# Causes of Multi-Mechanism Abnormal Formation Pressure in Offshore Oil and Gas Wells
## 3 Methods for Determining Causes of Abnormal Formation Pressure
### Mining Similar Abnormal Pressure Mechanism Samples Based on Cluster Analysis
#### 3.1.2 Establishing a Cause Mechanism Identification Model Based on BO-LightGBM
Second is uncertainty estimation: a Gaussian process can not only predict the expected value of the objective function but also provide an uncertainty estimate of the prediction results. Through a Gaussian process, the posterior probability distribution of each sample point can be obtained, allowing for modeling the confidence in the prediction  
Figure 4: Histogram optimization in LightGBM. results. This is very useful for optimization algorithms, as they can explore more samples in areas with high uncertainty to improve global optimization accuracy. Finally, sampling efficiency: a Gaussian process model can select the next sampling point based on confidence, making the sampling process more efficient. By using a Gaussian process model, Bayesian optimization algorithms can choose the most likely locations to achieve the global optimum, not just local optima. This reduces unnecessary sampling and finds better solutions within limited sampling times. The Gaussian distribution and sampling are shown in Figure 5.  
A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution. The function of a Gaussian process is defined as follows:  
\[f(x)\,\sim GP(m(x),k(x,x^{\prime})) \tag{1}\]  
where \(m(x)\) is the mean function, usually set to 0, and \(k(x,x^{\prime})\) is the covariance function, with \(x^{\prime}\) being a random variable. Considering the prior distribution with a mean of \(p(f\mid X,\beta)\):  
\[p(f\mid X,\ \beta)=N(0,\sum) \tag{2}\]  
where \(X\) is the training set, \(f\) is the set of function values of the unknown function, and \(K\) is the matrix formed by \(\sum\)(\(x,x^{\prime}\)). When observation noise \(\epsilon\) is present, \(p(\epsilon)=N(0,\,\sigma^{2})\), where \(\sigma^{2}\) is the variance, the likelihood distribution is given by:  
\[p(y\mid f)=N(f,\sigma^{2}I) \tag{3}\]  
where \(y\) is the set of observations, and \(I\) is the identity matrix. The marginal likelihood can be obtained as follows:  
\[p(y\mid X,\ \beta)=N(0,\sum+\sigma^{2}I) \tag{4}\]  
Figure 5: Gaussian distribution and sampling. Based on the properties of Gaussian processes, we can obtain the posterior distribution:  
\[\left[ \begin{array}{c}y\\ f_{*}\end{array} \right]\ \sim\ N\bigg{\{}0,\left[ \begin{array}{cc}\sum+\sigma^{2}I&K_{*}\\ K_{*}^{T}&K_{**}\end{array} \right]\bigg{\}} \tag{5}\]  
where \(f_{*}\) is the prediction value, \(K_{*}\) is the covariance matrix between the training inputs and the test inputs, \(K_{*}^{T}\) is the transpose of \(K_{*}\), and \(K_{**}\) is the covariance between the test inputs. The predictive distribution is given by:  
\[f_{*}=K_{*}^{T}(\sum+\sigma^{2}I)^{-1}y \tag{6}\]  
\[\text{cov}(f_{*})=K_{**}-K_{*}^{T}(\sum+\sigma^{2}I)^{-1}K_{*} \tag{7}\]  
The acquisition function is crucial for determining the next evaluation point. When using an upper confidence bound strategy, the next evaluation point is:  
\[x_{t+1}=\text{arg max }\mu_{t}(x)+\sqrt{\alpha_{t}}\sigma_{t}(x) \tag{8}\]  
where \(\alpha_{t}\) is a constant balancing exploration and exploitation, \(\mu_{t}(x)\) is the mean, and \(\sigma_{t}(x)\) is the standard deviation. The search process in Bayesian optimization is illustrated in Figure 6.  
1.3 Calculating Overlapping Weight Relationships Based on the Identification Results of Abnormal Pressure Mechanisms  
The trained model is utilized to identify the abnormal pressure causation categories for each depth data point within a certain section of the Rio del Rey Basin in Cameroon. By inferring through the model, each depth data point can be classified to determine its corresponding abnormal pressure causation category. Following this classification, the  
Figure 6: Bayesian optimization search process. data across the entire well section are statistically analyzed to compute the frequency of occurrence of each abnormal pressure causation category within the well section.