Everything the analysis computes comes from one binary relation: test i responds to fault j, or it does not. Call the matrix D, with dij = 1 where the response exists. The rest is reading it two ways and weighting by failure rate.
Reading down: signatures and ambiguity
A fault's signature is its column: the pattern of results the test set produces when that fault is present. Three cases exhaust the possibilities:
| Column | Meaning | Consequence |
|---|---|---|
| All zeroes | No test responds | The fault is undetectable by this test set |
| Unique | No other fault shares the pattern | The fault is isolated to one item |
| Shared with others | Identical to one or more columns | Those faults form an ambiguity group |
An ambiguity group is a property of the test set, never of the hardware. Two items that produce the same reading on every available measurement are indistinguishable no matter how skilled the technician is, and no threshold change will separate them: the fix is a measurement that responds to one and not the other.
Reading across: what a test is worth
A test's coverage is its row: the faults it responds to. Two rows that are identical describe two tests that are diagnostically the same, whatever they cost. A row that is a subset of another describes a test that adds nothing once the other is present. And a row of mostly zeroes describes a test that is cheap to justify and hard to defend.
The metrics, weighted by rate
Counting rows is the standard beginner's error, because a matrix has no idea that one fault happens twenty times as often as another. Every figure is a rate-weighted share, over the fault set F:
FD = Σ λj over detected faults ÷ Σ λj over all faults
FI(1) = Σ λj over faults in a group of one ÷ Σ λj over detected faults
FI(≤k) = Σ λj over faults in groups of size ≤ k ÷ Σ λj over detected faults
Two conventions travel with those definitions and both must be stated. Isolation is quoted as a fraction of what is detected, not of everything, because a fault that was never detected cannot be isolated; a report that mixes the two bases is comparing different quantities. And the group size is in replaceable items, not in components, since the point of the number is how many things come off the equipment.
The cost of a group, in removals
An ambiguity group of n worked through one item at a time costs, on average,
(n + 1) / 2 removals per fault
so a group of three costs two removals to fix one, and the extra one is a good unit taken off and sent to stores. That expression is the bridge between a testability metric and a logistics cost, and it is why isolation, not detection, is usually what the support budget cares about.
Detection and isolation are different purchases
This is the result the model exists to produce, and it is invisible without one. A test that responds to a fault nothing else sees buys detection. A test that responds to some members of an existing ambiguity group buys isolation and may detect nothing new at all. The two are bought with different tests, cost different money, and satisfy different requirements, and a programme with a single "testability" number cannot tell which it is short of.
Where the matrix comes from
The cells are engineering, not bookkeeping, and there are only three honest sources:
| Source | Confidence |
|---|---|
| The design: a signal path exists from the fault to the observation point | Strong, and checkable by inspection |
| Analysis or simulation of the response with the fault injected | Strong, and expensive |
| Engineering judgement | Weak, and the majority of most matrices |
A cell filled by judgement should be recorded as such, because the second most common way to overstate testability is a matrix full of optimistic ones, and the most common is the next section.
What the model cannot represent
- Partial or intermittent response. The cell is binary. A test that catches a fault only when it is hot, or only above a threshold, is either a one that overstates or a zero that understates, and neither is right.
- False alarms. The matrix says what a test responds to when a fault is present, and says nothing about what it does when nothing is wrong. False alarm rate is a separate analysis with separate data, and it is the figure that decides whether anybody believes the other two.
- Multiple simultaneous faults. Signatures are computed one fault at a time. Two faults together can produce a third signature that names an innocent item.
- Test sequence effects. A test that is only valid after another has passed, or that destroys the evidence for a later one, needs the strategy rather than the matrix.
- Wear and drift. A parametric fault that moves slowly crosses a threshold at some point; the matrix records the crossing, not the months before it.