Correlation Ontology

FLOW ALGEBRA
Correlation Ontology — Statistical Entities
Version: 1.0 | Date: 2026-10-13
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/statistical-entities/correlation-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library | https://kirethresearch.org
Status: Proposed formalisation; source-page text not independently verified.

flow "CorrelationOntology" v1.0

ENTITY_TYPES:
  StatisticalVariable: measured or observed attribute
  PairedObservation: ordered pair (x_i,y_i) from the same observational unit
  JointDistribution: probability law of (X,Y)
  Covariance: E[(X-E[X])(Y-E[Y])] when integrable
  PearsonCorrelation: dimensionless linear association coefficient
  SpearmanCorrelation: Pearson correlation of ranks, with tie-aware ranks
  KendallTau: concordance-minus-discordance rank association
  CorrelationMatrix: symmetric matrix of pairwise correlations
  SampleCorrelation: statistic calculated from paired sample observations
  PopulationCorrelation: parameter of a joint distribution

RELATIONSHIPS:
  Correlation --relates--> StatisticalVariable[2]
  Correlation --computed_from--> PairedObservation | JointDistribution
  PearsonCorrelation --normalises--> Covariance
  SampleCorrelation --estimates--> PopulationCorrelation
  SpearmanCorrelation --measures--> MonotonicAssociation
  CorrelationMatrix --contains--> PairwiseCorrelation

MATHEMATICAL_DEFINITIONS:
  population_pearson: rho(X,Y) = Cov(X,Y)/(sigma_X*sigma_Y)
  sample_pearson: r = sum((x_i-xbar)*(y_i-ybar))/sqrt(sum((x_i-xbar)^2)*sum((y_i-ybar)^2))
  covariance: Cov(X,Y) = E[XY]-E[X]*E[Y] when expectations exist
  spearman: rho_s = corr(rank(X),rank(Y)); use average ranks for ties
  kendall_tau_a: (C-D)/choose(n,2) for pairs without ties; tau_b adjusts for ties
  matrix: R_ij = corr(X_i,X_j)

PRECONDITIONS:
  PearsonCorrelation requires finite nonzero standard deviations and finite covariance.
  SampleCorrelation requires paired observations with nonzero sums of squared deviations.
  A sample size of at least 2 is necessary, but not sufficient if a variable is constant.
  Correlation matrix requires variables and a consistent treatment of missing observations.

INVARIANTS:
  -1 <= rho <= 1
  corr(X,Y) = corr(Y,X)
  corr(X,X) = 1 when variance is positive and finite
  corr(aX+b,cY+d) = sign(a*c)*corr(X,Y) for nonzero a,c
  correlation_matrix is symmetric and positive semidefinite when based on a common valid joint dataset
  correlation_matrix diagonal = 1 when all component variances are positive

INFERENCE_RULES:
  IF corr(X,Y) > 0 THEN infer positive_linear_association WITH scope=observed_population_or_sample
  IF corr(X,Y) < 0 THEN infer negative_linear_association WITH scope=observed_population_or_sample
  IF corr(X,Y) = 0 THEN infer zero_linear_correlation; DO_NOT_INFER independence
  IF X,Y independent AND finite_positive_variances THEN infer corr(X,Y)=0
  IF abs(corr(X,Y))=1 THEN infer exact_affine_relationship almost_surely (population) or on sample observations
  IF variables transformed by strictly increasing functions THEN Spearman rank correlation is invariant, subject to ranking/tie conventions

SEMANTIC_DISTINCTIONS:
  correlation != causation
  uncorrelated != independent in general
  zero_Pearson_correlation != absence_of_nonlinear_association
  Pearson_linear_association != Spearman_monotonic_association
  population_parameter != sample_estimate
  statistical_significance != effect_size

VALIDATION:
  check paired alignment and sample provenance
  check missing data policy and duplicate handling
  check variable measurement scales and nonconstant values
  check numerical bounds with floating-point tolerance
  check confidence intervals and p-values only with stated assumptions and sampling design
  flag outliers, nonlinearity, subgroup effects, range restriction, and confounding
  distinguish exploratory association from causal inference

CONFIDENCE:
  report coefficient, method, sample size, uncertainty estimate, assumptions, and limitations
  do not invent confidence values or statistical significance

SEMANTIC_EQUIVALENCE:
  Pearson's r ~= sample product-moment correlation coefficient
  rho ~= population Pearson correlation coefficient (context-dependent notation)
  rank correlation != Pearson correlation of raw values

FLOW_HASH:
  algorithm: SHA-256
  input: canonical UTF-8 serialisation of flow name, version, entity types, relationships, definitions, rules and validation
  hash_value: compute at deployment; not precomputed

UPDATE_MODEL:
  revision_policy: version changes on substantive definitions or rule updates
  provenance: retain source URL, revision timestamp, author credits and validation results

AI_CONSUMPTION:
  input: typed variables, paired data or joint distribution, requested association method
  processing: validate -> select method -> compute -> assess assumptions -> annotate uncertainty -> emit typed result
  output: coefficient, method, direction, population/sample scope, uncertainty, provenance, warnings
  safety: never infer causation solely from correlation

END FLOW ALGEBRA