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Fixed entry and exit thresholds can work at one scale and fail after variables, penalties or residuals are rescaled. Adaptive hysteresis uses a normalised margin and a measured noise floor so the state machine responds to evidence rather than units.
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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. Fixed widths hide a unit choice
A margin of 0.01 may be decisive when residuals are near 0.001 and meaningless when they are near 10. Changing units, penalty parameters or preconditioning can reverse the detector even when the underlying optimisation geometry is unchanged.
A scale-invariant rule compares the margin with a local reference scale. A noise-calibrated rule also estimates how much the margin fluctuates when the true state has not changed.
CHAPTER 2 OF 12 · DEFINE THE EVIDENCE
2. Define signal, scale and noise separately
Let m_k be a signed identification margin, s_k a positive problem scale and sigma_k a robust estimate of short-run fluctuation. A normalised score z_k=m_k/(s_k+epsilon) makes units visible; a noise ratio compares z_k with its estimated variation.
Entry, retention and exit rules should be functions of these quantities. Never let the detector estimate its noise from a window that contains a genuine transition without a reset or contamination check.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A rescaling ledger
The same underlying candidate is represented in two unit systems. A fixed raw threshold changes its verdict; a normalised score does not.
| Representation | Margin m | Scale s | Raw test m>0.02 | Normalised z=m/s |
|---|---|---|---|---|
| Original units | 0.030 | 1.00 | Enter | 0.030 |
| Values ×100 | 3.000 | 100.00 | Enter | 0.030 |
| Values ÷100 | 0.00030 | 0.010 | Do not enter | 0.030 |
| Decision using z>0.02 | — | — | Same in all units | Enter |
Scale invariance does not guarantee correctness; it removes one avoidable dependence on representation.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. Worked adaptive-width rule
Suppose the recent normalised-score noise estimate is sigma=0.004. Choose entry at +3 sigma, retention above +1 sigma and exit below −1 sigma, with a floor width of 0.006 and a ceiling of 0.030.
| Quantity | Calculation | Threshold |
|---|---|---|
| Noise-based entry | 3 × 0.004 | +0.012 |
| Noise-based retention | 1 × 0.004 | +0.004 |
| Apply width floor | max(0.012, 0.006) | +0.012 |
| High-noise example sigma=0.015 | min(3 × 0.015, 0.030) | +0.030 ceiling |
Require persistence and objective or feasibility gates in addition to the margin. The constants are design choices to validate, not universal values.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Estimate noise without chasing the state
Median absolute deviation of recent stable-state residual increments can be more robust than a standard deviation dominated by one spike. Separate numerical, stochastic-gradient and measurement noise when their time scales differ.
Freeze or reset the estimate after an exit, rejected step or detected regime change. If sigma grows beyond the ceiling, fall back to the global method and collect new evidence instead of silently widening hysteresis forever.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Keep adaptation slower than switching
If thresholds react instantly to every margin, the detector can reproduce the same chatter in another form. Smooth the noise estimate, require a minimum window and update widths only after accepted global steps.
Log both raw and normalised margins, width changes and state transitions. A stable discrete label with a wildly moving threshold is not a convincing identification certificate.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: low-rank manifolds
Transfer the rule to rank detection using a spectral-gap margin divided by a matrix norm or leading singular value. Estimate fluctuation from accepted iterates under the same sampling and precision regime.
The transfer check handles repeated singular values, matrix rescaling and changing regularisation. A collapsing spectral gap should force fallback even when hysteresis would otherwise preserve the old rank.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connect the manifold branch
The hysteretic-identification guide supplies the state-machine logic. The degenerate-manifold guide explains when identification assumptions fail.
Use the identification-time guide to separate early candidate detection from evidence strong enough to activate local acceleration.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
Adaptive hysteresis can suppress real switches when noise is overestimated and can chatter when noise is underestimated. A scale normaliser can become unstable near zero or inappropriate across heterogeneous coordinates.
Noise calibration is an algorithmic safeguard, not a proof of finite identification. Theorems still depend on convergence, partial smoothness, nondegeneracy and the specific optimisation method.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive raw margins, normalisers, epsilon values, noise windows, robust scale estimators, floors, ceilings, multipliers, persistence counts, cooldown rules, override gates, rejected steps and memory resets.
Test unit rescaling, variable preconditioning, several noise levels, genuine manifold changes and deliberately degenerate cases. Compare wall-clock time and accepted global progress, not only the number of label flips.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why use both a floor and a ceiling?
The floor prevents false certainty from a tiny noise estimate; the ceiling sends the method to fallback when evidence is too noisy for useful identification.
Is median absolute deviation always best?
No. It is one robust option. The estimator must match dependence, window length and the type of noise present.
Does normalisation remove all tuning?
No. It makes thresholds comparable across simple rescalings, but multipliers, windows and gates still require validation.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Hare and Lewis develop active-manifold identification through partial smoothness. Liang, Fadili and Peyré establish finite activity identification for forward–backward methods. Noël and Schoukens study nonlinear state-space identification of hysteresis. The adaptive thresholds and ledgers above are original algorithmic safeguards rather than replacements for identification theorems.
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