Correlation & Regression

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

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

BEGIN_FLOW

CorrelationRegression :=
 StatisticalAssociation ∧ DependenceMeasurement ∧ PredictiveModelling
 ∧ ParameterEstimation ∧ UncertaintyQuantification ∧ Validation

Population := TargetCollectionOrGeneratingProcess
Sample := {(X_i,Y_i)}_{i=1..n}
Predictor := X
Response := Y
ParameterVector := β
ErrorTerm := ε

Covariance := Cov(X,Y)=E[(X-E[X])(Y-E[Y])]
SampleCovariance := s_xy=Σ_i(x_i-x_bar)(y_i-y_bar)/(n-1)
Correlation := StandardisedCovariance
PearsonCorrelation := ρ_XY=Cov(X,Y)/(σ_Xσ_Y)
SamplePearsonCorrelation := r=Σ_i(x_i-x_bar)(y_i-y_bar)/sqrt(Σ_i(x_i-x_bar)^2Σ_i(y_i-y_bar)^2)
CorrelationRange := -1≤ρ≤1
PositiveCorrelation := ρ>0
NegativeCorrelation := ρ<0
ZeroLinearCorrelation := ρ=0
PerfectPositiveCorrelation := ρ=1
PerfectNegativeCorrelation := ρ=-1
CorrelationSymmetry := Corr(X,Y)=Corr(Y,X)
CorrelationDoesNotImplyCausation := StatisticalAssociation↛CausalRelationship
ZeroCorrelationConstraint := ZeroPearsonCorrelation↛StatisticalIndependence
IndependenceRelation := Independence⇒Covariance=0 when moments exist

SpearmanCorrelation := PearsonCorrelationOfRanks
KendallTau := ConcordanceMinusDiscordanceMeasure
PartialCorrelation := AssociationBetweenXAndYAfterLinearAdjustmentForCovariates
Autocorrelation := Corr(Y_t,Y_(t-k))
CrossCorrelation := Corr(X_t,Y_(t-k))
CorrelationMatrix := R=[ρ_jk]
CovarianceMatrix := Σ=[Cov(X_j,X_k)]
PositiveSemidefiniteConstraint := ValidCorrelationMatrixMustBeSymmetricPSDWithUnitDiagonal

ScatterplotRelation := PairedObservations→VisualAssociationPattern
OutlierSensitivity := PearsonCorrelationMayBeStronglyAffectedByExtremeObservations
NonlinearAssociation := StrongDependenceMayCoexistWithWeakPearsonCorrelation

Regression := ConditionalRelationshipModel ∧ ResponsePrediction ∧ ParameterEstimation
SimpleLinearRegression := Y=β0+β1X+ε
MultipleLinearRegression := Y=β0+β1X1+...+βpXp+ε
ConditionalMeanModel := E[Y|X]=β0+β1X
RegressionFunction := m(x)=E[Y|X=x]
LinearPredictor := η=Xβ

ObservedResponseVector := y
DesignMatrix := X
CoefficientVector := β
FittedCoefficientVector := β_hat
FittedValues := y_hat=Xβ_hat
ResidualVector := e=y-y_hat

OrdinaryLeastSquares := β_hat=argmin_β(y-Xβ)^T(y-Xβ)
NormalEquations := X^TXβ_hat=X^Ty
OLSClosedForm := β_hat=(X^TX)^(-1)X^Ty when X has full column rank
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
CorrelationSlopeRelation := b1=r(s_y/s_x)

CoefficientInterpretation :=
 β_j=ExpectedChangeInResponsePerUnitPredictorChange
 holding other included predictors fixed

Residual := e_i=y_i-y_hat_i
ResidualSumOfSquares := RSS=Σ_i e_i²
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
SimpleRegressionR2Relation := SimpleLinearRegressionWithIntercept⇒R²=r²
AdjustedR2 := 1-(1-R²)(n-1)/(n-p-1)

ResidualVarianceEstimator := s²=RSS/(n-k)
CoefficientCovariance := Var(β_hat|X)=σ²(X^TX)^(-1)
CoefficientStandardError := sqrt(diagonal(s²(X^TX)^(-1)))
RegressionTStatistic := t_j=(β_hat_j-β_j0)/SE(β_hat_j)
RegressionConfidenceInterval := β_hat_j±t_(1-α/2,df)SE(β_hat_j)

Prediction := y_hat_new=x_new^Tβ_hat
MeanResponseEstimate := E_hat[Y|x_new]=x_new^Tβ_hat
PredictionVariance := MeanResponseVariance+IrreducibleErrorVariance
PredictionInterval := PointPrediction±CriticalValue×PredictionSE

RegressionAssumptions := {
 CorrectFunctionalForm,
 IndependentOrCorrectlyModelledErrors,
 ZeroConditionalMean,
 FiniteVariance,
 NoPerfectMulticollinearity
}

Homoscedasticity := Var(ε|X)=σ²
Heteroscedasticity := Var(ε|X)DependsOnX
NormalErrorAssumption := ε|X~Normal(0,σ²)

GaussMarkovConditions :=
 LinearModel ∧ ZeroConditionalMean ∧ NoPerfectMulticollinearity
 ∧ HomoscedasticUncorrelatedErrors

GaussMarkovTheorem := GaussMarkovConditions⇒OLSIsBLUE

Multicollinearity := StrongLinearDependenceAmongPredictors
VarianceInflationFactor := VIF_j=1/(1-R_j²)
PerfectMulticollinearity := rank(X)<number_of_columns(X)
HatMatrix := H=X(X^TX)^(-1)X^T
Leverage := h_ii=H_ii
CookDistance := InfluenceMeasureCombiningResidualAndLeverage

ResidualDiagnostics := {
 ResidualVsFitted,QQAssessment,ScaleLocation,LeverageAssessment,
 InfluenceAssessment,AutocorrelationAssessment
}

HeteroscedasticityRobustCovariance := SandwichCovarianceEstimator
ClusterRobustCovariance := ClusterAggregatedSandwichCovariance
WeightedLeastSquares := β_hat_WLS=(X^TWX)^(-1)X^TWy
GeneralisedLeastSquares := β_hat_GLS=(X^TΩ^(-1)X)^(-1)X^TΩ^(-1)y

PolynomialRegression := LinearRegressionOnPolynomialBasisFunctions
InteractionTerm := ProductOrJointFunctionOfPredictorsRepresentingEffectModification
CategoricalPredictor := EncodedUsingIndicatorOrContrastVariables

LogisticRegression := logit(P(Y=1|X))=Xβ
LogisticProbability := P(Y=1|X)=1/(1+exp(-Xβ))
LogisticCoefficientRelation := exp(β_j)=ConditionalOddsRatioPerUnitChange

PoissonRegression := log(E[Y|X])=Xβ
PoissonCoefficientRelation := exp(β_j)=ConditionalRateRatio

GeneralisedLinearModel := RandomComponent ∧ LinearPredictor ∧ LinkFunction
LinkFunction := g(E[Y|X])=Xβ
NonlinearRegression := Y=f(X;β)+ε

RegularisedRegression := DataFitLoss+Penalty→CoefficientEstimate
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→TrainModel→EvaluateHeldOutPrediction
TrainingError := ErrorMeasuredOnFittedData
ValidationError := ErrorMeasuredOnHeldOutData
Overfitting := LowTrainingError∧PoorOutOfSamplePerformance
Underfitting := ModelTooSimpleToCaptureRelevantStructure
BiasVarianceTradeoff := ModelComplexityInfluencesPredictionBiasAndVariance

CorrelationRegressionDistinction :=
 CorrelationIsSymmetricAssociationMeasure
 whereas RegressionSpecifiesDirectionalResponsePredictorModel

RegressionDoesNotImplyCausation :=
 RegressionCoefficient↛CausalEffectWithoutIdentificationAssumptions

Confounding := CommonCauseOfPredictorAndResponseMayDistortAssociation
OmittedVariableBias := OmittedRelevantCorrelatedPredictor→BiasedCoefficientUnderSpecifiedConditions
Endogeneity := E[X^Tε]≠0
EndogeneityConsequence := OLSMayBeBiasedOrInconsistent
InstrumentalVariablesRelation := ValidInstrument∧EndogenousPredictor→AlternativeEstimation

MeasurementError := ErrorInPredictorOrResponse
MissingData := IncompletePredictorOrResponseValues→RequiresExplicitHandling
MultipleImputationRegression := ImputeMultipleDatasets→FitModelEach→PoolInference
SurveyRegression := ComplexSamplingDesign→WeightedOrDesignBasedRegressionInference
TimeSeriesRegression := TemporalDependence→ModelAutocorrelationAndDynamicStructure
SpatialRegression := SpatialDependence→SpatialErrorOrDependenceModel

CorrelationRegressionConstraint :=
 ConstraintOn(
  Variables,MeasurementScale,Population,SamplingDesign,FunctionalForm,
  DependenceStructure,Estimator,ErrorModel
 )

CorrelationRegressionConsistency :=
 Data∧Variables∧AssociationMetric∧RegressionModel∧SamplingDesign∧ErrorStructure
 jointly_compatible

InvalidCorrelationRegression :=
 UndefinedVariables
 ∨ InvalidMeasurementScale
 ∨ IncompatibleModel
 ∨ RankDeficiencyUnresolved
 ∨ InvalidReferenceDistribution
 ∨ IgnoredDependence
 ∨ UnsupportedCausalInterpretation

CorrelationRegressionValidation :=
 ResolveScientificQuestion
 → ResolvePopulationSampleAndVariables
 → ResolveMeasurementScales
 → ResolveAssociationOrRegressionIntent
 → ComputeCorrelationIfRequested
 → SpecifyRegressionModelIfRequested
 → ConstructDesignMatrix
 → EstimateParameters
 → ComputeFittedValuesAndResiduals
 → QuantifyUncertainty
 → DiagnoseAssumptionsAndInfluence
 → EvaluatePredictivePerformance
 → ValidateInterpretation
 → ValidateResult

CorrelationRegressionState := {
 Population,Sample,Predictors,Response,AssociationMetric,CorrelationEstimate,
 CovarianceEstimate,RegressionModel,DesignMatrix,CoefficientEstimates,
 FittedValues,Residuals,R2,StandardErrors,ConfidenceIntervals,
 PredictionIntervals,SamplingDesign,DiagnosticStatus,PredictivePerformance,
 Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(ModelA,ModelB) :=
 same(DataSemantics)
 ∧ same(TargetVariables)
 ∧ same(AssociationOrConditionalTarget)
 ∧ same(FunctionalFormOrEquivalentRepresentation)
 ∧ same(SamplingDesign)
 ∧ same(ErrorAndDependenceStructure)

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

CoreClasses := {
 Correlation,PearsonCorrelation,SpearmanCorrelation,PartialCorrelation,Covariance,
 Regression,SimpleLinearRegression,MultipleLinearRegression,RegressionCoefficient,
 Intercept,Slope,Residual,FittedValue,DesignMatrix,R2,Prediction,ConfidenceInterval,
 PredictionInterval,LogisticRegression,PoissonRegression,RegularisedRegression
}

CoreRelations := {
 correlatesWith,predicts,explains,computedFrom,estimates,hasCoefficient,
 hasResidual,hasFittedValue,hasAssociation,hasVariance,hasStandardError,
 diagnosedBy,adjustedFor,constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 Correlation --correlatesWith--> Variable
 RegressionModel --predicts--> Response
 Correlation --computedFrom--> Sample
 RegressionCoefficient --estimatedFrom--> Sample
 RegressionModel --hasCoefficient--> RegressionCoefficient
 RegressionModel --hasResidual--> Residual
 RegressionModel --hasFittedValue--> FittedValue
 VariablePair --hasAssociation--> Correlation
 RegressionCoefficient --hasStandardError--> StandardError
 RegressionModel --diagnosedBy--> ResidualDiagnostic
 RegressionModel --adjustedFor--> Covariate
 RegressionModel --tracesTo--> Provenance

InferenceRules := {
 Covariance∧PositiveStandardDeviations
  ⇒ Correlation=Covariance/(SD_X×SD_Y),

 Independence∧FiniteSecondMoments
  ⇒ Covariance=0,

 ZeroCorrelation
  ↛ Independence,

 SimpleLinearRegressionWithIntercept
  ⇒ R²=PearsonCorrelation²,

 FullRankDesignMatrix
  ⇒ UniqueOLSCoefficientVector,

 ZeroConditionalMean
  ⇒ OLSUnbiasedUnderCompatibleLinearModel,

 RegressionAssociation
  ↛ CausalEffect,

 InvalidDependenceModel
  ⇒ FlagInferenceValidityRisk
}

CanonicalCorrelationRegressionRepresentation :=
 ParseCorrelationRegressionRequest
 → ResolvePopulationSampleAndVariables
 → ResolveMeasurementScales
 → ResolveAssociationMetric
 → ComputeCovarianceAndCorrelation
 → ResolveRegressionTarget
 → SpecifyFunctionalForm
 → ConstructDesignMatrix
 → EstimateCoefficients
 → ComputeFittedValuesResidualsAndFitMeasures
 → ComputeUncertainty
 → DiagnoseModelAssumptions
 → EvaluatePrediction
 → ValidateInterpretation

CorrelationRegressionGraph := G=(V,E)

V := {
 Correlations,PearsonCorrelations,SpearmanCorrelations,PartialCorrelations,
 Covariances,RegressionModels,SimpleLinearRegressions,MultipleRegressions,
 LogisticRegressions,PoissonRegressions,RegularisedRegressions,Predictors,
 Responses,RegressionCoefficients,Residuals,FittedValues,DesignMatrices,
 R2Measures,StandardErrors,ConfidenceIntervals,PredictionIntervals,
 Diagnostics,Evidence,ProvenanceNodes
}

E := {
 correlatesWith,predicts,explains,computedFrom,estimates,hasCoefficient,
 hasResidual,hasFittedValue,hasAssociation,hasVariance,hasStandardError,
 diagnosedBy,adjustedFor,constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(CorrelationRegressionResolution) :=
 weighted_mean(
  VariableResolution,MeasurementScaleValidity,AssociationMetricValidity,
  RegressionModelValidity,SamplingDesignValidity,ErrorStructureValidity,
  DiagnosticQuality,PredictiveValidation,ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectCorrelationRegressionIntent
 → IdentifyVariablesPopulationAndSample
 → ResolveCorrelationAssociationPredictionOrCoefficientIntent
 → ResolveLinearGeneralisedNonlinearOrRegularisedModel
 → IdentifyRequestedEstimateInferenceDiagnosticOrPrediction

AI_REASONING_FLOW :=
 Query
 → ParseCorrelationRegressionProblem
 → ResolvePopulationSampleAndVariables
 → ResolveMeasurementScales
 → ComputeAssociationMetricIfRequested
 → ResolveResponseAndPredictors
 → SpecifyRegressionModel
 → ConstructDesignMatrix
 → EstimateParameters
 → ComputeFittedValuesAndResiduals
 → ComputeR2OrRelevantFitMeasures
 → QuantifyCoefficientUncertainty
 → DiagnoseResidualsDependenceMulticollinearityAndInfluence
 → EvaluatePredictivePerformance
 → ValidateCausalOrAssociationalInterpretation
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseCorrelationRegressionEntities,ParseTypedRelations,ResolvePopulations,
 ResolveSamples,ResolveVariables,ResolveMeasurementScales,ComputeCovariances,
 ComputePearsonCorrelations,ComputeSpearmanCorrelations,ComputePartialCorrelations,
 ConstructCorrelationMatrices,ResolveResponses,ResolvePredictors,
 ConstructDesignMatrices,FitSimpleLinearRegression,FitMultipleLinearRegression,
 FitWeightedLeastSquares,FitGeneralisedLeastSquares,FitLogisticRegression,
 FitPoissonRegression,FitNonlinearRegression,FitRidgeRegression,
 FitLassoRegression,ComputeRegressionCoefficients,ComputeFittedValues,
 ComputeResiduals,ComputeR2,ComputeAdjustedR2,ComputeStandardErrors,
 ComputeConfidenceIntervals,ComputePredictions,ComputePredictionIntervals,
 DiagnoseMulticollinearity,DiagnoseHeteroscedasticity,DiagnoseDependence,
 DiagnoseLeverageAndInfluence,ExecuteCrossValidation,
 ValidateCorrelationRegressionSemantics,CompareSemanticEquivalence,
 UpdateConfidence,PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataVariablesModelDesignOrEvidence
 → ValidateInput
 → IdentifyAffectedCorrelationRegressionClaims
 → UpdateCorrelationRegressionState
 → RecomputeAssociationMeasures
 → ReconstructDesignMatrixIfRequired
 → ReestimateRegressionParameters
 → RecomputeFittedValuesResidualsAndFitMeasures
 → RecomputeUncertainty
 → RerunDiagnosticsAndPredictiveValidation
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

DataUpdate :=
 ChangedObservedData
 ⇒ recompute(
  CovarianceEstimate,CorrelationEstimate,CoefficientEstimates,FittedValues,
  Residuals,FitMeasures,StandardErrors
 )

ModelUpdate :=
 ChangedFunctionalFormOrPredictors
 ⇒ reevaluate(
  DesignMatrix,CoefficientEstimates,ResidualDiagnostics,PredictivePerformance
 )

DesignUpdate :=
 ChangedSamplingOrDependenceDesign
 ⇒ recompute(
  VarianceEstimates,StandardErrors,ConfidenceIntervals,TestStatistics
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  CorrelationRegression,
  CorrelationRegressionValidation,
  CorrelationRegressionConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 CorrelationRegressionEntityResolution,Population,Sample,Predictors,Response,
 AssociationMetric,CovarianceEstimate,CorrelationEstimate,RegressionModel,
 DesignMatrix,CoefficientEstimates,FittedValues,Residuals,R2,AdjustedR2,
 StandardErrors,ConfidenceIntervals,Predictions,PredictionIntervals,
 SamplingDesign,DiagnosticStatus,PredictivePerformance,InterpretationStatus,
 ValidationStatus,Confidence,Provenance,Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA