\abstract{ This chapter defines what is meant by the terms
components, component fault modes and `unitary~state' component fault modes.
The application of Bayes theorem  in current methodologies, and
the unsuitability of the `null hypothesis' or p value statistical approach.
Mathematical constraints and definitions are made using set theory.
}

 
\section{Introduction}
When building a system from components, 
we should be able to find all known failure modes for each component.
For most common electrical and mechanical components, the failure modes
for a given type of part can be obtained from standard literature\cite{mil1991}
\cite{mech}. %The failure modes for a given component $K$ form a set $F$.


\subsection{Unitary State Component Failure Mode sets}

An important factor in defining a set of failure modes is that they
should be as clearly defined as possible.
%
It should not be possible for instance for
a component to have two or more failure modes active at once.

Having a set of failure modes where $N$ modes could be active simultaneously
would mean having to consider $2^N$ failure mode scenarios.
% 
Should a component be analysed and simultaneous failure mode cases exit,
the combinations could be represented by a new failure modes, or
the component should be considered from a fresh perspective,
perhaps considering it as several smaller components
within one package.

\begin{definition}
A set of failure modes where only one fault mode
can be active at a time is termed a `unitary~state' failure mode set.
This is termed the $U$ set thoughout this study.
This corresponds to the `mutually exclusive' definition in 
probability theory\cite{probandstat}.
\end{definition}

We can define a function $FM()$ to 
take a given component $K$ and return its set of failure modes $F$.

$$  FM : K \mapsto F $$

We can further define a set $U$ which is a set of sets of failure modes, where
the component failure modes in each of its members are unitary~state.
Thus if the failure modes of $F$ are unitary~state, we can say $F \in U$.


\subsection{Component failure modes : Unitary State example}

A component with simple ``unitary~state'' failure modes is the electrical resistor.

Electrical resistors can fail by going OPEN or SHORTED.
However they cannot fail with both conditions active. The conditions
OPEN and SHORT are mutually exclusive.
Because of this the failure mode set $F=FM(R)$ is `unitary~state'.


Thus

$$ R_{SHORTED} \cap R_{OPEN} = \emptyset $$


We can make this a general case by taking a set $C$ representing a collection
of component failure modes,
We can now state that


$$ c1 \cap c2 \neq \emptyset  | c1 \neq c2 \wedge c1,c2 \in C \wedge C \not\in U  $$

That is to say that if it is impossible that any pair of failure modes can be active at the same time
the failure mode set is not unitary~state and does not exist in the family of sets $U$

 Note where that are more than two failure~modes, by banning pairs from happening at the same time
 we have banned larger combinations as well



\subsection{Component Failure Modes and Statistical Sample Space}

A sample space is defined as the set of all possible outcomes.
When dealing with failure modes, we are not interested in 
the state where the compoent is working perfectly or `OK' (i.e. operating with no error).
We are interested only in ways in which it can fail.
By definition while all components in a system are `working perfectly' 
that system will not exhibit faulty behavuiour.
Thus the statistical sample space $\Omega$ for a component/sub-system K is
%$$ \Omega = {OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N} $$
$$ \Omega(K) = \{OK, failure\_mode_{1},failure\_mode_{2},failure\_mode_{3} ... failure\_mode_{N}\} $$
The failure mode set for a given component or sub-system $F$
is therefore
$$ F = \Omega(K) \backslash OK $$

\subsection{Bayes Theorem}

Describe application - likely hood of faults being the cause of symptoms -
probablistic approach - no direct causation paths to the higher~abstraction fault mode.
Often for instance a component in a module within a module within a module etc
that has a probability of causing a SYSTEM level fault.

Used in FTA\cite{NASA}\cite{NUK}. Problems, difficult to get reliable stats 
for probability to cause because of small sample numbers... 

FMMD approach can by traversing down the tree  use known component failure figures
to 
%$$ c1 \cap c2 \eq \emptyset  | c1 \neq c2 \wedge c1,c2 \in C \wedge C \in U  $$

%Thus if the failure~modes are pairwaise mutually exclusive they qualify for inclusion into the 
%unitary~state set family.

\subsection{Tests of Hypotheses and Significance}

In high reliability systems the fauls are often logged - strange occurances - 
processors resetting - what are the common factors - P values -
for instance very high voltage spikes can reset micro controllers -
but how do you corrollate that with unshielded suppressed contactors...

Maybe looking at the equipment and seeing if there is a 5\%
level of the error being caused ?
i.e. using it to search for these conditions ?