## 4 Construction Risk Assessment Method of Deep Foundation Pit Projects
### Objective Weight Calculation Based on the PP  
The principle of the PP is to find an optimal projection direction by analyzing the structure and characteristics of high-dimensional data and project the high-dimensional data into a one-dimensional to three-dimensional space in the optimal projection direction [11]. The PP has advantages for small sample sizes and high-dimensional data. In addition, the structural features (objective weights) of the data are obtained by projecting the data from a high-dimensional space to a low-dimensional space [17].  
The steps of calculating the index weight using the PP are as follows.  
Step 1. Projecting high-dimensional data.  
In the evaluation data \(\left[x_{ij}\right]_{n\times m}\), \(x_{ij}\) represents the \(j\)-th index value of the \(i\)-th evaluation object, \(n\) is the number of evaluation objects, and \(m\) is the number of indices. The evaluation data \(\left[x_{ij}\right]_{n\times m}\) are in the form of a high-dimensional nonlinear matrix, which is difficult to deal with when using traditional evaluation methods.  
The objective of the PP is to project \(\left[x_{ij}\right]_{n\times m}\) to \(\overset{\rightarrow}{\boldsymbol{\alpha}}=\{\alpha(1),\alpha(2),\cdots, \alpha(m)\}\) and obtain the projection value of the low-dimensional space \(y(i)\)[11]:  
\[y(i)=\sum_{j=1}^{m}\alpha(j)x_{ij}, \tag{1}\]  
where \(\sum_{j=1}^{m}\alpha^{2}(j)=1\), and \(0<\alpha(j)<1\).  
Step 2. Constructing the objective function.  
Different projection vectors correspond to different projection values. The optimal projection value of the low-dimensional space \(y^{*}(i)\) should reflect the characteristics of the high-dimensional space, i.e., the projection points in the low-dimensional space should be clustered, but the projection points should be dispersed. The standard deviation and local density of the projection points are used to define the optimal projection function [17]:  
\[maxQ(\alpha)=S_{y}\big{|}D_{y}\big{|}, \tag{2}\]  
\[S_{y}=\frac{\sum_{i=1}^{n}[y(i)-E(y)]}{\sqrt{n-1}}, \tag{3}\]\[D_{y}=\sum_{i=1}^{n}\sum_{j=1}^{m}\left(R-r_{ij}\right)u\left(R-r_{ij}\right), \tag{4}\]  
where \(S_{y}\) is the standard deviation of the projection values. The larger the \(S_{y}\), the more scattered the projection points are in the low-dimensional space. \(D_{y}\) is the local density of the projection values. The larger the \(D_{y}\), the higher the density of the projection points is in the low-dimensional space. \(E\) represents the average value of the projection values, \(r_{ij}\) is the distance between the projection points, \(u\left(R-r_{ij}\right)\) is the unit step function, and \(R\) is the radius of the observation window.  
Step 3. Solving the objective function based on the PSO.  
The PSO is selected to solve the objective function. It is a stochastic optimization algorithm based on the iterative updating of the population. The particle constantly adjusts its velocity (\(v\)) and position (\(s\)) by tracking the individual optimal solution (\(pbest\)) and the population optimal solution (\(gbest\)) to approach the optimal solution of the objective function [17]:  
\[v(t+1)=v(t)+c_{1}r(pbest-v(t))+c_{2}r(gbest-v(t)), \tag{5}\]  
\[s(t+1)=s(t)+v(t+1), \tag{6}\]  
where \(c_{1}\) and \(c_{2}\) are learning factors, and \(r\) is a random number in the range of \([0,1]\).  
Equation (2) is used as the fitness function of the PSO. The optimal solution is obtained using Equations (5) and (6), and the optimal projection vector (\(\overset{\rightarrow}{\boldsymbol{\alpha}^{*}}\)) is obtained after reaching the predetermined termination condition. The square of each element in the \(\overset{\rightarrow}{\boldsymbol{\alpha}^{*}}\) is the objective weight \(\left\{\omega_{j}\right\}\) of the corresponding index.