### 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Stochastic process with exogenous constraint

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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|Stochastic process with exogenous constraint

As shown in [WEB 04], a hidden Markov model (HMM) [RAB 89] can represent the degradation of components. The modeling of component degradation by HMM has also been used in [MOG 12, ROB 13, LE 14].

Time is usually considered as a conditional factor in component reliability, as shown in the previous section. It can be insufficient [SIN 95]. The conditions of use and the environmental context (like humidity, temperature, etc.) can alter the component reliability. All factors that alter component reliability are called co-variables or exogenous variables [BAG 01]. As described in [COX 55], the component reliability can be modeled precisely by taking into account the effects of exogenous variables.

In [WEB 04], several models of MC are defined according to the operational context of the component. A Markov switching model (MSM) is introduced to model the switching from one MC to another subsequent to the state variation of the exogenous variables. These models are also considered as conditional $\mathrm{MC}$ where the transitions are conditional to exogenous variables.

The MSM models are non-stationary because of the fast modifications of parameter values [BEN 99, p. 147]. A MSM represents the conditional distribution $P\left(x_{i}^{(k)}, u_{i}^{(k)}\right)$ given the input state sequence $\left(u_{i}^{(0)}, u_{i}^{(1)}, \ldots u_{i}^{(k)}\right)$, where $u_{i}^{(k)}$ represents the state of the exogenous variable. The simulation of the MSM is based on discontinuous changes of parameters associated with each modification of the exogenous variable state. It is very hard to obtain an analytical solution as with MC, and it is quite simple to use a simulation.

The modeling solution by a DBN is really simple [WEB 04]. One or several exogenous variables modeling the constraints or the operational conditions are added as new discrete variables $u_{i}^{(k)}$ in the time slice $k$. The CPT that defines the transition between two consecutive time steps, $x_{i}^{(k+1)} \mid x_{i}^{(k)}$, is defined conditional on $u_{i}^{(k)}$, as shown in Figure $4.5$ for one variable.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Model of a dynamic multi-state system

A DBN is particularly interesting when dealing with several components in a system. The DBN presented in section $4.2$ allows us to represent several multi-state stochastic processes in a system model. A multi-state model, as presented in Chapter 3 , can easily combine the models of dynamic multi-state components as presented in section $4.2$

to give a whole model of the dynamic multi-state system. The computation in a DBN with several stochastic processes is solved by different inference mechanisms well suited to this problem and to the conditions of use of the models.

The exact inference algorithms are based on a junction tree [JEN 96]. This mechanism is applied to unroll up models. If all time slices are defined in the model, then the usual inference algorithm can be used to compute the exact results (Figure 4.8) but with high computation costs. The models can be of high complexity with much dependence between the components (Figure 4.9) and thus are not practical for such a modeling step. Moreover, they are not adapted for a large time horizon.

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Discussion on dependent processes

The components of systems are not always independant. To decrease the model complexity in the case of dependent processes, it is possible to mix the dependent components in only one stochastic process that is combined with other independent processes by a multi-state BN, as shown previously. According to this method, the

DBN models only independent processes. The whole structure of the global system is then simplified, but the number of states of some variables increases.

Nevertheless if some dependence exists between stochastic processes, as in the roll up of DBN shown in Figure 4.9, it is necessary to use a specific inference algorithm that computes the joint distribution at each time step with significant computing and memory costs. The approximate inference algorithm proposed by [BOY 98, KOL 99] or particular filtering [KOL 00] can estimate the marginal distribution with a bounded error, which is often sufficient for dependability purposes.Unfortunately, another phenomenon introduces difficulties in computing the marginal distribution even in the case of the independent structure shown in Figure 4.10. In the analysis of the functioning scenarios, it is of interest to integrate observations or evidence like events in the DBN. If evidence about a component is introduced in the DBN for a state variable $x_{i}^{(k)}$ or an exogenous variable $u_{i}^{(k)}$, the inference is correct until the processes are independent. However, if evidence is introduced on a variable, for instance $y_{i}^{(k+1)}$, and this evidence introduces a dependence between the variables $x_{i}^{(k+1)}$, then a computing problem appears. This dependence requires the use of specific algorithms. So, it is necessary to be cautious when using DBN and evidence to compute the distributions correctly by considering the right hypothesis.

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Stochastic process with exogenous constraint

DBN 的建模解决方案非常简单 [WEB 04]。添加一个或多个对约束或操作条件进行建模的外生变量作为新的离散变量在一世(ķ)在时间片ķ. 定义两个连续时间步长之间转换的 CPT，X一世(ķ+1)∣X一世(ķ), 定义为在一世(ķ)，如图4.5对于一个变量。

## 统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Model of a dynamic multi-state system

DBN 在处理系统中的多个组件时特别有趣。部分介绍的 DBN4.2允许我们在系统模型中表示几个多状态随机过程。第 3 章中介绍的多状态模型可以很容易地组合第 3 章中介绍的动态多状态组件模型4.2

## 统统计代写|贝叶斯网络概率解释代写Probabilistic Reasoning With Bayesian Networks代考|Discussion on dependent processes

DBN 仅对独立进程建模。然后全局系统的整个结构被简化，但是一些变量的状态数量增加了。

## 广义线性模型代考

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