Covariance Matrix Ontology

flow "CovarianceMatrixOntology" v4.0

# ============================================================
# COVARIANCE MATRIX ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)
# ============================================================

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/multivariate-statistics/covariance-matrix-ontology.html"
  title: "Covariance Matrix Ontology"
  domain: "Probability and Statistics"
  category: "Multivariate Statistics"
  entity: COVARIANCE_MATRIX
  entity_type: STATISTICAL_MATRIX

DEFINE COVARIANCE_MATRIX:
  COVARIANCE_MATRIX :=
    MATRIX(
      VARIANCES,
      PAIRWISE_COVARIANCES,
      MULTIVARIATE_DEPENDENCE_STRUCTURE
    )

CORE_OBJECT:
  RANDOM_VECTOR:
    X := [X_1, X_2, ..., X_p]^T

  MEAN_VECTOR:
    mu := E[X]

  COVARIANCE_MATRIX:
    Sigma :=
      E[
        (X - mu)
        (X - mu)^T
      ]

  ELEMENT:
    sigma_ij :=
      Cov(X_i, X_j)

MATRIX_STRUCTURE:
  Sigma :=
    [
      Var(X_1)        Cov(X_1,X_2)  ... Cov(X_1,X_p)
      Cov(X_2,X_1)    Var(X_2)      ... Cov(X_2,X_p)
      ...
      Cov(X_p,X_1)    Cov(X_p,X_2)  ... Var(X_p)
    ]

  diagonal:
    sigma_ii := Var(X_i)

  off_diagonal:
    sigma_ij := Cov(X_i, X_j)

ENTITIES:
  RANDOM_VARIABLE:
    type: STATISTICAL_VARIABLE

  RANDOM_VECTOR:
    type: VECTOR_VALUED_RANDOM_VARIABLE

  MEAN_VECTOR:
    type: LOCATION_PARAMETER

  VARIANCE:
    type: DISPERSION_MEASURE

  COVARIANCE:
    type: PAIRWISE_LINEAR_DEPENDENCE_MEASURE

  COVARIANCE_MATRIX:
    type: SYMMETRIC_MATRIX
    symbol: Sigma

  SAMPLE_COVARIANCE_MATRIX:
    type: STATISTICAL_ESTIMATOR
    symbol: S

  CORRELATION_MATRIX:
    type: STANDARDISED_DEPENDENCE_MATRIX
    symbol: R

RELATIONSHIPS:
  RANDOM_VECTOR
    -> HAS
    -> MEAN_VECTOR

  RANDOM_VECTOR
    -> HAS
    -> COVARIANCE_MATRIX

  VARIABLE
    -> HAS
    -> VARIANCE

  VARIABLE_PAIR
    -> HAS
    -> COVARIANCE

  VARIANCES
    + PAIRWISE_COVARIANCES
    -> FORM
    -> COVARIANCE_MATRIX

  COVARIANCE_MATRIX
    -> REPRESENTS
    -> LINEAR_DEPENDENCE_STRUCTURE

  COVARIANCE_MATRIX
    -> STANDARDISES_TO
    -> CORRELATION_MATRIX

COVARIANCE:
  definition:
    Cov(X_i, X_j) :=
      E[
        (X_i - mu_i)
        (X_j - mu_j)
      ]

  equivalent_form:
    Cov(X_i, X_j) :=
      E[X_i X_j]
      -
      E[X_i]E[X_j]

  self_covariance:
    Cov(X_i, X_i) :=
      Var(X_i)

PROPERTIES:
  SYMMETRY:
    Cov(X_i, X_j)
      =
    Cov(X_j, X_i)

    therefore:
      Sigma^T = Sigma

  POSITIVE_SEMIDEFINITE:
    FOR_ANY_VECTOR a:
      a^T Sigma a >= 0

  DIAGONAL_NONNEGATIVITY:
    FOR_ALL_i:
      sigma_ii >= 0

  DIMENSION:
    IF:
      RANDOM_VECTOR_DIMENSION = p
    THEN:
      Sigma:
        dimensions = p x p

  UNITS:
    diagonal:
      SQUARED_UNITS_OF_VARIABLE
    off_diagonal:
      PRODUCT_OF_VARIABLE_UNITS

LINEAR_TRANSFORMATION:
  GIVEN:
    Y := A X + b

  THEN:
    E[Y] :=
      A mu + b

    Cov(Y) :=
      A Sigma A^T

  SPECIAL_CASE:
    Y := a^T X

    Var(Y) :=
      a^T Sigma a

SUM_AND_DIFFERENCE:
  FOR_RANDOM_VECTORS:
    Cov(X + Y) :=
      Cov(X)
      + Cov(Y)
      + Cov(X,Y)
      + Cov(Y,X)

  IF:
    X AND Y ARE_INDEPENDENT

  THEN:
    Cov(X,Y) := 0

    Cov(X + Y) :=
      Cov(X)
      + Cov(Y)

SAMPLE_COVARIANCE_MATRIX:
  GIVEN:
    observations:
      x_1, x_2, ..., x_n

  sample_mean:
    x_bar :=
      (1/n)
      * SUM[i=1..n] x_i

  estimator:
    S :=
      (1/(n-1))
      * SUM[i=1..n]
        (x_i - x_bar)
        (x_i - x_bar)^T

  element:
    s_jk :=
      (1/(n-1))
      * SUM[i=1..n]
        (x_ij - x_bar_j)
        (x_ik - x_bar_k)

CORRELATION_CONNECTION:
  standard_deviation:
    sigma_i :=
      sqrt(sigma_ii)

  correlation:
    rho_ij :=
      sigma_ij
      /
      (sigma_i * sigma_j)

  matrix_relation:
    R :=
      D^(-1/2)
      Sigma
      D^(-1/2)

  where:
    D :=
      diag(
        sigma_11,
        sigma_22,
        ...,
        sigma_pp
      )

  REQUIRE:
    positive_variances_for_standardisation

INDEPENDENCE_CONNECTION:
  GENERAL_RULE:
    INDEPENDENCE(X_i, X_j)
      ->
    Cov(X_i, X_j) = 0
    WHEN:
      COVARIANCE_EXISTS

  REVERSE_RULE:
    Cov(X_i, X_j) = 0
      DOES_NOT_GENERALLY_IMPLY
    INDEPENDENCE(X_i, X_j)

  MULTIVARIATE_NORMAL_EXCEPTION:
    IF:
      X IS_JOINTLY_MULTIVARIATE_NORMAL

    THEN:
      ZERO_COVARIANCE_BETWEEN_COMPONENTS_OR_BLOCKS
        ->
      INDEPENDENCE_OF_RELEVANT_COMPONENTS_OR_BLOCKS

EIGEN_STRUCTURE:
  decomposition:
    Sigma :=
      Q Lambda Q^T

  where:
    Q:
      columns:
        ORTHONORMAL_EIGENVECTORS

    Lambda:
      diagonal:
        NONNEGATIVE_EIGENVALUES

  interpretation:
    EIGENVECTORS:
      DIRECTIONS_OF_VARIATION

    EIGENVALUES:
      VARIANCE_ALONG_EIGENVECTOR_DIRECTIONS

PRINCIPAL_COMPONENT_CONNECTION:
  PCA:
    USES:
      COVARIANCE_MATRIX
      OR:
      CORRELATION_MATRIX

  eigen_relation:
    Sigma a_k :=
      lambda_k a_k

  principal_component:
    Z_k :=
      a_k^T X

  variance:
    Var(Z_k) :=
      lambda_k

MAHALANOBIS_DISTANCE:
  IF:
    Sigma IS_INVERTIBLE

  THEN:
    D_M(x, mu) :=
      sqrt(
        (x-mu)^T
        Sigma^(-1)
        (x-mu)
      )

  purpose:
    MEASURE_DISTANCE
    WHILE_ACCOUNTING_FOR:
      VARIABLE_SCALE
      COVARIANCE

INVERSE_COVARIANCE:
  name:
    PRECISION_MATRIX

  symbol:
    Omega := Sigma^(-1)

  REQUIRE:
    Sigma:
      NONSINGULAR

  role:
    REPRESENTS:
      CONDITIONAL_DEPENDENCE_STRUCTURE
    IN:
      MULTIVARIATE_GAUSSIAN_MODELS

  gaussian_property:
    IF:
      Omega_ij = 0
      AND:
      i != j

    THEN:
      X_i
        CONDITIONALLY_INDEPENDENT_OF
      X_j
      GIVEN:
        ALL_OTHER_COMPONENTS
    UNDER:
      MULTIVARIATE_GAUSSIAN_MODEL

SINGULARITY:
  COVARIANCE_MATRIX:
    SINGULAR
    IF:
      RANK(Sigma) < p

  possible_causes:
    - EXACT_LINEAR_DEPENDENCE
    - REDUNDANT_VARIABLES
    - ZERO_VARIANCE_VARIABLE
    - HIGH_DIMENSIONAL_SAMPLE_STRUCTURE

  consequence:
    ORDINARY_MATRIX_INVERSE:
      DOES_NOT_EXIST

  responses:
    - REMOVE_REDUNDANT_VARIABLES
    - DIMENSIONALITY_REDUCTION
    - REGULARISATION
    - GENERALISED_INVERSE_WHEN_APPROPRIATE

HIGH_DIMENSIONAL_CONTEXT:
  IF:
    p >= n
    FOR:
      STANDARD_SAMPLE_COVARIANCE_ESTIMATION

  THEN:
    SAMPLE_COVARIANCE_MATRIX:
      MAY_BE_SINGULAR

  methods:
    - SHRINKAGE_COVARIANCE
    - REGULARISED_COVARIANCE
    - SPARSE_COVARIANCE_ESTIMATION
    - FACTOR_MODELS

ESTIMATION:
  POPULATION_TARGET:
    Sigma

  SAMPLE_ESTIMATOR:
    S

  ALTERNATIVE_ESTIMATORS:
    - MAXIMUM_LIKELIHOOD_ESTIMATOR
    - SHRINKAGE_ESTIMATOR
    - ROBUST_COVARIANCE_ESTIMATOR
    - BAYESIAN_COVARIANCE_ESTIMATOR

  SELECT_BASED_ON:
    - SAMPLE_SIZE
    - DIMENSION
    - DISTRIBUTIONAL_ASSUMPTIONS
    - OUTLIERS
    - REGULARISATION_REQUIREMENTS

APPLICATIONS:
  COVARIANCE_MATRIX
    -> SUPPORTS
    -> MULTIVARIATE_NORMAL_MODELS

  COVARIANCE_MATRIX
    -> SUPPORTS
    -> PRINCIPAL_COMPONENT_ANALYSIS

  COVARIANCE_MATRIX
    -> SUPPORTS
    -> FACTOR_ANALYSIS

  COVARIANCE_MATRIX
    -> SUPPORTS
    -> DISCRIMINANT_ANALYSIS

  COVARIANCE_MATRIX
    -> SUPPORTS
    -> MAHALANOBIS_DISTANCE

  COVARIANCE_MATRIX
    -> SUPPORTS
    -> PORTFOLIO_MODELLING

  COVARIANCE_MATRIX
    -> SUPPORTS
    -> SIGNAL_PROCESSING

  COVARIANCE_MATRIX
    -> SUPPORTS
    -> MACHINE_LEARNING

MODEL_CONNECTIONS:
  COVARIANCE_MATRIX
    -> CONNECTS_TO
    -> VARIANCE

  COVARIANCE_MATRIX
    -> CONNECTS_TO
    -> CORRELATION_MATRIX

  COVARIANCE_MATRIX
    -> CONNECTS_TO
    -> MULTIVARIATE_STATISTICS

  COVARIANCE_MATRIX
    -> CONNECTS_TO
    -> LINEAR_ALGEBRA

  COVARIANCE_MATRIX
    -> CONNECTS_TO
    -> EIGENVALUES

  COVARIANCE_MATRIX
    -> CONNECTS_TO
    -> EIGENVECTORS

  COVARIANCE_MATRIX
    -> CONNECTS_TO
    -> PRECISION_MATRIX

  COVARIANCE_MATRIX
    -> CONNECTS_TO
    -> MULTIVARIATE_NORMAL_DISTRIBUTION

SEMANTIC_EQUIVALENCE:
  "covariance matrix"
    ~= "variance-covariance matrix"

  "sample covariance matrix"
    ~= "matrix of sample variances and covariances"

  "inverse covariance matrix"
    ~= "precision matrix"

  EXCLUDE_EQUIVALENCE:
    COVARIANCE_MATRIX != CORRELATION_MATRIX
    COVARIANCE != CORRELATION
    COVARIANCE != CAUSATION
    ZERO_COVARIANCE != INDEPENDENCE_BY_DEFAULT
    COVARIANCE_MATRIX != PRECISION_MATRIX

VALIDATION_RULES:
  RULE_1:
    COVARIANCE_MATRIX:
      MUST_BE:
        SQUARE
        SYMMETRIC
        POSITIVE_SEMIDEFINITE

  RULE_2:
    DIAGONAL_ELEMENTS:
      MUST_BE:
        NONNEGATIVE

  RULE_3:
    MATRIX_DIMENSION:
      MUST_MATCH:
        RANDOM_VECTOR_DIMENSION

  RULE_4:
    MATRIX_INVERSION:
      MUST_NOT_BE_APPLIED
      TO:
        SINGULAR_COVARIANCE_MATRIX
      WITHOUT:
        APPROPRIATE_GENERALIZED_OR_REGULARISED_METHOD

  RULE_5:
    CORRELATION_CONVERSION:
      REQUIRES:
        POSITIVE_STANDARD_DEVIATIONS
        FOR_VARIABLES_BEING_STANDARDISED

  RULE_6:
    COVARIANCE_INTERPRETATION:
      MUST_PRESERVE:
        VARIABLE_UNITS
        SCALE_DEPENDENCE

  RULE_7:
    ZERO_OFF_DIAGONAL_ENTRY:
      MUST_NOT_BE_INTERPRETED_AS:
        GENERAL_INDEPENDENCE
      WITHOUT:
        ADDITIONAL_ASSUMPTIONS

CONFIDENCE:
  ontology_identity: 0.99
  covariance_definition: 0.99
  matrix_properties: 0.99
  linear_transformation_structure: 0.99
  correlation_relationship: 0.99
  eigenstructure_relationship: 0.99
  precision_matrix_relationship: 0.98

UPDATE_MODEL:
  IF:
    NEW_OBSERVATIONS_ARRIVE

  THEN:
    UPDATE:
      SAMPLE_MEAN_VECTOR
      SAMPLE_COVARIANCE_MATRIX

    RECOMPUTE_IF_REQUIRED:
      CORRELATION_MATRIX
      EIGENVALUES
      EIGENVECTORS
      PRECISION_MATRIX
      PCA_COMPONENTS

  IF:
    NEW_VARIABLE_ADDED

  THEN:
    EXPAND:
      RANDOM_VECTOR
      COVARIANCE_MATRIX

    ADD:
      NEW_VARIANCE
      NEW_PAIRWISE_COVARIANCES

    REVALIDATE:
      DIMENSIONS
      RANK
      POSITIVE_SEMIDEFINITENESS

  IF:
    VARIABLE_TRANSFORMED

  THEN:
    RECOMPUTE:
      MEAN
      VARIANCE
      COVARIANCES
      MATRIX_STRUCTURE

  PRESERVE:
    VARIABLE_IDENTITIES
    VARIABLE_UNITS
    OBSERVATION_PROVENANCE
    ESTIMATION_METHOD
    TRANSFORMATION_HISTORY

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_RANDOM_VECTOR
    - IDENTIFY_VARIABLE_DIMENSION
    - IDENTIFY_OR_ESTIMATE_MEAN_VECTOR
    - COMPUTE_PAIRWISE_COVARIANCES
    - CONSTRUCT_COVARIANCE_MATRIX
    - VALIDATE_SYMMETRY
    - VALIDATE_POSITIVE_SEMIDEFINITENESS
    - CHECK_MATRIX_RANK
    - DERIVE_CORRELATION_OR_PRECISION_IF_REQUIRED
    - APPLY_EIGEN_ANALYSIS_IF_REQUIRED
    - RETURN_DEPENDENCE_STRUCTURE

  constraints:
    - PRESERVE_VARIABLE_ORDER
    - PRESERVE_VARIABLE_UNITS
    - DO_NOT_INFER_CAUSATION_FROM_COVARIANCE
    - DO_NOT_INFER_GENERAL_INDEPENDENCE_FROM_ZERO_COVARIANCE
    - CHECK_RANK_BEFORE_MATRIX_INVERSION
    - EXPOSE_ESTIMATION_METHOD
    - EXPOSE_STANDARDISATION_ASSUMPTIONS

FLOW_HASH:
  canonical_entity: "covariance-matrix"
  namespace: "probability-statistics/multivariate-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - VARIANCE_STRUCTURE
    - PAIRWISE_COVARIANCE_STRUCTURE
    - MATRIX_PROPERTIES

END_FLOW