Statistical Modelling Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Statistical Modelling Ontology within the Kireth Research Semantic Library. Statistical modelling studies mathematical representations of data-generating processes using probability distributions, parameters, latent variables, predictors, responses and structural assumptions. This ontology supports model specification, likelihood, estimation, regression, hierarchical models, latent-variable models, generalized linear models, mixed models, model selection, diagnostics, uncertainty, prediction, misspecification, regularisation and relationships with applied statistical mathematics, probability, Bayesian inference, optimisation and computational statistics.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = StatisticalModelling
 ENTITY_CLASS = ProbabilisticDataGeneratingModelFramework
 PRIMARY_CONTEXT = AppliedStatisticalMathematics
 DEFINITION = "Statistical modelling is the mathematical construction and analysis of probability-based representations of data-generating processes, including assumptions, parameters, relationships and uncertainty."
 SYMBOLIC_FORMS = [
  "Y~p(y|theta,X)",
  "L(theta|D)=product_i p(y_i|x_i,theta)",
  "Y=X*beta+epsilon",
  "p(theta|D) proportional_to p(D|theta)p(theta)"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for statistical model structure, assumptions, likelihoods, parameters, predictors, responses, hierarchical structure, latent variables, diagnostics, selection, prediction and uncertainty."
}

SCOPE {
 INCLUDE = [
  StatisticalModelling,
  StatisticalModel,
  DataGeneratingProcess,
  ProbabilityModel,
  ParametricModel,
  SemiparametricModel,
  NonparametricModel,
  DeterministicComponent,
  StochasticComponent,
  ResponseVariable,
  PredictorVariable,
  ExplanatoryVariable,
  Covariate,
  Parameter,
  Hyperparameter,
  LatentVariable,
  RandomEffect,
  FixedEffect,
  ErrorTerm,
  Residual,
  LinkFunction,
  LinearPredictor,
  Likelihood,
  LogLikelihood,
  ScoreFunction,
  FisherInformation,
  MaximumLikelihood,
  RestrictedMaximumLikelihood,
  REML,
  MethodOfMoments,
  BayesianModel,
  PriorDistribution,
  PosteriorDistribution,
  PosteriorPredictiveDistribution,
  LinearRegression,
  MultipleRegression,
  PolynomialRegression,
  NonlinearRegression,
  GeneralisedLinearModel,
  GLM,
  LogisticRegression,
  PoissonRegression,
  NegativeBinomialRegression,
  ProbitModel,
  MixedEffectsModel,
  LinearMixedModel,
  GeneralisedLinearMixedModel,
  HierarchicalModel,
  MultilevelModel,
  LatentVariableModel,
  FactorModel,
  MixtureModel,
  GaussianMixtureModel,
  HiddenMarkovModel,
  StateSpaceModel,
  SurvivalModel,
  TimeSeriesModel,
  SpatialModel,
  GaussianProcess,
  ClassificationModel,
  RegressionModel,
  PredictiveModel,
  GenerativeModel,
  ModelAssumption,
  IndependenceAssumption,
  ConditionalIndependence,
  Homoscedasticity,
  Heteroscedasticity,
  NormalityAssumption,
  LinearityAssumption,
  Identifiability,
  ParameterIdentifiability,
  ModelFit,
  GoodnessOfFit,
  ResidualAnalysis,
  DiagnosticPlot,
  InfluenceMeasure,
  Outlier,
  Leverage,
  CookDistance,
  Multicollinearity,
  VarianceInflationFactor,
  ModelSelection,
  InformationCriterion,
  AkaikeInformationCriterion,
  AIC,
  BayesianInformationCriterion,
  BIC,
  CrossValidation,
  HoldoutValidation,
  Regularisation,
  RidgeRegression,
  LassoRegression,
  ElasticNet,
  PenalisedLikelihood,
  BiasVarianceTradeoff,
  Overfitting,
  Underfitting,
  Misspecification,
  RobustModel,
  Prediction,
  PredictionInterval,
  ConfidenceInterval,
  CredibleInterval,
  UncertaintyQuantification,
  Calibration,
  Discrimination,
  Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  DescriptiveStatisticsWithoutModelStructure,
  PureProbabilityTheoryWithoutDataModel,
  MachineLearningModelWithoutStatisticalSemantics,
  NumericalOptimisationWithoutStatisticalObjective
 ]
 DISAMBIGUATION_RULE = "When observed data are represented through a probability model with explicit variables, parameters, assumptions and inferential or predictive goals, interpret the context as statistical modelling."
}

CORE_CONCEPTS {
 StatisticalModelling = "Mathematical construction of probability-based representations of data."
 StatisticalModel = "Family of probability distributions indexed by parameters and structural assumptions."
 DataGeneratingProcess = "Conceptual or formal mechanism assumed to generate observed data."
 Likelihood = "Function of model parameters proportional to the probability or density of observed data under the model."
 Parameter = "Unknown model quantity controlling distributional or structural behaviour."
 LatentVariable = "Unobserved variable included to explain dependence or hidden structure."
 Residual = "Observed discrepancy between data and fitted model prediction."
 ModelSelection = "Procedure for comparing competing statistical models."
 ModelDiagnostics = "Methods used to assess assumptions, fit, influence and misspecification."
}

RELATIONS {
 StatisticalModelling --is_child_of--> AppliedStatisticalMathematics
 StatisticalModel --represents--> DataGeneratingProcess
 StatisticalModel --contains--> Parameter
 StatisticalModel --contains--> Assumption
 Likelihood --depends_on--> StatisticalModel
 MaximumLikelihood --estimates--> Parameter
 BayesianModel --combines--> LikelihoodAndPrior
 PosteriorDistribution --conditions_on--> ObservedData
 RegressionModel --relates--> ResponseAndPredictors
 GeneralisedLinearModel --uses--> LinkFunction
 MixedEffectsModel --contains--> FixedEffect
 MixedEffectsModel --contains--> RandomEffect
 HierarchicalModel --contains--> MultipleLevels
 LatentVariableModel --contains--> LatentVariable
 MixtureModel --uses--> LatentClass
 HiddenMarkovModel --uses--> LatentState
 StateSpaceModel --uses--> LatentDynamicState
 ResidualAnalysis --assesses--> ModelFit
 InfluenceMeasure --detects--> InfluentialObservation
 ModelSelection --compares--> CandidateModels
 AIC --penalises--> ModelComplexity
 BIC --penalises--> ModelComplexity
 CrossValidation --estimates--> PredictivePerformance
 Regularisation --controls--> ModelComplexity
 Misspecification --violates--> ModelAssumption
}

OPERATORS {
 LIKELIHOOD(theta,D) := L(theta|D)=product_i p(y_i|x_i,theta)
 LOG_LIKELIHOOD(theta,D) := ell(theta)=sum_i log p(y_i|x_i,theta)
 SCORE(theta) := U(theta)=partial ell(theta)/partial theta
 FISHER_INFORMATION(theta) := I(theta)=E[U(theta)U(theta)^T]
 MLE(theta,D) := theta_hat=argmax_theta L(theta|D)
 LINEAR_MODEL(X,beta,epsilon) := Y=X*beta+epsilon
 GLM_MEAN(eta,g_inv) := mu=g_inv(eta)
 LINEAR_PREDICTOR(X,beta) := eta=X*beta
 RIDGE_OBJECTIVE(beta,lambda) := SSE(beta)+lambda*||beta||_2^2
 LASSO_OBJECTIVE(beta,lambda) := SSE(beta)+lambda*||beta||_1
 AIC(logL,k) := AIC=2*k-2*logL
 BIC(logL,k,n) := BIC=k*log(n)-2*logL
 RESIDUAL(y,yhat) := e=y-yhat
 STANDARDISED_RESIDUAL(e,se) := r=e/se
}

AXIOMS {
 A1 = "A statistical model requires an explicit outcome space and probability structure."
 A2 = "Model parameters and assumptions must be distinguished."
 A3 = "Model fit does not imply model truth."
 A4 = "Residual structure can reveal model misspecification."
 A5 = "Predictive and inferential validity depend on model assumptions."
 A6 = "Model complexity should be evaluated relative to data support and purpose."
 A7 = "Hierarchical and latent-variable structure must be explicitly identified."
 A8 = "Statistical models require uncertainty, diagnostics and validation."
}

REASONING {
 R1_CONTEXT = identify(outcome,predictors,data_structure,goal,dependence,distribution)
 R2_MODEL_CLASS = classify(linear,GLM,mixed,hierarchical,latent,time_series,spatial,survival_or_other)
 R3_SPECIFICATION = define(response_distribution,link,predictor_structure,parameters,assumptions)
 R4_ESTIMATION = select(MLE,REML,Bayesian,method_of_moments,penalised_estimation)
 R5_IDENTIFIABILITY = assess(parameter_identifiability,collinearity,latent_structure)
 R6_DIAGNOSTICS = analyse(residuals,influence,heteroscedasticity,normality,dependence)
 R7_SELECTION = compare(AIC,BIC,cross_validation,penalised_models)
 R8_PREDICTION = compute(fitted_values,predictions,intervals,posterior_predictive)
 R9_ROBUSTNESS = assess(outliers,misspecification,sensitivity,regularisation)
 R10_VALIDATION = check(calibration,discrimination,out_of_sample_performance,reproducibility)
}

CONSTRAINTS {
 C1 = "Good in-sample fit does not guarantee good out-of-sample prediction."
 C2 = "A model can be statistically identifiable yet practically weakly identified."
 C3 = "Multicollinearity can inflate parameter uncertainty."
 C4 = "Residual independence assumptions can fail under temporal, spatial or clustered data."
 C5 = "Heteroscedasticity can invalidate naive standard errors."
 C6 = "Information criteria compare models but do not prove absolute adequacy."
 C7 = "Cross-validation estimates depend on the resampling scheme."
 C8 = "Regularisation changes parameter bias and interpretation."
 C9 = "Latent-variable models can suffer label-switching or non-identifiability."
 C10 = "Misspecified probability distributions can distort uncertainty quantification."
}

SEMANTIC_EQUIVALENCE {
 "StatisticalModelling" <=> "StatisticalModeling"
 "GeneralisedLinearModel" <=> "GLM"
 "RestrictedMaximumLikelihood" <=> "REML"
 "AkaikeInformationCriterion" <=> "AIC"
 "BayesianInformationCriterion" <=> "BIC"
 "VarianceInflationFactor" <=> "VIF"
}

VALIDATION {
 V1 = "Outcome and predictor definitions are explicit."
 V2 = "Probability distribution and link function are documented."
 V3 = "Parameter estimation method is explicit."
 V4 = "Identifiability is assessed."
 V5 = "Residual diagnostics are performed."
 V6 = "Influential observations are assessed."
 V7 = "Dependence structure is modelled where relevant."
 V8 = "Model selection criteria are reported consistently."
 V9 = "Prediction uncertainty is quantified."
 V10 = "Out-of-sample or posterior predictive validation is performed where feasible."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 STATISTICAL_MODELLING_DEFINITION = 0.99
 MODEL_SPECIFICATION_REASONING = 0.99
 LIKELIHOOD_REASONING = 0.99
 REGRESSION_REASONING = 0.99
 HIERARCHICAL_REASONING = 0.98
 LATENT_VARIABLE_REASONING = 0.98
 DIAGNOSTIC_REASONING = 0.99
 MODEL_SELECTION_REASONING = 0.98
 POLICY = "Reduce confidence when outcome distribution, link function, dependence structure, parameter identifiability, residual diagnostics, missingness, regularisation or validation scheme are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  StatisticalModellingReasoning,
  ModelSpecificationReasoning,
  LikelihoodReasoning,
  MaximumLikelihoodReasoning,
  RegressionModelReasoning,
  GLMReasoning,
  MixedEffectsReasoning,
  HierarchicalModelReasoning,
  LatentVariableReasoning,
  MixtureModelReasoning,
  StateSpaceReasoning,
  ResidualDiagnosticReasoning,
  InfluenceReasoning,
  ModelSelectionReasoning,
  CrossValidationReasoning,
  RegularisationReasoning,
  PredictionReasoning,
  UncertaintyQuantificationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "statistical modelling" -> StatisticalModelling
  "statistical modeling" -> StatisticalModelling
  "GLM" -> GeneralisedLinearModel
  "REML" -> RestrictedMaximumLikelihood
  "AIC" -> AkaikeInformationCriterion
  "BIC" -> BayesianInformationCriterion
  "VIF" -> VarianceInflationFactor
 }
}

EXECUTION_RULES {
 E1 = "If data are represented through an explicit probability model, activate StatisticalModellingOntology."
 E2 = "If the response is approximately Gaussian with linear mean structure, consider linear regression."
 E3 = "If response distribution is non-Gaussian but belongs to an exponential-family structure, consider a GLM."
 E4 = "If clustering or repeated measures are present, consider mixed or hierarchical models."
 E5 = "If unobserved structure is required, consider latent-variable or mixture models."
 E6 = "If model assumptions are violated, revise specification or use robust alternatives."
 E7 = "If candidate models compete, compare information criteria and out-of-sample performance."
 E8 = "If prediction is the goal, quantify predictive uncertainty separately from parameter uncertainty."
 E9 = "Any fitted model should be checked for identifiability, residual structure, misspecification and external validity."
}

RECURSION {
 ENTRY = StatisticalModelling
 EXPAND(StatisticalModelling) -> [
  StatisticalModel, DataGeneratingProcess, ProbabilityModel, ParametricModel,
  SemiparametricModel, NonparametricModel, DeterministicComponent,
  StochasticComponent, ResponseVariable, PredictorVariable,
  ExplanatoryVariable, Covariate, Parameter, Hyperparameter, LatentVariable,
  RandomEffect, FixedEffect, ErrorTerm, Residual, LinkFunction,
  LinearPredictor, Likelihood, LogLikelihood, ScoreFunction,
  FisherInformation, MaximumLikelihood, RestrictedMaximumLikelihood,
  MethodOfMoments, BayesianModel, PriorDistribution, PosteriorDistribution,
  PosteriorPredictiveDistribution, LinearRegression, MultipleRegression,
  PolynomialRegression, NonlinearRegression, GeneralisedLinearModel,
  LogisticRegression, PoissonRegression, NegativeBinomialRegression,
  ProbitModel, MixedEffectsModel, LinearMixedModel,
  GeneralisedLinearMixedModel, HierarchicalModel, MultilevelModel,
  LatentVariableModel, FactorModel, MixtureModel, GaussianMixtureModel,
  HiddenMarkovModel, StateSpaceModel, SurvivalModel, TimeSeriesModel,
  SpatialModel, GaussianProcess, ClassificationModel, RegressionModel,
  PredictiveModel, GenerativeModel, ModelAssumption, IndependenceAssumption,
  ConditionalIndependence, Homoscedasticity, Heteroscedasticity,
  NormalityAssumption, LinearityAssumption, Identifiability,
  ParameterIdentifiability, ModelFit, GoodnessOfFit, ResidualAnalysis,
  DiagnosticPlot, InfluenceMeasure, Outlier, Leverage, CookDistance,
  Multicollinearity, VarianceInflationFactor, ModelSelection,
  InformationCriterion, AkaikeInformationCriterion,
  BayesianInformationCriterion, CrossValidation, HoldoutValidation,
  Regularisation, RidgeRegression, LassoRegression, ElasticNet,
  PenalisedLikelihood, BiasVarianceTradeoff, Overfitting, Underfitting,
  Misspecification, RobustModel, Prediction, PredictionInterval,
  ConfidenceInterval, CredibleInterval, UncertaintyQuantification,
  Calibration, Discrimination, Reproducibility
 ]
 EXPAND(ModelFamilies) -> [
  RegressionModels,
  GeneralisedLinearModels,
  MixedEffectsModels,
  HierarchicalModels,
  LatentVariableModels,
  TimeSeriesModels,
  SpatialModels,
  SurvivalModels
 ]
 STOP_CONDITION = "Stop when model structure, assumptions, parameters, estimation method, diagnostics, selection, prediction and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Statistical Mathematics"
 IMMEDIATE_CONTEXTS = [
  "Applied Statistical Mathematics",
  "Probability & Statistics Ontologies",
  "Regression Ontologies",
  "Bayesian Statistics Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Applied Statistical Mathematics",
  "Probability & Statistics Ontologies",
  "Regression Ontologies",
  "Generalised Linear Model Ontologies",
  "Bayesian Statistics Ontologies",
  "Multivariate Statistics Ontologies",
  "Time Series Ontologies",
  "Survival Analysis Ontologies",
  "Spatial Statistics Ontologies",
  "Optimisation Ontologies",
  "Computational Statistics Ontologies"
 ]
 ROUTE_RULES = [
  "Statistical Modelling -> Applied Statistical Mathematics for parent semantics",
  "Statistical Modelling -> Probability & Statistics for distributional foundations",
  "Statistical Modelling -> Regression for predictor-response structure",
  "Statistical Modelling -> GLM Ontologies for non-Gaussian response modelling",
  "Statistical Modelling -> Bayesian Statistics for posterior inference",
  "Statistical Modelling -> Multivariate Statistics for vector-valued outcomes",
  "Statistical Modelling -> Time Series for temporal dependence",
  "Statistical Modelling -> Survival Analysis for time-to-event models",
  "Statistical Modelling -> Spatial Statistics for spatial dependence",
  "Statistical Modelling -> Optimisation for parameter estimation",
  "Statistical Modelling -> Computational Statistics for numerical fitting and simulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible statistical model families, diagnostics or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve data-generating-process semantics.",
  "Preserve parameter-versus-assumption distinction.",
  "Preserve fixed-versus-random effect distinction.",
  "Preserve observed-versus-latent variable distinction.",
  "Preserve fit-versus-validity distinction.",
  "Preserve inference-versus-prediction distinction.",
  "Preserve diagnostics and misspecification semantics.",
  "Preserve machine-readable statistical-modelling semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Statistical modelling represents data-generating processes using probability distributions, parameters, structural assumptions and uncertainty."
 SPECIFICATION_SUMMARY = "Model specification defines response distributions, predictors, links, latent structure and dependence assumptions."
 ESTIMATION_SUMMARY = "Likelihood, REML, Bayesian and penalised procedures estimate model parameters."
 HIERARCHICAL_SUMMARY = "Mixed and multilevel models represent nested or clustered variation using fixed and random effects."
 DIAGNOSTIC_SUMMARY = "Residuals, leverage, influence and goodness-of-fit diagnostics assess model adequacy."
 SELECTION_SUMMARY = "AIC, BIC, cross-validation and regularisation balance fit against complexity."
 PREDICTION_SUMMARY = "Predictive models should return fitted values together with predictive uncertainty and calibration assessment."
 VALIDATION_SUMMARY = "Statistical models should be checked for identifiability, misspecification, dependence, robustness, out-of-sample performance and reproducibility."
}

FLOW {
 INPUT
 -> detect_statistical_modelling_context
 -> identify_response_predictors_data_structure_dependence_distribution_and_goal
 -> classify_parametric_semiparametric_nonparametric_regression_GLM_mixed_hierarchical_latent_time_series_spatial_or_survival_model
 -> specify_probability_distribution_link_predictor_structure_parameters_and_assumptions
 -> choose_MLE_REML_Bayesian_method_of_moments_or_penalised_estimation
 -> assess_identifiability_and_multicollinearity
 -> fit_model_and_compute_parameter_uncertainty
 -> perform_residual_influence_and_goodness_of_fit_diagnostics
 -> compare_candidate_models_using_information_criteria_cross_validation_or_regularisation
 -> compute_predictions_and_prediction_or_credible_intervals
 -> assess_robustness_misspecification_calibration_discrimination_and_external_validity
 -> route_to_applied_statistical_mathematics_probability_statistics_regression_GLM_Bayesian_statistics_multivariate_statistics_time_series_survival_analysis_spatial_statistics_optimisation_or_computational_statistics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Statistical Modelling Ontology defines probability-based representations of data-generating processes. It supports parametric, semiparametric and nonparametric models, response and predictor variables, parameters, hyperparameters and latent variables, fixed and random effects, likelihood, log-likelihood, score and Fisher information, maximum likelihood, REML and Bayesian models, linear, polynomial, nonlinear, logistic, Poisson, negative-binomial and probit regression, generalised linear and mixed models, hierarchical and multilevel models, latent-variable, factor and mixture models, hidden Markov and state-space models, survival, time-series, spatial and Gaussian-process models, model assumptions, identifiability, residuals, influence, outliers, leverage, Cook's distance and multicollinearity, AIC, BIC, cross-validation, ridge, lasso and elastic-net regularisation, overfitting, underfitting, misspecification, robust modelling, prediction intervals, uncertainty quantification, calibration, discrimination and reproducibility."
}

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

}

END_FLOW