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A clouding alarm, acoustic burst or spectral threshold is an observation, not the mechanism itself. When that noisy trigger also schedules the next inspection, its false positives and false negatives reshape both the labels and the times at which evidence is collected.
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Chapter contents
Choose the route closest to the decision, or read from Chapter 1. The chapters remain open; these grouped routes are a navigable index.
Route 1 · Define the evidence
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE THE EVIDENCE
1. Keep state, trigger and visit separate
Let the latent material state describe the physical regime, the trigger describe what a sensor reports and the visit process describe when the specimen is examined. Collapsing the three turns a noisy observation rule into a claimed mechanism.
A trigger can be wrong in both directions. A positive alarm may be instrumental noise; a quiet specimen may already be transitioning. Because an alarm often creates extra visits, the observed dataset over-represents trigger-positive histories even before any exit model is fitted.
CHAPTER 2 OF 12 · DEFINE THE EVIDENCE
2. Write the misclassification matrix explicitly
For a binary trigger, sensitivity is the chance of a positive signal in the target latent state; specificity is the chance of a negative signal outside it. Both may vary with temperature, thickness, operator or sensor version.
The matrix belongs inside the observation model. Treating the observed trigger as error-free forces every discordant physical outcome into the transition model and can make the mechanism map appear more abrupt or more persistent than the material process really is.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A validation confusion ledger
Suppose an independent assay classifies 200 specimen-windows, with 50 in the target state and 150 outside it.
| Reference state | Trigger + | Trigger − | Row total |
|---|---|---|---|
| Target state | 40 | 10 | 50 |
| Other state | 15 | 135 | 150 |
| Column total | 55 | 145 | 200 |
| Rate | Sensitivity 80% | Specificity 90% | — |
The trigger has reasonable operating characteristics, yet 15 of 55 positives are false positives and 10 of 50 true target windows are missed. The observed label cannot safely replace the latent state.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked predictive value under a new prevalence
Predictive value changes when the target state is more or less common. With sensitivity 0.80, specificity 0.90 and target-state prevalence 0.25, true-positive mass is 0.25×0.80=0.20 and false-positive mass is 0.75×0.10=0.075. The positive predictive value is 0.20/(0.20+0.075), about 73%.
| Component | Probability mass | Reading |
|---|---|---|
| True positive | 0.200 | Target state and trigger + |
| False positive | 0.075 | Other state but trigger + |
| All trigger positives | 0.275 | Observed positive mass |
| Positive predictive value | 0.727 | About 73% of positives are true |
A mechanism map built from positive visits alone would therefore contain about 27% non-target windows under these assumptions.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Join trigger error to informative visits
Model the visit intensity using recorded history and, when justified, latent state. The likelihood can combine visit times, trigger labels, validation measurements, interval-censored exits and cause labels. This preserves the evidence carried by an alarm without pretending the alarm is truth.
If validation occurs only after positive triggers, sensitivity cannot be learned directly because quiet target states remain hidden. A random or stratified validation sample is needed to observe both rows of the confusion matrix.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Map mechanisms with posterior responsibilities
Instead of assigning each visit to one mechanism, compute the probability that the latent state and exit cause take each value given the trigger history, visit process and validation parameters. Aggregate these responsibilities at the specimen level, not as if visits were independent specimens.
Show uncertainty around the map. A colourful maximum-probability label conceals near ties, and a visit-adjusted count is still conditional on the model linking states, triggers and visits.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: adhesive monitoring
An acoustic burst may trigger microscopy of an adhesive joint. Use a burst confusion matrix calibrated against destructive sectioning, and model burst-driven visit intensity separately from debonding transitions.
The transfer fails when the destructive validation changes future risk, when quiet cracking lies outside sensor support or when the calibration specimen differs materially from the production joint. Those cases require a multistate design or a targeted sensitivity analysis.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect observation and exit routes
The informative-inspection guide separates visit intensity from specimen risk. The joint timing-and-cause guide preserves interval and mechanism uncertainty.
Use uncertain exit causes and probabilistic attribution when the final mechanism label is also noisy.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and identifiability
Without validation, repeated measures or restrictions, state persistence, trigger error and visit dependence can imitate one another. A model may fit observed sequences well while assigning the explanation to the wrong layer.
Constant sensitivity and specificity are often too strong across temperature or ageing. Differential error tied to the future exit is especially consequential. Report sensitivity analyses that vary the confusion matrix and the latent link, not only standard errors under one fitted model.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive sensor firmware, thresholds, raw signals, planned and actual visits, trigger reasons, missed visits, validation selection, reference-assay uncertainty, exits and protocol changes. Match every measurement to the information available when the next visit was chosen.
Compare naive trigger counts, validation-adjusted counts, visit-weighted or joint models and specimen-level posterior maps. Check calibration by state prevalence, operating condition and replicate; inspect whether conclusions survive plausible false-negative rates.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Can a very specific trigger be treated as a mechanism?
No. High specificity reduces false positives but does not identify the physical cause, and sensitivity may still be poor.
Should triggered visits be deleted?
Usually not. Their timing is informative evidence; model why they occurred and preserve the common scheduled visits as a reference.
Does more validation always solve the problem?
Only if validation covers the relevant states and is itself credible. Positives-only validation leaves quiet target states unresolved.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Lange and colleagues develop a joint multistate and informative-observation model with misclassification. Han and colleagues model informative observation times and dropout. Nevo and colleagues study competing risks with uncertain event types. The ledgers above are original teaching examples.
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