Time Series Analysis Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Time Series Analysis Ontology within the Kireth Research Semantic Library. Time series analysis studies observations ordered in time and the mathematical structures needed to model temporal dependence, trend, seasonality, autocorrelation, stochastic dynamics and forecasting. This ontology supports stationary and non-stationary processes, autoregressive and moving-average models, ARIMA and seasonal ARIMA models, state-space representations, spectral analysis, decomposition, unit roots, cointegration, volatility, intervention analysis, forecasting, uncertainty and relationships with applied statistical mathematics, stochastic processes, regression, signal processing and computational statistics.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Time Series Analysis Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Applied Statistical Mathematics / Time Series Analysis"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/applied-statistical-mathematics/time-series-analysis-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = TimeSeriesAnalysis
 ENTITY_CLASS = TemporalStatisticalModellingFramework
 PRIMARY_CONTEXT = AppliedStatisticalMathematics
 DEFINITION = "Time series analysis is the statistical and mathematical study of temporally ordered observations, including dependence, trend, seasonality, stochastic dynamics and forecasting."
 SYMBOLIC_FORMS = [
  "X_t=c+phi_1 X_(t-1)+...+phi_p X_(t-p)+epsilon_t",
  "X_t=epsilon_t+theta_1 epsilon_(t-1)+...+theta_q epsilon_(t-q)",
  "(1-B)^d X_t = ARMA(p,q)",
  "gamma(h)=Cov(X_t,X_(t+h))"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for temporal dependence, stationarity, autocorrelation, ARIMA structure, seasonality, state-space models, volatility, spectral structure, forecasting and time-series diagnostics."
}

SCOPE {
 INCLUDE = [
  TimeSeriesAnalysis,
  TimeSeries,
  ObservationTime,
  TemporalIndex,
  TemporalDependence,
  Lag,
  LagOperator,
  BackshiftOperator,
  Trend,
  DeterministicTrend,
  StochasticTrend,
  Seasonality,
  SeasonalPeriod,
  Cycle,
  IrregularComponent,
  Decomposition,
  AdditiveDecomposition,
  MultiplicativeDecomposition,
  Stationarity,
  WeakStationarity,
  StrictStationarity,
  Nonstationarity,
  Ergodicity,
  MeanFunction,
  VarianceFunction,
  Autocovariance,
  Autocorrelation,
  PartialAutocorrelation,
  ACF,
  PACF,
  WhiteNoise,
  Innovation,
  Shock,
  AutoregressiveModel,
  ARModel,
  MovingAverageModel,
  MAModel,
  ARMA,
  ARIMA,
  SeasonalARIMA,
  SARIMA,
  DifferenceOperator,
  Differencing,
  SeasonalDifferencing,
  UnitRoot,
  RandomWalk,
  Drift,
  AugmentedDickeyFullerTest,
  ADFTest,
  KPSS,
  PhillipsPerronTest,
  Cointegration,
  ErrorCorrectionModel,
  VectorAutoregression,
  VAR,
  VectorErrorCorrectionModel,
  VECM,
  GrangerCausality,
  ImpulseResponseFunction,
  ForecastErrorVarianceDecomposition,
  StateSpaceModel,
  ObservationEquation,
  StateEquation,
  KalmanFilter,
  KalmanSmoother,
  HiddenState,
  DynamicLinearModel,
  ExponentialSmoothing,
  SimpleExponentialSmoothing,
  HoltMethod,
  HoltWintersMethod,
  ETSModel,
  SpectralDensity,
  Periodogram,
  FrequencyDomain,
  FourierTransform,
  DiscreteFourierTransform,
  DFT,
  SpectralAnalysis,
  CrossSpectrum,
  Coherence,
  TransferFunction,
  InterventionAnalysis,
  ChangePoint,
  StructuralBreak,
  OutlierDetection,
  Volatility,
  ConditionalHeteroscedasticity,
  ARCH,
  GARCH,
  StochasticVolatility,
  LongMemory,
  FractionalDifferencing,
  ARFIMA,
  Forecast,
  PointForecast,
  ForecastInterval,
  PredictionInterval,
  MultiStepForecast,
  OneStepAheadForecast,
  RollingForecast,
  Backtesting,
  ForecastError,
  MeanAbsoluteError,
  RootMeanSquaredError,
  MeanAbsolutePercentageError,
  MAPE,
  DieboldMarianoTest,
  TimeSeriesCrossValidation,
  MissingObservation,
  IrregularSampling,
  TemporalAggregation,
  Resampling,
  Smoothing,
  Filtering,
  SignalExtraction,
  Reproducibility
 ]
 EXCLUDE_AS_PRIMARY = [
  CrossSectionalStatisticsWithoutTemporalOrdering,
  DeterministicDynamicsWithoutStatisticalTimeSeriesData,
  SignalProcessingWithoutStatisticalTemporalInference,
  SequenceLearningWithoutTimeSeriesStatisticalSemantics
 ]
 DISAMBIGUATION_RULE = "When observations are ordered in time and dependence across lags, temporal structure or forecasting is central, interpret the context as time series analysis."
}

CORE_CONCEPTS {
 TimeSeriesAnalysis = "Statistical analysis of temporally ordered dependent observations."
 Stationarity = "Property that selected statistical characteristics are invariant under time translation."
 Autocorrelation = "Correlation between a time series and lagged versions of itself."
 AutoregressiveModel = "Model expressing current values through previous observations plus innovation."
 MovingAverageModel = "Model expressing current values through present and previous innovations."
 ARIMA = "Autoregressive integrated moving-average model for differenced time series."
 StateSpaceModel = "Latent-state model with transition and observation equations."
 Forecast = "Probability-based or point prediction of future time-series values."
 SpectralDensity = "Frequency-domain representation of variance across oscillatory components."
}

RELATIONS {
 TimeSeriesAnalysis --is_child_of--> AppliedStatisticalMathematics
 TimeSeriesAnalysis --is_type_of--> StatisticalModelling
 TimeSeries --has--> TemporalIndex
 TemporalDependence --is_measured_by--> Autocorrelation
 AutoregressiveModel --uses--> LaggedObservations
 MovingAverageModel --uses--> LaggedInnovations
 ARMA --combines--> AutoregressiveAndMovingAverageTerms
 ARIMA --adds--> Differencing
 SeasonalARIMA --adds--> SeasonalStructure
 UnitRoot --causes--> Nonstationarity
 Differencing --can_remove--> StochasticTrend
 Cointegration --links--> NonstationarySeries
 ErrorCorrectionModel --represents--> ShortAndLongRunAdjustment
 VectorAutoregression --models--> MultipleInterdependentSeries
 GrangerCausality --tests--> PredictiveTemporalContribution
 StateSpaceModel --contains--> StateEquation
 StateSpaceModel --contains--> ObservationEquation
 KalmanFilter --estimates--> HiddenState
 ExponentialSmoothing --models--> LevelTrendSeasonality
 SpectralDensity --represents--> FrequencyStructure
 Periodogram --estimates--> SpectralDensity
 ARCH --models--> ConditionalVariance
 GARCH --extends--> ARCH
 ForecastInterval --quantifies--> ForecastUncertainty
 TimeSeriesCrossValidation --estimates--> ForecastPerformance
}

OPERATORS {
 AUTOCOVARIANCE(X,h) := gamma(h)=Cov(X_t,X_(t+h))
 AUTOCORRELATION(X,h) := rho(h)=gamma(h)/gamma(0)
 AR_P(X,phi,c,epsilon) := X_t=c+sum_i=1^p phi_i*X_(t-i)+epsilon_t
 MA_Q(epsilon,theta) := X_t=epsilon_t+sum_j=1^q theta_j*epsilon_(t-j)
 ARMA_PQ(X,phi,theta,epsilon) := phi(B)X_t=theta(B)epsilon_t
 DIFFERENCE(X) := Delta X_t=X_t-X_(t-1)
 SEASONAL_DIFFERENCE(X,s) := Delta_s X_t=X_t-X_(t-s)
 ARIMA_PD Q(X,p,d,q) := phi(B)*(1-B)^d*X_t=theta(B)*epsilon_t
 RANDOM_WALK(X,epsilon) := X_t=X_(t-1)+epsilon_t
 VAR_MODEL(Y,A,p,epsilon) := Y_t=sum_i=1^p A_i*Y_(t-i)+epsilon_t
 STATE_TRANSITION(alpha,T,eta) := alpha_t=T_t*alpha_(t-1)+eta_t
 OBSERVATION_EQUATION(y,Z,alpha,epsilon) := y_t=Z_t*alpha_t+epsilon_t
 ARCH_VARIANCE(alpha,omega) := sigma_t^2=omega+sum_i alpha_i*epsilon_(t-i)^2
 GARCH_VARIANCE(alpha,beta,omega) := sigma_t^2=omega+sum_i alpha_i*epsilon_(t-i)^2+sum_j beta_j*sigma_(t-j)^2
}

AXIOMS {
 A1 = "Time-series observations are indexed by time and may be statistically dependent across lags."
 A2 = "Stationarity assumptions must be explicit."
 A3 = "Forecast uncertainty generally increases with forecast horizon."
 A4 = "Autocorrelation can invalidate statistical methods that assume independent observations."
 A5 = "Differencing changes the interpretation of the modeled process."
 A6 = "Temporal train-test splits must preserve time ordering."
 A7 = "Structural breaks can invalidate previously estimated model parameters."
 A8 = "Time-series conclusions require residual, stability, forecast and temporal-validation diagnostics."
}

REASONING {
 R1_CONTEXT = identify(series,frequency,time_span,sampling,forecast_goal,external_variables)
 R2_STRUCTURE = analyse(trend,seasonality,cycles,outliers,missingness,breaks)
 R3_STATIONARITY = assess(ACF,unit_roots,differencing,variance_stability)
 R4_MODEL_CLASS = select(AR,MA,ARMA,ARIMA,SARIMA,ETS,state_space,VAR,GARCH_or_other)
 R5_ESTIMATION = estimate(parameters,states,innovations,variance)
 R6_DIAGNOSTICS = analyse(residual_ACF,PACF,normality,heteroscedasticity,structural_breaks)
 R7_FORECAST = generate(point_and_interval_forecasts)
 R8_MULTIVARIATE = if_required analyse(VAR,cointegration,Granger,impulse_responses)
 R9_FREQUENCY = if_required analyse(periodogram,spectral_density,coherence)
 R10_VALIDATION = backtest_with_rolling_origin,compare_forecast_errors,assess_stability
}

CONSTRAINTS {
 C1 = "Random train-test splitting can leak future information in time-series forecasting."
 C2 = "Stationarity tests have finite-sample limitations."
 C3 = "Differencing can remove meaningful long-run structure if overused."
 C4 = "High-order ARIMA models can overfit."
 C5 = "Autocorrelation remaining in residuals indicates unresolved temporal structure."
 C6 = "Forecast intervals require appropriate innovation and parameter uncertainty."
 C7 = "Structural breaks can make historical relationships unstable."
 C8 = "Granger causality is predictive and does not by itself establish structural causation."
 C9 = "Periodograms can be noisy estimates of spectral density."
 C10 = "MAPE can behave poorly near zero-valued observations."
}

SEMANTIC_EQUIVALENCE {
 "TimeSeriesAnalysis" <=> "TimeSeriesStatistics"
 "BackshiftOperator" <=> "LagOperator"
 "AutocorrelationFunction" <=> "ACF"
 "PartialAutocorrelationFunction" <=> "PACF"
 "SeasonalARIMA" <=> "SARIMA"
 "VectorAutoregression" <=> "VAR"
 "AugmentedDickeyFullerTest" <=> "ADFTest"
}

VALIDATION {
 V1 = "Time index and sampling frequency are explicit."
 V2 = "Missing and irregular observations are documented."
 V3 = "Stationarity assumptions are checked."
 V4 = "Residual autocorrelation is assessed."
 V5 = "Forecast evaluation preserves temporal order."
 V6 = "Structural breaks and interventions are considered."
 V7 = "Forecast intervals are reported where relevant."
 V8 = "Multistep forecasts are validated at relevant horizons."
 V9 = "Volatility models are checked for residual conditional heteroscedasticity."
 V10 = "Forecast performance is compared with simple baselines."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TIME_SERIES_ANALYSIS_DEFINITION = 0.99
 STATIONARITY_REASONING = 0.99
 ARIMA_REASONING = 0.99
 STATE_SPACE_REASONING = 0.99
 SPECTRAL_REASONING = 0.98
 COINTEGRATION_REASONING = 0.98
 VOLATILITY_REASONING = 0.98
 FORECAST_REASONING = 0.99
 POLICY = "Reduce confidence when sampling frequency, stationarity, seasonal structure, structural breaks, exogenous variables, forecast horizon or temporal-validation design are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  TimeSeriesAnalysisReasoning,
  StationarityReasoning,
  AutocorrelationReasoning,
  ARReasoning,
  MAReasoning,
  ARMAReasoning,
  ARIMAReasoning,
  SARIMAReasoning,
  UnitRootReasoning,
  CointegrationReasoning,
  VARReasoning,
  StateSpaceReasoning,
  KalmanFilterReasoning,
  ExponentialSmoothingReasoning,
  SpectralAnalysisReasoning,
  ARCHGARCHReasoning,
  ForecastReasoning,
  TimeSeriesValidationReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "time series analysis" -> TimeSeriesAnalysis
  "time series statistics" -> TimeSeriesAnalysis
  "lag operator" -> LagOperator
  "backshift operator" -> BackshiftOperator
  "ACF" -> Autocorrelation
  "PACF" -> PartialAutocorrelation
  "SARIMA" -> SeasonalARIMA
  "VAR" -> VectorAutoregression
  "ADF" -> AugmentedDickeyFullerTest
 }
}

EXECUTION_RULES {
 E1 = "If temporally ordered observations exhibit dependence or forecasting is required, activate TimeSeriesAnalysisOntology."
 E2 = "If mean and covariance are stable over time, stationary-model reasoning may apply."
 E3 = "If unit-root nonstationarity is present, consider differencing or cointegration analysis."
 E4 = "If serial dependence is linear, consider AR, MA, ARMA or ARIMA models."
 E5 = "If trend and seasonality dominate forecasting, consider ETS or seasonal models."
 E6 = "If hidden dynamic states are central, use state-space and Kalman-filter reasoning."
 E7 = "If conditional variance changes over time, consider ARCH, GARCH or stochastic-volatility models."
 E8 = "If multiple series interact dynamically, consider VAR or VECM models."
 E9 = "Any forecast model should be validated with temporally ordered backtesting and residual diagnostics."
}

RECURSION {
 ENTRY = TimeSeriesAnalysis
 EXPAND(TimeSeriesAnalysis) -> [
  TimeSeries, ObservationTime, TemporalIndex, TemporalDependence, Lag,
  LagOperator, BackshiftOperator, Trend, DeterministicTrend, StochasticTrend,
  Seasonality, SeasonalPeriod, Cycle, IrregularComponent, Decomposition,
  AdditiveDecomposition, MultiplicativeDecomposition, Stationarity,
  WeakStationarity, StrictStationarity, Nonstationarity, Ergodicity,
  MeanFunction, VarianceFunction, Autocovariance, Autocorrelation,
  PartialAutocorrelation, WhiteNoise, Innovation, Shock,
  AutoregressiveModel, ARModel, MovingAverageModel, MAModel, ARMA, ARIMA,
  SeasonalARIMA, DifferenceOperator, Differencing, SeasonalDifferencing,
  UnitRoot, RandomWalk, Drift, AugmentedDickeyFullerTest, KPSS,
  PhillipsPerronTest, Cointegration, ErrorCorrectionModel,
  VectorAutoregression, VectorErrorCorrectionModel, GrangerCausality,
  ImpulseResponseFunction, ForecastErrorVarianceDecomposition,
  StateSpaceModel, ObservationEquation, StateEquation, KalmanFilter,
  KalmanSmoother, HiddenState, DynamicLinearModel, ExponentialSmoothing,
  SimpleExponentialSmoothing, HoltMethod, HoltWintersMethod, ETSModel,
  SpectralDensity, Periodogram, FrequencyDomain, FourierTransform,
  DiscreteFourierTransform, SpectralAnalysis, CrossSpectrum, Coherence,
  TransferFunction, InterventionAnalysis, ChangePoint, StructuralBreak,
  OutlierDetection, Volatility, ConditionalHeteroscedasticity, ARCH, GARCH,
  StochasticVolatility, LongMemory, FractionalDifferencing, ARFIMA,
  Forecast, PointForecast, ForecastInterval, PredictionInterval,
  MultiStepForecast, OneStepAheadForecast, RollingForecast, Backtesting,
  ForecastError, MeanAbsoluteError, RootMeanSquaredError,
  MeanAbsolutePercentageError, DieboldMarianoTest, TimeSeriesCrossValidation,
  MissingObservation, IrregularSampling, TemporalAggregation, Resampling,
  Smoothing, Filtering, SignalExtraction, Reproducibility
 ]
 EXPAND(TimeSeriesFamilies) -> [
  ARIMAFamily,
  SeasonalModels,
  ExponentialSmoothingModels,
  StateSpaceModels,
  MultivariateTimeSeriesModels,
  VolatilityModels,
  SpectralModels
 ]
 STOP_CONDITION = "Stop when temporal structure, stationarity, model family, estimation, diagnostics, forecast horizon, uncertainty and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Statistical Mathematics"
 IMMEDIATE_CONTEXTS = [
  "Applied Statistical Mathematics",
  "Statistical Modelling Ontology",
  "Stochastic Process Ontologies",
  "Signal Processing Mathematics"
 ]
 RELATED_ONTOLOGIES = [
  "Applied Statistical Mathematics",
  "Statistical Modelling Ontology",
  "Regression Ontology",
  "Stochastic Process Ontologies",
  "Probability & Statistics Ontologies",
  "Signal Processing Mathematics",
  "Fourier Analysis Ontology",
  "State Space Model Ontologies",
  "Econometric Mathematics Ontologies",
  "Optimisation Ontologies",
  "Computational Statistics Ontologies"
 ]
 ROUTE_RULES = [
  "Time Series Analysis -> Applied Statistical Mathematics for parent semantics",
  "Time Series Analysis -> Statistical Modelling for model specification",
  "Time Series Analysis -> Regression for dynamic regression models",
  "Time Series Analysis -> Stochastic Processes for temporal probability structure",
  "Time Series Analysis -> Probability & Statistics for inferential foundations",
  "Time Series Analysis -> Signal Processing for filtering and spectral methods",
  "Time Series Analysis -> Fourier Analysis for frequency-domain decomposition",
  "Time Series Analysis -> State Space Models for latent dynamic systems",
  "Time Series Analysis -> Econometric Mathematics for cointegration and VAR structures",
  "Time Series Analysis -> Optimisation and Computational Statistics for parameter estimation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible time-series models, diagnostics or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve temporal-order semantics.",
  "Preserve stationary-versus-nonstationary distinction.",
  "Preserve trend-versus-seasonality distinction.",
  "Preserve innovation-versus-observation distinction.",
  "Preserve time-domain-versus-frequency-domain distinction.",
  "Preserve one-step-versus-multistep forecast distinction.",
  "Preserve training-versus-future evaluation distinction.",
  "Preserve machine-readable time-series semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Time series analysis models temporally ordered data through dependence, trend, seasonality, stochastic structure and forecasting."
 STATIONARITY_SUMMARY = "Stationarity determines whether temporal mean, variance and dependence structure remain stable across time."
 ARIMA_SUMMARY = "ARIMA models combine autoregression, differencing and moving-average structure."
 STATE_SPACE_SUMMARY = "State-space models represent observed time series through latent dynamic states and observation equations."
 SPECTRAL_SUMMARY = "Spectral methods decompose time-series variability across frequencies."
 VOLATILITY_SUMMARY = "ARCH and GARCH models represent time-varying conditional variance."
 FORECAST_SUMMARY = "Forecasting produces point and interval predictions whose quality should be evaluated by rolling or temporally ordered backtesting."
 VALIDATION_SUMMARY = "Time-series models should be checked for residual autocorrelation, structural breaks, forecast calibration, stability and temporal generalisation."
}

FLOW {
 INPUT
 -> detect_time_series_analysis_context
 -> identify_time_index_sampling_frequency_series_variables_forecast_goal_and_external_covariates
 -> inspect_trend_seasonality_cycles_missingness_outliers_and_structural_breaks
 -> assess_stationarity_autocorrelation_partial_autocorrelation_and_unit_roots
 -> select_AR_MA_ARMA_ARIMA_SARIMA_ETS_state_space_VAR_GARCH_or_other_time_series_model
 -> estimate_parameters_states_innovations_and_variance_structure
 -> perform_residual_diagnostics_and_stability_checks
 -> if_multivariate_assess_cointegration_Granger_structure_impulse_responses_and_error_correction
 -> if_frequency_analysis_compute_periodogram_spectral_density_cross_spectrum_or_coherence
 -> generate_one_step_and_multistep_point_and_interval_forecasts
 -> backtest_using_rolling_origin_or_time_series_cross_validation
 -> compare_errors_against_baselines_and_competing_models
 -> route_to_applied_statistical_mathematics_statistical_modelling_regression_stochastic_processes_probability_statistics_signal_processing_Fourier_analysis_state_space_models_econometric_mathematics_optimisation_or_computational_statistics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Time Series Analysis Ontology defines statistical analysis of temporally ordered data. It supports temporal dependence and lags, trend, seasonality and decomposition, weak and strict stationarity, autocovariance, autocorrelation and partial autocorrelation, white noise and innovations, AR, MA, ARMA, ARIMA and seasonal ARIMA models, differencing, unit roots and random walks, ADF, KPSS and Phillips-Perron tests, cointegration and error-correction models, VAR and VECM systems, Granger causality, impulse responses and forecast-error decomposition, state-space models, Kalman filtering and smoothing, exponential smoothing, Holt and Holt-Winters methods, spectral density and periodograms, Fourier analysis, interventions and structural breaks, ARCH, GARCH and stochastic-volatility models, long-memory and ARFIMA processes, point and interval forecasts, rolling backtesting, forecast error metrics, Diebold-Mariano comparison, time-series cross-validation, missing and irregular observations, smoothing, filtering and signal extraction."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-TIME-SERIES-ANALYSIS-ONTOLOGY-V1"
}

}

END_FLOW