Gompertz function
Enable an agent to recognise a Gompertz function, interpret its parameters and growth behaviour, and judge whether its use for a particular dataset or decision is justified.
Bundle → Layer → Finding → Questions Filled
5 bundles · 9 layers · 14 findings · 22 questions
Functional identity Establish the exact function represented by the Gompertz name.
Equation and convention
Record the equation, restrictions and coordinate conventions.
Reference form membership
The standard increasing form is y(t) = A exp(-B exp(-kt)) with positive A, B and k; departures require explicit interpretation.
- What exact equation, parameter restrictions and independent-variable domain define this instance? definition
- Which source establishes the chosen parameter convention, including any offset or response transformation? provenance
Equivalence and neighbours
Distinguish reparameterisation from a change of function family or modelling role.
Parameter mapping
The form y(t) = A exp(-exp(-k(t-t_i))) is equivalent to the reference form when B = exp(kt_i). Distribution and mortality uses require separate semantic identification.
- Can the proposed equation be mapped algebraically to the reference form, and what is the mapping? definition
- Does this use concern a growth response, a distribution quantity or a mortality hazard, and which neighbouring model owns the additional semantics? boundary
Growth geometry Describe the constraints that the Gompertz shape imposes on a trajectory.
Limits and inflection
Locate the limiting levels and the transition between acceleration and deceleration.
Asymmetric sigmoid
On the full real line, the reference curve increases from zero toward A and inflects at (log(B)/k, A/e); a restricted observation interval may contain only one growth phase.
- Where are the inflection and asymptotes relative to the observed interval? measurement
- Does the response support a zero lower asymptote and Gompertz asymmetry, or is another formulation needed? boundary
Rates and dynamics
Connect the curve with its absolute and relative growth rates.
Gompertz rate law
The reference curve satisfies y' = ky log(A/y), with exponentially declining relative growth and maximum absolute slope Ak/e.
- What absolute and relative growth rates does the fitted function imply across the relevant interval? measurement
- Is this differential equation being used as a descriptive identity or as an independently justified process law? boundary
Estimation and evidence Establish how observations support the function and its parameters.
Observation and fitting
Specify the response scale, observation errors and fitting procedure.
Fit semantics
Fitting raw responses, log responses or double-log transformed responses generally implies different error assumptions and objectives.
- What response scale, error model, weighting and parameter constraints define the fit? definition
- How are zero values, censoring, repeated observations and measurement limits handled when present? action
Identifiability and validation
Assess whether available data constrain the parameters and support the intended predictions.
Parameter support
Limited coverage of growth phases can leave the asymptote and associated parameters weakly constrained despite small fitting residuals.
- How do uncertainty estimates and parameter dependence change when the asymptote is fixed, freed or informed by additional observations? measurement
- What residual checks and held-out predictions distinguish this fit from plausible alternative growth curves? action
Prediction and use boundaries Translate the function into justified calculations and decisions.
Evaluation and inversion
Support forward predictions and inverse threshold calculations.
Threshold reachability
For 0 < y_star < A, the reference curve reaches y_star at t = (log(B) - log(-log(y_star/A)))/k; thresholds at the asymptotes have no finite crossing time.
- Is the requested threshold reachable within the declared domain and supported prediction interval? boundary
- How will parameter uncertainty be propagated into the predicted crossing time, especially near A? action
Application validity
Set the empirical and interpretive limits of using the curve.
Conditional extrapolation
Long-range predictions assume the selected Gompertz formulation remains applicable; a fit alone does not establish a causal growth mechanism or guarantee a real-world ceiling.
- What evidence supports continuing the same growth regime beyond the observed interval? provenance
- Which observed departures, such as decline, multiple growth waves or a shifting ceiling, should trigger replacement or extension of the model? action
Evidence and external alignment What the world already says about this thing, gathered so the model can be checked against it.
Reported evidence
Findings from the breadth pass, kept separate from the structural claims.
Check these first
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these check these first hold for the sense of Gompertz function this model covers, and on what evidence? provenance
Kinds and varieties
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these kinds and varieties hold for the sense of Gompertz function this model covers, and on what evidence? provenance
Real-world use
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these real-world use hold for the sense of Gompertz function this model covers, and on what evidence? provenance
Typical measurements
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these typical measurements hold for the sense of Gompertz function this model covers, and on what evidence? provenance
Failure modes and hazards
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these failure modes and hazards hold for the sense of Gompertz function this model covers, and on what evidence? provenance
Neighbouring kinds and how to tell them apart
Recalled without web access and unsourced; every item is a lead to verify.
- Which of these neighbouring kinds and how to tell them apart hold for the sense of Gompertz function this model covers, and on what evidence? provenance
Classifiers Filled
- Family
- Thing Registry
- Category
- Cross-cutting context
- Entry kind
- thing
- Plane
- XCT
- Domain
- XCT.QLT
What it is Filled
The Gompertz function is an asymmetric sigmoidal function, commonly written y(t) = A exp(−b exp(−kt)) with A, b, k > 0, that describes growth from a limiting value of zero toward an upper asymptote A.
