Regression Ontology

FULL EXECUTABLE FLOW ALGEBRA — REGRESSION ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/correlation-regression/regression-ontology-2.html
FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Correlation & Regression"
ONTOLOGY := "Regression Ontology"
PRIMARY_ENTITY := Regression
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

Regression := StatisticalModelling ∧ ConditionalRelationshipEstimation ∧ ResponsePrediction ∧ ParameterEstimation ∧ UncertaintyQuantification ∧ ModelValidation
Population := TargetCollectionOrGeneratingProcess
Sample := {(X_i,Y_i)}_{i=1..n}
ResponseVariable := Y
PredictorVector := X
ParameterVector := β
ErrorTerm := ε
RegressionTarget := ConditionalDistribution(Y|X) ∨ ConditionalMean(E[Y|X])
RegressionFunction := m(x)=E[Y|X=x]
LinearRegression := E[Y|X]=Xβ
SimpleLinearRegression := Y=β0+β1X+ε
MultipleLinearRegression := Y=β0+β1X1+...+βpXp+ε
LinearPredictor := η=Xβ
DesignMatrix := X
ObservedResponseVector := y
FittedValues := y_hat=Xβ_hat
ResidualVector := e=y-y_hat
OrdinaryLeastSquares := β_hat=argmin_β(y-Xβ)^T(y-Xβ)
ResidualSumOfSquares := RSS=Σ_i(y_i-y_hat_i)²
NormalEquations := X^TXβ_hat=X^Ty
OLSClosedForm := β_hat=(X^TX)^(-1)X^Ty when FullColumnRank
MoorePenroseSolution := β_hat=X^+y
SimpleRegressionSlope := b1=Σ_i(x_i-x_bar)(y_i-y_bar)/Σ_i(x_i-x_bar)²
SimpleRegressionIntercept := b0=y_bar-b1x_bar
CovarianceSlopeRelation := b1=S_XY/S_X²
CorrelationSlopeRelation := b1=r_XY(s_Y/s_X)
ResidualOrthogonality := X^Te=0
ProjectionMatrix := H=X(X^TX)^(-1)X^T
FittedVector := y_hat=Hy
ResidualMaker := M=I-H
TotalSumOfSquares := TSS=Σ_i(y_i-y_bar)²
ExplainedSumOfSquares := ESS=Σ_i(y_hat_i-y_bar)²
SumSquaresDecomposition := ModelWithIntercept ⇒ TSS=ESS+RSS
CoefficientOfDetermination := R²=1-RSS/TSS
AdjustedR2 := 1-(1-R²)(n-1)/(n-k)
SimpleRegressionR2Relation := SimpleLinearRegressionWithIntercept ⇒ R²=r_XY²
ZeroConditionalMean := E[ε|X]=0
Homoscedasticity := Var(ε|X)=σ²I
Heteroscedasticity := Var(ε|X)=Ω
NoPerfectMulticollinearity := rank(X)=number_of_columns(X)
GaussMarkovConditions := LinearInParameters ∧ ZeroConditionalMean ∧ NoPerfectMulticollinearity ∧ HomoscedasticUncorrelatedErrors
GaussMarkovTheorem := GaussMarkovConditions ⇒ OLSIsBLUE
OLSUnbiasedness := E[β_hat|X]=β
OLSCovariance := Var(β_hat|X)=σ²(X^TX)^(-1)
ResidualVarianceEstimator := s²=RSS/(n-k)
CoefficientStandardError := SE(β_hat_j)=sqrt(s²[(X^TX)^(-1)]_jj)
RegressionTStatistic := t_j=(β_hat_j-β_j0)/SE(β_hat_j)
RegressionConfidenceInterval := β_hat_j±t_(1-α/2,n-k)SE(β_hat_j)
Prediction := y_hat_new=x_new^Tβ_hat
PredictionVariance := MeanResponseVariance+ErrorVariance
PredictionInterval := PointPrediction±CriticalValue×PredictionStandardError
ResidualDiagnostics := {ResidualVsFitted,NormalQQ,ScaleLocation,Leverage,Influence,Autocorrelation}
Leverage := h_ii=H_ii
CookDistance := InfluenceMeasureCombiningResidualAndLeverage
Multicollinearity := StrongLinearDependenceAmongPredictors
VarianceInflationFactor := VIF_j=1/(1-R_j²)
HeteroscedasticityRobustCovariance := SandwichEstimator → RobustStandardErrors
ClusterRobustCovariance := ClusterAggregatedScores → ClusterRobustStandardErrors
WeightedLeastSquares := β_hat_WLS=(X^TWX)^(-1)X^TWy
GeneralisedLeastSquares := β_hat_GLS=(X^TΩ^(-1)X)^(-1)X^TΩ^(-1)y
PolynomialRegression := PolynomialBasisExpansion → LinearRegressionInCoefficients
SplineRegression := SplineBasisExpansion → RegressionCoefficientEstimation
InteractionTerm := PredictorProductOrJointFunction → ConditionalEffectModification
CategoricalPredictor := IndicatorOrContrastEncoding → RegressionDesignColumns
GeneralisedLinearModel := RandomComponent ∧ LinearPredictor ∧ LinkFunction
GLMMeanRelation := g(E[Y|X])=Xβ
LogisticRegression := logit(P(Y=1|X))=Xβ
LogisticProbability := P(Y=1|X)=1/(1+exp(-Xβ))
LogisticCoefficientRelation := exp(β_j)=ConditionalOddsRatio
PoissonRegression := log(E[Y|X])=Xβ
PoissonCoefficientRelation := exp(β_j)=ConditionalRateRatio
NegativeBinomialRegression := CountRegressionWithOverdispersion
ProbitRegression := Φ^(-1)(P(Y=1|X))=Xβ
NonlinearRegression := Y=f(X;β)+ε
NonlinearLeastSquares := β_hat=argmin_βΣ_i[y_i-f(x_i;β)]²
RegularisedRegression := DataFitLoss+Penalty → RegularisedCoefficientEstimate
RidgeRegression := argmin_β{RSS+λ||β||²_2}
LassoRegression := argmin_β{RSS+λ||β||_1}
ElasticNetRegression := DataFitLoss+λ1||β||_1+λ2||β||²_2
CrossValidation := PartitionData → FitOnTraining → EvaluateOnValidation
Overfitting := LowTrainingLoss ∧ HighValidationLoss
Underfitting := ModelInsufficientlyFlexible → PoorFit
BiasVarianceTradeoff := ModelComplexity ↔ PredictionBiasAndVariance
RegressionVsCorrelation := RegressionDirectionalConditionalModel whereas CorrelationSymmetricAssociationMeasure
RegressionDoesNotImplyCausation := RegressionCoefficient ↛ CausalEffect without identification assumptions
Confounding := CommonCauseOfPredictorAndResponseMayDistortCoefficientInterpretation
OmittedVariableBias := OmittedRelevantCorrelatedVariable → CoefficientBias
Endogeneity := E[X^Tε]≠0
EndogeneityConsequence := OLSMayBeBiasedOrInconsistent
InstrumentalVariablesRegression := ValidInstrument → AlternativeIdentificationAndEstimation
TwoStageLeastSquares := FirstStagePredictEndogenousX → SecondStageRegressYOnPredictedX
MeasurementError := ErrorInPredictorOrResponse
MissingData := IncompletePredictorsOrResponses → ExplicitMissingDataMethod
MultipleImputationRegression := ImputeMultipleDatasets → FitEach → PoolInference
SurveyRegression := ComplexSamplingDesign ∧ Weights ∧ DesignVariance → SurveyAdjustedRegression
TimeSeriesRegression := RegressionWithTemporalDependence
SpatialRegression := RegressionWithSpatialDependence
MixedEffectsRegression := FixedEffects ∧ RandomEffects → HierarchicalConditionalModel
RandomInterceptModel := Y_ij=β0+u_j+X_ijβ+ε_ij
QuantileRegression := ConditionalQuantile(Y|X)=Xβ_τ
RobustRegression := RegressionEstimatorReducingOutlierSensitivity
BayesianRegression := Prior(β) ∧ Likelihood(Data|β) → Posterior(β|Data)
PosteriorPredictiveRegression := PosteriorParameters → PredictiveDistributionForNewResponse

RegressionConstraint := ConstraintOn(Response,Predictors,Population,SamplingDesign,FunctionalForm,ErrorDistribution,DependenceStructure,Estimator,Regularisation)
RegressionConsistency := Response ∧ Predictors ∧ FunctionalForm ∧ Estimator ∧ ErrorStructure ∧ SamplingDesign jointly_compatible
InvalidRegression := UndefinedResponse ∨ UndefinedPredictors ∨ IncompatibleDimensions ∨ UnresolvedRankDeficiency ∨ InvalidEstimator ∨ InvalidReferenceDistribution ∨ IgnoredDependence ∨ UnsupportedCausalInterpretation

RegressionValidation :=
 ResolveScientificQuestion → ResolvePopulationSampleResponseAndPredictors → ResolveMeasurementScales
 → SpecifyRegressionTarget → SelectModelFamily → SpecifyFunctionalForm → ConstructDesignMatrix
 → SelectEstimator → EstimateParameters → ComputeFittedValuesAndResiduals → QuantifyUncertainty
 → DiagnoseAssumptionsAndInfluence → EvaluatePredictivePerformance → ValidateInterpretation → ValidateResult

RegressionState := {Population,Sample,Response,Predictors,RegressionTarget,ModelFamily,FunctionalForm,DesignMatrix,ParameterVector,CoefficientEstimates,FittedValues,Residuals,FitMeasures,StandardErrors,ConfidenceIntervals,Predictions,PredictionIntervals,SamplingDesign,DiagnosticStatus,PredictivePerformance,Evidence,Confidence,Provenance}

SEMANTIC_EQUIVALENCE(RegressionA,RegressionB) :=
 same(ResponseSemantics) ∧ same(PredictorSemantics) ∧ same(ConditionalTarget)
 ∧ same(FunctionalFormOrEquivalentBasis) ∧ same(EstimatorObjective)
 ∧ same(SamplingDesign) ∧ same(ErrorAndDependenceStructure)

RegressionOntology := Classes ∧ Relations ∧ Constraints ∧ Axioms ∧ InferenceRules
CoreClasses := {Regression,SimpleLinearRegression,MultipleLinearRegression,GeneralisedLinearModel,LogisticRegression,PoissonRegression,NonlinearRegression,PolynomialRegression,SplineRegression,RidgeRegression,LassoRegression,ElasticNetRegression,MixedEffectsRegression,QuantileRegression,RobustRegression,BayesianRegression,RegressionCoefficient,Intercept,Slope,Residual,FittedValue,Prediction}
CoreRelations := {predicts,models,targets,computedFrom,estimates,hasCoefficient,hasIntercept,hasSlope,hasResidual,hasFittedValue,hasPrediction,hasUncertainty,diagnosedBy,adjustedFor,regularisedBy,constrainedBy,equivalentTo,tracesTo}

TypedRelations :=
 RegressionModel --predicts--> Response
 RegressionModel --models--> ConditionalRelationship
 RegressionModel --targets--> RegressionTarget
 RegressionModel --computedFrom--> Sample
 RegressionModel --estimates--> ParameterVector
 RegressionModel --hasCoefficient--> RegressionCoefficient
 RegressionModel --hasResidual--> Residual
 RegressionModel --hasFittedValue--> FittedValue
 RegressionModel --hasPrediction--> Prediction
 RegressionCoefficient --hasUncertainty--> ConfidenceInterval
 RegressionModel --diagnosedBy--> RegressionDiagnostic
 RegularisedRegression --regularisedBy--> Penalty
 RegressionModel --tracesTo--> Provenance

InferenceRules := {
 FullColumnRank ⇒ UniqueOLSParameterEstimate,
 ZeroConditionalMean ⇒ OLSUnbiasedUnderCompatibleLinearModel,
 GaussMarkovConditions ⇒ OLSIsBLUE,
 SimpleLinearRegressionWithIntercept ⇒ R²=PearsonCorrelation²,
 LogisticRegression ⇒ exp(Coefficient)=ConditionalOddsRatioUnderModel,
 PoissonRegression ⇒ exp(Coefficient)=ConditionalRateRatioUnderModel,
 RegressionAssociation ↛ CausalEffect,
 InvalidDependenceStructure ⇒ FlagRegressionInferenceRisk
}

CanonicalRegressionRepresentation :=
 ParseRegressionRequest → ResolvePopulationSampleResponseAndPredictors → ResolveRegressionTarget
 → ResolveMeasurementScales → SelectRegressionFamily → SpecifyFunctionalForm → ConstructDesignMatrix
 → ResolveEstimatorAndRegularisation → EstimateParameters → ComputeFittedValuesResidualsAndFitMeasures
 → ComputeCoefficientUncertainty → ComputePredictionsAndPredictionUncertainty
 → DiagnoseAssumptionsMulticollinearityAndInfluence → EvaluateOutOfSamplePerformance → ValidateRegression

RegressionGraph := G=(V,E)
V := {Regressions,SimpleLinearRegressions,MultipleLinearRegressions,GeneralisedLinearModels,LogisticRegressions,PoissonRegressions,NonlinearRegressions,PolynomialRegressions,SplineRegressions,RidgeRegressions,LassoRegressions,ElasticNetRegressions,MixedEffectsRegressions,QuantileRegressions,RobustRegressions,BayesianRegressions,Responses,Predictors,RegressionCoefficients,Residuals,FittedValues,Predictions,DesignMatrices,Diagnostics,Evidence,ProvenanceNodes}
E := {predicts,models,targets,computedFrom,estimates,hasCoefficient,hasIntercept,hasSlope,hasResidual,hasFittedValue,hasPrediction,hasUncertainty,diagnosedBy,adjustedFor,regularisedBy,constrainedBy,equivalentTo,tracesTo}

CONFIDENCE(RegressionResolution) := weighted_mean(ResponseResolution,PredictorResolution,RegressionTargetResolution,ModelFamilyValidity,FunctionalFormValidity,EstimatorValidity,SamplingDesignValidity,ErrorStructureValidity,DiagnosticQuality,PredictiveValidation,ProvenanceQuality)

AI_QUERY_INTERPRETATION :=
 Query → DetectRegressionIntent → IdentifyPopulationSampleResponseAndPredictors
 → ResolveEstimationPredictionInferenceOrCausalInterpretationIntent
 → ResolveLinearGeneralisedNonlinearRegularisedHierarchicalOrBayesianModel
 → IdentifyRequestedCoefficientsFitDiagnosticsPredictionsOrUncertainty

AI_REASONING_FLOW :=
 Query → ParseRegressionProblem → ResolvePopulationAndSample → ResolveResponseAndPredictors
 → ResolveRegressionTarget → ResolveMeasurementScales → SelectModelFamily → SpecifyFunctionalForm
 → ConstructDesignMatrix → SelectEstimatorAndRegularisation → EstimateParameters
 → ComputeFittedValuesAndResiduals → ComputeFitMeasures → QuantifyCoefficientUncertainty
 → ComputePredictionsAndPredictionIntervals → DiagnoseFunctionalFormVarianceDependenceMulticollinearityAndInfluence
 → EvaluatePredictivePerformance → ValidateAssociationalOrCausalInterpretation
 → UpdateConfidence → PreserveProvenance → GenerateExplanation

AI_CONSUMPTION := {ParseRegressionEntities,ParseTypedRelations,ResolvePopulations,ResolveSamples,ResolveResponses,ResolvePredictors,ResolveRegressionTargets,ResolveMeasurementScales,SelectRegressionFamilies,ConstructDesignMatrices,FitSimpleLinearRegression,FitMultipleLinearRegression,FitWeightedLeastSquares,FitGeneralisedLeastSquares,FitLogisticRegression,FitPoissonRegression,FitNegativeBinomialRegression,FitProbitRegression,FitNonlinearRegression,FitPolynomialRegression,FitSplineRegression,FitRidgeRegression,FitLassoRegression,FitElasticNetRegression,FitMixedEffectsRegression,FitQuantileRegression,FitRobustRegression,FitBayesianRegression,ComputeRegressionCoefficients,ComputeFittedValues,ComputeResiduals,ComputeR2,ComputeAdjustedR2,ComputeStandardErrors,ComputeConfidenceIntervals,ComputePredictions,ComputePredictionIntervals,DiagnoseMulticollinearity,DiagnoseHeteroscedasticity,DiagnoseDependence,DiagnoseLeverageAndInfluence,ExecuteCrossValidation,ValidateRegressionSemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}

UPDATE_MODEL :=
 NewDataPredictorsModelDesignOrEvidence → ValidateInput → IdentifyAffectedRegressionClaims
 → UpdateRegressionState → ReconstructDesignMatrixIfRequired → ReestimateParameters
 → RecomputeFittedValuesResidualsAndFitMeasures → RecomputeCoefficientAndPredictionUncertainty
 → RerunDiagnosticsAndPredictiveValidation → RecomputeConfidence → PreserveProvenance → VersionModel

DataUpdate := ChangedObservedData ⇒ recompute(CoefficientEstimates,FittedValues,Residuals,FitMeasures,StandardErrors,ConfidenceIntervals,Predictions)
ModelUpdate := ChangedModelFamilyOrFunctionalForm ⇒ reevaluate(DesignMatrix,Estimator,CoefficientEstimates,Diagnostics,PredictivePerformance)
PredictorUpdate := ChangedPredictorSet ⇒ recompute(DesignMatrix,Rank,CoefficientEstimates,MulticollinearityDiagnostics,FitMeasures)

FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,Regression,RegressionValidation,RegressionConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {RegressionEntityResolution,Population,Sample,Response,Predictors,RegressionTarget,ModelFamily,FunctionalForm,DesignMatrix,CoefficientEstimates,FittedValues,Residuals,FitMeasures,StandardErrors,ConfidenceIntervals,Predictions,PredictionIntervals,SamplingDesign,DiagnosticStatus,PredictivePerformance,InterpretationStatus,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA