Multivariate Statistics Ontology

flow "MultivariateStatisticsOntology" v4.0

# ============================================================
# MULTIVARIATE STATISTICS 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/multivariate-statistics-ontology.html"
  title: "Multivariate Statistics Ontology"
  domain: "Probability and Statistics"
  category: "Multivariate Statistics"
  entity: MULTIVARIATE_STATISTICS
  entity_type: STATISTICAL_ONTOLOGY

DEFINE MULTIVARIATE_STATISTICS:
  MULTIVARIATE_STATISTICS :=
    STATISTICAL_FRAMEWORK(
      MULTIPLE_VARIABLES,
      JOINT_DISTRIBUTIONS,
      DEPENDENCE_STRUCTURES,
      MATRIX_REPRESENTATIONS,
      MULTIVARIATE_INFERENCE
    )

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

  OBSERVATION_VECTOR:
    x_i := [x_i1, x_i2, ..., x_ip]^T

  DATA_MATRIX:
    symbol: X_data
    dimensions: n x p
    rows: OBSERVATION_VECTORS
    columns: VARIABLES

  MEAN_VECTOR:
    mu := E[X]

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

ENTITIES:
  VARIABLE:
    type: STATISTICAL_VARIABLE

  RANDOM_VECTOR:
    type: VECTOR_VALUED_RANDOM_VARIABLE

  OBSERVATION_VECTOR:
    type: DATA_VECTOR

  DATA_MATRIX:
    type: MATRIX

  JOINT_DISTRIBUTION:
    type: MULTIVARIATE_PROBABILITY_DISTRIBUTION

  MARGINAL_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  CONDITIONAL_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  MEAN_VECTOR:
    type: LOCATION_PARAMETER

  COVARIANCE_MATRIX:
    type: DEPENDENCE_MATRIX

  CORRELATION_MATRIX:
    type: STANDARDISED_DEPENDENCE_MATRIX

  EIGENVALUE:
    type: MATRIX_SPECTRAL_QUANTITY

  EIGENVECTOR:
    type: MATRIX_SPECTRAL_DIRECTION

RELATIONSHIPS:
  MULTIVARIATE_STATISTICS
    -> ANALYSES
    -> MULTIPLE_VARIABLES

  MULTIPLE_VARIABLES
    -> FORM
    -> RANDOM_VECTOR

  OBSERVATION_VECTORS
    -> FORM
    -> DATA_MATRIX

  RANDOM_VECTOR
    -> HAS
    -> JOINT_DISTRIBUTION

  RANDOM_VECTOR
    -> HAS
    -> MEAN_VECTOR

  RANDOM_VECTOR
    -> HAS
    -> COVARIANCE_MATRIX

  JOINT_DISTRIBUTION
    -> DEFINES
    -> MARGINAL_DISTRIBUTIONS

  JOINT_DISTRIBUTION
    -> DEFINES
    -> CONDITIONAL_DISTRIBUTIONS

  COVARIANCE_MATRIX
    -> STANDARDISES_TO
    -> CORRELATION_MATRIX

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

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

  SAMPLE_CORRELATION:
    r_jk :=
      s_jk / sqrt(s_jj * s_kk)

COVARIANCE_RULES:
  ELEMENT:
    sigma_jk := Cov(X_j, X_k)

  DIAGONAL:
    sigma_jj := Var(X_j)

  REQUIRE:
    COVARIANCE_MATRIX:
      SYMMETRIC = TRUE
      POSITIVE_SEMIDEFINITE = TRUE

DEPENDENCE_RULES:
  INDEPENDENCE:
    IMPLIES:
      ZERO_COVARIANCE
    WHEN:
      REQUIRED_MOMENTS_EXIST

  ZERO_COVARIANCE:
    DOES_NOT_GENERALLY_IMPLY:
      INDEPENDENCE

  UNDER_JOINT_MULTIVARIATE_NORMALITY:
    ZERO_COVARIANCE:
      IMPLIES:
        INDEPENDENCE
      FOR_RELEVANT_COMPONENTS_OR_BLOCKS

  CORRELATION:
    MEASURES:
      LINEAR_ASSOCIATION
    DOES_NOT_BY_ITSELF_ESTABLISH:
      CAUSATION

JOINT_DISTRIBUTION_STRUCTURE:
  JOINT_DENSITY_OR_MASS:
    f_X(x_1, x_2, ..., x_p)

  MARGINALISATION:
    JOINT_DISTRIBUTION
      -> SUM_OR_INTEGRATE_OVER
      -> OTHER_VARIABLES
      -> MARGINAL_DISTRIBUTION

  CONDITIONING:
    JOINT_DISTRIBUTION
      + OBSERVED_COMPONENTS
      -> CONDITIONAL_DISTRIBUTION

MULTIVARIATE_NORMAL:
  notation:
    X ~ N_p(mu, Sigma)

  parameters:
    mu := MEAN_VECTOR
    Sigma := COVARIANCE_MATRIX

  NONDEGENERATE_DENSITY:
    f(x) :=
      [1 / ((2*pi)^(p/2) * |Sigma|^(1/2))]
      *
      exp(
        -0.5
        * (x-mu)^T
        * Sigma^(-1)
        * (x-mu)
      )

  REQUIRE_FOR_NONDEGENERATE_DENSITY:
    Sigma:
      POSITIVE_DEFINITE

MAHALANOBIS_DISTANCE:
  definition:
    D_M(x, mu) :=
      sqrt(
        (x-mu)^T
        * Sigma^(-1)
        * (x-mu)
      )

  role:
    COVARIANCE_ADJUSTED_DISTANCE

PRINCIPAL_COMPONENT_ANALYSIS:
  alias: PCA
  type: DIMENSIONALITY_REDUCTION

  PROCESS:
    CENTER_OR_STANDARDISE(DATA)
      -> COMPUTE(COVARIANCE_OR_CORRELATION_MATRIX)
      -> EIGEN_DECOMPOSITION
      -> ORDER_BY_EIGENVALUE
      -> PROJECT_DATA

  eigen_relation:
    Sigma * a_k = lambda_k * a_k

  component:
    Z_k := a_k^T X

  variance:
    Var(Z_k) := lambda_k

  purpose:
    - REDUCE_DIMENSION
    - IDENTIFY_DOMINANT_VARIATION
    - REPRESENT_CORRELATED_VARIABLES_COMPACTLY

FACTOR_ANALYSIS:
  type: LATENT_VARIABLE_MODEL

  model:
    X := mu + Lambda*F + epsilon

  entities:
    Lambda := FACTOR_LOADING_MATRIX
    F := LATENT_FACTOR_VECTOR
    epsilon := UNIQUE_ERROR_VECTOR

  purpose:
    EXPLAIN_OBSERVED_COVARIANCE
    USING:
      LATENT_FACTORS

  distinction:
    FACTOR_ANALYSIS != PCA

MULTIVARIATE_REGRESSION:
  model:
    Y := X*B + E

  entities:
    Y := MULTIVARIATE_RESPONSE_MATRIX
    X := DESIGN_MATRIX
    B := COEFFICIENT_MATRIX
    E := ERROR_MATRIX

  purpose:
    MODEL_MULTIPLE_RESPONSES
    CONDITIONAL_ON:
      PREDICTORS

MANOVA:
  full_name:
    MULTIVARIATE_ANALYSIS_OF_VARIANCE

  purpose:
    TEST_GROUP_DIFFERENCES
    ACROSS:
      MULTIPLE_DEPENDENT_VARIABLES

  compares:
    BETWEEN_GROUP_VARIATION
    WITH:
      WITHIN_GROUP_VARIATION

DISCRIMINANT_ANALYSIS:
  purpose:
    CLASSIFY_OBSERVATIONS
    USING:
      MULTIVARIATE_FEATURES

  LDA:
    full_name: LINEAR_DISCRIMINANT_ANALYSIS
    typical_assumptions:
      - CLASS_CONDITIONAL_MULTIVARIATE_NORMALITY
      - COMMON_COVARIANCE_MATRIX

  QDA:
    full_name: QUADRATIC_DISCRIMINANT_ANALYSIS
    permits:
      CLASS_SPECIFIC_COVARIANCE_MATRICES

CANONICAL_CORRELATION:
  purpose:
    FIND_LINEAR_COMBINATIONS
    OF:
      TWO_VARIABLE_SETS
    THAT:
      MAXIMISE_CORRELATION

CLUSTER_ANALYSIS:
  type: UNSUPERVISED_MULTIVARIATE_ANALYSIS

  purpose:
    GROUP_OBSERVATIONS
    USING:
      MULTIVARIATE_SIMILARITY

  methods:
    - HIERARCHICAL_CLUSTERING
    - K_MEANS
    - MODEL_BASED_CLUSTERING

  depends_on:
    - VARIABLE_SCALE
    - DISTANCE_OR_SIMILARITY
    - ALGORITHM
    - MODEL_ASSUMPTIONS

HIGH_DIMENSIONAL_CONTEXT:
  IF:
    p IS_LARGE_RELATIVE_TO n

  THEN:
    POSSIBLE_PROBLEMS:
      - SINGULAR_COVARIANCE_MATRIX
      - OVERFITTING
      - UNSTABLE_ESTIMATION
      - HIGH_VARIANCE

    POSSIBLE_METHODS:
      - REGULARISATION
      - SHRINKAGE
      - SPARSE_MODELLING
      - DIMENSIONALITY_REDUCTION

STATISTICAL_INFERENCE:
  ESTIMATE:
    - MEAN_VECTOR
    - COVARIANCE_MATRIX
    - CORRELATION_MATRIX
    - MODEL_PARAMETERS
    - LATENT_STRUCTURE

  TEST:
    - MULTIVARIATE_MEAN_HYPOTHESES
    - GROUP_DIFFERENCES
    - COVARIANCE_STRUCTURE
    - DEPENDENCE_STRUCTURE

  PREDICT:
    - MULTIVARIATE_RESPONSES
    - CLASS_MEMBERSHIP
    - FUTURE_OBSERVATIONS

DATA_PREPARATION:
  CHECK:
    - MISSING_VALUES
    - OUTLIERS
    - VARIABLE_TYPES
    - VARIABLE_SCALE
    - DISTRIBUTIONAL_SHAPE
    - COLLINEARITY
    - SAMPLE_SIZE
    - MATRIX_RANK

  OPTIONAL:
    - CENTERING
    - STANDARDISATION
    - TRANSFORMATION
    - IMPUTATION

MODEL_CONNECTIONS:
  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> PROBABILITY_THEORY

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> LINEAR_ALGEBRA

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> MATRIX_ALGEBRA

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> REGRESSION_ANALYSIS

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> DIMENSIONALITY_REDUCTION

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> MACHINE_LEARNING

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> BAYESIAN_STATISTICS

APPLICATIONS:
  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> SCIENTIFIC_DATA_ANALYSIS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> BIOSTATISTICS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ECONOMETRICS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ENVIRONMENTAL_ANALYSIS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> PATTERN_RECOGNITION

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ARTIFICIAL_INTELLIGENCE

SEMANTIC_EQUIVALENCE:
  "multivariate statistics"
    ~= "joint statistical analysis of multiple variables"

  "random vector"
    ~= "vector-valued random variable"

  "variance-covariance matrix"
    ~= "covariance matrix"

  "PCA"
    ~= "principal component analysis"

  "MANOVA"
    ~= "multivariate analysis of variance"

  EXCLUDE_EQUIVALENCE:
    MULTIVARIATE_ANALYSIS != COLLECTION_OF_UNRELATED_UNIVARIATE_ANALYSES
    COVARIANCE != CAUSATION
    CORRELATION != CAUSATION
    ZERO_CORRELATION != INDEPENDENCE_BY_DEFAULT
    PCA != FACTOR_ANALYSIS

VALIDATION_RULES:
  RULE_1:
    DATA_MATRIX:
      MUST_HAVE_CONSISTENT_DIMENSIONS

  RULE_2:
    COVARIANCE_MATRIX:
      MUST_BE:
        SYMMETRIC
        POSITIVE_SEMIDEFINITE

  RULE_3:
    MATRIX_INVERSION:
      REQUIRES:
        APPROPRIATE_RANK
        AND
        NONSINGULARITY
      OTHERWISE:
        USE_APPROPRIATE_REGULARISED_OR_GENERALIZED_METHOD

  RULE_4:
    MODEL_ASSUMPTIONS:
      MUST_BE_CHECKED
      BEFORE:
        PARAMETRIC_INFERENCE

  RULE_5:
    VARIABLE_SCALING:
      MUST_BE_CONSIDERED
      FOR:
        DISTANCE_BASED_METHODS
        VARIANCE_BASED_METHODS

  RULE_6:
    SAMPLE_SIZE:
      MUST_BE_EVALUATED
      RELATIVE_TO:
        VARIABLE_DIMENSION
        MODEL_COMPLEXITY

  RULE_7:
    DEPENDENCE_STRUCTURE:
      MUST_BE_PRESERVED
      DURING:
        JOINT_ANALYSIS

CONFIDENCE:
  ontology_identity: 0.99
  multivariate_definition: 0.99
  covariance_structure: 0.99
  multivariate_normal_structure: 0.99
  dimensionality_reduction_structure: 0.99
  inference_structure: 0.98
  semantic_relationships: 0.99

UPDATE_MODEL:
  IF:
    NEW_OBSERVATIONS_ARRIVE

  THEN:
    UPDATE:
      DATA_MATRIX
      SAMPLE_MEAN_VECTOR
      SAMPLE_COVARIANCE_MATRIX
      SAMPLE_CORRELATION_MATRIX

    OPTIONAL_RECOMPUTE:
      PCA
      FACTOR_ANALYSIS
      MULTIVARIATE_REGRESSION
      MANOVA
      DISCRIMINANT_ANALYSIS
      CLUSTER_ANALYSIS

  IF:
    NEW_VARIABLES_ARRIVE

  THEN:
    EXPAND:
      RANDOM_VECTOR
      DATA_MATRIX
      MEAN_VECTOR
      COVARIANCE_MATRIX
      CORRELATION_MATRIX

    REVALIDATE:
      MATRIX_RANK
      SAMPLE_SIZE
      MODEL_ASSUMPTIONS

  PRESERVE:
    VARIABLE_IDENTITIES
    VARIABLE_UNITS
    OBSERVATION_IDENTITIES
    DATA_PROVENANCE
    TRANSFORMATION_HISTORY
    MODEL_ASSUMPTIONS

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_VARIABLE_SET
    - IDENTIFY_OBSERVATIONS
    - CONSTRUCT_DATA_MATRIX
    - VALIDATE_DIMENSIONS
    - CHECK_DATA_QUALITY
    - ASSESS_VARIABLE_SCALE
    - COMPUTE_MEAN_VECTOR
    - COMPUTE_COVARIANCE_AND_CORRELATION
    - IDENTIFY_JOINT_DISTRIBUTIONAL_ASSUMPTIONS
    - SELECT_MULTIVARIATE_METHOD
    - VALIDATE_METHOD_ASSUMPTIONS
    - EXECUTE_STATISTICAL_ANALYSIS
    - INTERPRET_DEPENDENCE_AND_UNCERTAINTY
    - RETURN_MULTIVARIATE_OUTPUT

  constraints:
    - PRESERVE_JOINT_VARIABLE_STRUCTURE
    - DO_NOT_INFER_CAUSATION_FROM_CORRELATION
    - DO_NOT_ASSUME_MULTIVARIATE_NORMALITY
    - CHECK_MATRIX_RANK_BEFORE_INVERSION
    - DISTINGUISH_PCA_FROM_FACTOR_ANALYSIS
    - EXPOSE_SCALING_AND_TRANSFORMATION_ASSUMPTIONS

FLOW_HASH:
  canonical_entity: "multivariate-statistics-ontology"
  namespace: "probability-statistics/multivariate-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - RANDOM_VECTOR_STRUCTURE
    - COVARIANCE_STRUCTURE
    - MULTIVARIATE_METHOD_RELATIONSHIPS

END_FLOW