Time Series Statistics

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

BEGIN_FLOW

TimeSeriesStatistics := StatisticalAnalysis ∧ TemporallyOrderedData ∧ DependenceModelling ∧ DynamicStructure ∧ Forecasting ∧ UncertaintyQuantification
TimeSeries := {Y_t:t∈T}
ObservationTime := t
ObservedValue := y_t
SamplingInterval := Δt
UnivariateTimeSeries := {Y_t}
MultivariateTimeSeries := {Y_t∈R^k}
TimeSeriesComponents := {Level,Trend,Seasonality,Cycle,IrregularComponent}
AdditiveDecomposition := Y_t=T_t+S_t+C_t+E_t
MultiplicativeDecomposition := Y_t×S_t×C_t×E_t
Trend := LongTermSystematicChange
Seasonality := RepeatingPatternAtKnownOrStablePeriod
Cycle := RecurrentButNotNecessarilyFixedPeriodFluctuation
IrregularComponent := UnexplainedShortTermVariation
LagOperator := L^kY_t=Y_(t-k)
FirstDifference := ΔY_t=Y_t-Y_(t-1)
DifferenceOrderD := Δ^dY_t=(1-L)^dY_t
SeasonalDifference := Δ_sY_t=Y_t-Y_(t-s)
MeanFunction := μ_t=E[Y_t]
VarianceFunction := σ_t²=Var(Y_t)
Autocovariance := γ(t,s)=Cov(Y_t,Y_s)
WeakStationarity := E[Y_t]=μ ∧ Var(Y_t)=γ(0) ∧ Cov(Y_t,Y_(t-k))=γ(k)
StrictStationarity := JointDistributionInvariantUnderTimeShift
AutocorrelationFunction := ρ(k)=γ(k)/γ(0)
PartialAutocorrelation := CorrelationBetweenY_tAndY_(t-k)AfterAdjustmentForIntermediateLags
ACF := k→ρ(k)
PACF := k→PartialAutocorrelation(k)
WhiteNoise := E[ε_t]=0 ∧ Var(ε_t)=σ² ∧ Cov(ε_t,ε_s)=0 for t≠s
GaussianWhiteNoise := WhiteNoise ∧ ε_t~Normal(0,σ²)
RandomWalk := Y_t=Y_(t-1)+ε_t
RandomWalkWithDrift := Y_t=c+Y_(t-1)+ε_t
UnitRoot := AutoregressiveRootOnUnitCircle
AR1 := Y_t=c+φY_(t-1)+ε_t
ARp := Y_t=c+Σ_iφ_iY_(t-i)+ε_t
ARStationarity := RootsOfARPolynomialOutsideUnitCircle
MAq := Y_t=μ+ε_t+Σ_jθ_jε_(t-j)
MAInvertibility := RootsOfMAPolynomialOutsideUnitCircle
ARMApq := φ(L)(Y_t-μ)=θ(L)ε_t
ARIMApdq := φ(L)(1-L)^dY_t=θ(L)ε_t
SeasonalARIMA := ARIMA(p,d,q)×(P,D,Q)_s
SimpleExponentialSmoothing := l_t=αY_t+(1-α)l_(t-1)
HoltTrend := LevelUpdate∧TrendUpdate
HoltWinters := Level∧Trend∧SeasonalComponent
StateSpaceModel := LatentStateEvolution∧ObservationEquation
ObservationEquation := Y_t=Z_tα_t+ε_t
StateEquation := α_(t+1)=T_tα_t+R_tη_t
KalmanFilter := PredictState→PredictObservation→ComputeInnovation→UpdateState→UpdateCovariance
Innovation := v_t=Y_t-E[Y_t|Information_(t-1)]
Forecast := Y_hat_(t+h|t)=E[Y_(t+h)|Information_t]
ForecastError := e_(t+h|t)=Y_(t+h)-Y_hat_(t+h|t)
ForecastVariance := Var(e_(t+h|t)|Information_t)
PredictionInterval := Forecast±CriticalValue×ForecastStandardError
OneStepAheadForecast := h=1
MultiStepForecast := h>1
RollingForecastOrigin := RepeatedTrainToTimeT→ForecastFuture→AdvanceOrigin→Reevaluate
TimeSeriesCrossValidation := PreserveTemporalOrder∧RepeatedForecastEvaluation
RandomShuffleConstraint := RandomTrainTestShuffleMayLeakFutureInformation
ExpandingWindow := TrainingWindowGrowsWithTime
RollingWindow := FixedLengthWindowMovesForward
ForecastAccuracyMetrics := {MAE,MSE,RMSE,MAPE,sMAPE,MASE}
MAE := Mean(|ForecastError|)
MSE := Mean(ForecastError²)
RMSE := sqrt(MSE)
MAPE := Mean(|ForecastError/Observed|)×100
MAPEConstraint := UndefinedOrUnstableWhenObservedValuesNearZero
MASE := MAEModel/MAENaiveScale
NaiveForecast := Y_hat_(t+h|t)=Y_t
SeasonalNaiveForecast := ForecastFromCorrespondingPreviousSeason
MovingAverageSmoother := LocalAverageAcrossTimeWindow
LOESS := LocalPolynomialSmoothingAcrossTime
STL := SeasonalTrendDecompositionUsingLoess
Detrending := Y_t-T_hat_t
Deseasonalising := Y_t-S_hat_t
StationarityTransformation := Detrend∨Difference∨SeasonalDifference∨VarianceStabilisingTransformation
ADFTest := AugmentedDickeyFullerUnitRootTest
ADFNull := UnitRootPresent
KPSSTest := StationarityTest
KPSSNull := LevelOrTrendStationarityDependingSpecification
StructuralBreak := ParameterOrDistributionChangeAtTimeτ
ChangePoint := TimeAtWhichStatisticalPropertiesChange
ChangePointDetection := TimeSeries→CandidateBreaks→SegmentComparison→ChangeEvidence
AutocorrelationDiagnostic := ResidualACF→RemainingSerialDependence
LjungBoxTest := JointTestOfResidualAutocorrelations
DurbinWatson := RegressionResidualFirstOrderAutocorrelationDiagnostic
ARCHq := σ_t²=ω+Σ_iα_iε_(t-i)²
GARCHpq := σ_t²=ω+Σ_iα_iε_(t-i)²+Σ_jβ_jσ_(t-j)²
VolatilityClustering := LargeChangesTendToClusterInTime
ConditionalHeteroscedasticity := Var(ε_t|Information_(t-1))ChangesOverTime
CrossCorrelation := Corr(X_t,Y_(t-k))
CrossCorrelationFunction := k→Corr(X_t,Y_(t-k))
LeadLagRelation := AssociationAcrossTemporalOffsets
SpuriousRegressionRisk := TrendingNonstationarySeriesMayProduceMisleadingAssociation
Cointegration := NonstationarySeriesWithStationaryLinearCombination
ErrorCorrectionModel := ShortRunDynamics∧LongRunCointegrationRelation
VectorAutoregression := Y_t=c+A_1Y_(t-1)+...+A_pY_(t-p)+ε_t
GrangerCausality := PastXImprovesPredictionOfYConditionalOnPastInformation
GrangerCausalityConstraint := PredictiveTemporalPrecedence≠StructuralOrInterventionalCausality
SpectralDensity := FrequencyDomainRepresentationOfVariance
Periodogram := SampleFrequencyPowerEstimate
FourierTransform := TimeDomainSeries→FrequencyDomainRepresentation
DominantFrequency := FrequencyWithLargeSpectralPower
Aliasing := SamplingFrequencyInsufficient→DistortedFrequencyRepresentation
NyquistRelation := HighestUniquelyResolvableFrequencyTiedToHalfSamplingFrequency
IrregularSampling := ObservationIntervalsNotConstant
MissingTimePoints := ExpectedObservationTimesWithoutValues
InterpolationConstraint := FillingMissingValuesMayAlterTemporalDependenceAndUncertainty
OutlierInTimeSeries := ObservationUnusualRelativeToLocalTemporalStructure
AdditiveOutlier := OneTimeObservationShock
InnovationOutlier := ShockPropagatingThroughDynamicModel