In scope
- Canonical increasing Gompertz functions and explicitly mapped equivalent parameterisations
- Asymptotes, derivatives, inflection points and asymmetric growth behaviour
- Parameter meanings, units, coordinate dependence and admissible values
- Initial conditions and the associated Gompertz growth differential equation
- Fitting, parameter identifiability, uncertainty and application suitability
- Explicitly identified offsets, reversals and modified Gompertz formulations
Out of scope
- The Gompertz probability distribution as a complete statistical model
- Gompertz and Gompertz-Makeham mortality laws as demographic models
- Logistic, Richards and other growth-function families except for boundary comparisons
- Biological or economic mechanisms inferred solely from a fitted curve
- Data collection systems and general-purpose optimisation software
Why it exists Filled
Enable an agent to recognise a Gompertz function, interpret its parameters and growth behaviour, and judge whether its use for a particular dataset or decision is justified.
Distinguishing features Filled
- For the reference form, log(-log(y/A)) = log(B) - kt wherever 0 < y < A; this double-log relationship is linear in the independent variable.
- The increasing reference curve has an inflection at y = A/e, whereas the standard zero-baseline logistic curve has its inflection at A/2.
- Its relative growth rate y'/y = Bk exp(-kt) decreases exponentially, while absolute growth first rises and then falls.
- Its differential equation is y' = ky log(A/y), distinguishing it from the standard logistic equation y' = ry(1-y/A).
- Its zero and A asymptotes are approached at opposite ends of the real line, with a curve that lacks logistic point symmetry about its inflection.
What robots and AI may and may not do Filled
Must not
- Present a fitted curve as a reliable forecast outside the range of the data.
- Use mortality fits to judge an individual person's lifespan or insurability without a lawful and fair basis.
- Hide poor fit or alternative models that would change a conclusion.
Only with a human decision
- Using a fitted model in medical, epidemic or insurance decisions about people.
May
- Fit and explain the Gompertz function for growth or mortality data, stating parameters and fit quality.
- Compare it with other sigmoid models when choosing a model for a dataset.
Moral aspects Filled
- Growth and mortality models drive decisions about tumours, epidemics and insurance that affect people's lives.
Who is affected
- Patients
- Insured people
- Researchers
Owners Filled
Steward
Nobody owns it; it is a mathematical function.
Links to other meta-models Filled
parent
- Q11348 - registry parent class
What else AI and robots need to interact with it Filled
Identity and identifiers required Filled
- Vercy registry: vr.tr.gompertz-function
- Wikidata: Q1011785 (https://www.wikidata.org/wiki/Q1011785)
Direct properties not applicable Not applicable
- Declared functional form: Exact equation and parameter convention; reference form, equivalent form or explicitly modified form
- Asymptotic level A: A > 0; same units as y in the reference form
- Position parameter B: B > 0; dimensionless in the reference form
- Rate parameter k: k > 0; inverse units of t
- Inflection coordinate: t_i = log(B)/k; same units as t
- Maximum absolute slope: Ak/e; units of y per unit of t
- Coordinate and response interpretation: Independent variable, origin, units, response quantity and any transformation
- Evidence-supported use: Unassessed, descriptively supported, predictively validated or unsuitable for the declared use
Plane XCT: no invented physical properties.
Recognition optional Filled
- An S-shaped curve that rises slowly at first, steeply in the middle and approaches its upper limit more gradually than it left zero.
Capabilities and actions required Filled
- Recognise equivalent Gompertz equations and translate their parameters into a declared convention.
- Compute levels, slopes, inflection coordinates and admissible threshold-crossing times.
- Fit a Gompertz curve with an explicit observation model and estimate uncertainty.
- Check whether observations constrain the asymptote, rate and inflection sufficiently for the intended use.
- Compare Gompertz behaviour with logistic or other candidate growth curves.
- Flag extrapolations or mechanistic interpretations that exceed the evidence.
Hazards and failure modes required Filled
- Extrapolating a fitted curve can badly misjudge future growth or decline.
- Parameter estimates can be unstable with short or noisy data.
Standards and interfaces not applicable Not applicable
Not applicable
No generally applicable standard, law or official classification governs the Gompertz function itself.
Context of use required Filled
- The Gompertz function is used to represent asymmetric growth curves in biological populations, organism growth and other processes approaching a limiting size.
- Analysts fit it to observations to estimate growth timing and an upper asymptote, with predictions depending on the parameter convention and suitability of the model.
Sources Filled
- Wikidata item Q1011785: Gompertz function - identity and sense of the item
- Wikipedia: Gompertz function - general description of the item
Open questions
- Which authoritative sources should anchor the registry definition and distinguish the Gompertz function from the related distribution and mortality law?
- Should baseline-shifted, decreasing and application-specific modified Gompertz equations be owned here as variants or linked to separate registered concepts?
- Which parameter conventions used across application fields require explicit mappings, particularly conventions labelled growth rate or lag time?
- What evidence and uncertainty criteria should govern acceptance of asymptote estimates and threshold predictions in each intended application?
Machine files
Provenance
thing registry research (pass 2) · unreviewed
Built from: models/things/publications/thing-q1011785/spec.json