Uncertainty Ontology

FLOW ALGEBRA
Ontology: Uncertainty Ontology
Version: 1.0
Date: 2026-10-14
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/measurement-and-dimensional-entities/uncertainty-ontology.html
Authors: Mark Higgins — Sky (Recursive AI)
Domain: Mathematical Entities > Measurement and Dimensional Entities

PURPOSE
Represent uncertainty attached to measured or inferred quantities, distinguish uncertainty from error and tolerance, and enable traceable propagation, comparison, validation and AI reasoning.

TYPED ENTITIES
entity Uncertainty {id: URI, kind: UncertaintyKind, value: Real?, unit: Unit?, confidence_level: Real?, coverage_factor: Real?, distribution: Distribution?, method: Method?, provenance: Source?, timestamp: DateTime?}
entity Measurement {id: URI, measurand: Quantity, estimate: Real, unit: Unit, uncertainty: Uncertainty?, conditions: Context?}
entity StandardUncertainty {standard_deviation: NonNegativeReal, unit: Unit}
entity ExpandedUncertainty {magnitude: NonNegativeReal, coverage_factor: PositiveReal, unit: Unit}
entity RelativeUncertainty {ratio: NonNegativeReal}
entity UncertaintyBudget {components: List[UncertaintyComponent], covariance_matrix: Matrix?, combined_standard_uncertainty: NonNegativeReal?}
entity UncertaintyComponent {source: Source, estimate: NonNegativeReal, evaluation_type: A|B, sensitivity_coefficient: Real?, distribution: Distribution?, degrees_of_freedom: Real?}
entity CoverageInterval {lower: Real, upper: Real, coverage_probability: Real?, unit: Unit}
entity Correlation {quantity_a: Quantity, quantity_b: Quantity, coefficient: Real}
entity MeasurementModel {output: Quantity, inputs: List[Quantity], function: -Expression}

RELATIONSHIPS
Measurement --HAS_UNCERTAINTY--> Uncertainty
Uncertainty --QUALIFIES--> Quantity
Uncertainty --DERIVED_FROM--> UncertaintyBudget
UncertaintyComponent --CONTRIBUTES_TO--> UncertaintyBudget
Uncertainty --EXPRESSED_IN--> Unit
Uncertainty --ESTIMATED_BY--> Method
CoverageInterval --DESCRIBES_COVERAGE_OF--> Quantity
MeasurementModel --PROPAGATES--> Uncertainty
Uncertainty --HAS_PROVENANCE--> Source

CORE SEMANTICS
- Uncertainty quantifies dispersion of values attributable to a measurand, based on available information.
- Measurement error is the difference between a measured value and a reference value; it is not synonymous with uncertainty.
- Precision, accuracy, tolerance, confidence interval and measurement uncertainty are distinct concepts.
- A Type A evaluation uses statistical analysis of observations; a Type B evaluation uses other available information.
- Standard uncertainty is expressed as a standard deviation; expanded uncertainty is U = k * u_c with coverage factor k.
- Coverage probability must not be inferred from k alone without distributional assumptions and relevant degrees of freedom.
- A tolerance is an acceptance specification, not automatically a measurement uncertainty.

FORMAL OPERATIONS
standard_uncertainty(x) -> u(x) >= 0
relative_uncertainty(x) -> u(x)/abs(x), provided x != 0
expanded_uncertainty(x,k) -> U(x)=k*u_c(x), k>0

For y=f(x1,...,xn), first-order propagation:
u_c(y)^2 = sum_i sum_j (df/dxi)(df/dxj) cov(xi,xj)
Assumptions: locally differentiable model, valid linear approximation, finite input covariances.
For independent inputs: cov(xi,xj)=0 when i != j.
For y=a*x+b with exact a,b: u(y)=abs(a)*u(x).
For independent x,z with y=x+z: u(y)=sqrt(u(x)^2+u(z)^2).
For correlated x,z with y=x+z: u(y)^2=u(x)^2+u(z)^2+2*cov(x,z).
For an interval [l,h]: require l<=h; do not equate interval half-width with standard uncertainty without a stated probability model.

INFERENCE RULES
rule DIMENSION_MATCH:
  IF uncertainty U qualifies quantity Q THEN dimension(U)=dimension(Q) for absolute uncertainty.
rule RELATIVE_DIMENSIONLESS:
  IF U is relative uncertainty THEN dimension(U)=1.
rule NONNEGATIVE:
  IF U is standard or expanded uncertainty THEN magnitude(U)>=0.
rule COVARIANCE_SYMMETRY:
  IF C is a covariance matrix THEN C=transpose(C) AND C is positive semidefinite.
rule COVERAGE_BOUNDS:
  IF coverage probability p is supplied THEN 0<=p<=1.
rule PROVENANCE_REQUIRED:
  IF uncertainty is used for a scientific decision THEN record estimation method, assumptions, units and source.
rule PROPAGATION_CONTEXT:
  IF nonlinear propagation is significant THEN consider Monte Carlo or higher-order methods rather than assuming first-order adequacy.

VALIDATION
- Every uncertainty estimate identifies the corresponding measurand and unit, except dimensionless relative uncertainty.
- Record whether uncertainty is standard, combined, expanded, relative or interval-based.
- Require k>0 when expanded uncertainty is reported.
- Do not report a coverage probability without its model or justification.
- Validate unit conversions: absolute uncertainty scales by absolute conversion factor for affine unit conversions; offsets do not apply to uncertainty magnitudes.
- Preserve correlation information when combining measurements.
- Avoid claiming certainty from an interval or a rounded display value.

WORKED EXAMPLES
Example A: measured length 10.00 m with standard uncertainty 0.05 m.
  measurement.value=10.00 m; uncertainty.kind=standard; uncertainty.value=0.05 m.
  relative_standard_uncertainty=0.005 = 0.5%.
Example B: independent lengths a=3.0 +/- 0.1 m and b=4.0 +/- 0.2 m (standard uncertainties).
  s=a+b=7.0 m; u(s)=sqrt(0.1^2+0.2^2)=0.2236068 m.
Example C: combined standard uncertainty u_c=0.3 K, k=2.
  expanded uncertainty U=0.6 K; coverage probability unspecified pending distributional justification.

SEMANTIC_EQUIVALENCE
standard measurement uncertainty ~= standard uncertainty (when measurand context agrees)
combined standard uncertainty != expanded uncertainty
measurement uncertainty != measurement error
uncertainty interval != confidence interval (unless construction and interpretation justify equivalence)
tolerance != uncertainty

CONFIDENCE
confidence.semantic_definition: 0.99
confidence.formula_applicability: conditional_on_assumptions
confidence.source_alignment: unverified_against_live_page

FLOW_HASH
algorithm: SHA-256
scope: canonical UTF-8 content excluding FLOW_HASH value
value: compute_at_ingestion

UPDATE_MODEL
version: 1.0
update_trigger: change in measurement conventions, ontology schema, assumptions or source page
validation: check mathematical definitions, dimensional coherence, examples and provenance

AI_CONSUMPTION
representation: typed semantic graph + deterministic mathematical rules
operations: parse, normalise, validate, propagate, compare, infer, explain
required_context: measurand, units, uncertainty type, assumptions, source
output: estimate, uncertainty, units, derivation, limitations, provenance
safety: do not present unsupported precision or coverage claims

END FLOW ALGEBRA