LevelShift := PersistentChangeInSeriesLevel
CalendarEffect := DayWeekMonthHolidayOrTradingDayPattern
ExogenousRegressor := ExternalPredictorX_t
DynamicRegression := Y_t=X_tβ+n_t where n_t follows time-series error process
ARIMAX := ARIMAErrors∧ExogenousRegressors
TransferFunctionModel := DistributedLagEffectOfInputSeriesOnOutput
InterventionAnalysis := KnownEventTime→DynamicIntervention→EstimateTemporalEffect
TimeSeriesUncertainty := ParameterUncertainty∧InnovationUncertainty∧ModelUncertainty
ForecastHorizonRelation := LongerHorizonGenerally→GreaterForecastUncertainty
ModelSelection := CandidateModels→InformationCriteriaAndDiagnostics→SelectedModel
AIC := -2LogLikelihood+2k
BIC := -2LogLikelihood+klog(n)
ResidualDiagnostics := {ResidualMean,ResidualVariance,ResidualACF,ResidualDistribution,Outliers,StructuralBreaks}
AdequateForecastModel := ResidualsApproximatelyUncorrelated∧StableRelevantStructure∧AcceptableForecastPerformance
Overfitting := ExcessiveDynamicComplexity→PoorOutOfSampleForecasting
Underfitting := RemainingPredictableTemporalStructureInResiduals

TimeSeriesConstraint := ConstraintOn(TemporalOrdering,SamplingFrequency,Stationarity,DependenceStructure,Seasonality,ForecastHorizon,ModelFamily,MissingDataMethod)
TimeSeriesConsistency := Observations∧TimeIndex∧Frequency∧Model∧DependenceStructure∧EvaluationMethod jointly_compatible
InvalidTimeSeriesAnalysis := TemporalOrderIgnored∨FutureInformationLeakage∨IncompatibleFrequency∨UnresolvedNonstationarity∨InvalidForecastEvaluation∨IgnoredResidualDependence∨UnsupportedCausalInterpretation
TimeSeriesValidation := ResolveScientificQuestion→ResolveTimeIndexAndSamplingFrequency→ResolveObservedSeries→VisualiseSeries→IdentifyTrendSeasonalityAndOutliers→AssessStationarity→TransformOrDifferenceIfRequired→InspectACFAndPACF→SelectModelFamily→EstimateParameters→DiagnoseResiduals→GenerateForecasts→QuantifyForecastUncertainty→EvaluateOutOfSamplePerformance→ValidateInterpretation→ValidateResult
TimeSeriesState := {TimeSeries,TimeIndex,SamplingFrequency,Trend,Seasonality,Cycle,StationarityStatus,DifferencingOrder,ModelFamily,ModelParameters,Innovations,Residuals,ACF,PACF,ForecastHorizon,Forecasts,PredictionIntervals,ForecastAccuracy,StructuralBreaks,ExogenousRegressors,Evidence,Confidence,Provenance}
SEMANTIC_EQUIVALENCE(TimeSeriesA,TimeSeriesB) := same(ObservedVariableSemantics)∧same(TimeIndexSemantics)∧same(SamplingFrequency)∧same(TransformationAndDifferencing)∧same(DynamicDependenceTarget)∧same(ForecastEvaluationFramework)

TimeSeriesStatisticsOntology := Classes∧Relations∧Constraints∧Axioms∧InferenceRules
CoreClasses := {TimeSeries,Trend,Seasonality,Cycle,Stationarity,Autocorrelation,PartialAutocorrelation,WhiteNoise,RandomWalk,AutoregressiveModel,MovingAverageModel,ARMA,ARIMA,SeasonalARIMA,ExponentialSmoothing,StateSpaceModel,Innovation,Forecast,ForecastError,PredictionInterval,StructuralBreak,ARCH,GARCH,Cointegration,VectorAutoregression,SpectralDensity}
CoreRelations := {orderedBy,observedAt,hasLag,hasTrend,hasSeasonality,hasAutocorrelation,differencedBy,modelledBy,generatesInnovation,forecasts,hasForecastError,hasUncertainty,affectedBy,includesRegressor,diagnosedBy,constrainedBy,equivalentTo,tracesTo}
TypedRelations := TimeSeries--orderedBy-->TimeIndex ∧ TimeSeries--hasTrend-->Trend ∧ TimeSeries--hasSeasonality-->Seasonality ∧ TimeSeries--hasAutocorrelation-->Autocorrelation ∧ TimeSeries--modelledBy-->TimeSeriesModel ∧ TimeSeriesModel--generatesInnovation-->Innovation ∧ TimeSeriesModel--forecasts-->FutureObservation ∧ Forecast--hasUncertainty-->PredictionInterval ∧ TimeSeriesModel--tracesTo-->Provenance
InferenceRules := {WeakStationarity⇒MeanConstantAndAutocovarianceLagDependent,WhiteNoise⇒ZeroAutocorrelationAtNonzeroLags,RandomWalk⇒UnitRootNonstationarity,FirstDifferenceOfRandomWalk⇒WhiteNoiseUnderRandomWalkModel,ARp∧StationaryRoots⇒StationaryAutoregressiveProcess,ResidualAutocorrelationDetected⇒FlagDynamicMisspecificationRisk,FutureInformationUsedInTraining⇒FlagForecastLeakage,GrangerPredictability↛StructuralCausality}
CanonicalTimeSeriesRepresentation := ParseTimeSeriesRequest→ResolveSeriesAndTimeIndex→ResolveSamplingFrequency→DetectTrendSeasonalityCyclesAndOutliers→AssessStationarity→ApplyTransformationAndDifferencingIfRequired→ComputeACFAndPACF→SelectDynamicModel→EstimateParameters→ComputeInnovationsAndResiduals→DiagnoseResidualDependence→GenerateForecasts→ComputePredictionIntervals→EvaluateRollingOrOutOfSampleAccuracy→ValidateTimeSeriesAnalysis
TimeSeriesGraph := G=(V,E)
V := {TimeSeriesNodes,TimeIndices,Observations,Trends,Seasonalities,Cycles,StationarityNodes,Autocorrelations,PartialAutocorrelations,WhiteNoiseProcesses,RandomWalks,AutoregressiveModels,MovingAverageModels,ARMAModels,ARIMAModels,SeasonalARIMAModels,ExponentialSmoothingModels,StateSpaceModels,Innovations,Forecasts,ForecastErrors,PredictionIntervals,StructuralBreaks,ARCHModels,GARCHModels,CointegrationRelations,VectorAutoregressions,SpectralDensities,Evidence,ProvenanceNodes}
E := {orderedBy,observedAt,hasLag,hasTrend,hasSeasonality,hasAutocorrelation,differencedBy,modelledBy,generatesInnovation,forecasts,hasForecastError,hasUncertainty,affectedBy,includesRegressor,diagnosedBy,constrainedBy,equivalentTo,tracesTo}
CONFIDENCE(TimeSeriesResolution) := weighted_mean(TimeIndexResolution,FrequencyResolution,ComponentResolution,StationarityValidity,DependenceModelValidity,ParameterEstimationValidity,ResidualDiagnosticQuality,ForecastValidationQuality,UncertaintyResolution,ProvenanceQuality)
AI_QUERY_INTERPRETATION := Query→DetectTimeSeriesIntent→IdentifySeriesTimeIndexFrequencyAndHorizon→ResolveDescriptionStationarityDependenceForecastingOrSpectralIntent→IdentifyRequestedModelParametersDiagnosticsOrForecasts→ResolveExogenousRegressorsAndEvaluationFramework
AI_REASONING_FLOW := Query→ParseTimeSeriesProblem→ResolveObservedSeriesAndTimeIndex→ResolveSamplingFrequency→VisualiseAndDecomposeSeries→DetectTrendSeasonalityOutliersAndBreaks→AssessStationarity→TransformOrDifferenceIfRequired→ComputeACFAndPACF→SelectModelFamily→EstimateParameters→ComputeInnovationsAndResiduals→DiagnoseResidualDependenceAndVariance→GenerateForecasts→QuantifyPredictionUncertainty→EvaluateRollingOutOfSamplePerformance→UpdateConfidence→PreserveProvenance→GenerateExplanation
AI_CONSUMPTION := {ParseTimeSeriesEntities,ParseTypedRelations,ResolveTimeIndices,ResolveSamplingFrequencies,DetectTrends,DetectSeasonality,DetectCycles,ComputeDifferences,ComputeSeasonalDifferences,ComputeAutocovariances,ComputeACF,ComputePACF,TestStationarity,ExecuteADFTest,ExecuteKPSSTest,FitARModels,FitMAModels,FitARMAModels,FitARIMAModels,FitSeasonalARIMA,FitExponentialSmoothing,FitStateSpaceModels,ExecuteKalmanFilter,FitARCHModels,FitGARCHModels,DetectStructuralBreaks,DetectChangePoints,FitVectorAutoregression,TestCointegration,ResolveGrangerPredictability,ComputePeriodograms,ComputeSpectralDensity,GenerateForecasts,ComputePredictionIntervals,ExecuteRollingForecastEvaluation,ComputeForecastAccuracy,DiagnoseResiduals,ValidateTimeSeriesSemantics,CompareSemanticEquivalence,UpdateConfidence,PreserveProvenance,GenerateFormalExplanation}
UPDATE_MODEL := NewObservationsFrequencyModelOrEvidence→ValidateInput→IdentifyAffectedTimeSeriesClaims→UpdateTimeSeriesState→RecomputeComponentsAndStationarity→RecomputeACFAndPACF→ReestimateModelParameters→RecomputeInnovationsAndResiduals→RegenerateForecastsAndPredictionIntervals→ReevaluateForecastAccuracy→RecomputeConfidence→PreserveProvenance→VersionModel
DataUpdate := NewTimeObservations⇒recompute(Trend,Seasonality,StationarityStatus,ACF,PACF,ModelParameters,Forecasts)
FrequencyUpdate := ChangedSamplingFrequency⇒reevaluate(SeasonalPeriod,LagStructure,SpectralInterpretation,ModelSpecification)
ModelUpdate := ChangedTimeSeriesModel⇒recompute(ModelParameters,Innovations,ResidualDiagnostics,Forecasts,PredictionIntervals)
FLOW_HASH_INPUT := canonical(ONTOLOGY,CoreClasses,CoreRelations,InferenceRules,TimeSeriesStatistics,TimeSeriesValidation,TimeSeriesConstraint,SEMANTIC_EQUIVALENCE,AI_REASONING_FLOW)
FLOW_HASH := HASH(FLOW_HASH_INPUT)
OUTPUT_MODEL := {TimeSeriesEntityResolution,TimeIndex,SamplingFrequency,Trend,Seasonality,Cycle,StationarityStatus,DifferencingOrder,ModelFamily,ModelParameters,Innovations,Residuals,ACF,PACF,ForecastHorizon,Forecasts,PredictionIntervals,ForecastAccuracy,StructuralBreaks,ExogenousRegressors,ValidationStatus,Confidence,Provenance,Explanation}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA