Shortened the FMEDA entry in fmmd_concept and

placed full version in survey.tex

Put bayes theorem into survey.tex
Need better refs for it

fmmd_concept/fmmd_concept.tex

@@ -317,7 +317,7 @@ FMEA described in this section (\ref{pfmea}) is sometimes called `production FME
 
 \subsection{FMECA}
 
-Failure mode, effects, and criticality analysis (FMECA) extends FMEA.
+Failure mode, effects, and criticality analysis (FMECA) extends FMEA adding a criticallity factor.
 This is a bottom up methodology, which takes component failure modes
 and traces them to the SYSTEM level failures. 
 %
@@ -359,12 +359,13 @@ Again this essentially produces a prioritised `todo' list.
 \begin{itemize}
 \item Possibility to miss the effects of failure modes at SYSTEM level.
 \item Possibility to miss environmental affects.
+\item The $\beta$ factor is based on heuristics and does not reflect any rigourous calculations.
 \item Complex component interaction effects can be missed.
 \item No possibility to model base component level double failure modes.
 \end{itemize}
 
 
-\subsection { FMEDA or Statistical Analyis }
+\subsection { FMEDA }
 
 Failure Modes, Effects, and Diagnostic Analysis (FMEDA)
 % This 

survey/survey.tex

@@ -34,6 +34,99 @@ presents the design considerations that motivated and provided the specification
 the FMMD methodology.
 %
 
+\section{Introduction}
+
+\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:bayes1}
+ P(S|B) = 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}
+
+$$ P(S|B) P(S)  = P(S \cap B)  $$
+
+\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 $B_n$
+
+$$
+P(S_k|B_n) = \frac{P(S_k) \; P(B_n | S_k) }{P(B_n)}
+%P(B|S_k) = \frac{P(S_k | B)\, P(B)}{P(S_k)}
+$$
+%%% because  the probability of $B_n$ in the sample space SS
+%%%is the sum of all probabilities off all failure modes in the indexed set $SS$
+%%%multiplied by the probability of each failure mode causing
+%%%the system failure mode $S_k$.
+%%%
+%%%$$ 
+%%%P(B_n) = {\sum_j^n P(B_j \cap S_k)} = {\sum_j^n P(B_n|A_i) P(A_i)}
+%%%$$
+%%%
+%%%we can express this as
+%%%\begin{equation}
+%%%\label{eqn:bayes2}
+%%% P(S_k|B) = \frac{P(S_k) \; P(B_n|S_k)}{ \sum__{j=1}^{n}  P(B_j)P(S_k | B_j). } 
+%%%\end{equation}
+
+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 }
 
 This, like all top~down methodologies introduces the very serious problem
@@ -98,7 +191,7 @@ FMEA described in this section (\ref{pfmea}) is sometimes called `production FME
 
 \subsection{FMECA}
 
-Failure mode, effects, and criticality analysis (FMECA) extends FMEA.
+Failure mode, effects, and criticality analysis (FMECA) extends FMEA and adds a failure outcome criticallity factor.
 This is a bottom up methodology, which takes component failure modes
 and traces them to the SYSTEM level failures. 
 %
@@ -139,6 +232,7 @@ Again this essentially produces a prioritised `todo' list.
 \subsubsection{ FMECA weaknesses }
 \begin{itemize}
 \item Possibility to miss the effects of failure modes at SYSTEM level.
+\item The $\beta$ factor is based on heuristics and does not reflect any rigourous calculations.
 \item Possibility to miss environmental affects.
 \item No possibility to model base component level double failure modes.
 \end{itemize}