Kurtosis Ontology

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

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

BEGIN_FLOW

Kurtosis :=
    DistributionShapeMeasure
    ∧ StandardisedFourthCentralMoment
    ∧ TailAndExtremeValueSensitivityMeasure

RandomVariable := X
Mean := μ=E[X]
Variance := σ²=Var(X)
StandardDeviation := σ=sqrt(Var(X))
FourthCentralMoment := μ4=E[(X-μ)^4]

PopulationKurtosis :=
    β2=μ4/σ^4
    where σ>0
    and fourth moment exists

ExcessKurtosis :=
    γ2=β2-3

NormalReferenceKurtosis :=
    NormalDistribution
    ⇒ β2=3 ∧ γ2=0

KurtosisExistence :=
    E[|X|^4]<∞
    ∧ σ>0
    ⇒ PopulationKurtosisFinite

UndefinedKurtosis :=
    UndefinedFourthMoment
    ∨ ZeroVariance
    ⇒ StandardMomentKurtosisUndefined

KurtosisDimension := Dimensionless

Mesokurtic := ExcessKurtosis=0
Leptokurtic := ExcessKurtosis>0
Platykurtic := ExcessKurtosis<0

KurtosisInterpretation :=
    HigherKurtosis
    generally reflects
    greater contribution from extreme deviations
    and tail behaviour

PeakednessCaution :=
    Kurtosis
    should_not be interpreted
    solely as peakedness

Dataset := {x_1,...,x_n}
SampleSize := n
SampleMean := X_bar=(1/n)Σ_i x_i
SampleCentralMoment2 := m2=(1/n)Σ_i(x_i-X_bar)^2
SampleCentralMoment4 := m4=(1/n)Σ_i(x_i-X_bar)^4

MomentSampleKurtosis :=
    b2=m4/(m2^2)
    where m2>0

MomentSampleExcessKurtosis :=
    g2=b2-3

AdjustedFisherExcessKurtosis :=
    G2=
    [(n-1)/((n-2)(n-3))]
    ×[(n+1)g2+6]
    for n>3

SampleKurtosisConvention :=
    MultipleDefinitionsExist
    ⇒ FormulaConventionMustBeRecorded

PearsonKurtosis := β2
FisherKurtosis := β2-3

PearsonVsFisherConvention :=
    PearsonKurtosis uses NormalReference=3
    whereas FisherExcessKurtosis uses NormalReference=0

TranslationInvariance :=
    Kurtosis(X+b)=Kurtosis(X)

ScaleInvariance :=
    a≠0
    ⇒ Kurtosis(aX)=Kurtosis(X)

AffineInvariance :=
    a≠0
    ⇒ Kurtosis(aX+b)=Kurtosis(X)

StandardisationRelation :=
    Z=(X-μ)/σ
    ⇒ Kurtosis(Z)=Kurtosis(X)

MinimumPopulationKurtosis :=
    β2≥1
    for nondegenerate distributions
    with finite fourth moment

MinimumExcessKurtosis :=
    γ2≥-2

KurtosisSkewnessRelation :=
    β2≥γ1²+1
    when relevant moments exist

TailSensitivity :=
    FourthPowerDeviations
    strongly weight extreme observations

OutlierSensitivity :=
    SingleExtremeObservation
    may substantially alter sample kurtosis

NormalDistributionKurtosis :=
    X~N(μ,σ²)
    ⇒ β2=3 ∧ γ2=0

UniformDistributionKurtosis :=
    X~Uniform(a,b)
    ⇒ β2=9/5 ∧ γ2=-6/5

LogisticDistributionKurtosis :=
    X~Logistic(μ,s)
    ⇒ β2=21/5 ∧ γ2=6/5

ExponentialDistributionKurtosis :=
    X~Exponential(λ)
    ⇒ β2=9 ∧ γ2=6

PoissonDistributionExcessKurtosis :=
    X~Poisson(λ)
    ⇒ γ2=1/λ

BinomialDistributionExcessKurtosis :=
    X~Binomial(n,p)
    ⇒
    γ2=[1-6p(1-p)]/[np(1-p)]

GammaDistributionExcessKurtosis :=
    X~Gamma(α,β)
    ⇒ γ2=6/α
    under shape-rate parameterisation

ChiSquaredExcessKurtosis :=
    X~χ²_k
    ⇒ γ2=12/k

StudentTDistributionKurtosis :=
    T~t_ν
    ∧ ν>4
    ⇒ γ2=6/(ν-4)

StudentTFourthMomentUndefined :=
    ν≤4
    ⇒ StandardMomentKurtosisUndefined

CauchyDistributionKurtosis :=
    StandardMomentKurtosisUndefined

LaplaceDistributionKurtosis :=
    X~Laplace(μ,b)
    ⇒ β2=6 ∧ γ2=3

LognormalDistributionKurtosis :=
    X~LogNormal(μ,σ²)
    ⇒
    γ2=
    exp(4σ²)
    +2exp(3σ²)
    +3exp(2σ²)
    -6

WeibullDistributionKurtosis :=
    X~Weibull(k,λ)
    ⇒
    Kurtosis derived_from
    Γ(1+1/k),
    Γ(1+2/k),
    Γ(1+3/k),
    Γ(1+4/k)

NonlinearTransformationEffect :=
    Kurtosis(g(X))
    generally not directly determined
    by Kurtosis(X)

KurtosisAndNormality :=
    ExcessKurtosis≈0
    ↛ NormalDistribution

NormalityAssessment :=
    Skewness
    ∧ Kurtosis
    ∧ GraphicalDiagnostics
    ∧ FormalTests
    → DistributionAssessment

JarqueBeraRelation :=
    SampleSkewness
    ∧ SampleExcessKurtosis
    → JarqueBeraStatistic

JarqueBeraStatistic :=
    JB=n/6[S^2+K^2/4]
    under conventional notation

SamplingDistributionOfKurtosis :=
    depends_on(
        ParentDistribution,
        EstimatorConvention,
        SampleSize
    )

NormalSampleExcessKurtosisSE :=
    ApproximateSE≈sqrt(24/n)
    for large n under normality

BootstrapKurtosis :=
    ObservedSample
    → BootstrapSamples
    → ComputeKurtosisReplicates
    → ApproximateSamplingDistribution

WeightedKurtosis :=
    WeightedFourthCentralMoment
    /
    WeightedVariance²
    under selected weighting convention

RobustTailMeasures := {
    QuantileTailRatios,
    PercentileBasedTailWeight,
    MoorsKurtosis
}

MomentVsQuantileKurtosis :=
    MomentKurtosis
    highly tail-sensitive
    whereas QuantileKurtosis
    is more robust

MissingDataKurtosis :=
    Kurtosis
    computed_on selected nonmissing observations
    under explicit missingness rule

EmptyDatasetKurtosis := Undefined

ConstantDatasetKurtosis :=
    Variance=0
    ⇒ KurtosisUndefined

MeasurementScaleConstraint :=
    MomentKurtosis
    generally requires
    quantitative interval_or_ratio semantics

NominalKurtosisInvalid :=
    NominalVariable
    ⇒ MomentKurtosisNotMeaningful

UnitInvariance :=
    Kurtosis
    dimensionless
    and invariant_to nonzero linear rescaling

KurtosisConstraint :=
    ConstraintOn(
        VariableType,
        MeasurementScale,
        Variance,
        FourthMoment,
        EstimatorConvention,
        Missingness,
        Weighting
    )

KurtosisConsistency :=
    DataScope
    ∧ MeanDefinition
    ∧ VarianceDefinition
    ∧ FourthMomentDefinition
    ∧ EstimatorConvention
    jointly_compatible

InvalidKurtosis :=
    ZeroVariance
    ∨ UndefinedFourthMoment
    ∨ IncompatibleVariableType
    ∨ InvalidEstimatorConvention
    ∨ UndocumentedMissingness
    ∨ UndocumentedWeighting

KurtosisValidation :=
    ResolveVariable
    → ResolveVariableType
    → ResolveMeasurementScale
    → ResolveDataScope
    → ResolveObservationsOrDistribution
    → ResolveMissingness
    → ResolveWeights
    → ResolveKurtosisDefinition
    → ComputeMean
    → ComputeVariance
    → ValidatePositiveVariance
    → ComputeFourthCentralMoment
    → ComputeKurtosis
    → ResolvePearsonOrFisherConvention
    → ValidateResult

KurtosisState := {
    Variable,
    DataScope,
    Observations,
    EffectiveCount,
    Mean,
    Variance,
    FourthCentralMoment,
    KurtosisDefinition,
    PearsonKurtosis,
    ExcessKurtosis,
    TailWeightClassification,
    MissingnessRule,
    WeightingRule,
    Evidence,
    Confidence,
    Provenance
}

SEMANTIC_EQUIVALENCE(KurtosisA,KurtosisB) :=
    same(DataScope)
    ∧ same(VariableSemantics)
    ∧ same(KurtosisDefinition)
    ∧ same(PearsonOrFisherConvention)
    ∧ same(MissingnessRule)
    ∧ same(WeightingRule)

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

CoreClasses := {
    Kurtosis,
    PopulationKurtosis,
    SampleKurtosis,
    ExcessKurtosis,
    Mesokurtic,
    Leptokurtic,
    Platykurtic,
    FourthCentralMoment,
    PearsonKurtosis,
    FisherKurtosis,
    QuantileKurtosis,
    MoorsKurtosis,
    DistributionShape,
    StatisticalVariable,
    StatisticalObservation,
    Dataset,
    Sample,
    Population,
    KurtosisConstraint
}

CoreRelations := {
    computedFrom,
    summarises,
    measuresTailWeightOf,
    hasFourthMoment,
    hasVariance,
    hasExcess,
    hasClassification,
    estimatedBy,
    transformedBy,
    comparedWith,
    constrainedBy,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    Kurtosis --computedFrom--> Observations
    Kurtosis --summarises--> StatisticalVariable
    Kurtosis --measuresTailWeightOf--> Distribution
    Kurtosis --hasFourthMoment--> FourthCentralMoment
    Kurtosis --hasVariance--> Variance
    PearsonKurtosis --hasExcess--> ExcessKurtosis
    Kurtosis --hasClassification--> TailWeightClassification
    PopulationKurtosis --estimatedBy--> SampleKurtosis
    Kurtosis --transformedBy--> AffineTransformation
    MomentKurtosis --comparedWith--> QuantileKurtosis
    Kurtosis --tracesTo--> Provenance

InferenceRules := {
    FiniteFourthMoment
        ∧ PositiveVariance
        ⇒
        PopulationKurtosis=E[(X-μ)^4]/σ^4,

    PopulationKurtosis
        ⇒
        ExcessKurtosis=PopulationKurtosis-3,

    NormalDistribution
        ⇒
        PopulationKurtosis=3
        ∧ ExcessKurtosis=0,

    ExcessKurtosis>0
        ⇒ Leptokurtic,

    ExcessKurtosis=0
        ⇒ Mesokurtic,

    ExcessKurtosis<0
        ⇒ Platykurtic,

    Kurtosis(X+b)=Kurtosis(X),

    a≠0
        ⇒ Kurtosis(aX)=Kurtosis(X),

    ZeroVariance
        ⇒ KurtosisUndefined,

    UndefinedFourthMoment
        ⇒ KurtosisUndefined
}

CanonicalKurtosisRepresentation :=
    ParseKurtosisRequest
    → ResolveVariable
    → ResolveDatasetSamplePopulationOrDistribution
    → ResolveVariableTypeAndMeasurementScale
    → ResolveObservationsOrProbabilityModel
    → ResolveMissingness
    → ResolveWeights
    → ResolveMomentOrQuantileKurtosisDefinition
    → ComputeOrResolveMean
    → ComputeOrResolveVariance
    → ValidatePositiveVariance
    → ComputeFourthCentralMomentOrAlternativeComponents
    → ComputePearsonKurtosis
    → ComputeExcessKurtosis
    → ResolveTailWeightClassification
    → ValidateKurtosis

KurtosisGraph :=
    G=(V,E)

V := {
    KurtosisMeasures,
    PopulationKurtoses,
    SampleKurtoses,
    ExcessKurtoses,
    MesokurticNodes,
    LeptokurticNodes,
    PlatykurticNodes,
    FourthCentralMoments,
    Variances,
    Means,
    QuantileKurtoses,
    MoorsKurtoses,
    Variables,
    Observations,
    Datasets,
    Samples,
    Populations,
    Evidence,
    ProvenanceNodes
}

E := {
    computedFrom,
    summarises,
    measuresTailWeightOf,
    hasFourthMoment,
    hasVariance,
    hasExcess,
    hasClassification,
    estimatedBy,
    transformedBy,
    comparedWith,
    constrainedBy,
    equivalentTo,
    tracesTo
}

CONFIDENCE(KurtosisResolution) :=
    weighted_mean(
        VariableResolution,
        DataScopeResolution,
        MeasurementScaleValidity,
        MomentExistenceValidity,
        KurtosisDefinitionResolution,
        MissingnessHandling,
        WeightingValidity,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectKurtosisIntent
    → IdentifyVariableAndDataScope
    → DeterminePopulationSampleOrDistributionContext
    → ResolvePearsonFisherMomentOrQuantileKurtosis
    → IdentifyRequestedClassificationComparisonOrInference

AI_REASONING_FLOW :=
    Query
    → ParseKurtosisProblem
    → ResolveDatasetSamplePopulationOrDistribution
    → ResolveVariable
    → ResolveVariableTypeAndMeasurementScale
    → ResolveObservationsOrProbabilityDistribution
    → ResolveMissingValues
    → ResolveWeights
    → ResolveKurtosisDefinitionAndConvention
    → ComputeOrResolveMean
    → ComputeOrResolveVariance
    → ValidatePositiveVariance
    → ComputeFourthCentralMomentOrQuantileComponents
    → ComputePearsonKurtosis
    → ComputeExcessKurtosis
    → DetermineTailWeightClassification
    → ResolveDistributionalOrSamplingRelationsIfRequested
    → ValidateKurtosisResult
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseKurtosisEntities,
    ParseTypedRelations,
    ResolveVariables,
    ResolveObservations,
    ResolveDatasets,
    ResolveSamples,
    ResolvePopulations,
    ResolveProbabilityDistributions,
    ResolveMeasurementScales,
    ResolveMissingness,
    ResolveWeights,
    ComputeMeans,
    ComputeVariances,
    ComputeFourthCentralMoments,
    ComputePopulationKurtosis,
    ComputeSampleKurtosis,
    ComputeExcessKurtosis,
    ComputeAdjustedKurtosis,
    ResolveQuantileKurtosis,
    ResolveMoorsKurtosis,
    ResolveDistributionKurtosisFormulae,
    ResolveSamplingDistributions,
    ComputeBootstrapKurtosis,
    ValidateMomentExistence,
    ValidateKurtosisSemantics,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewObservationScopeDefinitionWeightOrDistributionEvidence
    → ValidateInput
    → IdentifyAffectedKurtosisClaims
    → UpdateKurtosisState
    → RecomputeEffectiveObservationSet
    → RecomputeMean
    → RecomputeVariance
    → RecomputeFourthCentralMoment
    → RecomputeKurtosis
    → ReevaluateTailWeightClassification
    → RevalidateConstraints
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

ObservationUpdate :=
    AddedRemovedOrCorrectedObservation
    ⇒ recompute(
        EffectiveCount,
        Mean,
        Variance,
        FourthCentralMoment,
        PearsonKurtosis,
        ExcessKurtosis
    )

DefinitionUpdate :=
    ChangedKurtosisConvention
    ⇒ recompute(
        KurtosisValue,
        ExcessKurtosis,
        TailWeightClassification,
        SemanticEquivalence
    )

DistributionUpdate :=
    ChangedProbabilityDistribution
    ⇒ reevaluate(
        MomentExistence,
        TheoreticalKurtosis,
        SamplingProperties
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        Kurtosis,
        KurtosisValidation,
        KurtosisConstraint,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    KurtosisEntityResolution,
    Variable,
    DataScope,
    VariableType,
    MeasurementScale,
    ObservationCount,
    EffectiveObservationCount,
    Mean,
    Variance,
    FourthCentralMoment,
    KurtosisDefinition,
    PearsonKurtosis,
    ExcessKurtosis,
    TailWeightClassification,
    MissingnessRule,
    WeightingRule,
    DistributionRelation,
    SamplingProperties,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA