Autocorrelation Ontology

FULL EXECUTABLE FLOW ALGEBRA — AUTOCORRELATION ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/time-series-statistics/autocorrelation-ontology.html
FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Time Series Statistics"
ONTOLOGY := "Autocorrelation Ontology"
PRIMARY_ENTITY := Autocorrelation
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

Autocorrelation := TemporalDependenceMeasure ∧ LaggedSelfCorrelation ∧ TimeSeriesStatistic ∧ DynamicStructureDescriptor
TimeSeries := {Y_t:t∈T}
TimeIndex := t
Lag := k
CurrentValue := Y_t
LaggedValue := Y_(t-k)
Mean := μ=E[Y_t]
Variance := γ(0)=Var(Y_t)

Autocovariance := γ(k)=Cov(Y_t,Y_(t-k))
Autocorrelation := ρ(k)=Corr(Y_t,Y_(t-k))
AutocorrelationFunction := ρ(k)=γ(k)/γ(0)
ACF := k→ρ(k)
ZeroLagAutocorrelation := ρ(0)=1
SymmetryRelation := ρ(-k)=ρ(k) for weakly stationary real-valued series
AutocorrelationRange := -1≤ρ(k)≤1
PositiveAutocorrelation := ρ(k)>0
NegativeAutocorrelation := ρ(k)<0
ZeroAutocorrelation := ρ(k)=0

SampleMean := y_bar=(1/n)Σ_t y_t
SampleAutocovariance := c_k=(1/n)Σ_{t=k+1}^n(y_t-y_bar)(y_(t-k)-y_bar)
SampleAutocorrelation := r_k=Σ_{t=k+1}^n(y_t-y_bar)(y_(t-k)-y_bar)/Σ_{t=1}^n(y_t-y_bar)²
SampleACF := k→r_k
EffectivePairsAtLagK := n-k
LagConstraint := 0≤k<n

WeakStationarity := E[Y_t]=μ ∧ Cov(Y_t,Y_(t-k))=γ(k)
StationarityRequirement := ClassicalACFInterpretationTypicallyAssumesStableLagDependence
NonstationarityRisk := TrendOrChangingMeanMayCreatePersistentApparentAutocorrelation
Detrending := Y_t-T_hat_t
Differencing := ΔY_t=Y_t-Y_(t-1)
SeasonalDifferencing := Δ_sY_t=Y_t-Y_(t-s)
PreprocessingRelation := TrendSeasonalityOrVarianceChanges→TransformBeforeDependenceInterpretationWhenAppropriate

WhiteNoise := E[ε_t]=0 ∧ Var(ε_t)=σ² ∧ Cov(ε_t,ε_(t-k))=0 for k≠0
WhiteNoiseACF := ρ(0)=1 ∧ ρ(k)=0 for k≠0
RandomWalk := Y_t=Y_(t-1)+ε_t
RandomWalkACFPattern := SlowlyDecayingHighSampleAutocorrelations
AR1 := Y_t=φY_(t-1)+ε_t
AR1StationaryConstraint := |φ|<1
AR1ACF := ρ(k)=φ^k
ARp := Y_t=c+Σ_{i=1}^pφ_iY_(t-i)+ε_t
ARpACF := SatisfiesYuleWalkerRecursion
MAq := Y_t=μ+ε_t+Σ_{j=1}^qθ_jε_(t-j)
MAqACFPattern := AutocorrelationCutsOffAfterLagQ under ideal MA(q)
ARMApq := AutoregressiveMovingAverageProcess
ARMAACFPattern := TypicallyTailsOffRatherThanFiniteCutoff

YuleWalkerEquation := γ(k)=Σ_{i=1}^pφ_iγ(k-i) for AR(p) appropriate lags
YuleWalkerEstimation := SampleAutocorrelations→EstimateARParameters
PartialAutocorrelation := CorrelationAtLagKAfterAdjustmentForIntermediateLags
PACFRelation := ACFMeasuresTotalLagAssociationWhereasPACFMeasuresDirectLinearLagAssociationConditionalOnIntermediateLags
ARIdentificationHeuristic := AR(p)OftenShowsPACFCutoffAndACFTailoff
MAIdentificationHeuristic := MA(q)OftenShowsACFCutoffAndPACFTailoff
IdentificationConstraint := ACFPACFPatternsAreHeuristicsNotProofOfModelOrder

SeasonalAutocorrelation := Corr(Y_t,Y_(t-s))
SeasonalLag := s
SeasonalACFPattern := PeaksAtMultiplesOfSeasonalPeriod
MultipleSeasonality := PeaksAtMultipleDistinctLagFamilies
CalendarSeasonality := RepeatingCalendarStructureMayGenerateSeasonalAutocorrelation

ResidualAutocorrelation := Corr(e_t,e_(t-k))
ResidualACF := k→Corr(e_t,e_(t-k))
ModelAdequacyRelation := ResidualAutocorrelation→RemainingTemporalStructure
ResidualWhiteNoiseGoal := ResidualACFApproximatelyZeroAtNonzeroLags
LjungBoxTest := JointTestOfAutocorrelationsAcrossSelectedLags
LjungBoxNull := ρ(1)=...=ρ(m)=0
BoxPierceTest := AlternativePortmanteauTestForJointAutocorrelation
DurbinWatsonStatistic := DW=Σ_{t=2}^n(e_t-e_(t-1))²/Σ_t e_t²
DurbinWatsonApproxRelation := DW≈2(1-r_1) under common settings

ACFConfidenceBandApproximation := ±z_(1-α/2)/sqrt(n) under white-noise approximation
BartlettVariance := LagDependentApproximateVarianceForSampleAutocorrelations
ConfidenceBandConstraint := SimpleBandsAreApproximateAndDependOnNullAndSeriesStructure
AutocorrelationTest := H0:ρ(k)=0
AutocorrelationTestStatistic := SampleAutocorrelation/EstimatedStandardError

SerialDependence := NonzeroTemporalAssociationAcrossObservations
PositiveSerialDependence := SimilarValuesClusterAcrossTime
NegativeSerialDependence := AlternatingOrReversingTemporalPattern
Persistence := SlowDecayOfAutocorrelationAcrossLags
Memory := DurationAndStructureOfTemporalDependence
ShortMemory := AutocorrelationDecaysSufficientlyFast
LongMemory := AutocorrelationDecaysSlowlyWithNonSummableOrNearLongRangeStructureUnderSpecifiedDefinition
LongRangeDependence := PersistentCorrelationAcrossLargeLags

AutocorrelationAndVariance := TemporalDependenceAffectsVarianceOfAggregatesAndEstimators
EffectiveSampleSize := n_eff≈n/(1+2Σ_{k≥1}ρ(k)) under compatible asymptotic conditions
PositiveAutocorrelationEffect := PositiveDependenceOftenReducesEffectiveIndependentInformation
NegativeAutocorrelationEffect := NegativeDependenceMayIncreaseEffectiveInformationForSomeEstimands
EffectiveSampleSizeConstraint := FormulaRequiresSuitableStationarityAndConvergenceConditions

RegressionErrorAutocorrelation := Cov(ε_t,ε_(t-k))≠0
OLSConsequence := CoefficientsMayRemainUnbiasedUnderExogeneityButClassicalStandardErrorsMayBeInvalid
HACCovariance := HeteroscedasticityAutocorrelationConsistentCovarianceEstimator
NeweyWestRelation := WeightedLaggedResidualCovariances→HACStandardErrors
GLSAutocorrelation := ExplicitErrorCovarianceModel→GeneralisedLeastSquares

CrossAutocorrelationDistinction := AutocorrelationComparesSeriesWithItselfWhereasCrossCorrelationComparesDifferentSeries
SpatialAutocorrelationDistinction := TemporalAutocorrelationUsesTimeLagWhereasSpatialAutocorrelationUsesSpatialNeighbourhoodOrDistance

AutocorrelationPlot := LagOnXAxis ∧ SampleAutocorrelationOnYAxis
Correlogram := VisualRepresentationOfACF
ACFInterpretation := ExamineSignMagnitudeDecayOscillationAndSeasonalPeaks
ExponentialDecay := CandidateShortMemoryAutoregressiveStructure
DampedOscillation := CandidateComplexRootOrCyclicDynamicStructure
SlowDecay := CandidateNonstationarityOrLongMemory
SharpSeasonalPeaks := CandidateSeasonality

TrendInducedAutocorrelation := SharedLongTermMovement→HighLagCorrelationWithoutStationaryDynamicInterpretation
SeasonalityInducedAutocorrelation := RepeatedSeasonalPattern→CorrelationAtSeasonalLags
AggregationEffect := TemporalAggregationMayAlterAutocorrelationStructure
SamplingFrequencyEffect := LagMeaningDependsOnObservationInterval
IrregularSamplingConstraint := StandardDiscreteLagACFRequiresAdaptationForUnequalTimeIntervals
MissingDataConstraint := MissingObservationsChangeAvailableLagPairsAndMayBiasNaiveACF
InterpolationRisk := InterpolationMayArtificiallyIncreaseAutocorrelation
OutlierRisk := ExtremeObservationsMayDistortSampleAutocorrelations

NonlinearDependenceConstraint := ZeroLinearAutocorrelationDoesNotGuaranteeTemporalIndependence
SquaredReturnAutocorrelation := Corr(Y_t²,Y_(t-k)²)
AbsoluteReturnAutocorrelation := Corr(|Y_t|,|Y_(t-k)|)
VolatilityClusteringRelation := RawSeriesMayHaveLowACFWhileSquaredOrAbsoluteSeriesShowsDependence
ARCHDiagnosticRelation := AutocorrelationOfSquaredResidualsMayIndicateConditionalHeteroscedasticity

AutocorrelationDoesNotImplyCausation := TemporalAssociation↛CausalMechanism
CommonTrendConfounding := SharedTrendMayGenerateApparentLagAssociation
CommonSeasonalityConfounding := SharedSeasonalStructureMayGenerateLagAssociation

AutocorrelationConstraint := ConstraintOn(TimeSeries,TimeIndex,Lag,Stationarity,SamplingFrequency,MissingDataMethod,Transformation,DependenceModel)
AutocorrelationConsistency := Series∧LagDefinition∧Frequency∧StationarityFramework∧Estimator jointly_compatible
InvalidAutocorrelationAnalysis := UndefinedTimeOrder∨InvalidLag∨IncompatibleSamplingFrequency∨UnresolvedTrend∨UnresolvedSeasonality∨IgnoredMissingness∨UnsupportedIndependenceInterpretation

AutocorrelationValidation := ResolveTimeSeriesAndTimeIndex→ResolveSamplingFrequency→ValidateTemporalOrder→ResolveLagSet→AssessTrendSeasonalityAndStationarity→TransformOrDifferenceIfRequired→ComputeSampleACF→QuantifyUncertainty→InspectSeasonalAndDecayPatterns→TestResidualAutocorrelationIfModelBased→ValidateInterpretation→ValidateResult

AutocorrelationState := {TimeSeries,TimeIndex,SamplingFrequency,LagSet,Autocovariances,Autocorrelations,SampleACF,StationarityStatus,TrendStatus,SeasonalityStatus,ConfidenceBands,PortmanteauTest,ResidualACF,EffectiveSampleSize,MissingDataMethod,Evidence,Confidence,Provenance}

SEMANTIC_EQUIVALENCE(AutocorrelationA,AutocorrelationB) := same(SeriesSemantics)∧same(TimeIndexSemantics)∧same(SamplingFrequency)∧same(LagDefinition)∧same(TransformationAndDifferencing)∧same(EstimatorConvention)

AutocorrelationOntology := Classes∧Relations∧Constraints∧Axioms∧InferenceRules
CoreClasses := {Autocorrelation,AutocorrelationFunction,SampleAutocorrelation,Autocovariance,Lag,SeasonalAutocorrelation,ResidualAutocorrelation,WhiteNoise,SerialDependence,Persistence,LongRangeDependence,Correlogram,LjungBoxTest,BoxPierceTest,DurbinWatsonStatistic,EffectiveSampleSize,HACCovariance}
CoreRelations := {autocorrelationOf,hasLag,computedFrom,normalises,hasSign,hasMagnitude,hasDecay,hasSeasonalPeak,testedBy,representedBy,affects,diagnosedBy,constrainedBy,equivalentTo,tracesTo}

TypedRelations := Autocorrelation--autocorrelationOf-->TimeSeries ∧ Autocorrelation--hasLag-->Lag ∧ SampleAutocorrelation--computedFrom-->ObservedSeries ∧ Autocorrelation--normalises-->Autocovariance ∧ Autocorrelation--representedBy-->Correlogram ∧ ResidualAutocorrelation--testedBy-->LjungBoxTest ∧ Autocorrelation--affects-->EffectiveSampleSize ∧ TimeSeriesModel--diagnosedBy-->ResidualACF ∧ Autocorrelation--tracesTo-->Provenance

InferenceRules := {WeakStationarity⇒AutocorrelationDependsOnlyOnLag,WhiteNoise⇒AutocorrelationZeroAtNonzeroLags,AR1∧AbsPhiLessThanOne⇒ρ(k)=φ^k,MAq⇒ACFCutoffAfterLagQUnderIdealModel,SeasonalPattern⇒CandidatePeaksAtSeasonalLags,SlowACFDecay⇒FlagNonstationarityOrLongMemoryCandidate,ResidualAutocorrelation⇒FlagModelMisspecificationRisk,ZeroAutocorrelation↛TemporalIndependence,Autocorrelation↛Causation}

CanonicalAutocorrelationRepresentation := ParseAutocorrelationRequest→ResolveSeriesTimeIndexAndFrequency→ResolveLagSet→AssessStationarityTrendAndSeasonality→ApplyTransformationOrDifferencingIfRequired→ComputeAutocovariances→ComputeAutocorrelations→ConstructACF→ComputeConfidenceBands→IdentifyDecayOscillationAndSeasonalPatterns→ExecutePortmanteauTestsIfRequired→ResolveEffectiveSampleSizeIfRequested→ValidateAutocorrelation

AutocorrelationGraph := G=(V,E)
V := {Autocorrelations,AutocorrelationFunctions,SampleAutocorrelations,Autocovariances,Lags,SeasonalAutocorrelations,ResidualAutocorrelations,WhiteNoiseProcesses,SerialDependenceNodes,PersistenceNodes,LongRangeDependenceNodes,Correlograms,LjungBoxTests,BoxPierceTests,DurbinWatsonStatistics,EffectiveSampleSizes,HACCovariances,TimeSeriesNodes,Evidence,ProvenanceNodes}
E := {autocorrelationOf,hasLag,computedFrom,normalises,hasSign,hasMagnitude,hasDecay,hasSeasonalPeak,testedBy,representedBy,affects,diagnosedBy,constrainedBy,equivalentTo,tracesTo}

CONFIDENCE(AutocorrelationResolution) := weighted_mean(TimeSeriesResolution,TimeIndexValidity,FrequencyValidity,LagValidity,StationarityAssessment,TrendSeasonalityHandling,EstimatorValidity,UncertaintyResolution,ResidualDiagnosticQuality,ProvenanceQuality)

AI_QUERY_INTERPRETATION := Query→DetectAutocorrelationIntent→IdentifySeriesTimeIndexFrequencyAndLagSet→ResolveACFResidualSeasonalPersistenceOrTestingIntent→IdentifyRequestedLagCorrelationDecayConfidenceBandOrPortmanteauTest→ResolveStationarityTrendSeasonalityAndMissingnessContext

AI_REASONING_FLOW := Query→ParseAutocorrelationProblem→ResolveTimeSeriesAndTimeIndex→ResolveSamplingFrequency→ValidateTemporalOrdering→ResolveLagSet→AssessTrendSeasonalityAndStationarity→TransformOrDifferenceIfRequired→ComputeAutocovariances→ComputeSampleACF→QuantifyUncertainty→InterpretDecayOscillationAndSeasonalPeaks→DiagnoseResidualAutocorrelationIfModelBased→ComputeEffectiveSampleSizeIfRequested→ValidateNonCausalInterpretation→UpdateConfidence→PreserveProvenance→GenerateExplanation

AI_CONSUMPTION := {ParseAutocorrelationEntities,ParseTypedRelations,ResolveTimeSeries,ResolveTimeIndices,ResolveSamplingFrequencies,ResolveLagSets,ComputeSampleMeans,ComputeAutocovariances,ComputeSampleAutocorrelations,ConstructACF,ConstructCorrelograms,ComputeConfidenceBands,ComputeBartlettVariances,DetectSeasonalACFPeaks,DetectSlowDecay,DetectOscillatoryACF,ComputeResidualACF,ExecuteLjungBoxTests,ExecuteBoxPierceTests,ComputeDurbinWatson,EstimateEffectiveSampleSize,ComputeHACCovariance,DiagnoseTrendInducedAutocorrelation,DiagnoseSeasonalityInducedAutocorrelation,ResolveMissingDataEffects,ValidateAutocorrelationSemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}

UPDATE_MODEL := NewObservationLagFrequencyModelOrEvidence→ValidateInput→IdentifyAffectedAutocorrelationClaims→UpdateAutocorrelationState→ReassessTrendSeasonalityAndStationarity→RecomputeAutocovariancesAndACF→RecomputeConfidenceBandsAndTests→RecomputeResidualACFIfRequired→RecomputeEffectiveSampleSize→RecomputeConfidence→PreserveProvenance→VersionModel
DataUpdate := NewOrChangedObservation⇒recompute(Mean,Autocovariances,SampleACF,ConfidenceBands,ResidualACF)
LagUpdate := ChangedLagSet⇒recompute(Autocovariances,Autocorrelations,ACFPatterns,PortmanteauTests)
FrequencyUpdate := ChangedSamplingFrequency⇒reevaluate(LagSemantics,SeasonalLags,DependenceInterpretation)

FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,Autocorrelation,AutocorrelationValidation,AutocorrelationConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {AutocorrelationEntityResolution,TimeSeries,TimeIndex,SamplingFrequency,LagSet,Autocovariances,Autocorrelations,SampleACF,StationarityStatus,TrendStatus,SeasonalityStatus,ConfidenceBands,PortmanteauTest,ResidualACF,EffectiveSampleSize,MissingDataMethod,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA