# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 5 Case Studies  
Two databases were analyzed, database-1 of 27 projects and database-2 of one storage facility. The projects are construction projects; they are of different market segments such as infrastructure, industrial, residents, and utilities, ranging in cost from millions to billions. The databases are QC faults tables containing faults, with additional data supplied sparsely: dates, locations, time to closure, disciplines. The dataset was loaded to an off-site server service. The available data are a partial download of it. The number of faults in the database studied here is \(\sim\)10\({}^{4}\). The timeline of the database is 8 years. Biased data are a common occurrence in the construction industry, and a recurrent finding in our studies and here. Six of the projects (about 20%) have little or no QC entries, attesting to a phenomenon that QC does not always keep orderly, consistent, and scalable logs (Case Study 1). Another notation found to cause bias is that open fault entries were uploaded en bloc in repetitive time sequences. This caused recurrent formations in the _fft_ analysis and emergent formations in trending in Case Study 1. The significance of quantity of faults-_fft_ to the phenomena of faults is therefore questionable; It may represent notation practices. The partiality of databases posed some limitations. For one, where locations were not listed, avalanche locations and spatial trajectories could not be developed. The data quality varied also in time within projects; this behavior is salient in Case Study 2. The barriers here are codes and regulations, the contracts, inspection practices, testing labs, and **praxis** (=best practice). The analysis follows the methodological constituents: PDF, power function, DSP, statistics (variance and entropy trending), descriptive statistics analysis, IC assessments, network analysis, morphology, and trajectories. The order of presentation is dictated by the rendering needs. Case Study 1  
The Matlab distribution fitter app was applied to database-1. The resulting Figure 13 validates that the database is under a log-normal probability as provisioned. The extremity of variance noted in Figure 13 and the high R\({}^{2}\) are traits of functions resembling exponentials such as LN distributions and power functions. The inconsistencies of the data, apparent from the blue histogram in Figure 13, are a product of the inconsistent timeline of faults and inconsistencies between segments and within projects. These alleged inconsistencies are not random; their occurrence, causes and trajectories are investigated further on. The estimator for the magnitude of faults used here is open fault duration. Power functions were machine fitted to all projects (some are presented within the figures), validating the theoretical provisions that magnitude distributions are power functions and networks are scale-free. All project power functions were compared to Figure 3. The mathematical traits attributed to the phases presented next by the phase parameter are referred to from [23]. The range \(1<b<2\) is a range for high-quality projects, such as well-managed and quality-controlled civil works, such are project 2 and project 3 here. The \(2<b<3\) projects are expected to be the most abundant projects, this is \"\(b\)\" generated from the number of faults reported (Figure 4); \(2<b<3\) projects are of lesser quality, clusters overlap only partially, and there is a possibility for avalanches to happen on every scale, and also a possibility of a second IC. Such are project 5, project 7, project 11, project 15, project 17, project 19, and project 23, all heavy civil engineering works. Projects with \(3<b<4\) may have a theoretical mean and S.D., such as project 8 and project 10. The rest of the accounted for projects have \(b>4\). They number almost half of the projects. If this finding is representative, then the number of faults in the industry is much higher than that prevalent in the literature.