ARIMA Ontology

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

BEGIN_FLOW

ARIMA := AutoregressiveIntegratedMovingAverage ∧ TimeSeriesModel ∧ DifferenceStationaryProcess ∧ LinearDynamicModel ∧ ForecastingFramework
TimeSeries := {Y_t:t∈T}
TimeIndex := t
ObservedValue := y_t
BackshiftOperator := L
LagRelation := LY_t=Y_(t-1)
AROrder := p
IntegrationOrder := d
MAOrder := q
ARIMAOrder := (p,d,q)
Innovation := ε_t
InnovationVariance := σ²

DifferenceOperator := Δ=(1-L)
FirstDifference := ΔY_t=(1-L)Y_t=Y_t-Y_(t-1)
DthDifference := Δ^dY_t=(1-L)^dY_t
IntegratedSeries := OriginalSeriesRequiringDifferencingToReachStationaryRepresentation
IntegrationOrder := MinimumDifferencingOrderNeededUnderSpecifiedModel

ARPolynomial := φ(L)=1-φ1L-φ2L²-...-φpL^p
MAPolynomial := θ(L)=1+θ1L+θ2L²+...+θqL^q
ARIMAEquation := φ(L)(1-L)^dY_t=c+θ(L)ε_t
ZeroMeanARIMA := φ(L)(1-L)^dY_t=θ(L)ε_t
DifferencedSeries := W_t=(1-L)^dY_t
ARMARepresentation := φ(L)W_t=c+θ(L)ε_t

ARComponent := DependenceOnPastDifferencedValues
MAComponent := DependenceOnCurrentAndPastInnovations
IntegratedComponent := DifferencingAppliedToRemoveStochasticTrend
ARIMApdq := AR(p) ∧ I(d) ∧ MA(q)
ARIMA000 := WhiteNoise
ARIMA100 := AR1
ARIMA001 := MA1
ARIMA110 := RandomWalkWithARDriftStructureUnderParameterisation
ARIMA010 := RandomWalkWithoutDriftUnderStandardForm

Stationarity := DifferencedSeriesHasStableMeanVarianceAndLagCovariance
ARStationarityConstraint := RootsOfφ(z)=0LieOutsideUnitCircle
Invertibility := MARepresentationRecoverableFromObservedHistory
MAInvertibilityConstraint := RootsOfθ(z)=0LieOutsideUnitCircle
CausalityRepresentation := StationaryARMAComponentExpressibleAsInfiniteMAUnderRootConditions

UnitRoot := RootAtOneInAutoregressiveRepresentation
UnitRootRelation := UnitRoot→DifferencingCandidate
RandomWalk := Y_t=Y_(t-1)+ε_t
RandomWalkDifference := ΔY_t=ε_t
Drift := ConstantMeanChangePerPeriodInIntegratedProcess
ARIMAWithDrift := φ(L)(1-L)^dY_t=c+θ(L)ε_t under compatible d parameterisation
TrendRelation := DriftInDifferencedModelMayImplyPolynomialTrendInLevelsDependingOnD

ModelIdentification := Determine(p,d,q)
StationarityAssessment := Plot ∧ ACF ∧ UnitRootTests ∧ DomainKnowledge
ADFTest := AugmentedDickeyFullerUnitRootTest
KPSS := StationarityTest
DifferencingSelection := AssessNonstationarity→ApplyDifference→ReassessStationarity
OverDifferencing := dTooLarge→UnnecessaryNoiseAndMA-likeStructure
UnderDifferencing := dTooSmall→PersistentNonstationarity

ACF := AutocorrelationFunction
PACF := PartialAutocorrelationFunction
ARIdentificationHeuristic := PACFCutoff∧ACFTailoff→CandidateAROrder
MAIdentificationHeuristic := ACFCutoff∧PACFTailoff→CandidateMAOrder
ARIMAIdentificationConstraint := ACFPACFPatternsAreHeuristicsNotProof

ParameterVector := {φ1,...,φp,θ1,...,θq,c,σ²}
MaximumLikelihoodEstimation := β_hat=argmaxβ LogLikelihood(Data|β)
ConditionalSumSquares := ApproximateEstimationConditioningOnInitialValues
ExactLikelihood := LikelihoodAccountingForInitialStateDistributionUnderSpecifiedImplementation
StateSpaceARIMA := ARIMARepresentedAsLinearGaussianStateSpaceModel
KalmanFilterLikelihood := StateSpaceFiltering→InnovationLikelihood

