# Theory of Faults (ToF): Numerical Quality Management in Complex Systems
## 3 Development
### Power Function  
The quantity of faults of the log-normal variable is exponentially related to \(z\) (Equation (5)):  
\[x=e^{\mu+z\cdot\sigma}=e^{\mu}.e^{z\cdot\sigma}\sim e^{z\cdot\sigma} \tag{6}\]  
Say \(z\) has an exponential distribution such as a normal distribution N (0, 1):  
\[P_{z}(z)\sim e^{-\frac{1}{2}z^{2}} \tag{7}\]  
Thus,  
\[P(X)=P(z)\frac{dz}{dx}\sim\frac{e^{-\frac{1}{2}z^{2}}}{e^{\sigma z}}=x^{-1- \frac{\dot{\sigma}}{2}},\forall x\in X \tag{8}\]  
Figure 2: PDF(X) for \(\sigma=1\), different \(\mu\). Let \(b=1+\frac{z}{2\sigma}\)  
\[P(X)=ax^{-b},\forall x\in X \tag{10}\]  
Therefore, the probability distribution (Equation (10)) of the magnitude of LN distribution faults is a power function; In this chapter and in numerical experiment A.3, this is our prior (Bayesian) probability; in the case studies and numerical experiment A.4, it is the posterior probability. Ref. [45] reports the work of [46], where the probability estimator (likelihood) of words \"\(q\)\" and the error probability (\(1-q\)) is divided between \"m\" letters:  
\[P(x)=\frac{1-q}{m}\;\;x\in X \tag{11}\]  
To produce the power parameter,  
\[b=\frac{2 ln(m)-ln(1-q)}{ln(m)-ln(1-q)} \tag{12}\]  
Following [46]'s footsteps, we used the post factum project mean for the probability estimator \"\(1-q\)\". It is important to note that \"\(1-q\)\" is not a measure for aleatoric risk but rather of system response. \"\(1-q\)\" is an estimator for systemic risk. We interpreted \"m\" as the number of activities; thus, the risk is equally distributed between the activities, in concert with the MaxEntropy principle that suggests a uniform distribution for discrete systems [9; 47]. Figure 3 is generated for a project of a duration of n = 192 weeks (3.5 years) and \(X=\sum x=\)100 faults in m = 50 activities. The time-averaged probability of having a fault in any discrete time period is \(\frac{X}{n}=1-q=0.52\). Figure 3: Power function generated using Equations (10) and (12) and Fun (1) for m = 50, \(\frac{X}{N}=\) 0.52, y = a\(\cdot\)x\({}^{-\text{b}}\), a = 0.86, b = 1.84. \"b\" was calculated using Equation (12). The scaling parameter \"a\" was calculated by integrating Equation (10) by Fun 1 to produce Figure 3:  
Although synthetic, the power function and its graph (Figure 3) give us a first window into fault magnitude likelihood distribution functions. Power functions were machine fitted directly to case study data of all the different construction projects, validating that the degree distributions are power functions. We used the \"synthetic\" \"\(1-q\)\"-generated power functions as a yardstick for comparison against the actual curves (Case Study 1). The relation generated by Equation (12) between b and q is presented in Figure 4:  
For q \(<0\), there is a certainty (\(1-q\)? 1) of faults on every \(n\in N\). This is the prevalent case in design and construction [48]. Within the range \(-3\leq q\leq-1\), Figure 4 may represent a project of \(m=50\), n = 150 weeks, and a total between 300 and 600 faults; this is equal to \(2\leq 1-q\leq 4\) and \(2.2\leq b\leq 2.5\). This is the range in which one expects to find the mode of construction projects. \(1<b<2\) relates to well-managed projects and \(b>2\) to private sector projects such as residential buildings. Projects with \(b>2\) accumulate a long tail of small effect faults and a diminishing probability for events of low probability. According to [23], \(b>2\) implies two attractors: here, two ICs. For \(b>3\), there is a mathematical possibility for the mean and S.D. (Standard Deviation):  
\[P_{z}(z)\sim e^{-\frac{1}{2}z^{2}} \tag{7}\]0  
\(b>>\) is related to avalanched projects. \(0<b<1\) is machine fitted to many large infrastructure error logs [36]. This is a \"\(b\)\" characteristic of scale-free networks organized in one big cluster, such as design networks. As \(x_{min}\) goes to zero, the magnitude of faults goes to infinity. There is an implicit assumption that \(X\) may be dealt with as a continuous variable. \(X<1\) implies faults that may occur once in every few projects.