Descriptive Statistics

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

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

BEGIN_FLOW

DescriptiveStatistics :=
    ObservedData
    ∧ DataOrganisation
    ∧ DataSummarisation
    ∧ DistributionDescription
    ∧ CentralTendency
    ∧ Dispersion
    ∧ Position
    ∧ Shape
    ∧ Association
    ∧ VisualRepresentation

Dataset := StatisticalObservations ∧ Variables ∧ StatisticalUnits ∧ Metadata
Observation := RecordedValue associated_with StatisticalUnit and Variable
Variable := MeasurableOrRecordedCharacteristic

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

AnalysisScope := {Univariate,Bivariate,Multivariate,Grouped,Temporal}

Frequency := CountOfOccurrence
AbsoluteFrequency := n_i
RelativeFrequency := f_i=n_i/n
PercentageFrequency := 100×f_i
FrequencyDistribution := ValueOrCategory → Frequency
CumulativeFrequency := RunningSumOfFrequencies
CumulativeRelativeFrequency := RunningSumOfRelativeFrequencies

FrequencyTable := {
    ValuesOrCategories,
    Frequencies,
    RelativeFrequencies,
    CumulativeFrequencies
}

CentralTendency := {
    ArithmeticMean,
    WeightedMean,
    Median,
    Mode,
    GeometricMean,
    HarmonicMean
}

ArithmeticMean := X_bar=(1/n)Σ_iX_i
WeightedMean := X_bar_w=Σ_iw_iX_i/Σ_iw_i
Median := CentralOrderedValue under selected convention
Mode := MostFrequentValueOrCategory
GeometricMean := (Π_iX_i)^(1/n) for positive values
HarmonicMean := n/Σ_i(1/X_i) where values nonzero

CentralTendencySelection :=
    VariableType
    ∧ DistributionShape
    ∧ AnalyticalPurpose
    → AppropriateCentre

Dispersion := {
    Range,
    InterquartileRange,
    Variance,
    StandardDeviation,
    MeanAbsoluteDeviation,
    MedianAbsoluteDeviation,
    CoefficientOfVariation
}

Minimum := min_iX_i
Maximum := max_iX_i
Range := Maximum-Minimum
Quartiles := {Q1,Q2,Q3}
Q2 := Median
InterquartileRange := IQR=Q3-Q1

PopulationVariance := σ²=(1/N)Σ_i(X_i-μ)²
SampleVariance := S²=[1/(n-1)]Σ_i(X_i-X_bar)²
PopulationStandardDeviation := σ=sqrt(σ²)
SampleStandardDeviation := S=sqrt(S²)
MedianAbsoluteDeviation := MAD=Median(|X_i-Median(X)|)

CoefficientOfVariation :=
    CV=StandardDeviation/Mean
    where ratio interpretation is meaningful and Mean≠0

PositionMeasures := {Rank,Quantile,Quartile,Decile,Percentile,ZScore}
Rank := OrderedPositionOfObservation
Quantile := ValueCorrespondingToSpecifiedCumulativeProportion
Percentile := Quantile expressed_on percentage scale
ZScore := z=(x-Mean)/StandardDeviation where StandardDeviation>0

FiveNumberSummary := {Minimum,Q1,Median,Q3,Maximum}

DistributionShape := {
    Symmetry,
    Skewness,
    Kurtosis,
    Modality,
    TailBehaviour
}

Skewness := MeasureOfDistributionAsymmetry
PositiveSkew := ExtendedRightTail
NegativeSkew := ExtendedLeftTail
Kurtosis := StandardisedFourthMoment when finite
ExcessKurtosis := Kurtosis-3 under normal-reference convention
Modality := {Unimodal,Bimodal,Multimodal}

EmpiricalDistribution := ObservedData → EmpiricalProbabilityMeasure
EmpiricalProbability := Count(ValueOrCondition)/n
EmpiricalCDF := F_n(x)=(1/n)Σ_i1_(X_i≤x)

Outlier := Observation unusually extreme under specified criterion
IQRLowerFence := Q1-1.5IQR
IQRUpperFence := Q3+1.5IQR
PotentialIQROutlier := X<IQRLowerFence ∨ X>IQRUpperFence
OutlierDoesNotImplyError := Outlier ↛ InvalidObservation

Covariance :=
    S_XY=[1/(n-1)]Σ_i(X_i-X_bar)(Y_i-Y_bar)

Correlation :=
    r=S_XY/(S_XS_Y)
    where S_X,S_Y>0

CorrelationBounds := -1≤r≤1
CorrelationDoesNotImplyCausation := Correlation ↛ Causation

ContingencyTable := CrossClassificationOfCategoricalVariables
CellFrequency := n_ij
RowProportion := n_ij/RowTotal_i
ColumnProportion := n_ij/ColumnTotal_j
JointProportion := n_ij/GrandTotal
ConditionalDistribution := DistributionOfVariableWithinSpecifiedCondition

GroupedStatistics :=
    GroupBy(GroupingVariable)
    → ComputeDescriptiveStatistics

GroupMean := MeanWithinGroup
GroupMedian := MedianWithinGroup
GroupVariance := VarianceWithinGroup

Histogram := QuantitativeData → Bins → FrequencyOrDensityBars
BarChart := CategoricalData → CategoryBars
BoxPlot := FiveNumberSummary ∧ Whiskers ∧ PotentialOutliers
ScatterPlot := PairedNumericalObservations → CoordinatePoints
LineChart := OrderedOrTemporalObservations → ConnectedValues

VisualisationSelection :=
    VariableType
    ∧ AnalysisScope
    ∧ DescriptiveGoal
    → CompatibleVisualisation

TimeSeries := Observations indexed_by time
Trend := LongTermDirectionalPattern
Seasonality := RepeatingTemporalPattern
MovingAverage := Mean over moving window

MissingValue := IntendedObservationUnavailable
MissingCount := Count(MissingValues)
MissingRate := MissingCount/ExpectedObservationCount
CompleteCaseCount := Count(RecordsWithoutSelectedMissingValues)

DataQualitySummary := {
    Missingness,
    Duplicates,
    InvalidValues,
    RangeViolations,
    CategoryConsistency
}

Transformation := Y=g(X)
LogTransformation := Y=log(X) where X>0
SquareRootTransformation := Y=sqrt(X) where X≥0
Standardisation := Z=(X-X_bar)/S where S>0
MinMaxScaling := Y=(X-Min)/(Max-Min) where Max>Min
RobustScaling := Y=(X-Median)/IQR where IQR>0

WeightedData := Observations associated_with weights
WeightedFrequency := ΣWeights within category_or_value
WeightedMean := Σ_iw_iX_i/Σ_iw_i

SummaryStatistic :=
    Function mapping Dataset
    to concise descriptive quantity

CategoricalSummary := Frequency ∧ Proportion ∧ Mode
OrdinalSummary := Frequency ∧ Median ∧ Quantiles
QuantitativeSummary := Centre ∧ Dispersion ∧ Position ∧ Shape
RobustQuantitativeSummary := Median ∧ IQR ∧ MAD
MomentBasedSummary := Mean ∧ Variance ∧ Skewness ∧ Kurtosis

DescriptiveProfile :=
    Variable
    → {
        Type,
        Count,
        Missingness,
        Frequencies,
        Centre,
        Dispersion,
        Position,
        Shape
    }

DatasetProfile :=
    Dataset
    → SetOf(DescriptiveProfiles)
    ∧ CrossVariableSummaries

DescriptiveConstraint :=
    ConstraintOn(
        VariableType,
        MeasurementScale,
        Missingness,
        Weighting,
        SummaryDefinition,
        DataScope
    )

SummaryCompatibility :=
    Statistic
    compatible_with
    VariableType
    ∧ MeasurementScale

InvalidDescriptiveSummary :=
    IncompatibleVariableType
    ∨ UndefinedStatistic
    ∨ InvalidWeighting
    ∨ HiddenMissingDataTreatment
    ∨ MisleadingAggregation

DescriptiveValidation :=
    ResolveDataset
    → ResolveVariables
    → ClassifyVariableTypes
    → ResolveMeasurementScales
    → ResolveMissingness
    → ResolveWeights
    → SelectCompatibleSummaries
    → ComputeSummaries
    → ValidateCalculations
    → ValidateInterpretation

DescriptiveStatisticsState := {
    Dataset,
    Variables,
    VariableTypes,
    MeasurementScales,
    Frequencies,
    CentralTendency,
    Dispersion,
    Position,
    Shape,
    Association,
    Missingness,
    Visualisations,
    Evidence,
    Confidence,
    Provenance
}

SEMANTIC_EQUIVALENCE(DescriptiveSummaryA,DescriptiveSummaryB) :=
    same(DataScope)
    ∧ same(VariableSemantics)
    ∧ same(SummaryDefinitions)
    ∧ same(MissingnessRules)
    ∧ same(WeightingRules)

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

CoreClasses := {
    DescriptiveStatistics,
    Dataset,
    StatisticalObservation,
    StatisticalVariable,
    FrequencyDistribution,
    CentralTendency,
    Dispersion,
    Quantile,
    DistributionShape,
    EmpiricalDistribution,
    Covariance,
    Correlation,
    ContingencyTable,
    GroupedStatistics,
    DescriptiveVisualisation,
    DescriptiveProfile
}

CoreRelations := {
    describes,
    computedFrom,
    summarises,
    hasFrequency,
    hasCentre,
    hasDispersion,
    hasPosition,
    hasShape,
    associatedWith,
    groupedBy,
    visualisedBy,
    transformedBy,
    constrainedBy,
    equivalentTo,
    tracesTo
}

TypedRelations :=
    DescriptiveStatistic --computedFrom--> Dataset
    DescriptiveStatistic --describes--> StatisticalVariable
    FrequencyDistribution --hasFrequency--> Frequency
    VariableSummary --hasCentre--> CentralTendency
    VariableSummary --hasDispersion--> Dispersion
    VariableSummary --hasPosition--> Quantile
    VariableSummary --hasShape--> DistributionShape
    StatisticalVariable --associatedWith--> StatisticalVariable
    GroupedStatistics --groupedBy--> GroupingVariable
    Dataset --visualisedBy--> DescriptiveVisualisation
    StatisticalVariable --transformedBy--> Transformation
    DescriptiveResult --tracesTo--> Provenance

InferenceRules := {
    QuantitativeVariable(X)
        ⇒ EligibleFor(Mean,Median,Variance,StandardDeviation,Quantiles),

    NominalVariable(X)
        ⇒ EligibleFor(Frequency,Proportion,Mode),

    OrdinalVariable(X)
        ⇒ EligibleFor(Frequency,Median,Quantiles),

    IQR=Q3-Q1,

    SampleVariance
        ⇒ SampleStandardDeviation=sqrt(SampleVariance),

    Correlation(X,Y)
        ⇒ -1≤r≤1,

    Correlation
        ↛ Causation,

    Outlier
        ↛ InvalidObservation,

    InvalidSummaryVariablePair
        ⇒ FlagDescriptiveError
}

CanonicalDescriptiveRepresentation :=
    ParseDataset
    → ResolveVariables
    → ClassifyVariableTypes
    → ResolveMeasurementScales
    → ResolveDataScope
    → ResolveMissingnessAndWeights
    → ComputeFrequencyStructure
    → ComputeCentreAndDispersion
    → ComputePositionAndShape
    → ComputeAssociations
    → GenerateVisualisationSpecification
    → ValidateDescriptiveStatistics

DescriptiveStatisticsGraph :=
    G=(V,E)

V := {
    Datasets,
    Observations,
    Variables,
    VariableTypes,
    MeasurementScales,
    Frequencies,
    Means,
    Medians,
    Modes,
    Variances,
    StandardDeviations,
    Quantiles,
    SkewnessMeasures,
    KurtosisMeasures,
    Correlations,
    ContingencyTables,
    EmpiricalDistributions,
    GroupedSummaries,
    Visualisations,
    Evidence,
    ProvenanceNodes
}

E := {
    describes,
    computedFrom,
    summarises,
    hasFrequency,
    hasCentre,
    hasDispersion,
    hasPosition,
    hasShape,
    associatedWith,
    groupedBy,
    visualisedBy,
    transformedBy,
    constrainedBy,
    equivalentTo,
    tracesTo
}

CONFIDENCE(DescriptiveStatisticsResolution) :=
    weighted_mean(
        DatasetResolution,
        VariableResolution,
        VariableTypeClassification,
        MeasurementScaleResolution,
        MissingnessHandling,
        WeightingValidity,
        CalculationValidity,
        ProvenanceQuality
    )

AI_QUERY_INTERPRETATION :=
    Query
    → DetectDescriptiveStatisticsIntent
    → IdentifyDatasetAndVariables
    → ClassifyVariableTypesAndScales
    → IdentifyRequestedSummaryAssociationOrVisualisation
    → ResolveGroupingWeightingAndMissingnessRules

AI_REASONING_FLOW :=
    Query
    → ParseDescriptiveStatisticsProblem
    → ResolveDataset
    → ResolveVariables
    → ClassifyVariableTypes
    → ResolveMeasurementScales
    → ResolveMissingValues
    → ResolveWeightsIfPresent
    → ComputeFrequencyDistributions
    → ComputeCentralTendency
    → ComputeDispersion
    → ComputeQuantilesAndPosition
    → ComputeShapeMeasuresIfValid
    → ComputeAssociationsIfRequested
    → ComputeGroupedSummariesIfRequested
    → SpecifyVisualRepresentationIfRequested
    → ValidateSummaryCompatibility
    → UpdateConfidence
    → PreserveProvenance
    → GenerateExplanation

AI_CONSUMPTION := {
    ParseDescriptiveStatisticsEntities,
    ParseTypedRelations,
    ResolveDatasets,
    ResolveVariables,
    ClassifyVariableTypes,
    ResolveMeasurementScales,
    ComputeFrequencies,
    ComputeRelativeFrequencies,
    ComputeMeans,
    ComputeWeightedMeans,
    ComputeMedians,
    ComputeModes,
    ComputeRanges,
    ComputeVariances,
    ComputeStandardDeviations,
    ComputeIQRs,
    ComputeMADs,
    ComputeQuantiles,
    ComputeZScores,
    ComputeSkewness,
    ComputeKurtosis,
    ComputeCovariance,
    ComputeCorrelation,
    BuildContingencyTables,
    ComputeGroupedSummaries,
    EvaluateMissingness,
    SpecifyDescriptiveVisualisations,
    ValidateDescriptiveStatistics,
    CompareSemanticEquivalence,
    UpdateConfidence,
    PreserveProvenance,
    GenerateFormalExplanation
}

UPDATE_MODEL :=
    NewDataVariableWeightOrSummaryDefinition
    → ValidateInput
    → IdentifyAffectedDescriptiveClaims
    → UpdateDescriptiveState
    → RecomputeFrequencies
    → RecomputeCentreAndDispersion
    → RecomputePositionAndShape
    → RecomputeAssociations
    → RecomputeGroupedSummaries
    → RevalidateVisualisations
    → RecomputeConfidence
    → PreserveProvenance
    → VersionModel

FLOW_HASH_INPUT :=
    canonical(
        ONTOLOGY,
        CoreClasses,
        CoreRelations,
        InferenceRules,
        DescriptiveStatistics,
        DescriptiveValidation,
        DescriptiveProfile,
        SEMANTIC_EQUIVALENCE,
        AI_REASONING_FLOW
    )

FLOW_HASH :=
    HASH(FLOW_HASH_INPUT)

OUTPUT_MODEL := {
    DescriptiveStatisticsEntityResolution,
    Dataset,
    Variables,
    VariableTypes,
    MeasurementScales,
    Frequencies,
    CentralTendency,
    Dispersion,
    PositionMeasures,
    DistributionShape,
    Associations,
    MissingnessSummary,
    GroupedSummaries,
    VisualisationSpecification,
    ValidationStatus,
    Confidence,
    Provenance,
    Explanation
}

END_FLOW

END OF FULL EXECUTABLE FLOW ALGEBRA