# Multi-Source Heterogeneous Kernel Mapping in Software Defect Prediction
## 7 Discussion  
The MSHKM method proposed in this paper can effectively improve the performance of HDP. However, there are several problems and potential threats to the validity of our empirical study. Firstly, this article deals with the problem of heterogeneous feature space of different projects and proposes two strategies. However, two strategies have different time complexities for the MSHKM algorithm. Assuming there are \(n\) source projects, and each matrix has a dimension of \(m\times m\), the complexity for strategy 2 is O(\(nm^{3}\)), while strategy 1 requires O(\(n^{3}m^{3}\)). Therefore, for situations with fewer source projects, it is better to choose the more effective strategy, while, for situations with a large number of source projects, the time cost should be considered, and strategy 2 should be chosen as much as possible. Regardless \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Measure** & **Level** & **Against** & & & & \\ \hline  & & **WPDP** & **CCA+** & **HDP-KS** & **CTKCCA** & **MHCPDP** \\ \hline  & N & 3 & 0 & 0 & 6 & 0 \\ Pd & S & 0 & 0 & 1 & 1 & 0 \\  & M & 4 & 0 & 0 & 3 & 0 \\  & L & 21 & 28 & 27 & 18 & 28 \\ \hline  & N & 1 & 0 & 0 & 13 & 0 \\  & S & 2 & 0 & 0 & 0 & 0 \\ Pf & M & 3 & 0 & 0 & 1 & 0 \\  & L & 22 & 28 & 28 & 14 & 28 \\ \hline  & N & 4 & 0 & 0 & 1 & 0 \\  & S & 0 & 0 & 0 & 0 & 0 \\  & M & 2 & 0 & 1 & 2 & 0 \\  & L & 22 & 28 & 27 & 25 & 28 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The counts of effectiveness levels for MSHKM, with respect to Cliff’s delta test. \begin{table}
\begin{tabular}{c c} \hline \hline
**Cliff’s Delta (\(\delta\))** & **Effectiveness Levels** \\ \hline \(-1\leq\delta<0.147\) & Negligible (N) \\ \(0.147\leq\delta<0.33\) & Small (S) \\ \(0.33\leq\delta<0.474\) & Medium (M) \\ \(0.474\leq\delta\leq 1\) & Large (L) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Evaluation standard of Cliff’s delta. of which strategy is used, the algorithm's complexity will increase with an increase in the number of source projects, which is the main disadvantage of this paper. For performance measures, we used Pd (also known as recall), Pf, and GM to evaluate the performance of all methods, as they are commonly used performance measures in software defect prediction, especially on the baseline methods used in this paper. Additional comprehensive performance measures, such as AUC [9], Matthews correlation coefficient (MCC) [47; 48], and balance [49], were not used. However, we intend to use such measures for specific problems as future work, such as using AUC to evaluate classifiers without fixed thresholds. It should be noted that F1 was excluded, as it is biased and not suitable in the presence of the class imbalance problem [47; 48]. Similarly, precision is not used because it is also be affected by the imbalance of category distribution. In the case of class imbalance, the classifier may tend to predict the class with a large proportion, resulting in a high value of precision. Accuracy is also a metric leading to misleading results. For example, when 95% of samples are positive samples, as long as the classifier classifies all samples as positive samples, the accuracy can still be 0.95, but the performance of the model may not be good, as the model cannot predict negative samples. For the classifier, we employed logistic regression (LR) for all experiments in this paper, since it has been successfully and commonly used in prior HDP research [9; 14; 15; 16; 17; 34; 35]. Moreover, the LR classifier does not require parameter setting and hyperparameter adjustment, thus avoiding additional computational costs [43]. In addition, LR classification techniques tend to produce top-performing classifiers in software defect prediction [16; 50]. Despite the above reasons, other classifiers still need to be compared on the basis of the novel MSHKM algorithm. Therefore, the selection of a more suitable classifier for use with MSHKM will also be considered as future work.