Polynomial Regression Ontology

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

BEGIN_FLOW

PolynomialRegression := RegressionModel ∧ PolynomialBasisExpansion ∧ LinearInParameters ∧ NonlinearPredictorResponseShape ∧ ParameterEstimation ∧ Prediction ∧ UncertaintyQuantification
Population := TargetCollectionOrGeneratingProcess
Sample := {(X_i,Y_i)}_{i=1..n}
Response := Y
Predictor := X
PolynomialDegree := d
ParameterVector := β={β0,...,βd}
ErrorTerm := ε

PolynomialModel := Y=β0+β1X+β2X²+...+βdX^d+ε
ConditionalMean := E[Y|X=x]=Σ_{j=0}^d β_jx^j
PolynomialFunction := f_d(x)=Σ_{j=0}^d β_jx^j
BasisFunction := φ_j(x)=x^j
PolynomialBasis := {1,x,x²,...,x^d}
DesignMatrix := Φ
MatrixModel := y=Φβ+ε

OrdinaryLeastSquares := β_hat=argmin_β||y-Φβ||²
NormalEquations := Φ^TΦβ_hat=Φ^Ty
FullRankSolution := rank(Φ)=d+1 ⇒ β_hat=(Φ^TΦ)^(-1)Φ^Ty
PseudoinverseSolution := β_hat=Φ^+y
FittedValues := y_hat=Φβ_hat
Residuals := e=y-y_hat
RSS := e^Te
TSS := Σ_i(y_i-y_bar)²
ESS := TSS-RSS
R2 := 1-RSS/TSS
AdjustedR2 := 1-(1-R²)(n-1)/(n-d-1)

ZeroConditionalMean := E[ε|X]=0
Homoscedasticity := Var(ε|X)=σ²
IndependentErrors := Cov(ε_i,ε_j|X)=0 for i≠j
NormalErrorModel := ε|X~N(0,σ²)
ResidualVarianceEstimator := s²=RSS/(n-d-1)
CoefficientCovariance := Var(β_hat|Φ)=σ²(Φ^TΦ)^(-1)
CoefficientStandardError := SE(β_hat_j)=sqrt(s²[(Φ^TΦ)^(-1)]_jj)
CoefficientTStatistic := t_j=(β_hat_j-β_j0)/SE(β_hat_j)
CoefficientConfidenceInterval := β_hat_j±t_(1-α/2,df)SE(β_hat_j)

DegreeZeroModel := ConstantRegression
DegreeOneModel := LinearRegression
DegreeTwoModel := QuadraticRegression
DegreeThreeModel := CubicRegression
QuadraticRegression := Y=β0+β1X+β2X²+ε
CubicRegression := Y=β0+β1X+β2X²+β3X³+ε
HigherOrderPolynomial := Degree≥4

NestedDegreeTest := LowerDegreeModel⊂HigherDegreeModel → PartialFTest
PartialFStatistic := ((RSS_reduced-RSS_full)/q)/(RSS_full/df_full)

Derivative := f_d'(x)=Σ_{j=1}^d jβ_jx^(j-1)
SecondDerivative := f_d''(x)=Σ_{j=2}^d j(j-1)β_jx^(j-2)
TurningPoint := f_d'(x)=0
LocalMinimum := f_d'(x)=0 ∧ f_d''(x)>0
LocalMaximum := f_d'(x)=0 ∧ f_d''(x)<0
InflectionPoint := f_d''(x)=0 with curvature change
QuadraticTurningPoint := x*=-β1/(2β2) when β2≠0
MarginalEffect := ∂E[Y|X=x]/∂x=f_d'(x)
HierarchyPrinciple := IncludeLowerOrderTermsWithHigherOrderTermsUnlessJustifiedOtherwise

