## 3 Experimental Results and Analysis
### Analysis of Algorithm for Reconstructing Fingerprint Database  
1. The localization of fingerprint databases without a reconstruction algorithm (IRS-URFL) and the complexity of constructing database fingerprints are as follows: In the fingerprint database construction phase, when the matrix reconstruction algorithm is not used to construct the fingerprint database \(\mathbf{F}^{\prime}\), the total number of elements in the fingerprint database \(\mathbf{F}^{\prime}\) is \(NMW\), and the generation of each fingerprint data requires approximately \(W\cdot K\cdot L\cdot times\) floating-point computations, where \(times\) is the number of times each fingerprint datum is sampled. Therefore, the number of floating-point calculations for the process is approximated as \(O\big{(}W^{2}\cdot K\cdot N\cdot M\cdot L\cdot times\big{)}\). 2. Reconstructing the fingerprint database using the LMaFit algorithm for the indoor radio signal-based robust fingerprint localization (IRS-RFL) algorithm, the complexity of constructing database fingerprints is as follows: When using the matrix reconstruction algorithm for constructing the fingerprint database, we first collect 60% of the fingerprint database \(\mathbf{F}^{\prime}\), and the computational complexity required for this process is about \(O\big{(}0.6W^{2}\cdot K\cdot N\cdot M\cdot L\cdot times\big{)}\). Then, we use the LMaFit algorithm to complement the remaining 40% of the fingerprint data, and in this way, we reconstruct to obtain the complete fingerprint database \(\mathbf{X}^{\prime}\). The floating-point operations of the LMaFit algorithm are divided into two main parts: matrix factorization and matrix data reconstruction through multiple iterations. In this study, firstly, the number of floating- point operations in the matrix factorization part is \(O\big{(}N(MW)^{2}\big{)}\). Secondly, the part of the matrix data reconstruction through multiple iterations mainly involves calculating the update parameters and performing matrix multiplication operations, so the number of floating-point operations in this part is \(O\big{(}\text{step}\left(N(MW)^{2}\right)\big{)}\), where \(step\) is the number of iterations. Thus, the number of floating-point computations using the LMaFit matrix reconstruction algorithm can be approximated as \(O\Big{(}N(MW)^{2}+\text{step}\left(N(MW)^{2}\right)\Big{)}\). Therefore, the total number of floating-point operations when using the matrix reconstruction algorithm for constructing the fingerprint database \(\mathbf{X}^{\prime}\) is approximated by \(O\Big{(}0.6W^{2}\cdot K\cdot N\cdot M\cdot L\cdot times+N(MW)^{2}+\text{ step}\Big{(}N(MW)^{2}\Big{)}\Big{)}\). 3. Complexity comparison: Therefore, theoretically, \(M=5\), \(W=8\), \(K=8\), \(L=3\), \(times=21\), \(step=35\). At this time, the computational complexity of the IRS-URFL algorithm and IRS-RFL algorithm is shown below. In Figure 5, the horizontal coordinates represent the number of rows in the matrix, and the vertical coordinates represent the computational complexity of the algorithms; the blue line represents the computational complexity of constructing the fingerprint database when the localization (IRS-URFL) algorithm for fingerprint databases is used without \begin{table}
\begin{tabular}{c c c c c c c c} \hline  & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** \\ \hline
1 & 0 & 0 & 0 & 0 & 0 & 0 & 1.62\% & 0 \\
2 & 0 & 0.76\% & 0 & 0 & 0 & 0 & 0 \\
3 & 1.68\% & 0 & 1.38\% & 0.31\% & 0 & 0 & 0.48\% & 2.93\% \\
4 & 0 & 0 & 0 & 0 & 0 & 0 & 2.47\% & 1.48\% \\
5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.9\% \\
6 & 0 & 3.27\% & 0\% & 0.29\% & 0 & 0 & 0.88\% & 1.59\% \\ \hline \end{tabular}
\end{table} 
Table 4: Relative error data. the reconstruction algorithm; and the red line represents the computational complexity of constructing the fingerprint database when the localization (IRS-RFL) algorithm for fingerprint databases is reconstructed using the low-rank matrix fitting (LMaFit) algorithm. The computational complexity of the fingerprint database was also considered.