# Adaptive Bat Algorithm Optimization Strategy for Observation Matrix
## 3 Adaptive Bat Algorithm Optimizes Observation Matrix
### The Design of Adaptive Bat Algorithm Optimal Observation Matrix  
In this section, the flow chart of the adaptive bat algorithm based on observation matrix optimization is described. The adaptive bat algorithm, to realize the observation matrix optimization search method by dynamically adjusting the size of the optimization radius, is as shown in Algorithm 1.  
```
Input: The bat's loudness \(A\), flight frequency \(f\) and pulse transmission frequency \(r\), the maximum iterations number is max. Initialization parameters; While\(t<\) max  Calculate the fitness value by Equation (6) and find the best individual;  Select the optimization center \(f_{x_{plut}}\) and calculate the radius with Equation (7);  Update new position of bats individual; if\(\left|f_{x_{1}^{\prime}}-f_{x_{plut}^{\prime}}\right|\leq radius\)  Update the velocity and position with Equations (4) and (5); else  Individual escapes from the search center and randomly generates new positions; endif  Local disturbance: For each individual, generate a random number \(rand1\); if\(rand1>r_{i}^{\prime}\)  The position of the bat is disturbed near the optimal individual with Equation (9): else  Calculate the fitness value of the new position with Equation (6); endif  Generate random numbers \(rand2\) for each bat individual; if\(rand2<A_{i}^{l}\) and \(f(x_{input}^{l})<f(x_{i}^{l})\)  Accept new solution and update loudness and pulse frequency with Equations (10) and (11); else  Update the global optimal solution with Equation (5); endif endwhile Output: the observation matrix and the optimal fitness value (minimum reconstruction error).
```  
**Algorithm 1**Adaptive Bat AlgorithmAmong them, the position of the bat individual is disturbed near the optimal individual with Equation (9):  
\[x_{i\,new}^{t}=x_{gbest}+\rho\overline{A^{t}} \tag{9}\]  
where \(\rho\in[-1,\,1]\), \(\overline{A^{t}}\) represents the average pulse loudness of bat individual at time \(t\).  
And then, the updated pulse frequency and loudness equation are as follows:  
\[r_{i}^{t+1}=r_{i}^{t}(1-e^{-\gamma t}) \tag{10}\]  
\[A_{i}^{t+1}=\psi A_{i}^{t} \tag{11}\]  
where \(\gamma\) represents the growth rate of pulse transmission frequency, and the value is an integer with greater than 0, \(\psi\) represents the decrease rate of the pulse frequency.  
Algorithm 2 describes the process of the proposed adaptive bat algorithm for signal reconstruction based on transform base and OMP algorithm. Among them, the main process of reconstructing is introduced in Algorithm 2.  
```
Input: Original signal: \(X\), sparse transformation base: \(\Psi\). Initialization: Generate \(p\) column of random Gaussian distribution with size of \(M\times N\) dimension as the bat's initialization velocity \(v_{i}^{0}\) and position \(x_{i}^{0}\);  Select sparse transform base and sparse representation signal \(X\) with Equation (1);  Generate \(p\) random Gaussian observation matrices as the initial velocity and position;  Turn the position of each bat individual \(x_{i}^{t}\) to a matrix \(\overline{\Phi_{l}^{t}}\) of \(M\times N\);  While\(t<\) max  Compress observation original signal \(X\) with \(\overline{\Phi_{l}^{t}}\) matrix;  Generate observation signals with Equation (2);  The reconstructed signal is obtained by the OMP reconstruction algorithm (Algorithm 2); end(While) Output: The observation matrix and the optimal fitness value (minimum reconstruction error).
```  
**Algorithm 2** Reconstruction Process