Chi-Squared Distribution Ontology

FULL EXECUTABLE FLOW ALGEBRA — CHI-SQUARED DISTRIBUTION ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/probability-distributions/chi-squared-distribution-ontology.html

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

BEGIN_FLOW

ChiSquaredDistribution :=
    ContinuousProbabilityDistribution
    ∧ NonnegativeSupport
    ∧ DegreesOfFreedomParameter
    ∧ SumOfSquaredStandardNormals
    ∧ GammaSpecialCase

ProbabilitySpace :=
    (Ω,F,P)

ChiSquaredRandomVariable :=
    X~χ²_ν

DegreesOfFreedom :=
    ν

ParameterConstraint :=
    ν>0
    for gamma-form extension
    ∧ commonly ν∈PositiveIntegers
    for sum-of-squares construction

Support :=
    [0,∞)

StandardNormalComponents :=
    Z_1,...,Z_ν
    iid N(0,1)

SumOfSquaresConstruction :=
    X
    =
    Σ_{i=1..ν} Z_i²

IntegerDFChiSquared :=
    Z_i iid N(0,1)
    ⇒
    Σ_i Z_i²
    ~
    χ²_ν

GammaRelation :=
    χ²_ν
    ≡
    Gamma(
        shape=ν/2,
        rate=1/2
    )

GammaScaleRelation :=
    χ²_ν
    ≡
    Gamma(
        shape=ν/2,
        scale=2
    )

ChiSquaredPDF :=
    f_X(x)
    =
    1/
    [
        2^(ν/2)
        Γ(ν/2)
    ]
    x^(ν/2-1)
    e^(-x/2)
    for x>0

PDFOutsideSupport :=
    f_X(x)=0
    for x<0

PDFNonnegativity :=
    f_X(x)≥0

PDFNormalisation :=
    Integral_0^∞
    f_X(x)dx
    =
    1

ChiSquaredCDF :=
    F_X(x)
    =
    P(X≤x)

CDFGammaRelation :=
    F_X(x)
    =
    P(
        ν/2,
        x/2
    )
    where P is regularised lower incomplete gamma

SurvivalFunction :=
    S_X(x)
    =
    1-F_X(x)

UpperTailProbability :=
    P(X>x)
    =
    Q(
        ν/2,
        x/2
    )

Expectation :=
    E[X]
    =
    ν

Variance :=
    Var(X)
    =
    2ν

StandardDeviation :=
    sqrt(2ν)

Mode :=
    max(ν-2,0)
    under standard convention

Skewness :=
    sqrt(8/ν)

ExcessKurtosis :=
    12/ν

MomentGeneratingFunction :=
    M_X(t)
    =
    (1-2t)^(-ν/2)
    for t<1/2

CharacteristicFunction :=
    φ_X(t)
    =
    (1-2it)^(-ν/2)

Moment :=
    E[X^k]
    =
    2^k
    Γ(k+ν/2)
    /
    Γ(ν/2)

Additivity :=
    X~χ²_ν1
    ∧ Y~χ²_ν2
    ∧ X⟂Y
    ⇒
    X+Y
    ~
    χ²_(ν1+ν2)

FiniteAdditivity :=
    Independent(X_i~χ²_νi)
    ⇒
    Σ_iX_i
    ~
    χ²_(Σ_iν_i)

NormalSquareRelation :=
    Z~N(0,1)
    ⇒
    Z²~χ²_1

ScaledNormalSquare :=
    X~N(0,σ²)
    ⇒
    X²/σ²
    ~
    χ²_1

NormalSample :=
    X_1,...,X_n
    iid N(μ,σ²)

SampleMean :=
    X_bar
    =
    (1/n)Σ_iX_i

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

SampleVarianceDistribution :=
    (n-1)S²/σ²
    ~
    χ²_(n-1)

DegreesOfFreedomSampleVariance :=
    ν
    =
    n-1

NormalMeanVarianceIndependence :=
    X_bar
    ⟂
    S²

VarianceInference :=
    ChiSquaredDistribution
    supports
    inference_about
    normal-population variance

VarianceConfidenceInterval :=
    For normal sample:
    [
        (n-1)S²/χ²_(1-α/2,n-1),
        (n-1)S²/χ²_(α/2,n-1)
    ]
    under conventional quantile notation

VarianceHypothesisTest :=
    H0:σ²=σ0²

VarianceTestStatistic :=
    χ²
    =
    (n-1)S²/σ0²

VarianceTestDistribution :=
    Under H0
    and normality:
        χ²~χ²_(n-1)

GoodnessOfFitTest :=
    CompareObservedAndExpectedCounts

PearsonChiSquaredStatistic :=
    X²
    =
    Σ_i
    (O_i-E_i)²/E_i

GoodnessOfFitDegreesOfFreedom :=
    NumberOfCategories
    -1
    -NumberOfEstimatedParameters
    under standard regular conditions

IndependenceTest :=
    ContingencyTable
    → PearsonChiSquaredStatistic

ExpectedCellCount :=
    E_ij
    =
    RowTotal_i
    × ColumnTotal_j
    /
    GrandTotal

ContingencyStatistic :=
    X²
    =
    Σ_iΣ_j
    (O_ij-E_ij)²/E_ij

IndependenceTestDegreesOfFreedom :=
    (r-1)(c-1)

HomogeneityTest :=
    CompareCategoricalDistributions
    across populations
    using chi-squared statistic
    under test assumptions

LikelihoodRatioChiSquared :=
    G²
    =
    2Σ_i
    O_i log(O_i/E_i)
    under suitable conventions

AsymptoticChiSquaredRelation :=
    Many likelihood-ratio,
    score,
    and Wald-type statistics
    converge_to
    ChiSquaredDistribution
    under regularity conditions

WilksTheoremRelation :=
    -2log(
        RestrictedLikelihood/
        UnrestrictedLikelihood
    )
    → χ²_df
    asymptotically
    under theorem conditions

QuadraticForm :=
    Q
    =
    Z^TAZ

IdempotentQuadraticForm :=
    Z~N(0,I)
    ∧ A symmetric idempotent
    with rank r
    ⇒
    Z^TAZ
    ~
    χ²_r

MahalanobisDistanceRelation :=
    X~N_d(μ,Σ)
    with nonsingular Σ
    ⇒
    (X-μ)^TΣ^(-1)(X-μ)
    ~
    χ²_d

MultivariateNormalGeometry :=
    ConstantMahalanobisDistance
    defines
    ellipsoidal probability regions

StudentTRelation :=
    Z~N(0,1)
    ∧ V~χ²_ν
    ∧ Z⟂V
    ⇒
    T
    =
    Z/sqrt(V/ν)
    ~
    t_ν

FDistributionRelation :=
    U~χ²_ν1
    ∧ V~χ²_ν2
    ∧ U⟂V
    ⇒
    F
    =
    (U/ν1)/(V/ν2)
    ~
    F_(ν1,ν2)

ChiSquaredApproximation :=
    χ²_ν
    becomes increasingly symmetric
    as ν grows

NormalApproximation :=
    For large ν:
        χ²_ν
        approximately
        N(ν,2ν)

WilsonHilfertyTransformation :=
    If X~χ²_ν,
    then
    (X/ν)^(1/3)
    approximately normal
    for suitable ν

ChiSquaredQuantile :=
    q_p
    such_that
    P(X≤q_p)=p

CriticalValue :=
    χ²_(p,ν)

UpperTailCriticalRegion :=
    X²
    >
    χ²_(1-α,ν)
    for appropriate one-sided test

PValue :=
    TailProbability
    under null chi-squared distribution

NoncentralChiSquaredDistribution :=
    SumOfSquares
    of independent normal variables
    with nonzero means
    under standardised construction

NoncentralityParameter :=
    λ_nc
    =
    Σ_i μ_i²
    for unit-variance normal components

NoncentralChiSquared :=
    X~χ²_ν(λ_nc)

CentralChiSquaredSpecialCase :=
    λ_nc=0
    ⇒
    NoncentralChiSquared
    reduces_to
    ChiSquaredDistribution

NoncentralExpectation :=
    ν+λ_nc

NoncentralVariance :=
    2(ν+2λ_nc)

ChiSquaredEntropy :=
    DerivedFrom
    GammaEntropy
    with shape ν/2
    and scale 2

ChiSquaredConstraint :=
    ConstraintOn(
        DegreesOfFreedom,
        Support,
        PDF,
        Moments,
        Construction
    )

DegreesOfFreedomConstraint :=
    ν>0

SupportConstraint :=
    X≥0

NormalisationConstraint :=
    Integral_0^∞f_X(x)dx=1

ChiSquaredConsistency :=
    DegreesOfFreedom
    ∧ PDF
    ∧ CDF
    ∧ Moments
    ∧ Construction
    jointly_compatible

InvalidChiSquaredDistribution :=
    ν≤0
    ∨ SupportViolation
    ∨ PDFNormalisationFailure
    ∨ MomentConflict

ChiSquaredValidation :=
    ResolveDegreesOfFreedom
    → ValidateDegreesOfFreedom
    → ResolveSupport
    → ConstructPDF
    → ValidateNormalisation
    → ConstructCDF
    → ValidateMoments
    → ResolveConstructionContext

ChiSquaredDistributionState := {
    RandomVariable,
    DegreesOfFreedom,
    Support,
    PDF,
    CDF,
    SurvivalFunction,
    Mean,
    Variance,
    Construction,
    StatisticalTestContext,
    Evidence,
    Confidence,
    Provenance
}

ChiSquaredEvidence :=
    NormalSampleVariance
    ∨ CategoricalCountData
    ∨ QuadraticForm
    ∨ AsymptoticTestStatistic
    ∨ TheoreticalDerivation

ChiSquaredUpdate :=
    NewDataDegreesOfFreedomOrModelEvidence
    → UpdateState
    → RecomputePDF
    → RecomputeCDF
    → RecomputeCriticalValues
    → Revalidate

SEMANTIC_EQUIVALENCE(ChiSquaredA,ChiSquaredB) :=
    same(DegreesOfFreedom)
    ⇒
    same(CentralChiSquaredProbabilityMeasure)

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

CoreClasses := {
    ChiSquaredDistribution,
    ContinuousDistribution,
    ChiSquaredRandomVariable,
    DegreesOfFreedom,
    ChiSquaredPDF,
    ChiSquaredCDF,
    GammaDistribution,
    StandardNormal,
    SampleVariance,
    PearsonChiSquaredStatistic,
    QuadraticForm,
    StudentTDistribution,
    FDistribution,
    NoncentralChiSquaredDistribution,
    ChiSquaredConstraint
}

CoreRelations := {
    distributionOf,
    hasDegreesOfFreedom,
    hasSupport,
    hasPDF,
    hasCDF,
    hasMean,
    hasVariance,
    specialCaseOf,
    sumOfSquaresOf,
    derivedFrom,
    usedFor,
    relatedTo,
    approximatedBy,
    constrainedBy,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    ChiSquaredDistribution --distributionOf--> ChiSquaredRandomVariable
    ChiSquaredDistribution --hasDegreesOfFreedom--> DegreesOfFreedom
    ChiSquaredDistribution --hasSupport--> Support
    ChiSquaredDistribution --hasPDF--> ChiSquaredPDF
    ChiSquaredDistribution --hasCDF--> ChiSquaredCDF
    ChiSquaredDistribution --specialCaseOf--> GammaDistribution
    ChiSquaredRandomVariable --sumOfSquaresOf--> StandardNormal
    SampleVarianceStatistic --derivedFrom--> NormalSample
    PearsonChiSquaredStatistic --usedFor--> GoodnessOfFitOrIndependenceTest
    ChiSquaredDistribution --relatedTo--> StudentTDistribution
    ChiSquaredDistribution --relatedTo--> FDistribution
    ChiSquaredDistribution --tracesTo--> Provenance

InferenceRules := {
    Z_i iid N(0,1)
        ⇒
        Σ_iZ_i²~χ²_ν,

    X~χ²_ν
        ⇒
        E[X]=ν,

    X~χ²_ν
        ⇒
        Var(X)=2ν,

    Independent(
        X~χ²_ν1,
        Y~χ²_ν2
    )
        ⇒
        X+Y~χ²_(ν1+ν2),

    Xi iid N(μ,σ²)
        ⇒
        (n-1)S²/σ²~χ²_(n-1),

    χ²_ν
        ≡
        Gamma(ν/2,rate=1/2),

    ν≤0
        ⇒
        InvalidChiSquaredDistribution
}

CanonicalChiSquaredRepresentation :=
    ParseChiSquaredModel
    → ResolveRandomVariableOrStatistic
    → ResolveDegreesOfFreedom
    → ValidateDegreesOfFreedom
    → ResolveSupport
    → ConstructPDF
    → ConstructCDF
    → ComputeMoments
    → ResolveGammaNormalTOrFRelations
    → ResolveStatisticalTestContextIfPresent
    → ValidateChiSquaredDistribution

ChiSquaredDistributionGraph :=
    G=(V,E)

V := {
    ChiSquaredDistributions,
    ChiSquaredRandomVariables,
    DegreesOfFreedomNodes,
    PDFs,
    CDFs,
    Means,
    Variances,
    GammaDistributions,
    StandardNormals,
    SampleVariances,
    PearsonStatistics,
    QuadraticForms,
    StudentTDistributions,
    FDistributions,
    NoncentralChiSquaredDistributions,
    Evidence,
    ProvenanceNodes
}

E := {
    distributionOf,
    hasDegreesOfFreedom,
    hasSupport,
    hasPDF,
    hasCDF,
    hasMean,
    hasVariance,
    specialCaseOf,
    sumOfSquaresOf,
    derivedFrom,
    usedFor,
    relatedTo,
    approximatedBy,
    constrainedBy,
    equivalentTo,
    tracesTo
}

CONFIDENCE(ChiSquaredDistributionResolution) :=
    weighted_mean(
        RandomVariableOrStatisticResolution,
        DegreesOfFreedomResolution,
        SupportValidity,
        PDFConsistency,
        MomentConsistency,
        ConstructionValidity,
        StatisticalTestAssumptionQuality,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectChiSquaredDistributionIntent
    → IdentifyRandomVariableOrTestStatistic
    → ResolveDegreesOfFreedom
    → IdentifyRequestedProbabilityCriticalValueOrProperty
    → ResolveVarianceTestGoodnessOfFitOrIndependenceContext

AI_REASONING_FLOW :=
    Query
    → ParseChiSquaredProblem
    → ResolveProbabilitySpace
    → ResolveChiSquaredRandomVariableOrStatistic
    → ResolveDegreesOfFreedom
    → ValidateDegreesOfFreedom
    → ConstructChiSquaredPDF
    → ConstructOrEvaluateChiSquaredCDF
    → ComputeTailProbabilityOrCriticalValueIfRequired
    → ComputeMeanVarianceOrMomentIfRequired
    → ResolveNormalSampleVarianceRelationIfRelevant
    → ResolveGoodnessOfFitOrIndependenceTestIfRelevant
    → ResolveTOrFDistributionRelationIfRelevant
    → ValidateChiSquaredModelAndAssumptions
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseChiSquaredDistributionEntities,
    ParseTypedRelations,
    ResolveChiSquaredRandomVariables,
    ResolveDegreesOfFreedom,
    EvaluateChiSquaredPDFs,
    EvaluateChiSquaredCDFs,
    ComputeTailProbabilities,
    ComputeCriticalValues,
    ComputeMeans,
    ComputeVariances,
    ResolveGammaRelations,
    ResolveNormalSquareRelations,
    ResolveSampleVarianceDistributions,
    EvaluatePearsonChiSquaredStatistics,
    ResolveGoodnessOfFitTests,
    ResolveIndependenceTests,
    ResolveStudentTRelations,
    ResolveFDistributionRelations,
    ResolveNoncentralChiSquaredModels,
    ValidateChiSquaredDistributions,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataDegreesOfFreedomStatisticOrConstraint
    → ValidateInput
    → IdentifyAffectedChiSquaredClaims
    → UpdateChiSquaredState
    → RecomputePDF
    → RecomputeCDF
    → RecomputeTailProbabilitiesAndCriticalValues
    → RecomputeMoments
    → RevalidateTestRelations
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

DegreesOfFreedomUpdate :=
    ChangedDegreesOfFreedom
    ⇒ revalidate(ν>0)
    → recompute(
        PDF,
        CDF,
        Mean,
        Variance,
        CriticalValues
    )

StatisticUpdate :=
    ChangedObservedStatistic
    ⇒ recompute(
        TailProbability,
        PValue,
        DecisionRelevantEvidence
    )

DataUpdate :=
    NewSampleOrCountData
    ⇒ reevaluate(
        DegreesOfFreedom,
        TestStatistic,
        ModelAssumptions,
        Confidence
    )

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        ChiSquaredDistribution,
        SumOfSquaresConstruction,
        GammaRelation,
        ChiSquaredValidation,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    ChiSquaredDistributionEntityResolution,
    ProbabilitySpace,
    ChiSquaredRandomVariableOrStatistic,
    DegreesOfFreedom,
    Support,
    PDF,
    CDF,
    TailProbability,
    CriticalValue,
    Mean,
    Variance,
    GammaRelation,
    NormalSampleVarianceRelation,
    StatisticalTestContext,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA