skeleton paragraphs for bayes and hyp-sig-P

component_failure_modes_definition/component_failure_modes_definition.tex

@@ -1,6 +1,8 @@
 
 \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.
 }
 
@@ -12,6 +14,9 @@ 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.
 %
@@ -73,7 +78,34 @@ the failure mode set is not unitary~state and does not exist in the family of se
  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{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 ?

component_failure_modes_definition/component_failure_modes_definition.tex~

@@ -0,0 +1,108 @@
+
+\abstract{ This chapter defines what is meant by the terms
+components, component fault modess and `unitary~state' component fault modes.
+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 $F$.
+
+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 whhere $N$ 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.
+\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 exlusive.
+Because of this the failure mode set $F=FM(R)$ is `unitary~state'.
+%A more complex component, say a micro controller could have several
+%faults active. It could for instance have a broken I/O output
+%and an unstable ADC input. Here the faults cannot be considered `unitary~state'.
+
+% A set of failure modes, where only one or no failure modes
+% are active is termed an `unitary~state' failure mode set. This
+% will be donoted as set $A$.
+% 
+To define `unitary~state' using set theory we can define a function
+`active'.
+The function $active(f)$ deontes that the failure mode $f$ (where $f$ is an element of $F$) is currently active.
+
+Thus for the set $F$ to exist in $U$ the following condition must be true.
+
+\begin{equation}
+\label{unitarystate_def}
+ F \in U | f \in F \wedge active(f) \wedge f1 \in F \wedge  f1 \neq f  \wedge \neg active(f1) 
+\end{equation}
+
+As an example the resistor $R$ 
+has two failure modes $R_{open}$ and $R_{shorted}$. 
+
+$$ FM(R) =  F = \{ R_{open}, R_{shorted} \} $$
+ 
+Applying equation \ref{`unitarystate'_definition} to a resistor
+for both fault modes
+
+ $$ active(R_{short}) | R_{short} \in F \wedge R_{open} \in F \wedge R_{open} \neq R_{short}  \wedge \neg active(R_{open})  $$
+ $$ active(R_{open}) | R_{open} \in F \wedge R_{short} \in F \wedge R_{short} \neq R_{open}  \wedge \neg active(R_{short})  $$
+
+For the case of the resistor with only two failure modes the results above, being true,
+show that the failure modes for a resistor of $ F = \{ R_{open}, R_{shorted} \} $ are `unitary~state'
+component failure modes.
+
+Thus 
+     $$ FM(R) =  \{ R_{open}, R_{shorted} \} \in U $$
+
+
+A general case can be stated by taking equation \ref{unitary_state_def} and making it a function thus.
+
+
+\begin{equation}
+\label{`unitarystate'_def}
+ UnitaryState(F) =  \forall f \in F | active(f) \wedge  f1 \in F \wedge f1 \neq f  \wedge \neg active(f1) 
+\end{equation}
+
+%Which can be written
+
+%$$  UnitaryState(FM(K)) $$
+
+
+
+% should this be a paragraph in Symptom Abstraction ????
+
+
+

component_failure_modes_definition/component_failure_modes_definition_paper.tex

@@ -0,0 +1,79 @@
+
+\abstract{ This chapter defines what is meant by the terms
+components, component fault modes and `unitary~state' component fault modes.
+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 $F$.
+
+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
+
+%$$ c1 \cap c2 \eq \emptyset  | c1 \neq c2 \wedge c1,c2 \in C \wedge C \in U  $$
+
+%Thus if the failure~modes are pairwaise mutually exclusive they qualify for inclusion into the 
+%unitary~state set family.

component_failure_modes_definition/paper.aux

@@ -0,0 +1,8 @@
+\relax 
+\citation{mil1991}
+\citation{mech}
+\citation{probandstat}
+\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}}
+\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Component failure modes : Unitary State example}{1}}
+\bibstyle{plain}
+\bibdata{vmgbibliography,mybib}

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