Empirical Model Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Empirical Model Ontology within the Kireth Research Semantic Library. An empirical model is a mathematical representation whose structure, parameters or predictive relationships are derived primarily from observed data rather than from first-principles physical laws. This ontology supports regression, curve fitting, response surfaces, surrogate models, statistical learning, calibration, interpolation, extrapolation, residual analysis, goodness of fit, validation, uncertainty, overfitting control and relationships with mathematical modelling, statistics, machine learning and scientific measurement.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Empirical Model Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Modelling / Empirical Model"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-modelling/empirical-model-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = EmpiricalModel
 ENTITY_CLASS = DataDerivedMathematicalModel
 PRIMARY_CONTEXT = MathematicalModelling
 DEFINITION = "An empirical model is a mathematical representation whose functional form, parameters or predictive relationships are derived primarily from observed data."
 SEMANTIC_ROLE = "Provides the machine-readable framework for data-derived model structure, fitting, residual analysis, validation, interpolation, extrapolation, uncertainty and empirical prediction."
}

SCOPE {
 INCLUDE = [
  EmpiricalModel,
  ObservationalData,
  ExperimentalData,
  TrainingData,
  ValidationData,
  PredictorVariable,
  ResponseVariable,
  Feature,
  TargetVariable,
  Parameter,
  Coefficient,
  RegressionModel,
  LinearRegression,
  NonlinearRegression,
  PolynomialRegression,
  CurveFit,
  ResponseSurface,
  InterpolationModel,
  SurrogateModel,
  StatisticalModel,
  DataDrivenModel,
  MachineLearningModel,
  Residual,
  ErrorTerm,
  LossFunction,
  ObjectiveFunction,
  GoodnessOfFit,
  R2,
  RMSE,
  MAE,
  CrossValidation,
  Overfitting,
  Underfitting,
  Regularisation,
  BiasVarianceTradeoff,
  PredictionInterval,
  ConfidenceInterval,
  Extrapolation,
  Interpolation,
  ModelSelection,
  Validation,
  Uncertainty
 ]

 EXCLUDE_AS_PRIMARY = [
  FirstPrinciplesPhysicalModelUnlessEmpiricalCalibrationDominates,
  PureDescriptiveStatisticsWithoutPredictiveModel,
  RawDatasetWithoutModelStructure,
  NumericalApproximationOfKnownEquationWithoutDataFit
 ]

 DISAMBIGUATION_RULE =
  "When a mathematical relationship is fitted, estimated or learned primarily from observed data rather than derived from governing laws, interpret it as an empirical model."
}

CORE_CONCEPTS {

 EmpiricalModel {
  definition = "A model inferred primarily from data."
 }

 ObservationalData {
  definition = "Recorded values obtained without fully controlled intervention."
 }

 ExperimentalData {
  definition = "Recorded values obtained under controlled or designed experimental conditions."
 }

 PredictorVariable {
  definition = "An input variable used to explain or predict a response."
 }

 ResponseVariable {
  definition = "The output quantity modelled as a function of predictors."
 }

 RegressionModel {
  definition = "A model expressing the conditional relationship between response and predictor variables."
 }

 CurveFit {
  definition = "A fitted mathematical function approximating observed data."
 }

 Residual {
  definition = "Observed value minus model-predicted value under a stated sign convention."
 }

 LossFunction {
  definition = "A criterion measuring discrepancy between predictions and observations."
 }

 GoodnessOfFit {
  definition = "A quantitative assessment of how closely a fitted model matches data."
 }

 CrossValidation {
  definition = "A resampling procedure used to estimate predictive performance on unseen data."
 }

 Overfitting {
  definition = "Excessive fitting to training data that harms generalisation."
 }

 Regularisation {
  definition = "A penalty or constraint introduced to reduce model complexity or improve generalisation."
 }

 Extrapolation {
  definition = "Prediction outside the range or regime represented by training data."
 }
}

RELATIONS {
 EmpiricalModel --is_special_case_of--> MathematicalModel
 EmpiricalModel --is_derived_from--> Data
 EmpiricalModel --uses--> PredictorVariable
 EmpiricalModel --predicts--> ResponseVariable
 RegressionModel --is_special_case_of--> EmpiricalModel
 CurveFit --is_special_case_of--> EmpiricalModel
 SurrogateModel --may_be--> EmpiricalModel
 StatisticalModel --may_be--> EmpiricalModel
 MachineLearningModel --may_be--> EmpiricalModel
 LossFunction --guides--> ParameterEstimation
 Residual --measures--> PredictionError
 GoodnessOfFit --evaluates--> EmpiricalModel
 CrossValidation --estimates--> GeneralisationPerformance
 Regularisation --controls--> ModelComplexity
 Overfitting --reduces--> GeneralisationPerformance
 Interpolation --occurs_within--> DataDomain
 Extrapolation --occurs_outside--> DataDomain
 Validation --evaluates--> PredictiveAdequacy
}

OPERATORS {
 FIT_MODEL(data,model_family,loss) :=
  parameter_values_minimising_or_optimising loss

 PREDICT(model,x) :=
  model_response_at_x

 RESIDUAL(y_observed,y_predicted) :=
  y_observed-y_predicted

 SSE(model,data) :=
  sum_i residual_i^2

 MSE(model,data) :=
  (1/n)*sum_i residual_i^2

 RMSE(model,data) :=
  sqrt(MSE(model,data))

 MAE(model,data) :=
  (1/n)*sum_i abs(residual_i)

 R_SQUARED(model,data) :=
  1-SSE/SST
  WHEN SST>0

 CROSS_VALIDATE(model_family,data,k) :=
  average_validation_metric_over_k_folds

 REGULARISED_FIT(data,loss,penalty,lambda) :=
  argmin_theta loss(theta)+lambda*penalty(theta)

 INTERPOLATE(model,x) :=
  prediction_within_training_support_or_declared_data_range

 EXTRAPOLATE(model,x) :=
  prediction_outside_training_support_or_declared_data_range
}

DEFINITIONS {
 D1 = "Empirical models are primarily data-derived."
 D2 = "A fitted model is conditional on the data and model family used."
 D3 = "Residuals represent unexplained discrepancy under the fitted model."
 D4 = "Good fit to training data does not imply good generalisation."
 D5 = "Cross-validation estimates out-of-sample predictive performance."
 D6 = "Regularisation trades fit against complexity."
 D7 = "Interpolation is generally better supported than extrapolation."
 D8 = "Empirical correlation does not by itself establish causation."
 D9 = "Prediction uncertainty should increase when data support weakens."
 D10 = "Model selection should consider both fit and complexity."
}

AXIOMS {
 A1_DATA_DEPENDENCE:
  EmpiricalModel depends on observed or experimental data

 A2_FIT_CRITERION:
  Parameter estimation requires an explicit loss, likelihood or fitting criterion

 A3_GENERALISATION:
  Predictive adequacy must be assessed beyond training fit when future prediction is intended

 A4_RESIDUAL_ANALYSIS:
  Residual structure contains information about model misspecification

 A5_EXTRAPOLATION_RISK:
  Predictions outside the observed data domain require reduced confidence

 A6_OVERFITTING_CONTROL:
  Increasing model flexibility can increase overfitting risk

 A7_UNCERTAINTY:
  Empirical predictions carry uncertainty from noise, sampling and model specification

 A8_CAUSAL_LIMIT:
  Empirical association alone does not imply causal mechanism
}

REASONING {

 R1_EMPIRICAL_MODEL_IDENTIFICATION {
  INPUT = model_description
  IDENTIFY = data_source
  IDENTIFY = fitted_relationship
  IDENTIFY = predictors_and_response
  IDENTIFY = loss_or_estimation criterion
  RETURN = EmpiricalModel
 }

 R2_MODEL_FITTING {
  INPUT = data_model_family_loss
  OPTIMISE = parameters
  COMPUTE = fitted_values
  COMPUTE = residuals
  RETURN = fitted_model
 }

 R3_GOODNESS_OF_FIT {
  INPUT = fitted_model_and_data
  COMPUTE = SSE_MSE_RMSE_MAE_or_R2
  RETURN = fit_metrics
 }

 R4_RESIDUAL_ANALYSIS {
  INPUT = residuals_and_predictors
  TEST = nonlinearity_heteroscedasticity_autocorrelation_outliers
  RETURN = residual_diagnostics
 }

 R5_MODEL_SELECTION {
  INPUT = candidate_models
  COMPARE = validation_error_complexity_information_criteria_or_regularised objective
  RETURN = selected_model_with_tradeoffs
 }

 R6_CROSS_VALIDATION {
  INPUT = model_family_and_data
  SPLIT = folds_or_resamples
  FIT = training_subsets
  EVALUATE = held_out_subsets
  RETURN = generalisation_estimate
 }

 R7_INTERPOLATION_EXTRAPOLATION {
  INPUT = query_point_and_training_domain
  IF within_support
   LABEL = Interpolation
  ELSE
   LABEL = Extrapolation
   REDUCE = confidence
  RETURN = prediction_context
 }

 R8_UNCERTAINTY {
  INPUT = fitted_model_data_noise_model
  ESTIMATE = parameter_and_prediction_uncertainty
  RETURN = confidence_or_prediction_intervals
 }

 R9_OVERFITTING_CONTROL {
  INPUT = model_complexity_and_validation_performance
  IF training_error_low AND validation_error_high
   IDENTIFY = overfitting
   APPLY = regularisation_simplification_or_more_data
  RETURN = adjusted_model_strategy
 }
}

CONSTRAINTS {
 C1 = "Training fit must not be confused with predictive validation."
 C2 = "Data quality and sampling design affect model validity."
 C3 = "Correlation does not establish causation."
 C4 = "Residual independence and variance assumptions must be checked where required."
 C5 = "R-squared alone is insufficient for model adequacy."
 C6 = "Extrapolation must be explicitly labelled."
 C7 = "Complex models require stronger overfitting controls."
 C8 = "Prediction intervals and confidence intervals must not be conflated."
 C9 = "Categorical, censored, dependent or heteroscedastic data may require specialised models."
 C10 = "Model performance metrics must match the intended use."
}

SEMANTIC_EQUIVALENCE {
 "EmpiricalModel" <=> "DataDerivedMathematicalModel"
 "CurveFit" <=> "FittedFunctionalApproximation"
 "RegressionModel" <=> "ConditionalResponseModel"
 "Residual" <=> "ObservedMinusPredictedDiscrepancy"
 "CrossValidation" <=> "ResampledOutOfSampleEvaluation"
 "Regularisation" <=> "ComplexityPenaltyOrConstraint"
}

VALIDATION {
 V1 = "Training and validation data are distinguished."
 V2 = "Predictors and response are identified."
 V3 = "Loss or estimation criterion is specified."
 V4 = "Residuals are examined for systematic structure."
 V5 = "Generalisation is assessed when prediction is intended."
 V6 = "Overfitting risk is considered."
 V7 = "Extrapolation is flagged."
 V8 = "Uncertainty intervals are correctly interpreted."
 V9 = "Evaluation metric matches the prediction objective."
 V10 = "Causal claims are not inferred from association alone."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 EMPIRICAL_MODEL_DEFINITION = 0.99
 FITTING_REASONING = 0.99
 GOODNESS_OF_FIT_REASONING = 0.99
 RESIDUAL_REASONING = 0.98
 CROSS_VALIDATION_REASONING = 0.99
 MODEL_SELECTION_REASONING = 0.98
 EXTRAPOLATION_REASONING = 0.99
 UNCERTAINTY_REASONING = 0.97

 POLICY =
  "Reduce confidence when data provenance, sampling design, response definition, model family, loss function, validation regime, uncertainty model or extrapolation distance are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  EmpiricalModelReasoning,
  RegressionReasoning,
  CurveFittingReasoning,
  ResidualAnalysis,
  GoodnessOfFitReasoning,
  CrossValidationReasoning,
  ModelSelectionReasoning,
  RegularisationReasoning,
  OverfittingDetection,
  InterpolationReasoning,
  ExtrapolationReasoning,
  PredictionUncertainty,
  StatisticalLearningReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "empirical model" -> EmpiricalModel
  "data-driven model" -> EmpiricalModel
  "regression model" -> RegressionModel
  "curve fit" -> CurveFit
  "response surface" -> ResponseSurface
  "residual" -> Residual
  "goodness of fit" -> GoodnessOfFit
  "cross-validation" -> CrossValidation
  "overfitting" -> Overfitting
  "regularisation" -> Regularisation
  "regularization" -> Regularisation
  "extrapolation" -> Extrapolation
 }

 INTERPRETATION_ORDER = [
  detect_empirical_model_context,
  identify data_source_predictors_response_model_family_and_loss,
  identify fitting_validation_interpolation_extrapolation_or_uncertainty goal,
  fit_or_evaluate model,
  analyse residuals_complexity_and_generalisation,
  verify data_support_metric_and_uncertainty assumptions,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("empirical model")
  OR data_derived_model_context
  THEN activate(EmpiricalModelOntology)

 E2:
  IF model_fitting_requested
  THEN require data_model_family_and_loss_or_likelihood

 E3:
  IF predictive_claim_requested
  THEN evaluate out_of_sample performance where possible

 E4:
  IF residuals_show_structure
  THEN flag possible misspecification

 E5:
  IF model_complexity_high_relative_to_data
  THEN assess overfitting

 E6:
  IF query_point_outside_training_domain
  THEN classify as extrapolation_and_reduce confidence

 E7:
  IF uncertainty_requested
  THEN distinguish confidence_interval_from_prediction_interval

 E8:
  IF association_detected
  THEN do_not infer causation_without causal assumptions

 E9:
  IF multiple_models_fit similarly
  THEN prefer model_selection criteria aligned_with intended_use
}

RECURSION {
 ENTRY = EmpiricalModel

 EXPAND(EmpiricalModel) -> [
  ObservationalData,
  ExperimentalData,
  TrainingData,
  ValidationData,
  PredictorVariable,
  ResponseVariable,
  Feature,
  TargetVariable,
  Parameter,
  RegressionModel,
  LinearRegression,
  NonlinearRegression,
  PolynomialRegression,
  CurveFit,
  ResponseSurface,
  InterpolationModel,
  SurrogateModel,
  Residual,
  LossFunction,
  GoodnessOfFit,
  CrossValidation,
  Overfitting,
  Regularisation,
  PredictionInterval,
  ConfidenceInterval,
  Extrapolation,
  Interpolation,
  ModelSelection,
  Uncertainty
 ]

 EXPAND(RegressionModel) -> [
  LinearRegression,
  PolynomialRegression,
  NonlinearRegression,
  GeneralisedRegression,
  ResidualModel
 ]

 EXPAND(ModelEvaluation) -> [
  SSE,
  MSE,
  RMSE,
  MAE,
  R2,
  CrossValidation,
  InformationCriterion,
  ResidualDiagnostics
 ]

 EXPAND(GeneralisationControl) -> [
  Regularisation,
  ComplexityControl,
  CrossValidation,
  EarlyStopping,
  ModelSelection
 ]

 STOP_CONDITION =
  "Stop when the data-derived relationship, fitting method, residual behaviour, validation performance, uncertainty, interpolation or extrapolation status is sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Modelling"

 IMMEDIATE_CONTEXTS = [
  "Mathematical Model Ontology",
  "Mathematical Modelling Ontology",
  "Probability & Statistics Ontologies",
  "Machine Learning Ontologies"
 ]

 RELATED_ONTOLOGIES = [
  "Mathematical Model Ontology",
  "Mathematical Modelling Ontology",
  "Applied Mathematics Ontologies",
  "Probability & Statistics Ontologies",
  "Regression Ontologies",
  "Statistical Learning Ontologies",
  "Machine Learning Ontologies",
  "Optimisation Ontologies",
  "Scientific Measurement Ontologies",
  "Numerical Analysis Ontologies",
  "Data Analysis Ontologies"
 ]

 ROUTE_RULES = [
  "Empirical Model -> Mathematical Model Ontology for formal model semantics",
  "Empirical Model -> Mathematical Modelling Ontology for modelling-process semantics",
  "Empirical Model -> Probability & Statistics for estimation and uncertainty",
  "Empirical Model -> Regression Ontologies for fitted response relationships",
  "Empirical Model -> Machine Learning for flexible data-driven models",
  "Empirical Model -> Optimisation for parameter fitting",
  "Empirical Model -> Measurement Ontologies for observed data and error",
  "Empirical Model -> Numerical Analysis for interpolation and approximation",
  "Empirical Model -> Data Analysis for residual and validation workflows"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible empirical model classes, fitting methods or evaluation pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve data-derived model semantics.",
  "Preserve training-versus-validation distinction.",
  "Preserve residual semantics.",
  "Preserve interpolation-versus-extrapolation distinction.",
  "Preserve association-versus-causation distinction.",
  "Preserve overfitting and generalisation semantics.",
  "Preserve uncertainty-aware prediction.",
  "Preserve machine-readable empirical-model semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "An empirical model is a mathematical representation whose structure or parameters are inferred primarily from observed data."

 DATA_SUMMARY =
  "Empirical models depend directly on observational or experimental data."

 FIT_SUMMARY =
  "Model fitting estimates parameters by optimising a loss, likelihood or related criterion."

 RESIDUAL_SUMMARY =
  "Residuals quantify discrepancy between observed and predicted values."

 VALIDATION_SUMMARY =
  "Predictive adequacy should be assessed on data not used solely for fitting."

 OVERFITTING_SUMMARY =
  "Overfitting occurs when a model captures training-specific variation at the expense of generalisation."

 INTERPOLATION_SUMMARY =
  "Interpolation predicts within the supported data domain."

 EXTRAPOLATION_SUMMARY =
  "Extrapolation predicts outside the supported data domain and should carry reduced confidence."

 UNCERTAINTY_SUMMARY =
  "Empirical prediction uncertainty arises from noise, finite sampling, parameter estimation and model specification."
}

FLOW {
 INPUT
 -> detect_empirical_model_context
 -> identify data_provenance_predictors_response_model_family_loss_and_intended_use
 -> split or identify training_validation_and_test evidence
 -> fit model_parameters
 -> compute predictions_residuals_and_fit metrics
 -> evaluate generalisation_and_overfitting
 -> classify interpolation_or_extrapolation context
 -> estimate uncertainty
 -> verify metric_data_support_association_causation_and_validation constraints
 -> route to mathematical_model mathematical_modelling probability_statistics regression machine_learning optimisation measurement numerical_analysis_or_data_analysis ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Empirical Model Ontology defines data-derived mathematical models and supports observational and experimental data, predictors, responses, regression, curve fitting, response surfaces, surrogate models, residuals, loss functions, goodness of fit, cross-validation, overfitting control, regularisation, interpolation, extrapolation, prediction uncertainty and validation while preserving the distinction between association, prediction and causation."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-EMPIRICAL-MODEL-ONTOLOGY-V1"
}

}

END_FLOW