Residual := e_t=Observed_t-Fitted_t
ARIMAResidualGoal := ResidualsApproximatelyWhiteNoise
ResidualMeanDiagnostic := Mean(e_t)≈0
ResidualACFDiagnostic := Corr(e_t,e_(t-k))≈0 for k>0
LjungBoxTest := JointResidualAutocorrelationTest
LjungBoxNull := ResidualAutocorrelationsUpToLagMEqualZero
ResidualNormality := OptionalDistributionalDiagnosticForGaussianIntervals
ResidualHeteroscedasticity := TimeVaryingInnovationVarianceDiagnostic
ARCHEffectDiagnostic := AutocorrelationInSquaredResiduals

ModelSelection := CandidateARIMAOrders→Estimate→CompareFitAndDiagnostics→ValidateForecasts
AIC := -2LogLikelihood+2k
AICc := AIC+SmallSampleCorrection
BIC := -2LogLikelihood+klog(n)
InformationCriterionConstraint := CompareModelsFitToSameTargetDataAndLikelihoodBasis
Parsimony := PreferSimplerAdequateModelWhenPredictiveAndDiagnosticPerformanceComparable

OneStepForecast := Y_hat_(t+1|t)
MultiStepForecast := Y_hat_(t+h|t)
ForecastRecursion := EstimatedARMAEquation→FutureConditionalExpectations
FutureInnovationExpectation := E[ε_(t+h)|I_t]=0
ForecastError := Y_(t+h)-Y_hat_(t+h|t)
ForecastVariance := FunctionOfInnovationVarianceAndPsiWeights
PredictionInterval := PointForecast±CriticalValue×ForecastStandardErrorUnderCompatibleDistribution
ForecastHorizon := h
LongHorizonRelation := ForecastOfStationaryDifferencedComponentRevertsTowardItsLongRunMeanWhileIntegratedLevelUncertaintyAccumulates

PsiWeightRepresentation := W_t=μ+Σ_{j=0}∞ψ_jε_(t-j)
ForecastVarianceRelation := Var(ForecastError_h)=σ²Σ_{j=0}^{h-1}ψ_j² for compatible stationary ARMA component
BackTransformation := DifferencedForecasts→ReconstructOriginalScaleForecasts
IntegrationReconstruction := ForecastDifferences→CumulativeLevelForecast

RollingOriginValidation := RepeatedHistoricalForecastOrigins→OutOfSampleErrors
TemporalCrossValidation := PreserveTimeOrder∧EvaluateFuturePredictions
ForecastMetrics := {MAE,MSE,RMSE,MAPE,MASE}
Benchmark := NaiveForecast ∨ SeasonalNaiveForecast
ForecastSkill := CompareARIMAErrorsAgainstBenchmark
FutureLeakage := FutureDataUsedInModelSelectionOrTransformation
LeakageConstraint := FutureLeakage⇒InvalidForecastValidation

SeasonalARIMA := ARIMA(p,d,q)×(P,D,Q)_s
SeasonalARPolynomial := Φ(L^s)
SeasonalMAPolynomial := Θ(L^s)
SeasonalDifference := (1-L^s)^D
SARIMAEquation := φ(L)Φ(L^s)(1-L)^d(1-L^s)^DY_t=θ(L)Θ(L^s)ε_t
SeasonalPeriod := s
SeasonalARIMAConstraint := SeasonalStructureMustMatchSamplingFrequencyAndCalendarSemantics

ARIMAX := ARIMAErrors ∧ ExogenousRegressors
DynamicRegression := Y_t=X_tβ+n_t where n_t follows ARIMA
ExogenousForecastRequirement := FutureXValuesMustBeKnownForecastOrScenarioSpecified
InterventionARIMA := ARIMAErrors ∧ InterventionVariables
TransferFunctionModel := DynamicInputEffects ∧ ARIMAErrors

StructuralBreak := PersistentChangeInParametersOrGeneratingProcess
BreakRisk := FixedARIMAModelMayFailAcrossRegimeChange
RollingARIMA := ReestimateARIMAOnMovingWindow
AdaptiveARIMA := UpdateParametersOrModelOrderAsNewDataArrive
Outlier := ObservationShockPotentiallyDistortingEstimation
AdditiveOutlier := OneTimeObservationDistortion
InnovationOutlier := ShockEnteringInnovationAndPropagatingThroughModel
LevelShift := PersistentChangeInSeriesLevel

MissingData := MissingHistoricalObservation
StateSpaceMissingDataHandling := KalmanFilterCanHandleMissingObservationsUnderCompatibleModel
IrregularSamplingConstraint := StandardARIMAAssumesRegularDiscreteIntervals
ResamplingRisk := AggregationOrInterpolationMayAlterARIMADynamics

ARIMAVsARMA := ARIMAAddsDifferencingToARMA
ARIMAVsExponentialSmoothing := DifferentStateAndErrorRepresentationsCanYieldRelatedForecastStructures
ARIMAVsRegression := ARIMAModelsTemporalDependenceWhereasRegressionModelsConditionalPredictorRelations
ARIMADoesNotImplyCausation := TemporalPredictiveStructure↛CausalMechanism

ARIMAConstraint := ConstraintOn(TimeSeries,SamplingFrequency,AROrder,IntegrationOrder,MAOrder,Stationarity,Invertibility,InnovationStructure,EstimationMethod,ForecastHorizon)
ARIMAConsistency := Series∧Differencing∧ARPolynomial∧MAPolynomial∧InnovationModel∧Frequency jointly_compatible
InvalidARIMA := UndefinedTimeOrder∨IrregularSamplingUnresolved∨UnderDifferencing∨OverDifferencing∨NonstationaryARComponent∨NoninvertibleMAComponent∨ResidualAutocorrelationUnresolved∨FutureLeakage

ARIMAValidation := ResolveSeriesAndTimeIndex→ResolveSamplingFrequency→AssessStationarity→SelectDifferencingOrder→InspectACFAndPACF→GenerateCandidateOrders→EstimateParameters→ValidateStationarityAndInvertibility→ComputeResiduals→ExecuteResidualDiagnostics→CompareInformationCriteria→GenerateForecasts→QuantifyForecastUncertainty→ExecuteRollingOriginValidation→CompareAgainstBenchmarks→ValidateResult

ARIMAState := {TimeSeries,TimeIndex,SamplingFrequency,AROrder,IntegrationOrder,MAOrder,ARParameters,MAParameters,Drift,InnovationVariance,DifferencedSeries,StationarityStatus,InvertibilityStatus,Residuals,ResidualACF,LjungBoxStatus,AIC,AICc,BIC,ForecastHorizon,Forecasts,PredictionIntervals,ForecastAccuracy,Evidence,Confidence,Provenance}

SEMANTIC_EQUIVALENCE(ARIMAA,ARIMAB) := same(SeriesSemantics)∧same(SamplingFrequency)∧same(DifferencingOperator)∧same(ARPolynomial)∧same(MAPolynomial)∧same(DeterministicTerms)∧same(InnovationDistributionOrEquivalentForecastStructure)

ARIMAOntology := Classes∧Relations∧Constraints∧Axioms∧InferenceRules
CoreClasses := {ARIMA,AROrder,IntegrationOrder,MAOrder,DifferenceOperator,ARPolynomial,MAPolynomial,Innovation,Stationarity,Invertibility,UnitRoot,Residual,Forecast,PredictionInterval,SeasonalARIMA,ARIMAX,InterventionARIMA,StateSpaceARIMA}
CoreRelations := {models,hasOrder,hasAROrder,hasIntegrationOrder,hasMAOrder,differencedBy,hasARPolynomial,hasMAPolynomial,generatesInnovation,estimatedBy,diagnosedBy,forecasts,hasUncertainty,extendedBy,constrainedBy,equivalentTo,tracesTo}

TypedRelations := ARIMA--models-->TimeSeries ∧ ARIMA--hasAROrder-->AROrder ∧ ARIMA--hasIntegrationOrder-->IntegrationOrder ∧ ARIMA--hasMAOrder-->MAOrder ∧ TimeSeries--differencedBy-->DifferenceOperator ∧ ARIMA--hasARPolynomial-->ARPolynomial ∧ ARIMA--hasMAPolynomial-->MAPolynomial ∧ ARIMA--generatesInnovation-->Innovation ∧ ARIMA--estimatedBy-->MaximumLikelihoodEstimator ∧ ARIMA--diagnosedBy-->ResidualDiagnostic ∧ ARIMA--forecasts-->FutureObservation ∧ Forecast--hasUncertainty-->PredictionInterval ∧ ARIMA--extendedBy-->SeasonalOrExogenousStructure ∧ ARIMA--tracesTo-->Provenance

InferenceRules := {d=0⇒ARIMAEquivalentToARMA,ARIMA010⇒RandomWalkUnderStandardZeroDriftForm,ARPolynomialRootsOutsideUnitCircle⇒StationaryARComponent,MAPolynomialRootsOutsideUnitCircle⇒InvertibleMAComponent,ResidualAutocorrelationDetected⇒FlagModelMisspecificationRisk,FutureLeakage⇒FlagForecastValidationInvalid,SeasonalPattern⇒ConsiderSeasonalARIMAExtension,ARIMAPredictiveAssociation↛CausalEffect}

CanonicalARIMARepresentation := ParseARIMARequest→ResolveSeriesTimeIndexAndFrequency→AssessStationarityAndUnitRoots→ResolveDifferencingOrder→ComputeDifferencedSeries→InspectACFAndPACF→ResolveCandidateARAndMAOrders→EstimateARIMAParameters→ValidateStationarityAndInvertibility→ComputeResidualsAndInnovationVariance→ExecuteResidualDiagnostics→CompareCandidateModels→GenerateForecastsAndPredictionIntervals→ExecuteRollingOriginValidation→ValidateARIMA

ARIMAGraph := G=(V,E)
V := {ARIMAModels,AROrders,IntegrationOrders,MAOrders,DifferenceOperators,ARPolynomials,MAPolynomials,Innovations,StationarityNodes,InvertibilityNodes,UnitRoots,Residuals,Forecasts,PredictionIntervals,SeasonalARIMAModels,ARIMAXModels,InterventionARIMAModels,StateSpaceARIMAModels,TimeSeriesNodes,Evidence,ProvenanceNodes}
E := {models,hasOrder,hasAROrder,hasIntegrationOrder,hasMAOrder,differencedBy,hasARPolynomial,hasMAPolynomial,generatesInnovation,estimatedBy,diagnosedBy,forecasts,hasUncertainty,extendedBy,constrainedBy,equivalentTo,tracesTo}

CONFIDENCE(ARIMAResolution) := weighted_mean(TimeSeriesResolution,FrequencyValidity,DifferencingValidity,OrderResolution,StationarityValidity,InvertibilityValidity,ParameterEstimationValidity,ResidualDiagnosticQuality,ForecastValidationQuality,UncertaintyResolution,ProvenanceQuality)

AI_QUERY_INTERPRETATION := Query→DetectARIMAIntent→IdentifySeriesFrequencyAndForecastHorizon→ResolveOrderSelectionEstimationDiagnosticsOrForecastIntent→IdentifyRequestedPDQParametersResidualsCriteriaOrPredictions→ResolveSeasonalityExogenousVariablesAndValidationContext

AI_REASONING_FLOW := Query→ParseARIMAProblem→ResolveTimeSeriesAndTimeIndex→ResolveSamplingFrequency→AssessStationarity→SelectDifferencingOrder→ComputeDifferencedSeries→InspectACFAndPACF→GenerateCandidateARIMAOrders→EstimateParameters→ValidateARStationarityAndMAInvertibility→ComputeResiduals→ExecuteLjungBoxAndOtherDiagnostics→CompareAICAICcBIC→GenerateForecasts→ComputePredictionIntervals→ExecuteRollingOriginValidation→CompareAgainstBenchmarks→UpdateConfidence→PreserveProvenance→GenerateExplanation

AI_CONSUMPTION := {ParseARIMAEntities,ParseTypedRelations,ResolveTimeSeries,ResolveTimeIndices,ResolveSamplingFrequencies,TestStationarity,ExecuteADFTest,ExecuteKPSSTest,SelectDifferencingOrders,ComputeDifferences,ComputeACF,ComputePACF,GenerateCandidateARIMAOrders,FitARIMAModels,EstimateMaximumLikelihoodParameters,ValidateARRoots,ValidateMARoots,ComputeInnovations,ComputeResiduals,ComputeResidualACF,ExecuteLjungBoxTests,ComputeAIC,ComputeAICc,ComputeBIC,GenerateARIMAForecasts,ComputeForecastVariances,ComputePredictionIntervals,FitSeasonalARIMA,FitARIMAX,FitInterventionARIMA,ExecuteRollingOriginValidation,ComputeForecastAccuracy,CompareBenchmarks,ValidateARIMASemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}

UPDATE_MODEL := NewObservationOrderSeasonalityOrEvidence→ValidateInput→IdentifyAffectedARIMAClaims→UpdateARIMAState→ReassessStationarityAndDifferencing→ReestimateModelOrderOrParametersIfRequired→RecomputeResidualsAndDiagnostics→RegenerateForecastsAndPredictionIntervals→ReevaluateForecastAccuracy→RecomputeConfidence→PreserveProvenance→VersionModel
DataUpdate := NewOrChangedObservation⇒recompute(DifferencedSeries,ParameterEstimates,Residuals,AIC,AICc,BIC,Forecasts,PredictionIntervals)
OrderUpdate := ChangedPDQ⇒recompute(ARPolynomial,MAPolynomial,Parameters,ResidualDiagnostics,InformationCriteria,Forecasts)
FrequencyUpdate := ChangedSamplingFrequency⇒reevaluate(DifferencingSemantics,SeasonalPeriod,LagInterpretation,ModelCompatibility)

FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,ARIMA,ARIMAValidation,ARIMAConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {ARIMAEntityResolution,TimeSeries,TimeIndex,SamplingFrequency,AROrder,IntegrationOrder,MAOrder,ARParameters,MAParameters,Drift,InnovationVariance,DifferencedSeries,StationarityStatus,InvertibilityStatus,Residuals,ResidualACF,LjungBoxStatus,AIC,AICc,BIC,ForecastHorizon,Forecasts,PredictionIntervals,ForecastAccuracy,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA