\hat{\boldsymbol{\psi}}_{k,\Lambda_{q}},\quad\Lambda_{q}=\Lambda_{q-1} \cup\lambda_{q}. \tag{25}\]  
Define a cost function as follows at the \(q\)th inner iteration to find the best parameter estimation value  
\[J(\boldsymbol{\vartheta}_{k,\Lambda_{q}})=\|\boldsymbol{r}_{k,q}\|^{2}=\| \boldsymbol{Y}-\hat{\boldsymbol{\Psi}}_{k,\Lambda_{q}}\boldsymbol{\vartheta}_ {k,\Lambda_{q}}\|^{2}. \tag{26}\]  
Minimize Equation (26), and the LS estimate can be obtained  
\[\hat{\boldsymbol{\vartheta}}_{k,\Lambda_{q}}=[\hat{\boldsymbol{\Psi}}_{k, \Lambda_{q}}^{\mathrm{T}}\hat{\boldsymbol{\Psi}}_{k,\Lambda_{q}}]^{-1}\hat{ \boldsymbol{\Psi}}_{k,\Lambda_{q}}^{\mathrm{T}}Y. \tag{27}\]  
Note that \(\hat{\boldsymbol{\vartheta}}_{k,\Lambda_{q}}\in\mathbb{R}^{q}\), using \(\Lambda_{k,q}\) and the transformation matrix \(\hat{T}\), the over-parameterized sparse parameter vector can be obtained. Then, the steps of the greedy pursuit hierarchical iterative (GPHI) algorithm are listed as follows. 1.