## 4 Experiments and Results
### Results on Real-World Datasets  
The results of different algorithms on 18 real-world datasets are shown in Table 2, and the optimal aRRMSE values on different datasets have been indicated in bold. Among them, the mSVR and the SST can be used as benchmark algorithms for two types of MTR, the algorithm adaptation method, and the problem transformation method, respectively. mSVR can output multiple targets at the same time, but it only integrates the errors of different output dimensions through \(l_{2}\)-norm regularization without considering the correlation of different outputs, so its performance on the datasets is inferior to other algorithm adaptation methods (MMR, MSLR, OKL and MTR-MLS). The SST explores multi-target correlations by stacking the outputs, which is difficult to explore the correlations among multiple targets and is sensitive to the stacking order of the targets, so its performance is not as good as ERC and RLC that employ an integration approach or random linear target combinations. Moreover, the three MTR algorithms (MMR, MSLR and MTR-MLS) with multi-layer sparse latent variable regression structure maintain good performance on different data, which shows the adaptability of the structure to MTR problems in different domains. Meanwhile, to verify whether there is a significant difference between the performance of the proposed method and the other compared methods on the dataset, the best aRRMSE values of the feature selection methods on different datasets are ranked, and the average rank of different methods on all datasets is calculated. We execute the _Friedman test_[40] with the significant level \(\alpha=0.05\). The null hypothesis states that all methods are statistically equivalent in performance between our proposed method and the compared algorithms. If such a null hypothesis is rejected, we further utilize the _Bonferroni-Dunn test_[41] as a post-hoc test to further analysis of the comparison. The critical difference (CD) is calculated to measure the difference between the proposed method and other algorithms, corresponding to the minimum difference in average ranks required for two methods to be considered significantly different. The calculation of CD is as follows:  
\[CD=q_{\alpha}\sqrt{\frac{n(n+1)}{6T}} \tag{24}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Datasets** & **SST** & **RLC** & **ERC** & **OKL** & **mSVR** & **MMR** & **MSLR** & **MTR-MLS** \\ \hline ANDRO & 57.9 & 57.0 & 56.7 & 55.3 & 62.7 & 52.7 & 51.5 & **49.5** \\ ATP1D & 37.2 & 38.4 & 37.2 & 36.4 & 38.1 & **33.2** & 35.2 & 38.2 \\ ATP7D & 50.7 & 46.1 & 51.2 & 47.5 & 47.8 & 44.3 & 45.8 & **43.7** \\ EDM & 74.7 & 73.5 & 74.1 & 74.1 & 73.7 & 71.6 & 68.6 & **67.3** \\ ENB & 12.1 & 12.0 & 11.4 & 13.8 & 22.0 & **11.1** & 19.4 & **11.1** \\ JURA & 59.1 & 59.6 & 59.0 & 59.9 & 61.1 & **58.2** & 62.3 & 60.8 \\ OES10 & 42.1 & 41.9 & 42.0 & 43.2 & 44.7 & **40.3** & 40.7 & **40.3** \\ OES97 & 52.6 & 52.3 & 52.4 & 53.5 & 55.7 & 49.7 & 48.9 & **46.0** \\ OSALES & 72.6 & 74.1 & **71.3** & 71.8 & 77.8 & 74.5 & 73.6 & **71.3** \\ RF1 & 69.8 & 72.7 & 69.9 & 81.4 & 75.4 & 73.1 & 66.3 & **65.3** \\ RF2 & 69.9 & 70.4 & 69.8 & 81.8 & 83.6 & 74.6 & 67.1 & **66.7** \\ SCM1D & 47.0 & 45.7 & 46.6 & 47.6 & 54.3 & 44.7 & 43.3 & **42.2** \\ SCM20D & 77.7 & 74.7 & 76.0 & 76.4 & 76.3 & 75.8 & 74.0 & 73.7 \\ SCPF & 83.1 & 83.5 & 83.0 & 82.0 & 82.8 & **81.2** & 81.7 & 83.3 \\ SF1 & 106.8 & 116.3 & 108.8 & 105.9 & 102.1 & **95.8** & 104.2 & 105.4 \\ SF2 & 116.7 & 119.4 & 113.6 & 100.4 & 104.3 & **98.4** & 105.8 & 99.3 \\ SLUMP & 69.5 & 69.0 & 69.0 & 69.9 & 71.1 & 58.7 & **56.7** & 57.7 \\ WQ & 90.9 & 90.2 & 90.6 & 89.1 & 89.9 & 88.9 & 89.3 & **88.5** \\ AveRank & 5.50 & 5.11 & 4.72 & 5.22 & 6.17 & 2.67 & 3.00 & **2.17** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The comparison with state-of-the-art algorithms on 18 real-world datasets in terms of aRRMSE (%). where \(n\) is the number of algorithms compared, and \(T\) is the number of datasets.