### 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Modeling Formalism of the Structure

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Function of Multi-State Systems

In the case of multi-state systems, standard dependability methods, proposed in the literature are difficult to implement [LIS 03]. In this section, the methodology previously presented in the Boolean case is transposed to multi-state systems to prove that it is easy to obtain multi-state models with BN. Methods are presented for the construction of a model of multi-state systems. The methods are based on cut-sets, tie-sets or the principle of top-down analysis based on functional analysis. Section $3.2 .3$ explains the functional analysisbased method and explains how it provides an easy way to build an efficient model.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN models in the multi-state case

The first step when modeling a multi-state model for dependability analysis is to define the set of variables $x_{i}$ that represent the component states [SHU 10] as follows:

$x_{i}=0$ if the component $i$ is in normal working state: $x_{i}=1 \ldots\left(l_{i}-1\right)$ if the component $i$ is in a degraded state; [3.1] $x_{i}=l_{i} \ldots n_{i}$ if the component $i$ is in a failure state.
with $l_{i}$ being the first failure state, i.e. the component does not satisfy its functioning goals. States $1 \ldots\left(l_{i}-1\right)$ are degraded functioning states, i.e. the component is not fully functional but it does not compromise the system mission. States $l_{i} \ldots n_{i}$ are several failure states of the component that can have different consequences on the system state.
The system state is also defined by a multi-state variable with respect to different functioning and malfunctioning scenarios. This variable is denoted as $y$ and takes its values in the following states:
$y=0$ corresponds to the well-functioning state;
$y=1 \ldots(l-1)$ correspond to degraded states;
[3.2]
$y=l \ldots n$ correspond to malfunctioning states.
Regarding the complexity of scenarios in a multi-state system, it is difficult or impossible to model the system dysfunction using a FT or the functioning of the system by using a RBD. The analysis based on minimal cut-sets or minimal tie-sets remains efficient, and this method provides the definition of all the scenarios. The BN is an efficient modeling method by which to represent these scenarios. The purpose is to model the state of the system as a function of the components’ states by the multi-state function $\phi$. This function can be written as $y=\phi(x)$, where vector $x=\left(x_{1}, x_{2}, \ldots, x_{r}\right)$.

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN model of multi-state systems from tie-sets

The multi-state structure function is easily modeled by a BN. The variables mainly represent the components, the system and

the scenarios. The second step is to structure the BN to efficiently link the component variables to the system variable, to encode the functioning and the failure scenarios of the system.

A first solution consists of enumerating all the minimal tie-sets or minimal cut-sets. By applying the same approach as in the binary case, a $\mathrm{BN}$ is defined to represent the conditional dependencies linking the system functioning or the failure states of the system with the minimal cut-sets or the minimal tie-sets. For the system shown in Figure 1.2, seven functioning scenarios exist: one is the perfect functioning state and the others are degraded functioning scenarios. In this modeling problem, the degraded states of the system are not modeled; therefore, the system state is defined only by two states: when the system is functioning $y=0$; and $y=1$ otherwise. The minimal tie-sets are defined from the following combination of components’ states: 0 corresponds to $O k, 1$ corresponds to $R c$, and 2 corresponds to $R o$ as defined previously in Table $1.2$ of Chapter 1 :
\begin{aligned} &L_{1}=\left{x_{1}=0, x_{2}=0\right} \ &L_{2}=\left{x_{1}=0, x_{3}=0\right} \ &L_{3}=\left{x_{1}=0, x_{2}=2\right} \ &L_{4}=\left{x_{1}=0, x_{3}=2\right} \ &L_{5}=\left{x_{1}=2, x_{2}=0, x_{3}=0\right} \ &L_{6}=\left{x_{1}=2, x_{2}=1, x_{3}=0\right} \ &L_{7}=\left{x_{1}=2, x_{2}=0, x_{3}=1\right} \end{aligned}
Tie-set $L_{j}$ is said to have occurred (been realized) if the components are in the states that define the tie-set. The occurrence of a tie-set has defined by $\left(L_{j}=0\right)$. If at least one of the tie-sets is occurred, then the system is in the functioning state $y=0$. The BN structure is obtained by linking each of the tie-sets $L_{j}$ to the variables characterizing the components’ states $x_{i}$ involved in each tie-set (see Figure 3.1).

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN models in the multi-state case

X一世=0如果组件一世处于正常工作状态：X一世=1…(l一世−1)如果组件一世处于退化状态；[3.1]X一世=l一世…n一世如果组件一世处于故障状态。

[3.2]

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN model of multi-state systems from tie-sets

\begin{对齐} &L_{1}=\left{x_{1}=0, x_{2}=0\right} \ &L_{2}=\left{x_{1}=0, x_{3}= 0\right} \ &L_{3}=\left{x_{1}=0, x_{2}=2\right} \ &L_{4}=\left{x_{1}=0, x_{3}= 2\right} \ &L_{5}=\left{x_{1}=2, x_{2}=0, x_{3}=0\right} \ &L_{6}=\left{x_{1}= 2, x_{2}=1, x_{3}=0\right} \ &L_{7}=\left{x_{1}=2, x_{2}=0, x_{3}=1\right} \end{对齐}\begin{对齐} &L_{1}=\left{x_{1}=0, x_{2}=0\right} \ &L_{2}=\left{x_{1}=0, x_{3}= 0\right} \ &L_{3}=\left{x_{1}=0, x_{2}=2\right} \ &L_{4}=\left{x_{1}=0, x_{3}= 2\right} \ &L_{5}=\left{x_{1}=2, x_{2}=0, x_{3}=0\right} \ &L_{6}=\left{x_{1}= 2, x_{2}=1, x_{3}=0\right} \ &L_{7}=\left{x_{1}=2, x_{2}=0, x_{3}=1\right} \end{对齐}

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