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

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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|BN model from tie-sets

Let us consider the problem from the point of view of success. Three tie-sets can be extracted from Figure 2.1, as given by equation [2.3]. The BN model (Figure 2.4) is given by:
\begin{aligned} &L_{1}=\left{x_{1}, x_{2}\right} \ &L_{2}=\left{x_{1}, x_{3}\right} \ &L_{3}=\left{x_{1}, x_{2}, x_{3}\right} \end{aligned}
The reader can note directly that none of the tie-sets are minimal because $L_{1} \cup L_{2}=L_{3}$. Nevertheless, as in the case of cut-sets, the inference mechanism will work properly and give the correct result. Let us consider the two minimal tie-sets $L_{1}$ and $L_{2}$, for computing the probability distribution of the system states. The deterministic CPT of tie-set $L_{1}$, according to the states of $x_{1}$ and $x_{2}$, are given in Table 2.4. The CPT for $L_{2}$ is given in Table 2.5. The combination of the two minimal tie-sets is enough to compute the probability distribution of $y$ (Table 2.7), thanks to its CPT table (see Table 2.6).The logical behavior of the system failures induces deterministic CPT, which are equivalent to Boolean gates. The BN are modeling Boolean equations by probabilities equal to 0 or 1 .

统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|BN model from a top-down approach

In the case of large systems, the enumeration of all functioning or dysfunctioning scenarios is cumbersome. To solve this problem, the FT modeling is based on a descending approach. Starting from a top event that characterizes the undesired event, the analysis goes down the tree by the definition of intermediate events identified as direct causes of upper events, until elementary events are obtained. For example, Figure $2.5$ shows the FT of the flow distribution system, which is obviously quite simple.

If a FT is available, it is very simple to translate it into a BN by simple mapping. As shown earlier for tie-sets and cut-sets, deterministic CPT can map Boolean relations between variables with logical operators: AND and OR. Figure $2.6$ shows the mapping result of the FT shown in Figure $2.5$ into a BN. For each event in the FT, a variable is defined in the $\mathrm{BN}$.

For instance, the AND gate in Figure $2.5$ is such that $E_{2}=x_{2} \wedge x_{3}$, i.e. $E_{2}$ occurs if $x_{2}=1$ and $x_{3}=1$. The CPT of the $\mathrm{BN}$ is given in Table 2.8. The OR gate in Figure $2.5$ is such that $y=x_{1} \vee E_{2}$, i.e. $y=1$ for the failure of the system if $x_{1}=1$ or $E_{2}=1$. The CPT of the BN is defined in Table 2.9.

统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Standard Boolean gates CPT

All Boolean gates can be modeled by a BN (OR, AND, Koon, Exclusive OR, etc.). It is sufficient to directly map the Boolean equation inside the CPT [SIM 07, SIM 08].

An $n$-component system that functions (or works) if and only if at least $k$ of the $n$ components work is called a k-out-of-n:G system. An $n$ component system that fails if and only if at least $k$ of the $n$ components fail is called a k-out-of-n:F system. Both parallel and series systems are special cases of the k-out-of-n system. A series system is equivalent to a 1-out-of-n:F system and an n-out-of-n:G system, while a parallel system is equivalent to an $n$-out-of-n:F system and a 1-out-of-n:G system.
Let us define the CPT of a 2-out-of- 3 : G system, with the components $x_{1}, x_{2}$ and $x_{3}$. The BN structure is shown in Figure $2.8$. The system is functioning, $y=0$, if at least two components are available; $x_{i}=0$ and $x_{j}=0$, with $i \neq j$ and $i, j \in{1,2,3}$. The CPT of $y$ is defined in Table 2.12.

Unlike FT or RBD, BN can integrate topological constraints, for instance the linear or circular consecutive-koon system. Such systems cannot be modeled by FT or RBD because of the independencehypothesis of events. The BN solves this problem by computing conditional independence and gives a systematic modeling method [WEB 10, WEB 11].

Consecutive-koon systems have attracted considerable attention since they were first proposed by Kontoleon in 1980 [KON 80]. A consecutive-koon system can be classified according to the linear or circular arrangement of its components and the functioning or malfunctioning principle. Thus, four types of $k$-out-of-n can be enumerated: linear consecutive-koon:F, linear consecutive-koon:G, circular consecutive-koon:F and circular consecutive-koon:G. A consecutive-koon:F system consists of a set of $n$ ordered components that compose a chain such that the system fails if at least $k$ consecutive components fail. A consecutive koon:G system is a chain of $n$ components such that the system works if at least $k$ consecutive components work. An illustration of these specific structures can be found in telecommunication systems with $n$ relay stations that can be modeled as a linear consecutive-koon:G system if the signal transmitted from each station is strong enough to reach the next $k$ stations. An oil pipeline system for transporting oil from point to point with $n$ spaced pump stations is another example of a linearconsecutive-koon system. A closed recurring water supply system with $n$ water pumps in a thermo-electric plant is a good example of a circular system. The system ensures its mission if each pump is powerful enough to pump water and steam to the next $k$ consecutive pumps [YAM 03].

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

\begin{对齐} &L_{1}=\left{x_{1}, x_{2}\right} \ &L_{2}=\left{x_{1}, x_{3}\right} \ &L_{3 }=\left{x_{1}, x_{2}, x_{3}\right} \end{aligned}\begin{对齐} &L_{1}=\left{x_{1}, x_{2}\right} \ &L_{2}=\left{x_{1}, x_{3}\right} \ &L_{3 }=\left{x_{1}, x_{2}, x_{3}\right} \end{aligned}

广义线性模型代考

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