# 统计代写|贝叶斯分析代写Bayesian Analysis代考|Learning from Similar Systems with Imperfect Data

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Learning from Similar Systems with Imperfect Data

Here we show how a hierarchical Bayesian data analysis can be carried out to compute the unknown failure rate of a new system, $A$, using historical data gathered from tests conducted on different sets of units of five types of systems, $B_1, \ldots, B_5$, with similar failure behavior to system $A$. Thus, the untested system $A$ is considered to be exchangeable with the tested systems.

In order to assess the failure distribution of the similar systems, we assume that a series of reliability tests have been conducted under the same operational settings for the first four systems, $B_1, \ldots, B_4$, but that the failure data from component $B_5$ is right censored. In the case of $B_1, \ldots, B_4$ the data resulting from these tests consist of the observed TTFs, $\left{t_{i j}\right}, i=1, \ldots, 4$ and $j=1, \ldots, n_i$, after a fixed period of testing time, of $n_i$ items of type $i$. In the case of $B_5$ the data consists of time to failure intervals $\left{t_{5, j}>2000\right}$ and $j=1, \ldots, n_5$, that is, the tests were suspended after 2000 time units. For nonrepairable systems the order of the data is immaterial.

Our aim is to use the sequence of observed failure times and intervals to assess the failure distributions of each one of the similar systems, from which we wish to estimate the posterior predictive failure distribution for $A$. Thus, if $n_i$ independent tests were conducted on components of type $i$ for a defined period of time, $T$, the data result in $n_i$ independent TTF, with underlying Exponential population distribution:
$$\left{t_{i j}\right}_{i=1}^{n_i} \sim \exp \left(\lambda_i\right), \quad i=1, \ldots, 5$$
The unknown failure rates, $\lambda_i$, of the $B_i$ similar systems are assumed exchangeable in their joint distribution, reflecting the lack of information-other than data – about the failure distribution of the systems. The parameters $\lambda_i$ are thus considered a sample from the conjugate Gamma prior distribution, governed by unknown hyperparameters $(\alpha, \beta)$ :
$$\left{\lambda_i\right}_{i=1}^5 \sim \operatorname{Gamma}(\alpha, \beta)$$
To complete the specification of the hierarchical model, we need to assign a prior probability distribution to the hyperparameters $(\alpha, \beta)$. Since no joint conjugate prior is available when $\alpha$ and $\beta$ are both assumed unknown, their prior distributions are specified independently.

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Dynamic Fault Trees

Most reliability analysis methods are based on parametric and nonparametric statistical models of TTF data and associated metrics. The underlying assumption of these methods is that a coherent, statistical model of system failure time can be developed that will prove stable enough to accurately predict a system’s behavior over its lifetime. However, given the increasing complexity of the component dependencies and failure behaviors of today’s real-time safety-critical systems, the statistical models may not be feasible to build or computationally tractable. This has led to an increasing interest in more flexible modeling frameworks for reliability analysis. The most notable such framework, dynamic fault trees (DFTs), extends FTs by incorporating event dependent behaviors (sequence-dependent failures, functional dependencies, and stand-by spares) of fault-tolerant systems.

Here a DFT is represented by an equivalent “event-based” BN with continuous random variables representing the TTF of the components of the system. These can be either the TTF of elementary components of the system, or the TTF of the fault tree constructs. In the latter case, the nodes in the $\mathrm{BN}$ are connected by means of incoming arcs to several components’ TTFs and are defined as deterministic functions of the corresponding input components’ TTF as shown in the example in Figure 14.11.

In order to specify the probability distribution of the $\mathrm{BN}$, we must give the marginal probability density functions of all root nodes and the NPTs and functions of all nonroot nodes. If the TTF nodes corresponding to elementary components of the system (or some subsystem) are assumed statistically independent (as is the case in standard static FT analysis, or dynamic gates with independent inputs), these are characterized by their marginal probability distributions. The marginal TTF distributions of the root nodes are generally given by standard probability density functions. For example, in Figure 14.11 there are two root nodes corresponding to components $A$, and $B$. The TTF distributions for component $A$ is defined as an exponential distribution with parameter $\lambda_A$, while the TTF distribution for component $B$ is defined as a Weibull distribution with shape parameter $\beta_C$ and scale parameter $\mu_C$. The values of the parameters of these density functions can be either obtained as prior information according to expert knowledge or estimated in a previous reliability data analysis step if some failure data is available.

# 贝叶斯分析代考

## 统计代写|贝叶斯分析代写Bayesian Analysis代考|Dynamic Fault Trees

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