Bayes theorem. Need more crit, esp secondary

failure mode effects and
component failure rates rather than component failure modes
used by FMEA FMECA FMEDA

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 %%
+\subsection{Bayes Theorm in Relation to Failure Modes}
-%%- 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
 
-% 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