### 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Integrating Environmental

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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|Operating Constraints in Reliability

A DBN can take into account the temporal dimension of the system states’ evolution along their lifetime by factorizing and discretizing the state space of each independent random variable at each time instant. A stochastic process is represented at time $k$ by a variable $x_{i}^{k}$ with a finite number of states $\left(h_{1}^{x}, \ldots h_{n}^{x}\right)$. The state of variables with the same value of $k$ constitutes the time slice $k$ [HUN 99, BOU 99].

A DBN can model the evolution of discrete random variables by defining the conditional dependence of a time slice $k+1$, given the states of the random variables at the previous time slice $k$. The definition of the dependence linking the variables at different time slices can model various complex stochastic processes. This time-based stochastic process is modeled by a CPT. Figure $4.1$ shows a particular case where a variable, $x_{i}^{k}$, is defined conditionally to itself at the previous time slide $x_{i}^{k-1}$. This is the Markovian case.

From an observed situation at any time $k$ or from the initial conditions with $k=0$, the inference mechanism in the DBN allows us to compute the state probability distribution of all variables for each time slice. To compute this, it is necessary to memorize the state probability distribution of all the variables in all time slices. The solution consists of developing the time slices for the entire desired time horizon, i.e. to duplicate all the variables for each time period. However, the BN size increases proportionally to the computing horizon [KJA 95]. This solution is not convenient for system dependability analysis because the process should be studied for a large time horizon. It conducts to a combinatory explosion of variables that cannot be handled by current inference mechanisms.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|DBN model of a MC

In the case of Markovian processes, the Markov property is used to simplify the inference mechanism. For instance, in a Markovian process, the CPT is time invariant. The inference can be realized iteratively without explicitly defining a variable for each time slice. The DBN model is then compact and only two successive time slices are modeled, as shown in Figure 4.2. A DBN with two time slices noted 2-TBN [BOY 98] allows us to define all the necessary parameters to model the MC. The first slice contains the variables at the current time step $k$, while the second allows us to compute the distribution of variables at the time step $(k+1)$ by inference. A variable $x_{i}^{(k+1)}$ is defined conditionally to its states in the current time step $x_{i}^{(k)}$. The CPT, $x_{i}^{(k+1) \mid x_{i}^{(k)}}$, is constant whatever the value of $k$ (Table 4.1). This CPT is defined from the transition probability matrix between the states of the MC. With this model, the future states at $(k+1)$ are conditionally independent of the past given the present states at time $(k)$. The CPT clearly shows a MC [KJA 95].

After the first inference, the distribution $P\left(x_{i}^{(k+1)}\right)$ is injected as the a priori distribution for $x_{i}^{(k)}$. The next inference allows us to compute the distribution for the next time step. An exact inference computes the probability distribution of the random variable for the time step $k+1$, from the distribution at time step $k$. The probability distribution for the next time steps $k+2, k+3, \ldots$ are computed by successive inferences [WEL 00]. For a time horizon of size $h, h$ inferences are necessary. This computing method is equivalent to the Chapman-Kolmogorov equation.

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|DBN model of non-homogeneous MC

The extension to non-homogeneous $\mathrm{MC}$ is possible by introducing time-indexed CPT. By working with the Bayesia company (http:// www.bayesia.com/) this possibility has been introduced in the BayesiaLab software. The parameters defined in the CPT can be indexed to an exogenous variable $k$ that represents time.

Here we illustrate the concept. Let us consider valves with three states: a normal functioning state and two failure states, i.e. a remained closed state ${1}$ and a remained open state ${2}$. In the case of varying parameters, the principle is illustrated by combining two Weibull laws for the valve $x_{1}$. The failure rates are time varying and defined according to Weibull laws with the following parameters:

• for the transition to the remained closed state ${1}$, the failure rate is defined as follows:
$\lambda_{11}=\frac{\beta \times k^{(\beta-1)}}{\alpha^{\beta}}$ with $\beta=3$ and $\alpha=500$
• for the remained open state, the failure rate is defined as follows:
$\lambda_{12}=\frac{\beta \times k^{(\beta-1)}}{\alpha^{\beta}}$ with $\beta=2.5$ and $\alpha=700$
The DBN model of the valve $x_{1}$ is shown in Figure 4.3. The probability distribution on the valve states is computed over 1,000 hours with 1,000 iterations, i.e. with a time step of 1 hour, as shown in Figure 4.4.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Operating Constraints in Reliability

DBN 可以通过在每个时刻对每个独立随机变量的状态空间进行分解和离散化，从而考虑系统状态沿其生命周期演化的时间维度。一个随机过程在时间表示ķ通过变量X一世ķ具有有限数量的状态(H1X,…HnX). 具有相同值的变量的状态ķ构成时间片ķ[匈奴 99，博 99]。

DBN 可以通过定义时间片的条件依赖性来模拟离散随机变量的演变ķ+1，给定前一个时间片的随机变量的状态ķ. 定义连接不同时间片变量的依赖关系可以模拟各种复杂的随机过程。这种基于时间的随机过程由 CPT 建模。数字4.1显示了一个特殊情况，其中一个变量，X一世ķ, 在上一次幻灯片中对其自身有条件地定义X一世ķ−1. 这就是马尔可夫案例。

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|DBN model of non-homogeneous MC

• 用于过渡到保持关闭状态1，故障率定义如下：
λ11=b×ķ(b−1)一种b和b=3和一种=500
• 对于保持打开状态，故障率定义如下：
λ12=b×ķ(b−1)一种b和b=2.5和一种=700
阀门的DBN型号X1如图 4.3 所示。阀门状态的概率分布是在 1000 小时内用 1000 次迭代计算的，即时间步长为 1 小时，如图 4.4 所示。

## 广义线性模型代考

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## MATLAB代写

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