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Bayes theorem. Need more crit, esp secondary

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failure mode effects and
component failure rates rather than component failure modes
used by FMEA FMECA FMEDA
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Robin Clark 2011-01-09 12:41:39 +00:00
parent 925d550890
commit 08d9242711
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@ -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