(misc_10_6)= # Use of IOR data and/or tests results As presented, a reliability model can be built based on IOR data and/or on {term}`test data ` (manufacturers, user) or combined with IOR data and/or on test data (manufacturers, user). The basic failure rate or the basic probability of failure in {numref}`misc-table_10_8` is based on such IOR data. The data was compiled from the IOR background in the NRPM study. Per {term}`miscellaneous items `, the anomalies are collected regarding representative products (quality, range of operational conditions, orbit as relevant) and the failures are classified in {term}`random failures ` (RF) and {term}`systematic failures ` (SF). To avoid duplication of systematic anomalies, they are only counted once. That means that the derived failure rates include as relevant systematic failure contribution. The IOR data recorded are: * Reference: miscellaneous item number, * Satellite subsystem * Miscellaneous item description (e.g. Li-ion battery), * Cumulated time in-orbit and/or tests for the relevant sample of items, * Number of failures, * Type of failure: random, systematic, * Point estimate: to provide a point of comparison with one-sided interval estimator, * File: the file addresses the events (failures) listed to derive a failure rate * Description of the event, * Classification of the failure, either random or systematic, * Time to failure, * Statistical weight: representing the percentage of the items fleet at the time of the failure (number of items operational at failure time divided by the number of the items in the fleet). * Example: the percentage is nearly 100% for low values of $TTF$ and the percentage decreases as the $TTF$ increases in time. (misc_10_6_1)= ## Failure rate estimator The basic failure rate estimation is a one-sided upper bound interval estimation, see {ref}`Part 2 - Methods ` of this handbook for details. (misc_10_6_2)= ## Proportion of failure estimator To determine a proportion of failures based on the observation of failures within a sample of n elements (e.g. {term}`IOR` data for “one-shot” device), it is possible to use: * Point estimate, * Bayesian estimator, * Estimation by interval. It is recommended to use the interval estimate, see {ref}`Part 2 - Methods ` for details. (misc_10_6_3)= ## Gamma Bayesian estimator The failure rate is no longer considered as a real but as a random variable $\Delta$. In the case of the exponential distribution, a natural prior distribution is the gamma distribution (conjugate of the exponential distribution), see {ref}`Part 2 - Methods ` for details. The {term}`Bayesian inference ` estimator is defined as the expected value of the random variable $\Delta$ knowing that a failure has been observed at $t_{1}$, $t_{2}$ … $t_{n}$). Then, with a sample encompassing $n$ items and $n$ associated times to failure: ````{admonition} Equation :class: equation ```{math} :label: Equation_5_9 E(\Delta | T_{1} = t_{1}, T_{2} = t_{2}, ..., T_{n} = t_{n}) = \frac{\alpha + n}{\beta + \sum_{i} (t_{i})} = \frac{\frac{\alpha}{n}}{\frac{\beta}{n} + \frac{\sum_{i} (t_{i})}{n}} ``` ```` The parameters of the gamma prior distribution are selected as per the knowledge of the item. Let us assume for a certain miscellaneous item that a basic failure rate is calculated. Let us assume that there are some items already in flight with a certain cumulated time, not sufficient enough to provide a result in the order of magnitude of the specification (without prior knowledge). Let us consider $\lambda_{0}$ as a first estimation of the failure rate (e.g. provided by Current document or existing model). Let us consider a Gamma ($\alpha$, $\beta$) distribution assumed as “prior” distribution of the failure rate $\Delta$. It is necessary to provide rules for determining the two parameters $\alpha$ and $\beta$ with two equations. We determine $\alpha$ so that the expected value of the gamma distribution is $\lambda_{0}$. ````{admonition} Equation :class: equation ```{math} :label: Equation_5_10 E(\Delta) = \frac{\alpha}{\beta} = \lambda_{0} ``` ```` This provides a first equation with $\alpha$ and $\beta$. A second equation is provided by the level of confidence" $a$" attributed to the value $\lambda_{0}$: ````{admonition} Equation :class: equation ```{math} :label: Equation_5_11 F_{\Delta}(\lambda_{0}) = P(\Delta \leq \lambda_{0}) = \int_{0}^{\lambda_{0}} \frac{\beta^{\alpha} e^{- \lambda \beta} \lambda^{\alpha - 1}}{\Gamma (\alpha)} = \frac{1}{\Gamma (\alpha)} \gamma (\alpha, \alpha) = a ``` ```` Ideally, the prior should be defined based on two different estimates for the failure rate $\lambda_{0}$, one representing an “average” value (point estimate, entering Eq. {eq}`Equation_5_10`) and one representing an estimate at a given {term}`level of confidence ` (e.g. conservative estimate, entering Eq. {eq}`Equation_5_11`). Once $\alpha$ and $\beta$ are determined it allows to derive the {term}`Bayesian inference ` estimator with $n$ being the number of failures and $\sum t_{i}$ the cumulated time.