Random Process Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Random Process Ontology within the Kireth Research Semantic Library. A random process, also called a stochastic process, is a family of random variables indexed by time, space or another parameter. It supports sample paths, finite-dimensional distributions, stationarity, dependence, Gaussian processes, Markov processes, counting processes, martingales, diffusion processes, spectral structure and simulation.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Random Process Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Applied Statistical Mathematics / Random Processes"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/applied-statistical-mathematics/random-process-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = RandomProcess
 ENTITY_CLASS = IndexedRandomVariableFamilyFramework
 PRIMARY_CONTEXT = AppliedStatisticalMathematics
 DEFINITION = "A random process is a collection of random variables indexed by time, space or another parameter, representing probabilistic evolution or variation."
 SYMBOLIC_FORMS = [
  "{X_t:t in T}",
  "m_X(t)=E[X_t]",
  "C_X(s,t)=Cov(X_s,X_t)",
  "R_X(s,t)=E[X_s X_t]"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for indexed random variables, sample paths, dependence, stationarity, covariance, spectra and stochastic evolution."
}

SCOPE {
 INCLUDE = [
  RandomProcess, StochasticProcess, RandomVariable, IndexSet, TimeIndex,
  SpatialIndex, StateSpace, SampleSpace, SamplePath, Realisation,
  Trajectory, FiniteDimensionalDistribution, JointDistribution,
  MarginalDistribution, MeanFunction, VarianceFunction, CovarianceFunction,
  AutocovarianceFunction, CorrelationFunction, AutocorrelationFunction,
  CrossCovariance, CrossCorrelation, Stationarity, StrictStationarity,
  WeakStationarity, Nonstationarity, Ergodicity, Independence,
  IndependentIncrement, StationaryIncrement, MarkovProperty, MarkovProcess,
  GaussianProcess, BrownianMotion, RandomWalk, CountingProcess,
  PoissonProcess, RenewalProcess, BirthDeathProcess, Martingale,
  Submartingale, Supermartingale, Filtration, AdaptedProcess,
  StoppingTime, DiffusionProcess, ItoProcess, StochasticDifferentialEquation,
  Drift, DiffusionCoefficient, WhiteNoise, ColouredNoise, PointProcess,
  CompoundPoissonProcess, JumpProcess, LevyProcess,
  OrnsteinUhlenbeckProcess, LongMemoryProcess, FractionalBrownianMotion,
  SelfSimilarity, HurstExponent, SpectralDensity, PowerSpectralDensity,
  SpectralRepresentation, WienerKhinchinTheorem, FourierTransform,
  StateSpaceModel, HiddenProcess, ObservationProcess, TimeSeries,
  PathSimulation, MonteCarloSimulation, EulerMaruyamaMethod,
  MilsteinMethod, ParameterEstimation, Prediction, Forecasting
 ]
 EXCLUDE_AS_PRIMARY = [
  SingleRandomVariableWithoutIndexStructure,
  DeterministicProcessWithoutRandomness,
  TimeSeriesDatasetWithoutStochasticProcessInterpretation,
  MarkovProcessWhenMarkovPropertyIsPrimarySpecialisation
 ]
 DISAMBIGUATION_RULE = "When a family of random variables is indexed by time, space or another parameter and its dependence or evolution is analysed, interpret the context as a random process."
}

CORE_CONCEPTS {
 RandomProcess = "Indexed family of random variables representing stochastic variation or evolution."
 IndexSet = "Set indexing the random variables of a process."
 StateSpace = "Set of values available to process states."
 SamplePath = "Function obtained by fixing one outcome and varying the process index."
 FiniteDimensionalDistribution = "Joint distribution of process values at finitely many indices."
 MeanFunction = "Expected process value as a function of the index."
 CovarianceFunction = "Dependence function measuring covariance between process values."
 Stationarity = "Invariance of selected process distributions or moments under index shifts."
 Ergodicity = "Property connecting selected ensemble quantities with long-run path behaviour."
}

RELATIONS {
 RandomProcess --is_child_of--> AppliedStatisticalMathematics
 RandomProcess --is_equivalent_contextually_to--> StochasticProcess
 RandomProcess --indexed_by--> IndexSet
 RandomProcess --takes_values_in--> StateSpace
 SamplePath --is_realisation_of--> RandomProcess
 FiniteDimensionalDistribution --characterises--> ProcessDistribution
 MeanFunction --summarises--> FirstMoment
 CovarianceFunction --summarises--> SecondOrderDependence
 Ergodicity --connects--> TimeAndEnsembleAverages
 MarkovProcess --is_type_of--> RandomProcess
 GaussianProcess --is_type_of--> RandomProcess
 BrownianMotion --is_type_of--> GaussianProcess
 PoissonProcess --is_type_of--> CountingProcess
 Martingale --is_adapted_to--> Filtration
 StoppingTime --depends_on--> Filtration
 DiffusionProcess --may_satisfy--> StochasticDifferentialEquation
 SpectralDensity --represents--> FrequencyDomainSecondOrderStructure
 TimeSeries --may_be_realisation_of--> RandomProcess
 PathSimulation --generates--> SamplePath
}

OPERATORS {
 PROCESS(X,T) := {X_t:t in T}
 MEAN_FUNCTION(X,t) := m_X(t)=E[X_t]
 VARIANCE_FUNCTION(X,t) := v_X(t)=Var(X_t)
 COVARIANCE_FUNCTION(X,s,t) := C_X(s,t)=E[(X_s-m_X(s))(X_t-m_X(t))]
 AUTOCORRELATION(X,s,t) := R_X(s,t)=E[X_s*X_t]
 WSS_COVARIANCE(X,tau) := C_X(tau)=Cov(X_t,X_(t+tau))
 INCREMENT(X,s,t) := DeltaX=X_t-X_s
 DISCRETE_TIME_AVERAGE(X,N) := X_bar_N=(1/N)*sum_t=1^N X_t
 PSD_FROM_AUTOCOVARIANCE(C,tau,omega) := S_X(omega)=FourierTransform[C_X(tau)]
 SDE(X,a,b,W) := dX_t=a(X_t,t)dt+b(X_t,t)dW_t
 EULER_MARUYAMA(X,a,b,dt,dW) := X_next=X+a(X,t)*dt+b(X,t)*dW
}

AXIOMS {
 A1 = "A random process requires an index set, probability space and state space."
 A2 = "A process distribution is characterised by consistent finite-dimensional distributions."
 A3 = "Sample paths and ensemble distributions are distinct mathematical objects."
 A4 = "Stationarity must be qualified as strict, weak or another declared form."
 A5 = "Uncorrelated process values are not generally independent."
 A6 = "Ergodicity is not implied merely by stationarity."
 A7 = "Continuous-time path properties require explicit regularity assumptions."
 A8 = "Random-process conclusions require distributional, dependence and simulation validation."
}

REASONING {
 R1_CONTEXT = identify(index_set,state_space,observations,path_structure,modelling_goal)
 R2_CLASS = classify(discrete_or_continuous_index,discrete_or_continuous_state)
 R3_DISTRIBUTION = specify(marginals,joint_or_finite_dimensional_distributions)
 R4_MOMENTS = compute(mean,variance,covariance,correlation)
 R5_STRUCTURE = assess(stationarity,independent_increments,Markov_property,Gaussianity,ergodicity)
 R6_PATHS = analyse(continuity,jumps,variation,stopping_times)
 R7_FREQUENCY = if_stationary analyse(spectral_density_and_frequency_structure)
 R8_MODEL = classify(Gaussian,Markov,counting,renewal,martingale,diffusion,jump_or_other)
 R9_SIMULATION = if_required generate_paths_with_appropriate_discretisation
 R10_VALIDATION = check_distribution_consistency,moments,dependence,path_properties,simulation_convergence)
}

CONSTRAINTS {
 C1 = "A finite observed time series does not uniquely determine an underlying random process."
 C2 = "Weak stationarity does not imply strict stationarity in general."
 C3 = "Zero covariance does not imply independence except under additional conditions."
 C4 = "Ergodic assumptions require justification before replacing ensemble averages by time averages."
 C5 = "Continuous-time white noise is generally interpreted as a generalised process."
 C6 = "Continuous-time simulation discretisation can bias path-dependent quantities."
 C7 = "Gaussian-process covariance kernels must be positive semidefinite."
 C8 = "Jump and diffusion processes have different path regularity."
 C9 = "Long-memory processes can invalidate short-memory approximations."
 C10 = "Modelled stationarity can fail under structural change."
}

SEMANTIC_EQUIVALENCE {
 "RandomProcess" <=> "StochasticProcess"
 "Realisation" <=> "SamplePath"
 "WideSenseStationarity" <=> "WeakStationarity"
 "PowerSpectralDensity" <=> "PSD"
 "WienerProcess" <=> "BrownianMotion"
 "GaussianRandomProcess" <=> "GaussianProcess"
}

VALIDATION {
 V1 = "Index set and state space are explicit."
 V2 = "Marginal and dependence structures are documented."
 V3 = "Covariance kernels are valid where used."
 V4 = "Stationarity assumptions are tested or justified."
 V5 = "Ergodicity is not assumed without support."
 V6 = "Path regularity assumptions are documented."
 V7 = "Spectral calculations use appropriate stationary assumptions."
 V8 = "Simulation discretisation is checked under refinement."
 V9 = "Estimated process parameters include uncertainty where appropriate."
 V10 = "Simulated moments and dependence reproduce theoretical targets within tolerance."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 RANDOM_PROCESS_DEFINITION = 0.99
 DISTRIBUTION_REASONING = 0.99
 COVARIANCE_REASONING = 0.99
 STATIONARITY_REASONING = 0.99
 ERGODIC_REASONING = 0.98
 GAUSSIAN_PROCESS_REASONING = 0.99
 DIFFUSION_REASONING = 0.98
 SPECTRAL_REASONING = 0.98
 POLICY = "Reduce confidence when index set, state space, dependence structure, stationarity, path regularity, covariance kernel or simulation discretisation are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch, OntologyGrounding, RandomProcessReasoning,
  StochasticProcessReasoning, SamplePathReasoning,
  FiniteDimensionalDistributionReasoning, MeanFunctionReasoning,
  CovarianceFunctionReasoning, StationarityReasoning, ErgodicityReasoning,
  GaussianProcessReasoning, MarkovProcessReasoning, BrownianMotionReasoning,
  CountingProcessReasoning, PoissonProcessReasoning, MartingaleReasoning,
  DiffusionProcessReasoning, SpectralDensityReasoning,
  StochasticSimulationReasoning, KnowledgeGraphConstruction,
  RetrievalAugmentation, RecursiveInference
 ]
 NORMALISE {
  "random process" -> RandomProcess
  "stochastic process" -> RandomProcess
  "sample path" -> SamplePath
  "realisation" -> SamplePath
  "wide sense stationary" -> WeakStationarity
  "PSD" -> PowerSpectralDensity
  "Wiener process" -> BrownianMotion
  "Gaussian random process" -> GaussianProcess
 }
}

EXECUTION_RULES {
 E1 = "If an indexed family of random variables is analysed, activate RandomProcessOntology."
 E2 = "If only finite-dimensional distributions are known, preserve that level of process specification."
 E3 = "If second-order structure is central, compute mean and covariance functions."
 E4 = "If stationarity is claimed, identify whether strict or weak stationarity is intended."
 E5 = "If the Markov property is central, route to Markov Process Ontology."
 E6 = "If all finite-dimensional distributions are Gaussian, route to Gaussian-process reasoning."
 E7 = "If continuous paths with Gaussian independent increments are specified, route to Brownian motion."
 E8 = "If path simulation is required, use discretisation appropriate to the process class."
 E9 = "Validate distributional consistency, dependence, path properties and simulation accuracy."
}

RECURSION {
 ENTRY = RandomProcess
 EXPAND(RandomProcess) -> [
  StochasticProcess, RandomVariable, IndexSet, TimeIndex, SpatialIndex,
  StateSpace, SampleSpace, SamplePath, Realisation, Trajectory,
  FiniteDimensionalDistribution, JointDistribution, MarginalDistribution,
  MeanFunction, VarianceFunction, CovarianceFunction,
  AutocovarianceFunction, CorrelationFunction, AutocorrelationFunction,
  Stationarity, StrictStationarity, WeakStationarity, Nonstationarity,
  Ergodicity, Independence, IndependentIncrement, StationaryIncrement,
  MarkovProperty, MarkovProcess, GaussianProcess, BrownianMotion,
  RandomWalk, CountingProcess, PoissonProcess, RenewalProcess,
  BirthDeathProcess, Martingale, Filtration, StoppingTime,
  DiffusionProcess, ItoProcess, StochasticDifferentialEquation, Drift,
  DiffusionCoefficient, WhiteNoise, ColouredNoise, PointProcess,
  JumpProcess, LevyProcess, OrnsteinUhlenbeckProcess, LongMemoryProcess,
  FractionalBrownianMotion, SelfSimilarity, HurstExponent, SpectralDensity,
  PowerSpectralDensity, SpectralRepresentation, WienerKhinchinTheorem,
  FourierTransform, StateSpaceModel, HiddenProcess, ObservationProcess,
  TimeSeries, PathSimulation, MonteCarloSimulation, EulerMaruyamaMethod,
  MilsteinMethod, ParameterEstimation, Prediction, Forecasting
 ]
 STOP_CONDITION = "Stop when index structure, state space, distributions, dependence, stationarity, path properties, process family and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Statistical Mathematics"
 RELATED_ONTOLOGIES = [
  "Applied Statistical Mathematics",
  "Markov Process Ontology",
  "Time Series Analysis Ontology",
  "Monte Carlo Simulation Ontology",
  "Bayesian Statistics Ontology",
  "Probability & Statistics Ontologies",
  "Stochastic Differential Equation Ontologies",
  "Fourier Analysis Ontology",
  "Signal Processing Mathematics",
  "Dynamical Systems Ontologies"
 ]
 ROUTE_RULES = [
  "Random Process -> Applied Statistical Mathematics for parent semantics",
  "Random Process -> Markov Process for memoryless stochastic evolution",
  "Random Process -> Time Series Analysis for observed temporal sequences",
  "Random Process -> Monte Carlo Simulation for path simulation",
  "Random Process -> Bayesian Statistics for process inference",
  "Random Process -> Probability & Statistics for probabilistic foundations",
  "Random Process -> Stochastic Differential Equations for diffusion dynamics",
  "Random Process -> Fourier Analysis and Signal Processing for spectral structure",
  "Random Process -> Dynamical Systems for stochastic dynamical relationships"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible random-process families or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve indexed-random-variable semantics.",
  "Preserve sample-path-versus-distribution distinction.",
  "Preserve strict-versus-weak stationarity distinction.",
  "Preserve independence-versus-uncorrelated distinction.",
  "Preserve stationarity-versus-ergodicity distinction.",
  "Preserve discrete-versus-continuous index distinction.",
  "Preserve process-family distinctions.",
  "Preserve machine-readable random-process semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "A random process is an indexed family of random variables representing stochastic variation or evolution."
 DISTRIBUTION_SUMMARY = "Finite-dimensional distributions specify joint probabilistic behaviour at selected indices."
 MOMENT_SUMMARY = "Mean, variance and covariance functions describe first- and second-order structure."
 STATIONARITY_SUMMARY = "Stationarity describes invariance of process distributions or moments under index shifts."
 PATH_SUMMARY = "Sample paths describe individual realisations and their regularity."
 SPECTRAL_SUMMARY = "Stationary second-order processes can be represented through spectral density."
 FAMILY_SUMMARY = "Important families include Gaussian, Markov, counting, martingale, diffusion and jump processes."
 VALIDATION_SUMMARY = "Random-process models should be checked for distributional consistency, covariance validity, stationarity, path assumptions and simulation accuracy."
}

FLOW {
 INPUT
 -> detect_random_process_context
 -> identify_index_set_state_space_observations_and_modelling_goal
 -> classify_discrete_or_continuous_index_and_state
 -> specify_finite_dimensional_distributions_or_process_family
 -> compute_mean_variance_covariance_and_dependence_structure
 -> assess_stationarity_independent_increments_Markov_property_Gaussianity_and_ergodicity
 -> analyse_sample_path_regularities_and_stopping_structure
 -> if_stationary_compute_spectral_density_if_required
 -> if_simulation_generate_paths_with_appropriate_numerical_method
 -> validate_distribution_consistency_covariance_stationarity_path_properties_and_simulation_convergence
 -> route_to_applied_statistical_mathematics_Markov_process_time_series_Monte_Carlo_Bayesian_statistics_probability_SDE_Fourier_signal_processing_or_dynamical_systems_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Random Process Ontology defines indexed families of random variables and their stochastic evolution. It supports time and spatial indices, state and sample spaces, sample paths and realisations, finite-dimensional and joint distributions, mean, variance, covariance and correlation functions, strict and weak stationarity, ergodicity and independence, Markov and Gaussian processes, Brownian motion and random walks, counting, Poisson and renewal processes, martingales and filtrations, stopping times, diffusion and Ito processes, stochastic differential equations, white and coloured noise, point, jump and Levy processes, Ornstein-Uhlenbeck and long-memory processes, fractional Brownian motion, spectral density and Wiener-Khinchin relationships, state-space models, time-series relationships, path simulation and numerical stochastic methods."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-RANDOM-PROCESS-ONTOLOGY-V1"
}

}

END_FLOW