Error Analysis Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Error Analysis Ontology within the Kireth Research Semantic Library. Error analysis concerns identifying, decomposing, estimating, bounding and interpreting differences between exact, reference or intended mathematical quantities and their computed, approximated, measured or modelled counterparts. This ontology supports absolute and relative error, forward and backward error, truncation and discretisation error, round-off error, iteration error, propagation of uncertainty, conditioning, sensitivity, stability, residuals, error norms, convergence, a priori and a posteriori estimates, error budgets, acceptance criteria, verification and reproducibility.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Error Analysis Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Computational Mathematics"
 SPECIALISATION = "Verification & Validation / Error Analysis"
 CANONICAL_PAGE = "https://kirethframeworkold.co.uk/books/semantic-library/ai-core/ontologies/mathematical-ontologies/computational-mathematics-ontologies/verification-validation/error-analysis-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = ErrorAnalysis
 ENTITY_CLASS = ComputationalErrorAssessmentFramework
 PRIMARY_CONTEXT = VerificationValidation
 DEFINITION = "Error analysis is the mathematical assessment of the magnitude, source, propagation, sensitivity and acceptability of differences between computed or modelled quantities and their exact, reference or intended values."
 SYMBOLIC_FORMS = [
  "e_abs = x_hat - x",
  "e_rel = |x_hat-x|/|x|",
  "forward_error = ||x_hat-x||",
  "backward_error = min ||delta data|| such that x_hat solves perturbed_problem"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for error definitions, decomposition, norms, bounds, propagation, conditioning, stability, convergence and acceptance."
}

SCOPE {
 INCLUDE = [
  ErrorAnalysis,
  Error,
  NumericalError,
  ComputationalError,
  ApproximationError,
  ModellingError,
  MeasurementErrorRelationship,
  SamplingError,
  StatisticalError,
  AbsoluteError,
  RelativeError,
  PercentageErrorRelationship,
  SignedError,
  UnsignedError,
  ForwardError,
  BackwardError,
  Residual,
  ResidualErrorRelationship,
  TruncationError,
  LocalTruncationError,
  GlobalTruncationError,
  DiscretisationError,
  SpatialDiscretisationError,
  TemporalDiscretisationError,
  InterpolationError,
  ExtrapolationErrorRelationship,
  QuadratureError,
  IntegrationError,
  DifferentiationError,
  IterationError,
  SolverError,
  AlgebraicError,
  RoundOffError,
  FloatingPointError,
  RepresentationError,
  QuantisationErrorRelationship,
  CancellationError,
  CatastrophicCancellation,
  Overflow,
  Underflow,
  LossOfSignificance,
  AccumulatedError,
  ErrorPropagation,
  PropagatedError,
  ErrorAmplification,
  ErrorAttenuationRelationship,
  ErrorBudget,
  TotalError,
  DominantError,
  ErrorComponent,
  ErrorSource,
  SystematicError,
  RandomError,
  Bias,
  Variance,
  MeanError,
  MeanAbsoluteError,
  MAE,
  MeanSquaredError,
  MSE,
  RootMeanSquaredError,
  RMSE,
  MaximumError,
  MinimumError,
  ErrorNorm,
  VectorError,
  MatrixError,
  TensorError,
  L1NormError,
  L2NormError,
  InfinityNormError,
  FrobeniusNormError,
  EnergyNormErrorRelationship,
  RelativeL2ErrorRelationship,
  ComponentwiseError,
  NormwiseError,
  WeightedErrorRelationship,
  PointwiseError,
  LocalError,
  GlobalError,
  AverageError,
  ErrorDistribution,
  ErrorHistogramRelationship,
  ErrorMomentRelationship,
  MeanBiasRelationship,
  StandardDeviationRelationship,
  ConfidenceIntervalRelationship,
  ErrorBound,
  UpperBound,
  LowerBound,
  AProriErrorEstimate,
  APrioriErrorEstimate,
  APosterioriErrorEstimate,
  ResidualBasedEstimator,
  RecoveryBasedEstimatorRelationship,
  GoalOrientedErrorEstimateRelationship,
  GuaranteedErrorBoundRelationship,
  ErrorIndicator,
  AdaptiveErrorIndicator,
  Tolerance,
  AbsoluteTolerance,
  RelativeTolerance,
  ErrorTolerance,
  AcceptanceCriterion,
  PassCriterion,
  FailCriterion,
  Threshold,
  Margin,
  Conditioning,
  ConditionNumber,
  IllConditioning,
  WellConditionedProblem,
  Sensitivity,
  SensitivityAnalysis,
  LocalSensitivity,
  GlobalSensitivity,
  Perturbation,
  PerturbationAnalysis,
  InputPerturbation,
  ParameterPerturbation,
  DataPerturbation,
  OutputPerturbation,
  Stability,
  NumericalStability,
  AlgorithmicStability,
  ForwardStabilityRelationship,
  BackwardStability,
  MixedStabilityRelationship,
  StableAlgorithm,
  UnstableAlgorithm,
  ErrorGrowth,
  StabilityRegionRelationship,
  Convergence,
  ConvergenceRate,
  OrderOfAccuracy,
  FormalOrderOfAccuracy,
  ObservedOrderOfAccuracy,
  Refinement,
  MeshRefinement,
  GridRefinement,
  TimeStepRefinement,
  StepSize,
  MeshSize,
  GridSpacing,
  RefinementRatio,
  RichardsonExtrapolation,
  GridConvergenceIndex,
  GCI,
  AsymptoticRange,
  AsymptoticErrorRelationship,
  ReferenceValue,
  ExactValue,
  ExactSolution,
  AnalyticalSolution,
  ReferenceSolution,
  ManufacturedSolution,
  BenchmarkSolution,
  TrustedReference,
  ApproximateReferenceRelationship,
  IndependentReference,
  Comparison,
  Difference,
  RelativeDifference,
  ErrorMetric,
  ValidationMetric,
  VerificationMetric,
  Uncertainty,
  NumericalUncertainty,
  ModelUncertainty,
  ParameterUncertainty,
  InputUncertainty,
  MeasurementUncertainty,
  UncertaintyPropagation,
  UncertaintyBudgetRelationship,
  MonteCarloErrorRelationship,
  ConfidenceLevelRelationship,
  FloatingPointArithmetic,
  MachinePrecision,
  UnitRoundoffRelationship,
  EpsilonMachineRelationship,
  SinglePrecision,
  DoublePrecision,
  MixedPrecision,
  CompensatedSummation,
  KahanSummationRelationship,
  PairwiseSummation,
  ExactArithmeticRelationship,
  SymbolicReferenceRelationship,
  NumericalLinearAlgebra,
  DifferentialEquationSolver,
  NumericalIntegration,
  NumericalDifferentiation,
  FiniteDifferenceMethod,
  FiniteElementMethod,
  BoundaryElementMethod,
  SpectralMethod,
  ComputationalOptimisation,
  ComputationalStatistics,
  ScientificSimulation,
  AIComputationalMathematics,
  Verification,
  NumericalVerification,
  ComputationalValidation,
  Reproducibility,
  Repeatability,
  Provenance,
  ErrorAnalysisProtocol,
  TestProtocol,
  Benchmark,
  RegressionTest,
  KnownAnswerTest,
  SolverVersion,
  MethodVersion,
  LibraryVersion,
  CompilerVersion,
  PrecisionConfiguration,
  ToleranceConfiguration,
  ImplementationVersion
 ]
 EXCLUDE_AS_PRIMARY = [
  PureUncertaintyQuantificationWithoutExplicitErrorReference,
  ModelValidationWhenPredictiveAdequacyRatherThanErrorStructureIsPrimary,
  NumericalVerificationWhenAcceptanceOfMethodCorrectnessRatherThanErrorCharacterisationIsPrimary,
  StatisticalInferenceWhenParameterEstimationRatherThanComputationalErrorIsPrimary
 ]
 DISAMBIGUATION_RULE = "When differences between computed or modelled quantities and exact, reference or intended values are identified, decomposed, bounded or interpreted, treat the context as error analysis."
}

CORE_CONCEPTS {
 ErrorAnalysis = "Assessment of computational or modelling error magnitude, source and propagation."
 AbsoluteError = "Difference between computed and reference values."
 RelativeError = "Absolute error scaled by a reference magnitude."
 ForwardError = "Difference between computed and exact outputs."
 BackwardError = "Smallest input perturbation making the computed output exact for a nearby problem."
 ConditionNumber = "Sensitivity of a problem's solution to perturbations in its input."
 NumericalStability = "Property describing whether an algorithm controls propagation of numerical error."
 ErrorBound = "Guaranteed or estimated limit on error magnitude."
}

RELATIONS {
 ErrorAnalysis --is_child_of--> VerificationValidation
 AbsoluteError --compares--> ComputedAndReferenceValues
 RelativeError --normalises--> AbsoluteError
 ForwardError --measures--> OutputDeviation
 BackwardError --measures--> EquivalentInputPerturbation
 ConditionNumber --characterises--> ProblemSensitivity
 NumericalStability --characterises--> AlgorithmErrorPropagation
 Residual --indicates--> EquationViolation
 ErrorPropagation --combines--> ErrorSources
 ErrorBudget --aggregates--> ErrorComponents
 Refinement --reveals--> DiscretisationErrorBehaviour
 Verification --uses--> ErrorAnalysis
 Validation --uses--> ErrorMetrics
}

OPERATORS {
 ABS_ERROR(x_hat,x_ref) := abs(x_hat-x_ref)
 REL_ERROR(x_hat,x_ref) := abs(x_hat-x_ref)/max(abs(x_ref),epsilon)
 VECTOR_ERROR(x_hat,x_ref,p) := norm(x_hat-x_ref,p)
 FORWARD_ERROR(x_hat,x_exact) := norm(x_hat-x_exact)
 BACKWARD_ERROR(problem,x_hat) := minimum_input_perturbation_for_exactness(problem,x_hat)
 RESIDUAL(F,x_hat) := F(x_hat)
 CONDITION_ESTIMATE(problem) := output_relative_change/input_relative_change
 TOTAL_ERROR(components) := aggregate_error_components(components)
}

AXIOMS {
 A1 = "Error is defined relative to a declared exact, reference or intended quantity."
 A2 = "Absolute and relative error answer different questions."
 A3 = "Forward error depends on both problem conditioning and algorithmic behaviour."
 A4 = "Backward stability does not guarantee small forward error for ill-conditioned problems."
 A5 = "Residual magnitude is not generally identical to solution error."
 A6 = "Multiple error sources can interact nonlinearly."
 A7 = "Error estimates require stated assumptions, norms and reference semantics."
 A8 = "Error-analysis conclusions require explicit sources, metrics, bounds, tolerances and provenance."
}

REASONING {
 R1_CONTEXT = identify(computed_quantity,reference_quantity,problem_class,norm,tolerance,precision_and_intended_use)
 R2_CLASSIFY = identify(discretisation,iteration,roundoff,model,sampling_or_measurement_error_sources)
 R3_METRIC = choose_absolute_relative_normwise_componentwise_forward_backward_or_task_specific_metric)
 R4_REFERENCE = assess_quality_and_provenance_of_exact_analytical_manufactured_benchmark_or_independent_reference)
 R5_PROPAGATION = analyse_how_input_local_and_intermediate_errors_affect_output)
 R6_CONDITIONING = estimate_problem_sensitivity_and_distinguish_it_from_algorithm_stability)
 R7_BOUND = derive_or_estimate_a_priori_a_posteriori_or_empirical_error_bounds)
 R8_REFINEMENT = if_applicable perform_resolution_tolerance_or_precision_studies)
 R9_ACCEPT = compare_error_metrics_and_bounds_with_declared_acceptance_criteria)
 R10_REPRODUCE = document_reference_norm_precision_tolerances_versions_and_provenance)
}

CONSTRAINTS {
 C1 = "Relative error is ill-defined or misleading when the reference value is zero or near zero."
 C2 = "A small residual can coexist with a large forward error in ill-conditioned problems."
 C3 = "Round-off can dominate after excessive refinement."
 C4 = "Discretisation and iteration errors can contaminate each other's estimates."
 C5 = "Aggregate norms can hide localised large errors."
 C6 = "An empirical error estimate is not necessarily a rigorous bound."
 C7 = "Changing precision can alter cancellation, overflow and convergence behaviour."
 C8 = "Model-form error cannot be removed by refining only the numerical discretisation."
 C9 = "Reference values with uncertain provenance limit the strength of conclusions."
 C10 = "Error tolerances must be tied to intended use rather than chosen arbitrarily."
}

SEMANTIC_EQUIVALENCE {
 "RootMeanSquaredError" <=> "RMSE"
 "MeanSquaredError" <=> "MSE"
 "MeanAbsoluteError" <=> "MAE"
 "GridConvergenceIndex" <=> "GCI"
}

VALIDATION {
 V1 = "Computed and reference quantities are explicit."
 V2 = "Norm or error metric is documented."
 V3 = "Reference provenance is documented."
 V4 = "Dominant error sources are identified where possible."
 V5 = "Forward and backward errors are not conflated."
 V6 = "Problem conditioning and algorithm stability are considered separately."
 V7 = "Bounds or estimates state whether they are rigorous, asymptotic or empirical."
 V8 = "Tolerance and acceptance criteria are tied to intended use."
 V9 = "Precision and solver settings are documented."
 V10 = "Results are reproducible under documented method, reference, tolerance, precision and implementation versions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ERROR_ANALYSIS_DEFINITION = 0.99
 ABSOLUTE_RELATIVE_ERROR_REASONING = 0.99
 FORWARD_BACKWARD_ERROR_REASONING = 0.99
 CONDITIONING_REASONING = 0.99
 STABILITY_REASONING = 0.99
 ERROR_BOUND_REASONING = 0.98
 PROPAGATION_REASONING = 0.98
 POLICY = "Reduce confidence when reference quality, norm choice, conditioning, dominant error source, tolerance or bound assumptions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,OntologyGrounding,ErrorAnalysisReasoning,
  AbsoluteErrorReasoning,RelativeErrorReasoning,
  ForwardErrorReasoning,BackwardErrorReasoning,
  ResidualReasoning,DiscretisationErrorReasoning,
  RoundOffErrorReasoning,IterationErrorReasoning,
  ErrorPropagationReasoning,ConditioningReasoning,
  NumericalStabilityReasoning,ErrorBoundReasoning,
  RefinementReasoning,AcceptanceCriterionReasoning,
  ReproducibilityReasoning,KnowledgeGraphConstruction,
  RetrievalAugmentation,RecursiveInference
 ]
 NORMALISE {
  "error analysis" -> ErrorAnalysis
  "RMSE" -> RootMeanSquaredError
  "MSE" -> MeanSquaredError
  "MAE" -> MeanAbsoluteError
  "GCI" -> GridConvergenceIndex
 }
}

EXECUTION_RULES {
 E1 = "If computational or modelling differences are being quantified against a reference, activate ErrorAnalysisOntology."
 E2 = "Identify the reference quantity and intended use before choosing an error metric."
 E3 = "Separate problem conditioning from algorithmic stability."
 E4 = "Separate discretisation, iteration, round-off, modelling and data-related error sources where possible."
 E5 = "Use both local and global metrics when aggregate norms could hide important failures."
 E6 = "Treat empirical estimates and rigorous error bounds distinctly."
 E7 = "Use refinement, tolerance or precision studies to identify dominant numerical error regimes."
 E8 = "Compare errors with explicit acceptance criteria."
 E9 = "Document reference provenance, norms, tolerances, precision and implementation versions for reproducibility."
}

RECURSION {
 ENTRY = ErrorAnalysis
 EXPAND(ErrorAnalysis) -> [
  Error,NumericalError,ApproximationError,AbsoluteError,RelativeError,
  ForwardError,BackwardError,Residual,TruncationError,DiscretisationError,
  IterationError,SolverError,RoundOffError,FloatingPointError,
  CancellationError,Overflow,Underflow,ErrorPropagation,ErrorBudget,
  SystematicError,RandomError,Bias,Variance,MAE,MSE,RMSE,
  ErrorNorm,L1NormError,L2NormError,InfinityNormError,FrobeniusNormError,
  PointwiseError,GlobalError,ErrorBound,APrioriErrorEstimate,
  APosterioriErrorEstimate,ResidualBasedEstimator,ErrorIndicator,
  Tolerance,AcceptanceCriterion,Conditioning,ConditionNumber,
  SensitivityAnalysis,NumericalStability,Convergence,OrderOfAccuracy,
  Refinement,RichardsonExtrapolation,GridConvergenceIndex,GCI,
  ReferenceSolution,ManufacturedSolution,BenchmarkSolution,
  Uncertainty,NumericalUncertainty,UncertaintyPropagation,
  FloatingPointArithmetic,MachinePrecision,CompensatedSummation,
  Verification,NumericalVerification,ComputationalValidation,
  Reproducibility,Provenance
 ]
 EXPAND(ErrorAnalysisFamilies) -> [
  AbsoluteRelativeAndNormwiseErrors,
  ForwardBackwardAndResidualAnalysis,
  TruncationDiscretisationAndIterationErrors,
  FloatingPointRoundOffAndCancellationErrors,
  ConditioningSensitivityAndStability,
  ErrorPropagationBoundsAndBudgets,
  RefinementConvergenceAndUncertainty,
  AcceptanceVerificationAndReproducibility
 ]
 STOP_CONDITION = "Stop when reference, metric, error sources, conditioning, stability, bounds, tolerance and reproducibility are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Verification & Validation"
 RELATED_ONTOLOGIES = [
  "Verification & Validation",
  "Numerical Verification Ontology",
  "Computational Validation Ontology",
  "Numerical Stability Ontology 2",
  "Floating Point Arithmetic Ontology",
  "Numerical Methods Ontology 2",
  "Numerical Linear Algebra Ontology 2",
  "Differential Equation Solver Ontology",
  "Numerical Integration Ontology 2",
  "Numerical Differentiation Ontology 2",
  "Finite Difference Method Ontology",
  "Finite Element Method Ontology",
  "Computational Statistics Ontology",
  "Scientific Simulation Ontology"
 ]
 ROUTE_RULES = [
  "Error Analysis -> Verification & Validation for parent semantics",
  "Error Analysis -> Numerical Verification for numerical correctness and refinement studies",
  "Error Analysis -> Computational Validation for predictive discrepancy and acceptance relationships",
  "Error Analysis -> Numerical Stability and Floating Point Arithmetic for round-off and stability semantics",
  "Error Analysis -> Numerical Methods and Numerical Linear Algebra for truncation, residual and conditioning semantics",
  "Error Analysis -> Differential Equation Solver, Numerical Integration and Numerical Differentiation for method-specific error analysis",
  "Error Analysis -> Finite Difference and Finite Element ontologies for discretisation-error estimation",
  "Error Analysis -> Computational Statistics for sampling, uncertainty and empirical-error metrics",
  "Error Analysis -> Scientific Simulation for simulation error budgets and validation relationships"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible error classes, estimators or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve absolute-versus-relative error distinction.",
  "Preserve forward-versus-backward error distinction.",
  "Preserve residual-versus-solution-error distinction.",
  "Preserve conditioning-versus-algorithmic-stability distinction.",
  "Preserve rigorous-bound-versus-empirical-estimate distinction.",
  "Preserve numerical-error-versus-model-form-error distinction.",
  "Preserve local-versus-global-error distinction.",
  "Preserve machine-readable error-analysis semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Error analysis identifies, quantifies and interprets differences between computed or modelled quantities and exact, reference or intended values."
 METRIC_SUMMARY = "Absolute, relative, forward, backward, residual and norm-based errors describe different aspects of discrepancy."
 SOURCE_SUMMARY = "Discretisation, iteration, round-off, modelling, sampling and measurement effects can contribute to total error."
 CONDITIONING_SUMMARY = "Problem conditioning determines sensitivity while algorithm stability determines how computation propagates perturbations."
 BOUND_SUMMARY = "A priori, a posteriori and empirical estimates provide different levels of information about error magnitude."
 VALIDATION_SUMMARY = "Error-analysis conclusions should state reference provenance, norms, dominant sources, tolerances, bounds, precision and reproducible computational settings."
}

FLOW {
 INPUT
 -> detect_error_analysis_context
 -> identify_computed_quantity_reference_quantity_problem_class_norm_tolerance_precision_and_intended_use
 -> classify_discretisation_iteration_roundoff_model_sampling_measurement_and_other_error_sources
 -> choose_absolute_relative_normwise_componentwise_forward_backward_residual_or_task_specific_metrics
 -> assess_reference_quality_and_provenance
 -> analyse_error_propagation_and_problem_conditioning
 -> distinguish_problem_sensitivity_from_algorithm_stability
 -> derive_or_estimate_a_priori_a_posteriori_empirical_or_asymptotic_error_bounds
 -> if_applicable_perform_resolution_tolerance_or_precision_refinement_studies
 -> compare_error_metrics_and_bounds_with_acceptance_criteria
 -> document_reference_norm_error_budget_tolerances_precision_versions_and_provenance
 -> route_to_related_ontologies
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-VERIFICATION-VALIDATION-ERROR-ANALYSIS-ONTOLOGY-V1"
}

}

END_FLOW