## 3 Materials and Methods
### Research Method
#### 3.3.3 Time-at-Risk (TaR) and Monte Carlo Simulation  
As for the Time-at-Risk, [PERSON] and [PERSON] [42] proposed the concept of time@risk in the context of building a particular diagnostic system that provides continuous updates of possible scenarios and their probabilistic weights reflecting the diagnosis of hazardous regimes to target different types of instabilities. Such a system, based on the time@risk approach, could be used in many areas, e.g., to signal the possible occurrence of a crisis, provide insights for the adoption of appropriate policy measures and allow the assessment of future scenarios according to the chosen policy. The Time-at-Risk (TaR) concept proposed by [PERSON] and [PERSON] [41] differs from the concept proposed by [PERSON] and [PERSON] [42]. It is a quantile-based approach  
Figure 5: Correlation matrix reflecting dependencies between different phases and stages of the Fort Bema (Parkowo-Lesne Housing Estate) project. Figure 6: Heatmap reflecting dependencies between different phases and stages of the Fort Bema project. that refers to the Value-at-Risk (VaR) concept known from the financial world and is based on a probability distribution function (PDF) of the return times of peaks above the threshold value. The method on which the study is based uses a function that simulates random variables with a PERT distribution (where each project stage has an assumed duration given in the original project documentation). The appropriateness of the use of PERT distributions, which are a special case of beta distributions, has already been addressed in other studies [16; 84; 85]. Naturally, researchers polemicize on this issue. Some justify the superiority of Weibull distributions [67; 86]. Without going into special details, we choose PERT-distributions since they are easier to implement in the R package and they give the possibility to individually define the extremes and the most likely value (i.e., mode), which has already been defined for the Fort Bema project (see Tables 2 and 3). PERT belongs to the group of continuous probability distributions that are characterised by its minimum (_a_), most likely (_b_) and maximum (_c_) values that a variable can take. PERT is derived from the transformation of a four-parameter beta distribution for which the expected value is: \(\mathrm{E}[X]=\frac{a+b+c}{6}=\mu\). The PERT distribution has the characteristic property that its mean (of the distribution) is weighted by the (minimum and maximum) extrema values and the most likely value (mode), with the latter weighted (four times) more heavily than the former [87]. The original rationale for applying this distribution was to address the impact of the uncertainty surrounding the duration of tasks on the performance of a project schedule evaluated using the Programme Evaluation and Review Technique (PERT), one of the most popular methods for scheduling [87]. Unlike the triangular distribution, the PERT distribution uses the minimum, maximum and most likely parameters to create a smooth curve (Figure 7). From a mathematical perspective, the standard deviation of PERT distribution is equal to about 1/6 of a range [87; 88]. It is well suited for any risk assessment that seeks to address (reflect) uncertainties in the value of some measure, specifically when one relies on some subjective estimates to quantify such a measure. In this regard, the minimum, maximum and most likely (mode) values are somehow intuitive, and they enable definition of the distribution at the researcher's discretion. To simulate the PERT distributions in the study, we use R packages (such distributional and ggdist) [89], and the simulation itself is performed with the report function. The PERT distribution is also often compared with the triangular distribution (since it relies on the same parameters), but in contrast to the latter, the former is characterised by a smoother curve shape. In the case of a triangular distribution, the most likely value is not weighted in any particular way.