An important research area of Spectrum-Based Fault Localization (SBFL) is the effectiveness of risk evaluation formulas. Most previous studies have adopted an empirical approach, which can hardly be considered as sufficiently comprehensive because of the huge number of combinations of various factors in SBFL. Though some studies aimed at overcoming the limitations of the empirical approach, none of them has provided a completely satisfactory solution. Therefore, we provide a theoretical investigation on the effectiveness of risk evaluation formulas. We define two types of relations between formulas, namely, equivalent and better. To identify the relations between formulas, we develop an innovative framework for the theoretical investigation. Our framework is based on the concept that the determinant for the effectiveness of a formula is the number of statements with risk values higher than the risk value of the faulty statement. We group all program statements into three disjoint sets with risk values higher than, equal to, and lower than the risk value of the faulty statement, respectively. For different formulas, the sizes of their sets are compared using the notion of subset. We use this framework to identify the maximal formulas which should be the only formulas to be used in SBFL.

Machine Learning algorithms have provided core functionality to many application domains - such as bioinformatics, computational linguistics, etc. However, it is difficult to detect faults in such applications because often there is no “test oracle” to verify the correctness of the computed outputs. To help address the software quality, in this paper we present a technique for testing the implementations of machine learning classification algorithms which support such applications. Our approach is based on the technique “metamorphic testing”, which has been shown to be effective to alleviate the oracle problem. Also presented include a case study on a real-world machine learning application framework, and a discussion of how programmers implementing machine learning algorithms can avoid the common pitfalls discovered in our study. We also conduct mutation analysis and cross-validation, which reveal that our method has high effectiveness in killing mutants, and that observing expected cross-validation result alone is not sufficiently effective to detect faults in a supervised classification program. The effectiveness of metamorphic testing is further confirmed by the detection of real faults in a popular open-source classification program.

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