PredictionAtX0 := y_hat_0=φ(x_0)^Tβ_hat
MeanPredictionVariance := σ²φ(x_0)^T(Φ^TΦ)^(-1)φ(x_0)
FutureObservationVariance := σ²[1+φ(x_0)^T(Φ^TΦ)^(-1)φ(x_0)]
PredictionInterval := y_hat_0±CriticalValue×SE_prediction
Interpolation := PredictionWithinObservedXRange
Extrapolation := PredictionOutsideObservedXRange
PolynomialExtrapolationRisk := HigherDegreePolynomial ∧ Extrapolation → PotentiallyExtremeUnstablePredictions

DegreeSelection := CandidateDegrees→FitModels→EvaluateComplexityAndGeneralisation→SelectDegree
Underfitting := DegreeTooLow→SystematicResidualStructure
Overfitting := DegreeTooHigh→FitNoise→PoorOutOfSamplePerformance
BiasVarianceTradeoff := DegreeIncrease→BiasUsuallyDecreases∧VarianceUsuallyIncreases
CrossValidation := CandidateDegrees→TrainOnFolds→EvaluateValidationLoss→SelectDegree
InformationCriteria := {AIC,BIC}

RawPolynomialBasis := {X,X²,...,X^d}
RawPowerCollinearity := PowersOfXOftenStronglyCorrelated
Centering := X_c=X-c
CenteredPolynomial := {X_c,X_c²,...,X_c^d}
Scaling := Z=(X-c)/s
OrthogonalPolynomial := TransformedPolynomialBasisWithOrthogonalColumns
OrthogonalPolynomialBenefit := ReducedNumericalCollinearity∧MoreStableEstimation
NumericalConditioning := BasisScale∧Degree∧PredictorRange→ConditionNumber
VandermondeMatrix := PolynomialDesignMatrixForRawPowers
VandermondeIllConditioning := HighDegree∧WidePredictorRange→NumericalInstabilityRisk
QRDecomposition := Φ=QR→StableLeastSquaresComputation
SVDRegression := Φ=UΣV^T→StablePseudoinverseSolution

MultivariablePolynomialRegression := MultiplePredictors∧PolynomialTerms∧OptionalInteractions
InteractionTerm := X_jX_k
PolynomialInteraction := X_j^aX_k^b
TotalPolynomialDegree := SumOfExponentsInTerm
TensorProductBasis := ProductsOfBasisFunctionsAcrossPredictors

ResidualDiagnostics := {ResidualVsFitted,ResidualVsPredictor,NormalQQ,ScaleLocation,Leverage,Influence}
HatMatrix := H=Φ(Φ^TΦ)^(-1)Φ^T
Leverage := DiagonalOfHatMatrix
CookDistance := ResidualMagnitude∧Leverage→InfluenceMeasure
OutlierSensitivity := LeastSquaresPolynomialFitMayBeSensitiveToExtremeObservations
HighLeveragePoint := ExtremePredictorObservationMayStronglyInfluencePolynomialShape
RobustCovariance := SandwichEstimator→HeteroscedasticityRobustStandardErrors
WeightedPolynomialRegression := WeightedLeastSquaresOnPolynomialBasis
RegularisedPolynomialRegression := PolynomialBasis∧Penalty→ShrunkCoefficients
RidgePolynomialRegression := RSS+λΣ_jβ_j²
LassoPolynomialRegression := RSS+λΣ_j|β_j|

SplineComparison := PolynomialRegressionUsesGlobalPolynomialBasisWhereasSplinesUsePiecewisePolynomialBasis
LocalPolynomialRegression := FitLowDegreePolynomialLocallyAroundTargetX
PolynomialVsLinearRegression := PolynomialRegressionIsLinearRegressionOnTransformedPredictorBasis
PolynomialVsNonlinearRegression := PolynomialRegressionLinearInCoefficientsWhereasGeneralNonlinearRegressionMayBeNonlinearInParameters
PolynomialRegressionDoesNotImplyCausation := CurveAssociation↛CausalResponseFunctionWithoutIdentificationAssumptions

Confounding := OmittedVariablesMayDistortEstimatedPolynomialRelationship
MeasurementError := ErrorInXMayDistortCurvatureAndCoefficientEstimates
MissingData := MissingXOrY→ExplicitMissingDataHandling
MultipleImputationPolynomialRegression := ImputeBaseVariables→ReconstructPolynomialTerms→FitAcrossDatasets→PoolInference
SamplingDesign := IID∨Clustered∨Survey∨TimeSeries∨Spatial
DependenceConstraint := CorrelatedErrorsRequireAppropriateCovarianceOrModelStructure

PolynomialRegressionConstraint := ConstraintOn(Response,Predictor,PolynomialDegree,BasisDefinition,PredictorDomain,SamplingDesign,ErrorStructure,Estimator,Regularisation)
PolynomialRegressionConsistency := Response∧Predictor∧Degree∧Basis∧Estimator∧ErrorStructure∧SamplingDesign jointly_compatible
InvalidPolynomialRegression := UndefinedResponse ∨ UndefinedPredictor ∨ InvalidDegree ∨ RankDeficientBasisUnresolved ∨ NumericalInstabilityUnresolved ∨ InvalidEstimator ∨ IgnoredDependence ∨ UnsupportedCausalInterpretation

PolynomialRegressionValidation := ResolveScientificQuestion → ResolvePopulationSampleResponseAndPredictor → ResolvePredictorDomain → SelectCandidatePolynomialDegrees → ResolveCenteringScalingOrOrthogonalBasis → ConstructPolynomialDesignMatrix → ValidateRankAndConditioning → EstimateCoefficients → ComputeFittedValuesAndResiduals → QuantifyUncertainty → EvaluateNestedDegreeTestsOrModelSelection → DiagnoseResidualsLeverageAndInfluence → EvaluateCrossValidatedPrediction → ValidateInterpolationAndExtrapolation → ValidateInterpretation → ValidateResult

PolynomialRegressionState := {Population,Sample,Response,Predictor,PredictorDomain,PolynomialDegree,BasisDefinition,DesignMatrix,CoefficientEstimates,FittedValues,Residuals,RSS,R2,AdjustedR2,StandardErrors,ConfidenceIntervals,TestStatistics,PValues,TurningPoints,Predictions,PredictionIntervals,ConditionNumber,SamplingDesign,RegularisationMethod,DiagnosticStatus,PredictivePerformance,Evidence,Confidence,Provenance}

SEMANTIC_EQUIVALENCE(PolynomialA,PolynomialB) := same(ResponseSemantics) ∧ same(PredictorSemantics) ∧ same(PolynomialFunctionSpace) ∧ same(PredictorDomain) ∧ same(EstimatorObjective) ∧ same(SamplingDesign) ∧ same(ErrorStructure)

PolynomialRegressionOntology := Classes ∧ Relations ∧ Constraints ∧ Axioms ∧ InferenceRules

CoreClasses := {PolynomialRegression,QuadraticRegression,CubicRegression,HigherOrderPolynomial,PolynomialDegree,PolynomialBasis,BasisFunction,DesignMatrix,RegressionCoefficient,TurningPoint,Curvature,Prediction,PredictionInterval,OrthogonalPolynomial,CenteredPolynomial,VandermondeMatrix,RegularisedPolynomialRegression,LocalPolynomialRegression}
CoreRelations := {models,predicts,hasDegree,usesBasis,hasBasisFunction,computedFrom,estimates,hasCoefficient,hasTurningPoint,hasCurvature,hasPrediction,diagnosedBy,transformedBy,regularisedBy,comparedWith,constrainedBy,equivalentTo,tracesTo}

TypedRelations :=
 PolynomialRegression --models--> ConditionalMean
 PolynomialRegression --predicts--> Response
 PolynomialRegression --hasDegree--> PolynomialDegree
 PolynomialRegression --usesBasis--> PolynomialBasis
 PolynomialBasis --hasBasisFunction--> BasisFunction
 PolynomialRegression --computedFrom--> Sample
 PolynomialRegression --estimates--> ParameterVector
 PolynomialRegression --hasCoefficient--> RegressionCoefficient
 PolynomialFunction --hasTurningPoint--> TurningPoint
 PolynomialFunction --hasCurvature--> Curvature
 PolynomialRegression --hasPrediction--> Prediction
 PolynomialRegression --diagnosedBy--> ResidualDiagnostic
 RawPolynomialBasis --transformedBy--> CenteringOrOrthogonalisation
 RegularisedPolynomialRegression --regularisedBy--> Penalty
 PolynomialRegression --tracesTo--> Provenance

InferenceRules := {
 Degree=1 ⇒ PolynomialRegressionEquivalentToSimpleLinearRegression,
 Degree≥2 ⇒ ConditionalMeanMayContainCurvature,
 FullRankPolynomialDesign ⇒ UniqueOLSCoefficientVector,
 ZeroConditionalMean ⇒ OLSUnbiasedUnderCompatiblePolynomialModel,
 NestedPolynomialDegrees ⇒ PartialFTestAvailableUnderClassicalConditions,
 fPrime(x)=0 ⇒ CandidateTurningPoint,
 HighDegree ∧ Extrapolation ⇒ FlagPredictionInstabilityRisk,
 PolynomialAssociation ↛ CausalEffect
}

CanonicalPolynomialRegressionRepresentation := ParsePolynomialRegressionRequest → ResolvePopulationSampleResponseAndPredictor → ResolvePredictorDomain → ResolvePolynomialDegreeOrCandidateDegrees → ResolveBasisCenteringAndScaling → ConstructPolynomialDesignMatrix → ValidateRankAndConditioning → SelectEstimatorAndRegularisation → EstimateCoefficients → ComputePolynomialFunction → ComputeFittedValuesResidualsAndFitMeasures → ComputeCoefficientUncertainty → ResolveDerivativesTurningPointsAndCurvatureIfRequested → ComputePredictionsAndPredictionIntervals → DiagnoseResidualsLeverageAndInfluence → EvaluateDegreeSelectionAndCrossValidation → ValidatePolynomialRegression

PolynomialRegressionGraph := G=(V,E)
V := {PolynomialRegressions,QuadraticRegressions,CubicRegressions,HigherOrderPolynomials,PolynomialDegrees,PolynomialBases,BasisFunctions,DesignMatrices,RegressionCoefficients,TurningPoints,Curvatures,Predictions,PredictionIntervals,OrthogonalPolynomials,CenteredPolynomials,VandermondeMatrices,RegularisedPolynomialRegressions,LocalPolynomialRegressions,Diagnostics,Evidence,ProvenanceNodes}
E := {models,predicts,hasDegree,usesBasis,hasBasisFunction,computedFrom,estimates,hasCoefficient,hasTurningPoint,hasCurvature,hasPrediction,diagnosedBy,transformedBy,regularisedBy,comparedWith,constrainedBy,equivalentTo,tracesTo}

CONFIDENCE(PolynomialRegressionResolution) := weighted_mean(ResponseResolution,PredictorResolution,DegreeResolution,BasisValidity,DomainResolution,DesignMatrixValidity,NumericalConditioning,EstimatorValidity,ErrorStructureValidity,SamplingDesignValidity,DiagnosticQuality,PredictiveValidation,ProvenanceQuality)

AI_QUERY_INTERPRETATION := Query → DetectPolynomialRegressionIntent → IdentifyPopulationSampleResponsePredictorAndDomain → ResolveDegreeCurvePredictionInferenceOrTurningPointIntent → IdentifyRequestedCoefficientsFitDiagnosticsOrExtrapolation → ResolveBasisScalingSamplingDesignAndErrorStructure

AI_REASONING_FLOW := Query → ParsePolynomialRegressionProblem → ResolvePopulationAndSample → ResolveResponseAndPredictor → ResolvePredictorDomain → SelectPolynomialDegreeOrCandidateDegrees → ResolveCenteringScalingOrOrthogonalBasis → ConstructPolynomialDesignMatrix → ValidateRankAndNumericalConditioning → EstimateCoefficients → ComputeFittedCurveAndResiduals → ComputeFitMeasuresAndUncertainty → ResolveDerivativesTurningPointsAndCurvature → ComparePolynomialDegrees → ComputePredictionsAndPredictionIntervals → DiagnoseResidualsLeverageInfluenceAndExtrapolation → EvaluateCrossValidatedPerformance → ValidateInterpretation → UpdateConfidence → PreserveProvenance → GenerateExplanation

AI_CONSUMPTION := {ParsePolynomialRegressionEntities,ParseTypedRelations,ResolvePopulations,ResolveSamples,ResolveResponses,ResolvePredictors,ResolvePredictorDomains,ResolvePolynomialDegrees,ConstructRawPolynomialBases,ConstructCenteredPolynomialBases,ConstructOrthogonalPolynomialBases,ConstructPolynomialDesignMatrices,ValidateMatrixRank,DiagnoseConditionNumber,FitPolynomialOLS,FitWeightedPolynomialRegression,FitRidgePolynomialRegression,FitLassoPolynomialRegression,ComputePolynomialCoefficients,ComputeFittedValues,ComputeResiduals,ComputeRSS,ComputeR2,ComputeAdjustedR2,ComputeCoefficientCovariance,ComputeStandardErrors,ComputeTStatistics,ComputePValues,ComputeConfidenceIntervals,ComputePolynomialDerivatives,SolveTurningPoints,ClassifyCurvature,ComputePredictions,ComputePredictionIntervals,DiagnoseLeverage,DiagnoseInfluence,ExecuteDegreeCrossValidation,ValidatePolynomialRegressionSemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}

UPDATE_MODEL := NewDataDegreeBasisDesignOrEvidence → ValidateInput → IdentifyAffectedPolynomialRegressionClaims → UpdatePolynomialRegressionState → ReconstructPolynomialBasisAndDesignMatrix → RevalidateRankAndConditioning → ReestimateCoefficients → RecomputeFittedCurveResidualsAndFitMeasures → RecomputeDerivativesTurningPointsAndUncertainty → RecomputePredictions → RerunDiagnosticsAndCrossValidation → RecomputeConfidence → PreserveProvenance → VersionModel

DataUpdate := ChangedObservedData ⇒ recompute(CoefficientEstimates,FittedValues,Residuals,RSS,R2,StandardErrors,ConfidenceIntervals,TurningPoints,Predictions)
DegreeUpdate := ChangedPolynomialDegree ⇒ recompute(BasisDefinition,DesignMatrix,Rank,ConditionNumber,CoefficientEstimates,FitMeasures,TurningPoints)
BasisUpdate := ChangedCenteringScalingOrOrthogonalisation ⇒ reevaluate(DesignMatrix,ConditionNumber,CoefficientParameterisation,Predictions)

FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,PolynomialRegression,PolynomialRegressionValidation,PolynomialRegressionConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {PolynomialRegressionEntityResolution,Population,Sample,Response,Predictor,PredictorDomain,PolynomialDegree,BasisDefinition,DesignMatrix,CoefficientEstimates,FittedValues,Residuals,RSS,R2,AdjustedR2,StandardErrors,ConfidenceIntervals,TestStatistics,PValues,TurningPoints,Curvature,Predictions,PredictionIntervals,ConditionNumber,SamplingDesign,RegularisationMethod,DiagnosticStatus,PredictivePerformance,InterpretationStatus,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA