Decay Curve Fitting¶
Statistical model — instantiates Activation Decay Measurement
Fits delayed-probe observations to a practical decay curve and half-life, turning scattered readings into a timing model good enough to act on.
A handful of probe readings at scattered delays is not yet a timing model — you cannot read off "how long until it's half gone" from three dots. Decay Curve Fitting is the step that draws a line through those dots: it fits the delayed-probe observations to a decay function and extracts a compact summary — most usefully an activation half-life, the interval over which the effect drops to half its fresh value. Its defining move is interpolation into a continuous shape: it takes discrete "how much at t" measurements and produces a curve you can evaluate at any time in between, plus a single characteristic number that makes different primes comparable. It deliberately stops short of judgement — it says how fast the thing fades, not whether the remaining level is still enough, and not when to act.
Example¶
A language-learning app wants to know how fast newly taught vocabulary fades so it can time review. From its probe data it has recall accuracy for a word set at 1 day (92%), 4 days (74%), 10 days (58%), and 21 days (41%). Decay Curve Fitting takes these four points and fits an exponential-style forgetting curve — the shape long associated with Ebbinghaus's retention work[n1] — and reports the fit: recall halves roughly every eleven days, with the curve steepest in the first few days and flattening after.
The output is a function, R(t), and one headline number: an ≈11-day half-life. That is all it produces. It does not declare that 60% recall is "good enough," and it does not schedule a review — those belong to thresholding and cadence. What it has done is turn four lonely measurements into a model the rest of the pipeline can query at any delay.
How it works¶
- Pick a functional form proportionate to the data — exponential decay for a single fading trace, a two-term form when a fast and a slow component are both present, or a plain piecewise-linear rule when the points are few.
- Fit to the probe observations, weighting points by how trustworthy each probe was.
- Extract the half-life (or an equivalent time-constant) as the model's portable summary.
- Report the fit quality and the valid range, flagging that the curve is only trustworthy inside the interval that was actually probed.
Tuning parameters¶
- Model family — exponential vs. power-law vs. piecewise. Richer families fit better but overfit sparse data and invite false precision.
- Number of parameters — a one-parameter half-life is robust and portable; a multi-term fit captures shape but needs many probe points.
- Extrapolation horizon — how far past the last probe you allow the curve to be read. Wider extrapolation is convenient and dangerous.
- Fit-quality gate — the minimum goodness-of-fit before the curve is trusted rather than treated as a guess.
- Uncertainty band — a point curve vs. a fitted curve carrying confidence bands the downstream rule can respect.
When it helps, and when it misleads¶
Its strength is compression: a whole decay trajectory collapses into one number that lets you compare primes ("this campaign's half-life is a week; that lesson's is a month") and query persistence at delays you never actually probed. Its failure mode is extrapolation beyond the data — a curve fit to three-week-old observations says nothing reliable about three months, yet the smooth line invites exactly that reach, and a tidy half-life lends false precision to what may be a two-point guess. The classic misuse is assuming a single universal forgetting curve applies to every prime and audience, ignoring that shape varies with material, load, and context. The guarding discipline is to fit no more parameters than the probe points support, mark the valid range explicitly, and carry the uncertainty band forward instead of collapsing to a single confident line.
How it implements the components¶
decay_curve_model— its primary product: the fitted, evaluable-at-any-t representation of how activation declines.activation_half_life_estimator— extracts the half-life (or time-constant) that summarises the curve and makes primes comparable.
It models shape, not sufficiency: it does NOT implement usable_activation_threshold or intervention_window_rule — deciding what level still counts as usable and when to act on the curve is Activation Window Thresholding, its nearest twin, which consumes this fitted curve.
Related¶
- Instantiates: Activation Decay Measurement — this model is the archetype's curve-and-half-life core.
- Consumes: Time-Lagged Activation Probe supplies the delayed observations the fit is built on.
- Sibling mechanisms: Time-Lagged Activation Probe · Activation Window Thresholding · Refresh Cadence Adaptation · Contextual Reactivation · Decay Segment Comparison · Staleness Boundary Enforcement
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Decay Curve Fitting operates as a computation, comparison, model, or analytic representation used to infer, estimate, or choose because it fits delayed-probe observations to a practical decay curve and half-life, turning scattered readings into a timing model good enough to act on.
Independent corroboration: The frozen evidence defines Decay Curve Fitting as 'Fits delayed-probe observations to a practical decay curve and half-life, turning scattered readings into a timing model good enough to act on', so its operative form is Analysis, Modeling & Optimization.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Psychology
Origin pattern: Convergent development
Present-day reach: Multi-domain
Rationale: Experimental memory research cohered forgetting curves that fit retention over delay and summarize fading with a characteristic rate or half-life.
Related originating lineages:
- Cognitive Science — Activation and retrieval models supplied continuous accounts of cue and memory decay.
- Statistics & Experimental Design — Regression and nonlinear curve fitting supplied parameter estimation and uncertainty from discrete probe observations.
Review resolution: Experimental memory research cohered forgetting curves that fit retention over delay and summarize fading with a characteristic rate or half-life.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] The Ebbinghaus forgetting curve is the classic finding that retention of newly learned material drops sharply at first and then levels off, often modelled as roughly exponential. It is the canonical shape a decay fit reaches for, but it is a starting hypothesis to fit against, not a law to assume. ↩