Multiple Regression Ontology

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

BEGIN_FLOW

MultipleRegression := RegressionModel ∧ MultiplePredictors ∧ ConditionalRelationshipEstimation ∧ ParameterEstimation ∧ Prediction ∧ UncertaintyQuantification
Population := TargetCollectionOrGeneratingProcess
Sample := {(X_i1,...,X_ip,Y_i)}_{i=1..n}
ResponseVariable := Y
Predictors := {X1,...,Xp}
ParameterVector := β={β0,β1,...,βp}
ErrorTerm := ε
MultipleLinearRegression := Y=β0+β1X1+...+βpXp+ε
MatrixModel := y=Xβ+ε
ConditionalMean := E[Y|X1,...,Xp]=β0+Σ_jβ_jX_j
LinearPredictor := η=Xβ
DesignMatrix := X
EstimatedCoefficientVector := β_hat
FittedVector := y_hat=Xβ_hat
ResidualVector := e=y-y_hat
PartialRegressionCoefficient := ConditionalExpectedChangeInYPerUnitChangeInX_jHoldingOtherIncludedPredictorsFixed
OrdinaryLeastSquares := β_hat=argmin_β(y-Xβ)^T(y-Xβ)
NormalEquations := X^TXβ_hat=X^Ty
FullRankCondition := rank(X)=k
OLSClosedForm := FullRankCondition ⇒ β_hat=(X^TX)^(-1)X^Ty
ResidualOrthogonality := X^Te=0
HatMatrix := H=X(X^TX)^(-1)X^T
FittedValues := y_hat=Hy
ResidualMaker := M=I-H
Residuals := e=My
TSS := Σ_i(y_i-y_bar)²
RSS := Σ_i(y_i-y_hat_i)²
ESS := Σ_i(y_hat_i-y_bar)²
SumSquaresDecomposition := ModelWithIntercept ⇒ TSS=ESS+RSS
R2 := 1-RSS/TSS
AdjustedR2 := 1-(1-R²)(n-1)/(n-k)
MultipleCorrelation := Corr(Y,Y_hat)
MultipleCorrelationRelation := ModelWithIntercept ⇒ R²=Corr(Y,Y_hat)²
ZeroConditionalMean := E[ε|X]=0
Homoscedasticity := Var(ε|X)=σ²I
NoPerfectMulticollinearity := rank(X)=k
GaussMarkovConditions := LinearInParameters ∧ ZeroConditionalMean ∧ FullColumnRank ∧ Homoscedasticity ∧ UncorrelatedErrors
GaussMarkovTheorem := GaussMarkovConditions ⇒ OLSIsBLUE
OLSUnbiasedness := E[β_hat|X]=β
OLSCovariance := Var(β_hat|X)=σ²(X^TX)^(-1)
ResidualVarianceEstimator := s²=RSS/(n-k)
EstimatedCoefficientCovariance := s²(X^TX)^(-1)
CoefficientStandardError := SE(β_hat_j)=sqrt(s²[(X^TX)^(-1)]_jj)
CoefficientTStatistic := t_j=(β_hat_j-β_j0)/SE(β_hat_j)
CoefficientConfidenceInterval := β_hat_j±t_(1-α/2,n-k)SE(β_hat_j)
JointNullHypothesis := H0:Rβ=r
OverallRegressionNull := H0:β1=β2=...=βp=0
OverallFStatistic := (ESS/p)/(RSS/(n-k))
NestedModelFTest := ((RSS_R-RSS_U)/q)/(RSS_U/(n-k_U))
PartialFTest := TestSubsetOfRegressionCoefficientsConditionalOnRemainingTerms
Prediction := y_hat_new=x_0^Tβ_hat
PredictionVariance := σ²[1+x_0^T(X^TX)^(-1)x_0]
PredictionInterval := Prediction±CriticalValue×PredictionStandardError

Multicollinearity := StrongLinearAssociationAmongPredictors
PerfectMulticollinearity := ExactLinearDependenceAmongDesignColumns
VarianceInflationFactor := VIF_j=1/(1-R_j²)
Tolerance := 1/VIF_j
ConditionNumber := MeasureOfDesignMatrixIllConditioning
MulticollinearityEffect := IncreasedCoefficientVariance ∧ UnstableIndividualCoefficientEstimates
PartialCorrelation := AssociationBetweenYAndX_jAfterLinearAdjustmentForOtherPredictors
SemipartialCorrelation := AssociationBetweenYAndResidualisedX_j
PartialR2 := IncrementalExplainedVariationConditionalOnReducedModel
PartialR2Formula := (RSS_R-RSS_F)/RSS_R
IncrementalR2 := R²_full-R²_reduced
StandardisedCoefficient := RegressionCoefficientAfterStandardisingVariables
InteractionTerm := X_jX_k
InteractionModel := Y=β0+β1X1+β2X2+β3X1X2+ε
CategoricalPredictor := IndicatorOrContrastEncodedVariable
Centering := X_jc=X_j-c_j
Standardisation := Z_j=(X_j-MeanX_j)/SD_X_j
Confounding := OmittedOrIncludedVariablesMayAlterCoefficientInterpretation
OmittedVariableBias := OmittedRelevantCorrelatedVariable → BiasInIncludedCoefficient
Endogeneity := E[X^Tε]≠0
EndogeneityConsequence := OLSMayBeBiasedOrInconsistent
MultipleRegressionDoesNotImplyCausation := ConditionalAssociationCoefficients ↛ CausalEffectsWithoutIdentificationAssumptions
ResidualDiagnostics := {ResidualVsFitted,NormalQQ,ScaleLocation,ResidualVsLeverage,InfluenceAssessment,DependenceAssessment}
Leverage := h_ii=H_ii
CookDistance := InfluenceMeasureCombiningResidualMagnitudeAndLeverage
DFBETAS := StandardisedChangeInCoefficientAfterDeletingObservation
DFFITS := StandardisedChangeInFittedValueAfterDeletingObservation
Heteroscedasticity := Var(ε_i|X)NotConstant
HeteroscedasticityRobustCovariance := SandwichCovarianceEstimator
ClusteredErrors := WithinClusterDependence
ClusterRobustCovariance := ClusterAggregatedSandwichEstimator
WeightedLeastSquares := β_hat_WLS=(X^TWX)^(-1)X^TWy
GeneralisedLeastSquares := β_hat_GLS=(X^TΩ^(-1)X)^(-1)X^TΩ^(-1)y
MissingData := MissingResponseOrPredictorValues
CompleteCaseRegression := FitUsingCompleteRequiredRows
MultipleImputationRegression := GenerateMultipleCompletedDatasets → FitModelPerDataset → PoolCoefficientsAndVariance
SurveyMultipleRegression := ComplexSamplingDesign ∧ SurveyWeights ∧ DesignBasedVariance → SurveyAdjustedMultipleRegression
HierarchicalRegression := PredictorsEnteredInPrespecifiedBlocks → EvaluateIncrementalFit
StepwiseRegression := DataDrivenSequentialVariableSelection
StepwiseConstraint := NaivePostSelectionPValuesAndIntervalsMayBeInvalid
RegularisedMultipleRegression := MultiplePredictorLoss+Penalty
RidgeRegression := argmin_β{RSS+λΣ_jβ_j²}
LassoRegression := argmin_β{RSS+λΣ_j|β_j|}
ElasticNetRegression := argmin_β{RSS+λ1Σ_j|β_j|+λ2Σ_jβ_j²}
CrossValidation := PartitionData → FitCandidateModels → EvaluateHeldOutLoss → SelectModelOrPenalty
Overfitting := ModelCapturesSampleSpecificNoise → PoorGeneralisation
HighDimensionalRegression := pLargeRelativeToN → OLSInstabilityOrNonidentifiability
PGreaterThanN := p≥n → X^TXNotInvertibleUnderOrdinaryFullRankOLS

MultipleRegressionConstraint := ConstraintOn(Response,PredictorSet,DesignMatrix,FunctionalForm,SamplingDesign,ErrorStructure,Estimator,VariableSelectionMethod,Regularisation)
MultipleRegressionConsistency := Response ∧ PredictorSet ∧ DesignMatrix ∧ ConditionalTarget ∧ Estimator ∧ ErrorStructure ∧ SamplingDesign jointly_compatible
InvalidMultipleRegression := UndefinedResponse ∨ UndefinedPredictors ∨ IncompatibleDimensions ∨ UnresolvedPerfectMulticollinearity ∨ InvalidEstimator ∨ IgnoredDependence ∨ InvalidPostSelectionInference ∨ UnsupportedCausalInterpretation

MultipleRegressionValidation :=
 ResolveScientificQuestion → ResolvePopulationSampleResponseAndPredictors → ResolveConditionalTarget
 → ResolveMeasurementScales → SpecifyPredictorSetAndFunctionalForms → ConstructDesignMatrix
 → ValidateRankAndCollinearity → SelectEstimator → EstimateCoefficients
 → ComputeFittedValuesResidualsAndFitMeasures → EstimateCoefficientCovariance
 → QuantifyIndividualAndJointUncertainty → DiagnoseResidualsLeverageInfluenceAndDependence
 → EvaluateIncrementalAndPredictivePerformance → ValidateInterpretation → ValidateResult

MultipleRegressionState := {Population,Sample,Response,Predictors,DesignMatrix,ParameterVector,CoefficientEstimates,StandardisedCoefficients,FittedValues,Residuals,RSS,TSS,ESS,R2,AdjustedR2,PartialR2,StandardErrors,ConfidenceIntervals,TStatistics,FStatistics,PValues,Predictions,PredictionIntervals,VIF,SamplingDesign,VariableSelectionMethod,RegularisationMethod,DiagnosticStatus,PredictivePerformance,Evidence,Confidence,Provenance}

SEMANTIC_EQUIVALENCE(MultipleRegressionA,MultipleRegressionB) := same(ResponseSemantics) ∧ same(PredictorColumnSpace) ∧ same(ConditionalTarget) ∧ same(EstimatorObjective) ∧ same(SamplingDesign) ∧ same(ErrorAndDependenceStructure) ∧ same(SelectionAndRegularisationFramework)

MultipleRegressionOntology := Classes ∧ Relations ∧ Constraints ∧ Axioms ∧ InferenceRules
CoreClasses := {MultipleRegression,MultipleLinearRegression,PredictorSet,DesignMatrix,RegressionCoefficient,PartialRegressionCoefficient,StandardisedCoefficient,InteractionTerm,PartialCorrelation,PartialR2,R2,AdjustedR2,Multicollinearity,VarianceInflationFactor,Residual,Leverage,InfluenceMeasure,WeightedLeastSquares,GeneralisedLeastSquares,RidgeRegression,LassoRegression,ElasticNetRegression}
CoreRelations := {predicts,models,includesPredictor,estimates,hasCoefficient,adjustsFor,hasInteraction,hasPartialAssociation,hasFitMeasure,hasResidual,hasDiagnostic,affectedBy,regularisedBy,selectedBy,constrainedBy,equivalentTo,tracesTo}

TypedRelations :=
 MultipleRegression --predicts--> Response
 MultipleRegression --models--> ConditionalMean
 MultipleRegression --includesPredictor--> Predictor
 MultipleRegression --estimates--> ParameterVector
 MultipleRegression --hasCoefficient--> RegressionCoefficient
 RegressionCoefficient --adjustsFor--> OtherPredictors
 MultipleRegression --hasInteraction--> InteractionTerm
 Predictor --hasPartialAssociation--> PartialCorrelation
 MultipleRegression --hasFitMeasure--> R2
 MultipleRegression --hasResidual--> Residual
 MultipleRegression --hasDiagnostic--> VIF
 MultipleRegression --affectedBy--> Multicollinearity
 RegularisedMultipleRegression --regularisedBy--> Penalty
 MultipleRegression --selectedBy--> ModelSelectionMethod
 MultipleRegression --tracesTo--> Provenance

InferenceRules := {
 FullColumnRank ⇒ UniqueOLSParameterEstimate,
 ZeroConditionalMean ⇒ OLSUnbiasedUnderCompatibleModel,
 GaussMarkovConditions ⇒ OLSIsBLUE,
 VIF_j=1/(1-R_j²),
 FullModelNestedWithinComparison ⇒ IncrementalFitAssessableByPartialFTest,
 PerfectMulticollinearity ⇒ OLSCoefficientVectorNotUniquelyIdentified,
 ConditionalAssociation ↛ CausalEffect,
 DataDrivenSelection ⇒ RequireSelectionAwareInferenceForNaiveConfirmatoryClaims
}

CanonicalMultipleRegressionRepresentation :=
 ParseMultipleRegressionRequest → ResolvePopulationSampleResponseAndPredictors
 → ResolveConditionalTarget → SpecifyPredictorSetFunctionalFormsAndInteractions
 → ConstructDesignMatrix → ValidateRankAndMulticollinearity → SelectEstimatorAndRegularisation
 → EstimateCoefficients → ComputeFittedValuesResidualsAndFitMeasures
 → ComputePartialAndIncrementalEffects → EstimateCoefficientCovariance
 → ComputeIndividualAndJointTestsAndIntervals → DiagnoseInfluenceVarianceAndDependence
 → EvaluatePredictionAndCrossValidation → ValidateMultipleRegression

MultipleRegressionGraph := G=(V,E)
V := {MultipleRegressions,MultipleLinearRegressions,PredictorSets,DesignMatrices,RegressionCoefficients,PartialRegressionCoefficients,StandardisedCoefficients,InteractionTerms,PartialCorrelations,PartialR2Measures,R2Measures,AdjustedR2Measures,MulticollinearityNodes,VarianceInflationFactors,Residuals,Leverages,InfluenceMeasures,WeightedLeastSquaresModels,GeneralisedLeastSquaresModels,RidgeRegressions,LassoRegressions,ElasticNetRegressions,Diagnostics,Evidence,ProvenanceNodes}
E := {predicts,models,includesPredictor,estimates,hasCoefficient,adjustsFor,hasInteraction,hasPartialAssociation,hasFitMeasure,hasResidual,hasDiagnostic,affectedBy,regularisedBy,selectedBy,constrainedBy,equivalentTo,tracesTo}

CONFIDENCE(MultipleRegressionResolution) := weighted_mean(ResponseResolution,PredictorResolution,ConditionalTargetResolution,DesignMatrixValidity,RankValidity,MulticollinearityAssessment,EstimatorValidity,ErrorStructureValidity,SamplingDesignValidity,SelectionValidity,PredictiveValidation,ProvenanceQuality)

AI_QUERY_INTERPRETATION := Query → DetectMultipleRegressionIntent → IdentifyPopulationSampleResponseAndPredictorSet → ResolveCoefficientAdjustmentPredictionModelComparisonOrSelectionIntent → IdentifyRequestedPartialEffectsInteractionsFitDiagnosticsOrPredictions → ResolveEstimatorRegularisationSamplingDesignAndErrorStructure

AI_REASONING_FLOW := Query → ParseMultipleRegressionProblem → ResolvePopulationAndSample → ResolveResponseAndPredictors → ResolveConditionalTarget → ResolveMeasurementScales → SpecifyPredictorSetFunctionalFormsAndInteractions → ConstructDesignMatrix → ValidateRankAndMulticollinearity → SelectEstimatorAndRegularisation → EstimateCoefficients → ComputeFittedValuesAndResiduals → ComputeR2AdjustedR2AndPartialR2 → EstimateCoefficientCovariance → ComputeTTestsFTestsAndConfidenceIntervals → DiagnoseResidualsLeverageInfluenceVarianceAndDependence → EvaluatePredictionAndCrossValidation → ValidateAssociationalOrCausalInterpretation → UpdateConfidence → PreserveProvenance → GenerateExplanation

AI_CONSUMPTION := {ParseMultipleRegressionEntities,ParseTypedRelations,ResolvePopulations,ResolveSamples,ResolveResponses,ResolvePredictorSets,ResolveConditionalTargets,ConstructDesignMatrices,ValidateMatrixRank,DiagnoseMulticollinearity,ComputeVarianceInflationFactors,FitMultipleLinearRegression,FitWeightedLeastSquares,FitGeneralisedLeastSquares,FitRidgeRegression,FitLassoRegression,FitElasticNetRegression,ComputeRegressionCoefficients,ComputeStandardisedCoefficients,ComputeFittedValues,ComputeResiduals,ComputeRSS,ComputeTSS,ComputeESS,ComputeR2,ComputeAdjustedR2,ComputePartialR2,ComputePartialCorrelations,ComputeCoefficientCovariance,ComputeStandardErrors,ComputeTStatistics,ComputeFStatistics,ComputePValues,ComputeConfidenceIntervals,ComputePredictions,ComputePredictionIntervals,ComputeLeverage,ComputeCookDistance,ComputeDFBETAS,ExecuteCrossValidation,ValidateMultipleRegressionSemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}

UPDATE_MODEL := NewDataPredictorsModelDesignOrEvidence → ValidateInput → IdentifyAffectedMultipleRegressionClaims → UpdateMultipleRegressionState → ReconstructDesignMatrixIfRequired → RevalidateRankAndMulticollinearity → ReestimateCoefficients → RecomputeFittedValuesResidualsAndFitMeasures → RecomputePartialEffectsAndCoefficientUncertainty → RerunDiagnosticsAndPredictiveValidation → RecomputeConfidence → PreserveProvenance → VersionModel
DataUpdate := ChangedObservedData ⇒ recompute(CoefficientEstimates,FittedValues,Residuals,R2,AdjustedR2,PartialR2,StandardErrors,ConfidenceIntervals,Predictions)
PredictorUpdate := ChangedPredictorSet ⇒ recompute(DesignMatrix,Rank,VIF,CoefficientEstimates,PartialEffects,FitMeasures)
SelectionUpdate := ChangedVariableSelectionMethod ⇒ reevaluate(SelectedPredictors,PostSelectionInferenceValidity,PredictivePerformance)

FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,MultipleRegression,MultipleRegressionValidation,MultipleRegressionConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {MultipleRegressionEntityResolution,Population,Sample,Response,Predictors,DesignMatrix,CoefficientEstimates,StandardisedCoefficients,FittedValues,Residuals,R2,AdjustedR2,PartialR2,StandardErrors,ConfidenceIntervals,TStatistics,FStatistics,PValues,Predictions,PredictionIntervals,VIF,SamplingDesign,VariableSelectionMethod,RegularisationMethod,DiagnosticStatus,PredictivePerformance,InterpretationStatus,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA