Statistics Ontology

FULL EXECUTABLE FLOW ALGEBRA — STATISTICS ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/statistical-foundations/statistics-ontology.html

FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Statistical Foundations"
ONTOLOGY := "Statistics Ontology"
PRIMARY_ENTITY := Statistics
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

Statistics :=
    ScienceOfData
    ∧ DataCollection
    ∧ DataOrganisation
    ∧ DataAnalysis
    ∧ StatisticalInference
    ∧ UncertaintyQuantification
    ∧ EvidenceInterpretation

StatisticalQuestion :=
    Question
    about Population
    ∨ Process
    ∨ Relationship
    ∨ Parameter
    ∨ Prediction

Population :=
    CompleteSetOfRelevantUnitsOrOutcomes

StatisticalUnit :=
    Entity
    on which observations are made

Sample :=
    ObservedSubsetOrSequence
    drawn_or_obtained_from Population

SampleSize :=
    n

Observation :=
    RecordedMeasurementOrCategory
    for StatisticalUnit

Dataset :=
    {
        Observations,
        Variables,
        Metadata,
        SamplingContext
    }

Variable :=
    RecordedOrMeasurableCharacteristic

VariableType := {
    Categorical,
    Quantitative,
    Nominal,
    Ordinal,
    Binary,
    Discrete,
    Continuous
}

CategoricalVariable :=
    Variable
    taking category values

QuantitativeVariable :=
    Variable
    taking numerical values

Parameter :=
    PopulationOrModelQuantity

Statistic :=
    FunctionOfSampleData

Estimator :=
    Statistic
    intended_to estimate Parameter

Estimate :=
    RealisedEstimatorValue

DescriptiveStatistics :=
    Statistics
    summarising observed data

InferentialStatistics :=
    Methods
    drawing conclusions
    about population_or_model
    from sample data

ExploratoryDataAnalysis :=
    Data
    → Patterns
    → Distributions
    → Relationships
    → Anomalies

ConfirmatoryAnalysis :=
    PreSpecifiedHypothesisOrModel
    → StatisticalEvaluation

Frequency :=
    CountOfOccurrence

RelativeFrequency :=
    Frequency/SampleSize

Proportion :=
    CountMeetingCondition/SampleSize

Percentage :=
    100×Proportion

ArithmeticMean :=
    X_bar
    =
    (1/n)Σ_iX_i

WeightedMean :=
    Σ_iw_iX_i
    /
    Σ_iw_i

Median :=
    CentralOrderedValue
    under selected convention

Mode :=
    MostFrequentValueOrCategory

Range :=
    Maximum-Minimum

Variance :=
    AverageSquaredDeviation
    under population_or_sample convention

PopulationVariance :=
    σ²
    =
    E[(X-μ)²]

SampleVariance :=
    S²
    =
    [1/(n-1)]
    Σ_i(X_i-X_bar)²

StandardDeviation :=
    sqrt(Variance)

InterquartileRange :=
    Q3-Q1

MedianAbsoluteDeviation :=
    Median(
        |X_i-Median(X)|
    )

Quantile :=
    Value
    dividing distribution
    according_to cumulative probability

Percentile :=
    Quantile
    expressed_on
    percentage scale

ZScore :=
    z
    =
    (x-μ)/σ

Standardisation :=
    Value
    → ZScore

Distribution :=
    ProbabilityLaw
    describing variable_or_statistic

EmpiricalDistribution :=
    SampleData
    → EmpiricalProbabilityMeasure

EmpiricalCDF :=
    F_n(x)
    =
    (1/n)Σ_i1_(X_i≤x)

FrequencyTable :=
    ValuesOrCategories
    ∧ Counts
    ∧ RelativeFrequencies

ContingencyTable :=
    CrossClassification
    of categorical variables

Histogram :=
    BinnedNumericalData
    → FrequencyOrDensityRepresentation

BoxPlot :=
    QuantileSummary
    ∧ PotentialOutlierDisplay

ScatterPlot :=
    PairedNumericalObservations
    → RelationshipVisualisation

ProbabilityModel :=
    MathematicalModel
    assigning probabilities
    to possible data

RandomVariable :=
    Mapping
    from random outcomes
    to measurable values

ExpectedValue :=
    E[X]

Covariance :=
    Cov(X,Y)
    =
    E[(X-E[X])(Y-E[Y])]

Correlation :=
    Corr(X,Y)
    =
    Cov(X,Y)/(SD(X)SD(Y))
    where standard deviations positive

CorrelationConstraint :=
    -1≤Corr(X,Y)≤1

CorrelationDoesNotImplyCausation :=
    Correlation
    ↛ Causation

SamplingDistribution :=
    ProbabilityDistribution
    of Statistic
    under repeated sampling

StandardError :=
    StandardDeviation
    of SamplingDistribution
    or estimate thereof

SampleMeanStandardError :=
    σ/√n
    when population σ known
    under iid assumptions

EstimatedMeanStandardError :=
    S/√n

CentralLimitTheorem :=
    StandardisedSampleAggregate
    → NormalDistribution
    in distribution
    under theorem conditions

LawOfLargeNumbers :=
    SampleAverage
    → ExpectedValue
    under theorem conditions

PointEstimation :=
    Data
    → SingleParameterEstimate

IntervalEstimation :=
    Data
    → IntervalForParameter

ConfidenceInterval :=
    RandomIntervalProcedure
    with specified coverage
    under repeated sampling model

ConfidenceLevel :=
    1-α

MarginOfError :=
    CriticalValue
    × StandardError

HypothesisTesting :=
    NullHypothesis
    ∧ AlternativeHypothesis
    ∧ TestStatistic
    ∧ ReferenceDistribution
    → EvidenceAssessment

NullHypothesis := H0
AlternativeHypothesis := H1

TestStatistic :=
    FunctionOfData
    chosen_for hypothesis test

PValue :=
    ProbabilityUnderH0
    of statistic at least as incompatible
    with H0 as observed
    according_to test definition

SignificanceLevel :=
    α

DecisionRule :=
    PValue≤α
    ⇒ RejectH0
    under conventional procedure

TypeIError :=
    RejectH0WhenH0True

TypeIIError :=
    FailToRejectH0WhenSpecifiedAlternativeTrue

Power :=
    1-β

EffectSize :=
    MagnitudeOfDifferenceOrRelationship

StatisticalSignificance :=
    ResultRelativeToStatisticalThreshold

PracticalSignificance :=
    RealWorldOrDomainImportance

SignificanceDistinction :=
    StatisticalSignificance
    ≠
    PracticalSignificance

ParametricStatistics :=
    Methods
    using specified parametric model

NonparametricStatistics :=
    Methods
    with fewer_or_different distributional assumptions

RobustStatistics :=
    Methods
    designed_to reduce sensitivity
    to outliers_or_model deviations

ResamplingStatistics := {
    Bootstrap,
    Permutation,
    CrossValidation
}

Bootstrap :=
    ResampleWithReplacement
    from observed sample
    → ApproximateSamplingDistribution

PermutationMethod :=
    ReassignData
    under exchangeability assumptions
    → NullReferenceDistribution

Regression :=
    StatisticalModel
    relating response
    to predictors

LinearRegression :=
    Y
    =
    Xβ+ε

RegressionCoefficient :=
    β_j

Residual :=
    ObservedResponse-FittedResponse

CoefficientOfDetermination :=
    R²
    =
    ProportionOfSpecifiedVariationExplained
    under model definition

ANOVA :=
    DecomposeVariation
    → CompareMeanSquares
    → FStatistic

ChiSquaredMethods :=
    CountData
    → GoodnessOfFitOrAssociationTests

TMethods :=
    MeanOrCoefficientInference
    using StudentTDistribution
    under relevant assumptions

FMethods :=
    VarianceRatioOrModelComparison
    using FDistribution
    under relevant assumptions

Likelihood :=
    L(θ|Data)

MaximumLikelihoodEstimate :=
    θ_hat
    =
    argmax_θL(θ|Data)

BayesianStatistics :=
    Prior
    ∧ Likelihood
    → Posterior

PosteriorInference :=
    PosteriorDistribution
    → Estimates
    ∧ CredibleIntervals
    ∧ Predictions

PredictiveStatistics :=
    Data
    ∧ Model
    → PredictionForNewObservation

PredictionInterval :=
    Interval
    for future observation
    under predictive model

SamplingDesign :=
    Procedure
    governing sample selection

ExperimentalDesign :=
    Procedure
    governing interventions
    and data collection

RandomSampling :=
    RandomMechanism
    used_to select observations

RandomAssignment :=
    RandomMechanism
    used_to assign treatments

Confounding :=
    ThirdVariableOrStructure
    distorting association_or_causal interpretation

Bias :=
    SystematicDifference
    between expected estimator
    and target parameter
    or systematic study distortion
    depending context

EstimatorBias :=
    E[T]-θ

MeasurementBias :=
    SystematicMeasurementDistortion

SelectionBias :=
    SystematicSelectionDistortion

MissingData :=
    IntendedValues
    unavailable

Outlier :=
    Observation
    unusually extreme
    under reference model_or_data pattern

DataQuality :=
    Accuracy
    ∧ Completeness
    ∧ Consistency
    ∧ Validity
    ∧ Provenance

StatisticalAssumption :=
    Condition
    required_by model_or_method

AssumptionSet := {
    Independence,
    DistributionalForm,
    Homoscedasticity,
    Linearity,
    RandomSampling,
    Exchangeability,
    Identifiability
}

ModelDiagnostics :=
    Data
    ∧ Residuals
    ∧ AssumptionChecks
    → ModelAssessment

StatisticalWorkflow :=
    DefineQuestion
    → IdentifyPopulation
    → DefineVariables
    → DesignStudy
    → CollectData
    → ValidateData
    → DescribeData
    → ExplorePatterns
    → SpecifyModel
    → EstimateParameters
    → QuantifyUncertainty
    → TestHypothesesOrCompareModels
    → DiagnoseModel
    → InterpretEvidence
    → PreserveProvenance

StatisticalResult :=
    {
        Estimate,
        Uncertainty,
        Model,
        Assumptions,
        Evidence,
        Interpretation
    }

StatisticalClaim :=
    Statement
    supported_or_evaluated
    by StatisticalEvidence

StatisticalEvidence :=
    Data
    ∧ Method
    ∧ Model
    ∧ Assumptions
    ∧ Uncertainty

StatisticalValidity :=
    MethodAppropriate
    ∧ AssumptionsAdequatelyAddressed
    ∧ CalculationsCorrect
    ∧ InterpretationMatchesDesign

StatisticalConstraint :=
    ConstraintOn(
        DataType,
        SamplingDesign,
        Model,
        Parameter,
        Statistic,
        Inference
    )

InvalidStatisticalInference :=
    InappropriateMethod
    ∨ ViolatedCriticalAssumption
    ∨ InvalidCalculation
    ∨ UnsupportedGeneralisation
    ∨ UnsupportedCausalInterpretation

StatisticsState := {
    Question,
    Population,
    Sample,
    Variables,
    Dataset,
    SamplingDesign,
    Model,
    Parameters,
    Statistics,
    Estimates,
    Uncertainty,
    Evidence,
    Confidence,
    Provenance
}

StatisticsValidation :=
    ResolveQuestion
    → ResolvePopulation
    → ResolveSample
    → ResolveVariableTypes
    → ResolveStudyDesign
    → ResolveMethod
    → ValidateAssumptions
    → ValidateCalculations
    → ValidateInference
    → ValidateInterpretation

SEMANTIC_EQUIVALENCE(StatisticalAnalysisA,StatisticalAnalysisB) :=
    same(TargetQuestion)
    ∧ same(EstimandOrParameter)
    ∧ same(DataSemantics)
    ∧ same(InferentialScope)
    under semantically equivalent models_and_assumptions

StatisticsOntology :=
    Classes
    ∧ Relations
    ∧ Constraints
    ∧ Axioms
    ∧ InferenceRules

CoreClasses := {
    Statistics,
    Population,
    StatisticalUnit,
    Sample,
    Observation,
    Dataset,
    Variable,
    Parameter,
    Statistic,
    Estimator,
    Estimate,
    Distribution,
    SamplingDistribution,
    DescriptiveStatistics,
    InferentialStatistics,
    ConfidenceInterval,
    HypothesisTest,
    Regression,
    StatisticalModel,
    StatisticalEvidence
}

CoreRelations := {
    sampledFrom,
    observedOn,
    containsVariable,
    describes,
    estimates,
    hasParameter,
    hasStatistic,
    distributedAs,
    tests,
    predicts,
    assumes,
    supports,
    contradicts,
    derivedFrom,
    dependsOn,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    Sample --sampledFrom--> Population
    Observation --observedOn--> StatisticalUnit
    Dataset --containsVariable--> Variable
    DescriptiveStatistic --describes--> Dataset
    Estimator --estimates--> Parameter
    StatisticalModel --hasParameter--> Parameter
    Dataset --hasStatistic--> Statistic
    Statistic --distributedAs--> SamplingDistribution
    HypothesisTest --tests--> Hypothesis
    StatisticalModel --predicts--> Outcome
    StatisticalModel --assumes--> StatisticalAssumption
    StatisticalEvidence --supports--> StatisticalClaim
    StatisticalResult --tracesTo--> Provenance

InferenceRules := {
    Estimator(T,θ)
        ⇒
        Estimate=T(Data),

    UnbiasedEstimator(T,θ)
        ⇒
        E[T]=θ,

    IIDSample
        ∧ finite_mean
        ⇒
        SampleMean
        converges_to PopulationMean
        under LLN conditions,

    SuitableCLTConditions
        ⇒
        StandardisedAggregate
        converges_in_distribution
        to NormalDistribution,

    PValue≤α
        ⇒
        RejectH0
        under specified testing procedure,

    Correlation
        ↛ Causation,

    StatisticalSignificance
        ↛ PracticalSignificance,

    CriticalAssumptionViolation
        ⇒
        FlagInferenceRisk
}

CanonicalStatisticsRepresentation :=
    ParseStatisticalQuestion
    → ResolvePopulation
    → ResolveSample
    → ResolveVariables
    → ClassifyVariableTypes
    → ResolveStudyDesign
    → ResolveStatisticalMethod
    → ResolveParametersAndStatistics
    → ResolveUncertainty
    → ValidateAssumptions
    → CanonicaliseAnalysis

StatisticsGraph :=
    G=(V,E)

V := {
    Populations,
    StatisticalUnits,
    Samples,
    Observations,
    Datasets,
    Variables,
    Parameters,
    StatisticsNodes,
    Estimators,
    Estimates,
    Distributions,
    SamplingDistributions,
    Models,
    Hypotheses,
    Tests,
    Intervals,
    Evidence,
    ProvenanceNodes
}

E := {
    sampledFrom,
    observedOn,
    containsVariable,
    describes,
    estimates,
    hasParameter,
    hasStatistic,
    distributedAs,
    tests,
    predicts,
    assumes,
    supports,
    contradicts,
    derivedFrom,
    dependsOn,
    equivalentTo,
    tracesTo
}

CONFIDENCE(StatisticalResult) :=
    weighted_mean(
        QuestionResolution,
        PopulationSampleResolution,
        DataQuality,
        StudyDesignQuality,
        MethodAppropriateness,
        AssumptionValidity,
        UncertaintyQuality,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectStatisticalIntent
    → IdentifyPopulationSampleAndVariables
    → ClassifyVariableTypes
    → ResolveTargetParameterOrQuestion
    → IdentifyRequestedSummaryTestModelOrPrediction

AI_REASONING_FLOW :=
    Query
    → ParseStatisticalProblem
    → ResolvePopulation
    → ResolveSample
    → ResolveVariablesAndTypes
    → ResolveStudyAndSamplingDesign
    → ValidateDataQuality
    → SelectDescriptiveOrInferentialFramework
    → ResolveProbabilityModel
    → ComputeStatisticsOrEstimates
    → QuantifyUncertainty
    → EvaluateHypothesisOrModelIfRequired
    → DiagnoseAssumptions
    → DistinguishAssociationFromCausation
    → ValidateInterpretation
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseStatisticsEntities,
    ParseTypedRelations,
    ResolvePopulations,
    ResolveSamples,
    ResolveDatasets,
    ResolveVariables,
    ClassifyVariableTypes,
    ComputeDescriptiveStatistics,
    ResolveProbabilityDistributions,
    ResolveSamplingDistributions,
    ComputePointEstimates,
    ComputeStandardErrors,
    ComputeConfidenceIntervals,
    EvaluateHypothesisTests,
    ResolveRegressionModels,
    ResolveANOVAModels,
    ResolveChiSquaredMethods,
    ResolveTMethods,
    ResolveFMethods,
    EvaluateLikelihoods,
    ResolveBayesianInference,
    ValidateStatisticalAssumptions,
    ValidateStatisticalInference,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataDesignModelMethodOrEvidence
    → ValidateInput
    → IdentifyAffectedStatisticalClaims
    → UpdateStatisticsState
    → RecomputeDescriptiveStatistics
    → RecomputeEstimates
    → RecomputeUncertainty
    → ReevaluateTestsOrModels
    → RevalidateAssumptions
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

DataUpdate :=
    ChangedDataset
    ⇒ recompute(
        Summaries,
        Estimates,
        TestStatistics,
        ModelFit
    )

DesignUpdate :=
    ChangedSamplingOrExperimentalDesign
    ⇒ reevaluate(
        BiasRisk,
        Generalisability,
        CausalInterpretation
    )

ModelUpdate :=
    ChangedStatisticalModel
    ⇒ reevaluate(
        Parameters,
        Estimates,
        Uncertainty,
        Diagnostics,
        Predictions
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        Statistics,
        StatisticalWorkflow,
        StatisticsValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    StatisticsEntityResolution,
    StatisticalQuestion,
    Population,
    Sample,
    Variables,
    VariableTypes,
    Dataset,
    StudyDesign,
    StatisticalMethod,
    ProbabilityModel,
    Parameters,
    Statistics,
    Estimates,
    Uncertainty,
    HypothesisOrPrediction,
    AssumptionStatus,
    StatisticalValidity,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA