diff --git a/survey/partition.dia b/survey/partition.dia new file mode 100644 index 0000000..343407e Binary files /dev/null and b/survey/partition.dia differ diff --git a/survey/partition.jpg b/survey/partition.jpg new file mode 100644 index 0000000..2058e77 Binary files /dev/null and b/survey/partition.jpg differ diff --git a/survey/partition2.dia b/survey/partition2.dia new file mode 100644 index 0000000..57e51ef Binary files /dev/null and b/survey/partition2.dia differ diff --git a/survey/partition2.jpg b/survey/partition2.jpg new file mode 100644 index 0000000..132a19e Binary files /dev/null and b/survey/partition2.jpg differ diff --git a/survey/survey.tex b/survey/survey.tex index 2e3d3db..dd6b4a0 100644 --- a/survey/survey.tex +++ b/survey/survey.tex @@ -39,108 +39,6 @@ the FMMD methodology. \subsection{Failure Modes and System Failure Symptoms} describe briefly what a base component failure mode is and what a system level failure mode is. -\subsection{Bayes Theorm in Relation to Failure Modes} - -\paragraph{Conditional Probability} -Bayes theorem describes the probability of causes. - -In the context of failure modes in components -we are interested in how they may affect a SYSTEM. -The SYSTEM failure modes can be seen as symptoms of the failure modes of base -components. -For example, let $B$ be a base component failure mode -abd let $S$ be a system level failure mode. - -We can say that the conditional probability of $S$ given $B$ is denoted as -\begin{equation} -\label{eqn:condprob} - P(S|B) = \frac{P(S \cap B)}{P(S)} -\end{equation} - -%Or in other words we can say that the probability of $B$ and $S$ occurring -%divided by the probability of $S$ occurring due to any cause, is the probability -%the $B$ caused $S$. -We can call this the {\em conditional probability} of $S$ given $B$. -Re-arranging \ref{eqn:bayes1} - -$$ P(S) P(S|B) = P(S \cap B) $$ - -The inverse condition, $B$ given $S$ is - -$$ P(B) P(B|S) = P(S \cap B) $$ - -As for one being the cause of the other, both equations must be equal, -we can state, - -$$ P(B) P(B|S) = P(S \cap B) = P(S) P(S|B) $$ - -we can now re-arrange the equation to remove the intersection $P(S \cap B)$ term -thus - -\begin{equation} -\label{eqn:bayes1} - P(S|B) = \frac{P(S) P(B|S)}{P(B)} . -\end{equation} - - -\paragraph{Multiple Events and conditional Probability} - -\paragraph{Bayes Theorem} - -Consider a SYSTEM error that has several potential base component causes. -Because a SYSTEM typically has a number of high level errors let us consider -a specific one and label it $S_k$. -We can call $P(S_k)$ the prior probability of the SYSTEM error. That is to -say the iprobability od $S_k$ occuring with no information about possible causes for it. - Consider a number of possible -base component `potential cause' events as $B_n$ where $n$ is an index. -Our sample space $SS$, for investigating the system failure mode/symptom -$S_k$ is thus $ SS = \{B_1 ... B_n\} $. -Thus if B is any event, we can apply bayes theorem -to determine the statistical likelihood that a given failure mode $B_n$ -will cause the system level error $S_k$ - -%IN ENGLEEEESH Inverse causality..... -%Prob $B_n$ caused $S_k$ is the prob $S_k$ caused by $B_n$ divided by prob of $B_n$ - -$$ -P(S_k|B_n) = \frac{P(S_k) \; P(B_n | S_k) }{P(B_n)} -$$ - -For example were we to have a component that has a failure mode $B_n$ with an MTTF of $10^{-7}$ hours -and its associated system failure mode $S_k$ has a MTTF of $5.10^{-8}$ hours, and given that -when the system error $S_k$ occurs, there is a 10\% probability that $B_n$ had occured, we can determine -the probability that $S_k$ is caused by $B_n$ thus - - -$$ -P(S_k|B_n) = \frac{5.10^{-8} \; 0.1 }{ 10^{-7}} = 0.05 = 5\% -$$ - - - -RESTRICTIONS: - -Because this uses conditional probability for multiple independent events -complications such as operational states or environmental conditions -cannot be represented by the Bayesian model. -% consider 747 engines and a volcanic ash cloud.... -\subsubsection{Proportional area Euler diagram example} - -show using area propostional Euler Diagrams the failure modes and their -possible sdystem level failure outcomes. - -Discuss unused sections of hardware in a product. - -Discuss protection devices like VDR's and capacitors for smoothing - -Discuss microprocessor watchdog and CRC ROM schemes - -Discuss hardware failsafes (good example over pressure saefty values). - -Keep relating these back to bayes theorem. - - \section {Four Current Failure Mode Analysis Methodologies} \subsection { FTA } @@ -646,20 +544,181 @@ FROM INTERBET HISTORY OF FTA \end{figure} -%%- RE_PHRASE %% -%%- RE_PHRASE %% Fault tree analysis (FTA) is a tool originally developed in -%% RE_PHRASE %% 1962 by Bell Labs for use in studying failure modes in the -%% RE_PHRASE %% launch control system of the Minuteman missile project. The tool now -%% RE_PHRASE %% finds wide use in numerous applications, from accident investigation to design -%% RE_PHRASE %% prototyping, and is also finding use for protection and control related -%% RE_PHRASE %% applications. This paper provides an elementary background to the application of -%% RE_PHRASE %% FTA for use in protection applications. The construction of the fault -%% RE_PHRASE %% tree as well as the use of reliability data is considered. -%% RE_PHRASE %% A simple example is presented. The intention is to provide a -%% RE_PHRASE %% brief introduction to the concept, to allow users to at least -%% RE_PHRASE %% understand how a fault tree is constructed and what can be done -%% RE_PHRASE %% with it. -% read exita doc and ref it +\subsection{Bayes Theorm in Relation to Failure Modes} -% typeset in {\Huge \LaTeX} \today +\paragraph{Conditional Probability} +Bayes theorem describes the probability of causes. + +In the context of failure modes in components +we are interested in how they may affect a SYSTEM. +The SYSTEM failure modes can be seen as symptoms of the failure modes of base +components. +For example, let $B$ be a base component failure mode +abd let $S$ be a system level failure mode. + +We can say that the conditional probability of $S$ given $B$ is denoted as +\begin{equation} +\label{eqn:condprob} + P(S|B) = \frac{P(S \cap B)}{P(S)} +\end{equation} + +%\paragraph{Multiple Events and conditional Probability} +% +%add copy, describe probabilities for multiple events..... + + +%Or in other words we can say that the probability of $B$ and $S$ occurring +%divided by the probability of $S$ occurring due to any cause, is the probability +%the $B$ caused $S$. +We can call this the {\em conditional probability} of $S$ given $B$. +Re-arranging \ref{eqn:bayes1} + +$$ P(S) P(S|B) = P(S \cap B) $$ + +The inverse condition, $B$ given $S$ is + +$$ P(B) P(B|S) = P(S \cap B) $$ + +As for one being the cause of the other, both equations must be equal, +we can state, + +$$ P(B) P(B|S) = P(S \cap B) = P(S) P(S|B). $$ + +We can now re-arrange the equation~\cite{probstat} to remove the intersection $P(S \cap B)$ term +thus + +\begin{equation} +\label{eqn:bayes1} + P(S|B) = \frac{P(S) P(B|S)} {P(B)} . +\end{equation} + +Equation \ref{eqn:bayes1} means, given the event $B$ what is the probability it was caused by $S$. +Because we are interested in what base component failure modes could have caused $S$ +we need to re-arrange this + +\begin{equation} +\label{eqn:bayes2} + P(B|S) = \frac{P(B) P(S|B)} {P(S)} . +\end{equation} + +Equation \ref{eqn:bayes2} can be read as given the system failure mode $S$ + +Typically a system level failure will have a number of possible causes, or base component failure +modes. Some base component failure modes may not be able to cause given system failures. +We can represent the the base component failure modes as a partioned set~\cite{nucfta}[fig VI-7], and overlay +a given system failure mode on it. + +\begin{figure}[h] + \centering + \includegraphics[width=350pt,keepaspectratio=true]{./survey/partition.jpg} + % partition.jpg: 510x264 pixel, 72dpi, 17.99x9.31 cm, bb=0 0 510 264 + \caption{Base Component Failure Modes represented as partitioned sets} + \label{fig:partitionbcfm} +\end{figure} + + +Figure \ref{fig:partitionbcfm} represents a small theoretical system +with nine base component failure modes. These are represented as partitions +in a set theoretic model of the systems possible failure mode causes. + +\begin{figure}[h] + \centering + \includegraphics[width=350pt,keepaspectratio=true]{./survey/partition2.jpg} + % partition.jpg: 510x264 pixel, 72dpi, 17.99x9.31 cm, bb=0 0 510 264 + \caption{Base Component Failure Modes with Overlaid System Error} + \label{fig:partitionbcfm2} +\end{figure} + +Figure \ref{fig:partitionbcfm2} represents the case where we are looking at a particular +system level failure $S_k$. Looking at the diagram we can see that this system failure +could be, but is not necessarily caused by base component failure modes $B_1, B_2 \; or \; B_4$. +Should any other base component failure mode (causation event occur) according to the diagram +it will not be able to cause the system failure $S_k$. + +\paragraph{Bayes Theorem} + +Consider a SYSTEM error that has several potential base component causes. +Because a SYSTEM typically has a number of high level errors let us consider +a specific one and label it $S_k$. +We can call $P(S_k)$ the prior probability of the SYSTEM error. That is to +say the iprobability od $S_k$ occuring with no information about possible causes for it. + Consider a number of possible +base component `potential cause' events as $B_n$ where $n$ is an index. +Our sample space $SS$, for investigating the system failure mode/symptom +$S_k$ is thus $ SS = \{B_1 ... B_n\} $. +Thus if B is any event, we can apply bayes theorem +to determine the statistical likelihood that a given failure mode $B_n$ +will cause the system level error $S_k$ + +%IN ENGLEEEESH Inverse causality..... +%Prob $B_n$ caused $S_k$ is the prob $S_k$ caused by $B_n$ divided by prob of $B_n$ + +$$ +% P(S_k|B_n) = \frac{P(S_k) \; P(B_n | S_k) }{P(B_n)} alternate form of no use to MEEEEEE +P(B_n|S_k) = \frac{P(B_n) \; P(S_k | B_n) }{P(S_k)} +$$ + +For example were we to have a component that has a failure mode $B_n$ with an MTTF of $10^{-7}$ hours +and its associated system failure mode $S_k$ has a MTTF of $5.10^{-8}$ hours, and given that +when the system error $S_k$ occurs, there is a 10\% probability that $B_n$ had occured (i.e. $P(S_k | B_n) = 0.1$), we can determine +the probability that $S_k$ is caused by $B_n$ thus + + +$$ +P(S_k|B_n) = \frac{5.10^{-8} \; 0.1 }{ 10^{-7}} = 0.05 = 5\% +$$ + + +To take an example from the diagram (see figure \ref{fig:partitionbcfm2}), where the base component fault cannot +lead to the system failure $S_k$. Taking say $B_9$ which does not overlap with $S_k$ +we can see that $P(S_k | B_9) = 0$. +Bayes theorem applied to $B_9$ becomes $P(S_k|B_9) = \frac{P(B_9) \; 0 }{ 10^{-7}}$ +As this is a factor in the numerator, +the application of bayes theorem to $B_9$ being a cause for $S_k$ has a probability +of zero, as we would expect. + +Because we are interested in finding the probability of $S_k$ for all +base component failure modes, it is helpful to re-define +$P(B_n)$. + + +% +% here derive the trad version of bayes with the summation as the denominator +% + +RESTRICTIONS: + +Because this uses conditional probability for multiple independent events +complications such as operational states or envi1ronmental conditions +cannot be represented by the Bayesian model. +% consider 747 engines and a volcanic ash cloud.... + +\paragraph{mutually independent events and base component failure statistics} + +FMEA, FTA, FMECA and to a great extent FMEDA, apply bayesian +concepts to individual base~components failure rates, rather than +using base~component failure modes, for the events under +investigation. +This means a lack of precision in interpretting the base failure +modes as statistically independent events. +Typically, a base component may fail in more than one way, +and usually once it has it stays in that failure mode. +This violates the principle of the events being statistically independent. + +show using area propostional Euler Diagrams the failure modes and their +possible sdystem level failure outcomes. + +Discuss unused sections of hardware in a product. + +Discuss protection devices like VDR's and capacitors for smoothing + +Discuss microprocessor watchdog and CRC ROM schemes + +Discuss hardware failsafes (good example over pressure saefty values). + +Keep relating these back to bayes theorem. + + + +typeset in {\Huge \LaTeX} \today