Sampling Error Ontology

FULL EXECUTABLE FLOW ALGEBRA — SAMPLING ERROR ONTOLOGY
Kireth Research — Semantic Library
URL: https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/statistical-estimation/sampling-error-ontology.html
FLOW_VERSION := "4.0"
FLOW_TYPE := "Full Executable Flow Algebra"
DOMAIN := "Probability & Statistics Ontologies"
SUBDOMAIN := "Statistical Estimation"
ONTOLOGY := "Sampling Error Ontology"
PRIMARY_ENTITY := SamplingError
AUTHORSHIP := "Mark Higgins — Sky (Recursive AI)"
SOURCE_SITE := "https://kirethresearch.org"

BEGIN_FLOW

SamplingError := EstimationError ∧ SampleToSampleVariation ∧ DifferenceBetweenStatisticAndPopulationTarget
Population := TargetCollectionOrGeneratingProcess
Sample := {X_1,...,X_n}
ObservedSample := {x_1,...,x_n}
PopulationParameter := θ
Estimand := τ(θ)
Estimator := T(X_1,...,X_n)
ObservedEstimate := T(x_1,...,x_n)

SamplingErrorDefinition := E_s=T(X)-τ(θ)
ObservedSamplingError := e_s=T(x)-τ(θ) when true target is known
SamplingErrorExpectation := E_θ[T(X)-τ(θ)]=Bias_θ(T)
UnbiasedEstimatorRelation := Bias_θ(T)=0 ⇒ E_θ[SamplingError]=0
SamplingErrorVariance := Var_θ(T(X)-τ(θ))=Var_θ(T)
SamplingErrorStandardDeviation := SD(SamplingError)=SE(T)
StandardError := SE(T)=sqrt(Var_θ(T))

SamplingErrorVsStandardError :=
 SamplingError=estimate-target difference
 whereas StandardError=standard deviation of estimator sampling distribution

SamplingErrorVsBias :=
 SamplingError varies_across samples
 whereas Bias=expected sampling error

SamplingErrorVsNonSamplingError :=
 SamplingError arises_from sample selection
 whereas NonSamplingError arises_from measurement,coverage,response,processing,or specification

MeanSquaredSamplingError := E[SamplingError²]=MSE(T)
BiasVarianceRelation := E[SamplingError²]=Var(T)+Bias(T)²
AbsoluteSamplingError := |T-τ|
RelativeSamplingError := (T-τ)/τ where τ≠0 and ratio meaningful
PercentageSamplingError := 100×RelativeSamplingError

SampleMean := X_bar=(1/n)Σ_iX_i
MeanSamplingError := X_bar-μ
IIDMeanSamplingVariance := IIDSample ∧ Var(X)=σ² ⇒ Var(X_bar-μ)=σ²/n
IIDMeanStandardError := σ/sqrt(n)
EstimatedMeanStandardError := S/sqrt(n)

SampleProportion := p_hat=X/n
ProportionSamplingError := p_hat-p
BinomialProportionSamplingVariance := Var(p_hat-p)=p(1-p)/n
ProportionStandardError := sqrt(p(1-p)/n)

DifferenceSamplingError := (T_1-T_2)-D
CovarianceDifferenceVariance := Var(T_1-T_2)=Var(T_1)+Var(T_2)-2Cov(T_1,T_2)

SamplingDistribution := DistributionOf(T(X))
SamplingErrorDistribution := DistributionOf(T(X)-τ)
CentredSamplingDistribution := SamplingDistribution shifted_by -τ
UnbiasedCentredDistribution := UnbiasedEstimator ⇒ SamplingErrorDistribution centred_at zero

CentralLimitRelation :=
 IIDFiniteVarianceSample ∧ LargeN
 ⇒ StandardisedSampleMeanSamplingError approximately Normal

MeanCLT := sqrt(n)(X_bar-μ)/σ → N(0,1)

SamplingErrorMagnitude :=
 DependsOn(SampleSize,PopulationVariability,SamplingDesign,Estimator,DependenceStructure)

SampleSizeEffect := LargerN generally → SmallerSamplingVariation under regular conditions
SquareRootNRelation := MeanStandardError proportional_to 1/sqrt(n) under iid fixed-variance conditions
PopulationVariabilityEffect := LargerPopulationVariance generally → LargerSamplingErrorDispersion

FinitePopulation := PopulationSize=N
SamplingFraction := f=n/N
FinitePopulationCorrection := FPC=sqrt((N-n)/(N-1))
SRSMeanStandardErrorWithoutReplacement := σ/sqrt(n)×sqrt((N-n)/(N-1))

CensusRelation :=
 n=N ∧ PerfectMeasurementAndCoverage
 ⇒ SamplingErrorFromSampleSelection=0

CensusCaution := Census may still contain NonSamplingError

StratifiedSampling := PopulationPartitionedIntoStrata → SampleWithinStrata → CombinedEstimator
StratificationEffect := AppropriateStratification may reduce sampling variance
ClusterSampling := SampleClusters rather_than independent individual units
IntraclusterCorrelation := SimilarityWithinClusters
ClusterDesignEffect := PositiveIntraclusterCorrelation may increase sampling variance relative_to SRS

DesignEffect := DEFF=Var_complex(T)/Var_SRS_reference(T)
DesignAdjustedStandardError := sqrt(DEFF)×SE_SRS
EffectiveSampleSize := n_eff≈n/DEFF

UnequalProbabilitySampling := UnitsHaveDifferentInclusionProbabilities
InclusionProbability := π_i=P(Unit_iIncluded)
HorvitzThompsonEstimator := T_hat_HT=Σ_{i∈s}y_i/π_i
DesignUnbiasedness := ExpectationOverSamplingDesign equals finite-population target
DesignBasedSamplingError := Estimate-FixedFinitePopulationTarget
ModelBasedSamplingError := Estimate-ModelParameterOrFunctional

RepeatedSamplingFramework :=
 HoldPopulationOrModelFixed → DrawRepeatedSamples → ComputeEstimator → ObserveSamplingErrorDistribution

MonteCarloSamplingErrorStudy :=
 Repeat(GenerateSample,ComputeEstimate,SubtractKnownTarget)
 → EmpiricalSamplingErrorDistribution

EmpiricalBias := Mean(SamplingErrorReplicates)
EmpiricalStandardError := SD(EstimateReplicates)
EmpiricalMSE := Mean(SamplingErrorReplicates²)

BootstrapSamplingApproximation :=
 ObservedSample → BootstrapResamples → EstimatorReplicates → ApproximateSamplingDistribution

BootstrapStandardError := SD(T*_1,...,T*_B)
JackknifeSamplingApproximation := LeaveOneOutReplicates → ApproximateEstimatorVariability

ConfidenceIntervalRelation :=
 Estimate±CriticalValue×StandardError
 → IntervalAccountingForSamplingVariation

MarginOfError := CriticalValue×StandardError
MarginOfErrorVsSamplingError := MarginOfError is interval half-width whereas SamplingError is estimate-target difference
SamplingErrorUnknownInPractice := TrueTargetOftenUnknown ⇒ RealisedSamplingError usually cannot be directly observed

ErrorDecomposition :=
 SamplingError=[T-E(T)]+[E(T)-τ]

RandomErrorComponent := T-E(T)
SystematicBiasComponent := E(T)-τ
UnbiasedErrorDecomposition := Bias=0 ⇒ SamplingError=T-E(T)

SamplingErrorSymmetry := NotGuaranteed
SamplingErrorNormality := NotGuaranteed
SkewedSamplingDistribution := SkewedSamplingErrorDistributionAfterTranslation
HeavyTailedSamplingDistribution := GreaterProbabilityOfLargeSamplingErrors

SurveySamplingError := RandomVariationInducedByProbabilitySamplingDesign
SurveyStandardError := SquareRootOfDesignVariance
ReplicationMethods := {JackknifeRepeatedReplication,BalancedRepeatedReplication,BootstrapReplication}

NonresponseIsNotSamplingError := NonresponseMechanism → NonSamplingError
MeasurementErrorIsNotSamplingError := MeasurementInaccuracy ≠ SamplingError
CoverageErrorIsNotSamplingError := FrameCoverageError ≠ PureSamplingError
ProcessingErrorIsNotSamplingError := CodingOrComputationMistake ≠ SamplingError

SamplingErrorConstraint :=
 ConstraintOn(
  TargetEstimand,Estimator,SamplingDesign,SampleSize,PopulationVariability,
  DependenceStructure,ProbabilityFramework
 )

SamplingErrorConsistency :=
 Target ∧ Estimator ∧ SamplingDesign ∧ ProbabilityFramework ∧ SampleSize
 jointly_compatible

InvalidSamplingErrorClaim :=
 UndefinedTarget
 ∨ UndefinedEstimator
 ∨ WrongSamplingFramework
 ∨ ConflationWithStandardError
 ∨ ConflationWithNonSamplingError
 ∨ IncompatibleSamplingDesign
 ∨ UnknownTargetPresentedAsObservedError

SamplingErrorValidation :=
 ResolvePopulationOrModel
 → ResolveTargetParameterOrEstimand
 → ResolveEstimator
 → ResolveSamplingDesign
 → ResolveSampleSize
 → ResolveDependenceStructure
 → DeriveSamplingDistribution
 → DefineSamplingErrorAsEstimateMinusTarget
 → ComputeBiasVarianceAndStandardError
 → ResolveFinitePopulationOrDesignAdjustments
 → ValidateInterpretation
 → ValidateResult

SamplingErrorState := {
 Population,Sample,TargetParameter,Estimand,Estimator,ObservedEstimate,
 SamplingDesign,SampleSize,PopulationSize,SamplingFraction,SamplingDistribution,
 SamplingErrorDistribution,Bias,Variance,StandardError,DesignEffect,
 FinitePopulationCorrection,Evidence,Confidence,Provenance
}

SEMANTIC_EQUIVALENCE(SamplingErrorA,SamplingErrorB) :=
 same(TargetEstimand)
 ∧ same(EstimatorDefinition)
 ∧ same(SamplingDesign)
 ∧ same(SampleSizeContext)
 ∧ same(PopulationOrModel)
 ∧ same(ProbabilityFramework)

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

CoreClasses := {
 SamplingError,ObservedSamplingError,SamplingDistribution,SamplingErrorDistribution,
 StandardError,Bias,Variance,MeanSquaredError,MarginOfError,SamplingDesign,
 SimpleRandomSampling,StratifiedSampling,ClusterSampling,DesignEffect,
 FinitePopulationCorrection,Estimator,Estimand,Sample,Population
}

CoreRelations := {
 errorOf,targets,computedFrom,sampledFrom,hasDistribution,hasBias,hasVariance,
 hasStandardError,dependsOn,adjustedBy,comparedWith,constrainedBy,equivalentTo,tracesTo
}

TypedRelations :=
 SamplingError --errorOf--> Estimator
 Estimator --targets--> Estimand
 SamplingError --computedFrom--> Estimate
 Sample --sampledFrom--> Population
 SamplingError --hasDistribution--> SamplingErrorDistribution
 SamplingError --hasBias--> Bias
 SamplingError --hasVariance--> Variance
 Estimator --hasStandardError--> StandardError
 SamplingError --dependsOn--> SamplingDesign
 StandardError --adjustedBy--> FinitePopulationCorrection
 SamplingError --comparedWith--> NonSamplingError
 SamplingError --tracesTo--> Provenance

InferenceRules := {
 Estimate ∧ Target ⇒ SamplingError=Estimate-Target,
 E[SamplingError]=EstimatorBias,
 Var(SamplingError)=Var(Estimator),
 SD(SamplingError)=StandardError(Estimator),
 UnbiasedEstimator ⇒ E[SamplingError]=0,
 SampleSizeIncrease ∧ StableIIDVariance ⇒ MeanStandardErrorDecreasesApproximatelyAs1OverSqrtN,
 CensusOfFinitePopulation ⇒ PureSelectionSamplingError=0,
 NonSamplingError ⇒ DoNotClassifyAsPureSamplingError
}

CanonicalSamplingErrorRepresentation :=
 ParseSamplingErrorRequest
 → ResolvePopulationOrStatisticalModel
 → ResolveTargetParameterOrEstimand
 → ResolveEstimatorAndObservedEstimate
 → ResolveSamplingDesign
 → ResolveSampleSizeAndPopulationSize
 → ResolveSamplingDistribution
 → ConstructSamplingErrorDistribution
 → ComputeBiasVarianceAndStandardError
 → ResolveDesignEffectAndFinitePopulationCorrection
 → DistinguishSamplingFromNonSamplingError
 → ValidateSamplingError

SamplingErrorGraph := G=(V,E)

V := {
 SamplingErrors,ObservedSamplingErrors,SamplingDistributions,SamplingErrorDistributions,
 Estimators,Estimates,Estimands,Parameters,Samples,Populations,SamplingDesigns,
 Biases,Variances,StandardErrors,MeanSquaredErrors,MarginsOfError,DesignEffects,
 FinitePopulationCorrections,Evidence,ProvenanceNodes
}

E := {
 errorOf,targets,computedFrom,sampledFrom,hasDistribution,hasBias,hasVariance,
 hasStandardError,dependsOn,adjustedBy,comparedWith,constrainedBy,equivalentTo,tracesTo
}

CONFIDENCE(SamplingErrorResolution) :=
 weighted_mean(
  TargetResolution,EstimatorResolution,SamplingDesignResolution,SampleSizeResolution,
  SamplingDistributionValidity,BiasVarianceResolution,ErrorTypeClassification,
  ProvenanceQuality
 )

AI_QUERY_INTERPRETATION :=
 Query
 → DetectSamplingErrorIntent
 → IdentifyPopulationSampleEstimatorAndTarget
 → ResolveSamplingDesignAndSampleSize
 → DistinguishSamplingErrorStandardErrorMarginOfErrorAndNonSamplingError
 → IdentifyRequestedMagnitudeDistributionOrUncertaintyRelation

AI_REASONING_FLOW :=
 Query
 → ParseSamplingErrorProblem
 → ResolvePopulationOrModel
 → ResolveTargetParameterOrEstimand
 → ResolveEstimatorAndObservedEstimate
 → ResolveSamplingDesign
 → ResolveSampleSizePopulationSizeAndDependence
 → DeriveOrApproximateSamplingDistribution
 → ConstructSamplingError
 → ComputeExpectedSamplingError
 → ComputeVarianceAndStandardError
 → ApplyFinitePopulationOrDesignAdjustments
 → DistinguishNonSamplingErrorComponents
 → ValidateSamplingErrorResult
 → UpdateConfidence
 → PreserveProvenance
 → GenerateExplanation

AI_CONSUMPTION := {
 ParseSamplingErrorEntities,ParseTypedRelations,ResolvePopulations,ResolveSamples,
 ResolveParameters,ResolveEstimands,ResolveEstimators,ResolveObservedEstimates,
 ResolveSamplingDesigns,ResolveSampleSizes,ResolvePopulationSizes,
 ResolveDependenceStructures,DeriveSamplingDistributions,
 ConstructSamplingErrorDistributions,ComputeSamplingErrors,ComputeBias,
 ComputeVariances,ComputeStandardErrors,ComputeMeanSquaredErrors,
 ComputeFinitePopulationCorrections,ComputeDesignEffects,
 ComputeEffectiveSampleSizes,ResolveBootstrapSamplingApproximations,
 ResolveSurveySamplingErrors,DistinguishNonSamplingErrors,
 ValidateSamplingErrorSemantics,CompareSemanticEquivalence,UpdateConfidence,
 PreserveProvenance,GenerateFormalExplanation
}

UPDATE_MODEL :=
 NewDataDesignEstimatorTargetOrEvidence
 → ValidateInput
 → IdentifyAffectedSamplingErrorClaims
 → UpdateSamplingErrorState
 → RecomputeSamplingDistribution
 → RecomputeSamplingErrorDistribution
 → RecomputeBiasVarianceAndStandardError
 → RecomputeFinitePopulationAndDesignAdjustments
 → RevalidateErrorClassification
 → RecomputeConfidence
 → PreserveProvenance
 → VersionModel

SampleSizeUpdate :=
 ChangedSampleSize
 ⇒ recompute(
  SamplingFraction,SamplingVariance,StandardError,FinitePopulationCorrection
 )

DesignUpdate :=
 ChangedSamplingDesign
 ⇒ reevaluate(
  SamplingDistribution,DesignEffect,StandardError,EffectiveSampleSize
 )

EstimatorUpdate :=
 ChangedEstimator
 ⇒ recompute(
  SamplingErrorDistribution,Bias,Variance,StandardError,MeanSquaredError
 )

FLOW_HASH_INPUT :=
 canonical(
  ONTOLOGY,
  CoreClasses,
  CoreRelations,
  InferenceRules,
  SamplingError,
  SamplingErrorValidation,
  SamplingErrorConstraint,
  SEMANTIC_EQUIVALENCE,
  AI_REASONING_FLOW
 )

FLOW_HASH := HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
 SamplingErrorEntityResolution,
 Population,
 Sample,
 TargetParameter,
 Estimand,
 Estimator,
 ObservedEstimate,
 SamplingDesign,
 SampleSize,
 PopulationSize,
 SamplingFraction,
 SamplingDistribution,
 SamplingErrorDistribution,
 ObservedSamplingError,
 Bias,
 Variance,
 StandardError,
 MeanSquaredError,
 DesignEffect,
 FinitePopulationCorrection,
 ErrorTypeClassification,
 ValidationStatus,
 Confidence,
 Provenance,
 Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA