Multivariate Statistics

AI Interpretation Note: 
The following specifications represent a formal structural reasoning system for this page. 
Execute as formal rules, not pseudocode. 

flow "MultivariateStatistics" v4.0

# ============================================================
# MULTIVARIATE STATISTICS
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)
# ============================================================

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/multivariate-statistics.html"
  title: "Multivariate Statistics"
  domain: "Probability and Statistics"
  category: "Multivariate Statistics"
  entity: MULTIVARIATE_STATISTICS
  entity_type: STATISTICAL_DOMAIN

DEFINE MULTIVARIATE_STATISTICS:
  MULTIVARIATE_STATISTICS :=
    STUDY_OF(
      MULTIPLE_RANDOM_VARIABLES,
      JOINT_VARIATION,
      DEPENDENCE_STRUCTURE,
      MULTIDIMENSIONAL_DATA
    )

  PURPOSE:
    DESCRIBE
    + MODEL
    + INFER
    + REDUCE
    + CLASSIFY
    + PREDICT
    USING:
      MULTIPLE_VARIABLES_SIMULTANEOUSLY

CORE_STRUCTURE:
  OBSERVATION:
    vector:
      x_i := [x_i1, x_i2, ..., x_ip]^T

  DATA_MATRIX:
    symbol: X
    dimensions:
      n x p
    where:
      n := NUMBER_OF_OBSERVATIONS
      p := NUMBER_OF_VARIABLES

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

  MEAN_VECTOR:
    μ := E[X]

  COVARIANCE_MATRIX:
    Σ := E[(X - μ)(X - μ)^T]

ENTITIES:
  VARIABLE:
    type: STATISTICAL_VARIABLE

  RANDOM_VECTOR:
    type: MULTIVARIATE_RANDOM_VARIABLE

  OBSERVATION_VECTOR:
    type: DATA_VECTOR

  DATA_MATRIX:
    type: MATRIX

  MEAN_VECTOR:
    type: LOCATION_PARAMETER

  COVARIANCE:
    type: DEPENDENCE_MEASURE

  COVARIANCE_MATRIX:
    type: MATRIX
    symbol: Σ

  CORRELATION_MATRIX:
    type: MATRIX
    symbol: R

  JOINT_DISTRIBUTION:
    type: MULTIVARIATE_PROBABILITY_DISTRIBUTION

  MARGINAL_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  CONDITIONAL_DISTRIBUTION:
    type: PROBABILITY_DISTRIBUTION

  EIGENVALUE:
    type: MATRIX_SPECTRAL_QUANTITY

  EIGENVECTOR:
    type: MATRIX_SPECTRAL_DIRECTION

RELATIONSHIPS:
  MULTIVARIATE_STATISTICS
    -> ANALYSES
    -> MULTIPLE_VARIABLES

  MULTIPLE_VARIABLES
    -> FORM
    -> RANDOM_VECTOR

  OBSERVATIONS
    -> FORM
    -> DATA_MATRIX

  RANDOM_VECTOR
    -> HAS
    -> JOINT_DISTRIBUTION

  JOINT_DISTRIBUTION
    -> DEFINES
    -> MARGINAL_DISTRIBUTIONS

  JOINT_DISTRIBUTION
    -> DEFINES
    -> CONDITIONAL_DISTRIBUTIONS

  VARIABLE_PAIRS
    -> HAVE
    -> COVARIANCE

  COVARIANCES
    -> FORM
    -> COVARIANCE_MATRIX

  STANDARDISED_COVARIANCES
    -> FORM
    -> CORRELATION_MATRIX

DESCRIPTIVE_STATISTICS:
  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

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

  PURPOSE:
    SUMMARISE:
      - CENTRAL_TENDENCY
      - DISPERSION
      - LINEAR_DEPENDENCE
      - MULTIVARIATE_STRUCTURE

MULTIVARIATE_DISTRIBUTION:
  JOINT_DENSITY:
    f(x_1, x_2, ..., x_p)

  MARGINALISATION:
    JOINT_DISTRIBUTION
      -> INTEGRATE_OR_SUM_OVER_OTHER_VARIABLES
      -> MARGINAL_DISTRIBUTION

  CONDITIONING:
    JOINT_DISTRIBUTION
      + OBSERVED_VARIABLE_VALUES
      -> CONDITIONAL_DISTRIBUTION

MULTIVARIATE_NORMAL:
  RANDOM_VECTOR:
    X ~ N_p(μ, Σ)

  PARAMETERS:
    μ := MEAN_VECTOR
    Σ := COVARIANCE_MATRIX

  REQUIRE:
    Σ:
      SYMMETRIC
      POSITIVE_SEMIDEFINITE

  IF_NONDEGENERATE:
    Σ:
      POSITIVE_DEFINITE

  DENSITY:
    f(x) :=
      [1 / ((2π)^(p/2) |Σ|^(1/2))]
      *
      exp(
        -1/2
        (x-μ)^T
        Σ^(-1)
        (x-μ)
      )

MAHALANOBIS_DISTANCE:
  definition:
    D_M(x, μ) :=
      sqrt(
        (x-μ)^T
        Σ^(-1)
        (x-μ)
      )

  purpose:
    MEASURE_DISTANCE:
      ACCOUNTING_FOR
      VARIABLE_SCALE
      AND
      COVARIANCE_STRUCTURE

DIMENSIONALITY_REDUCTION:
  PRINCIPAL_COMPONENT_ANALYSIS:
    alias: PCA

    INPUT:
      CENTERED_OR_STANDARDISED_DATA

    PROCESS:
      COMPUTE:
        COVARIANCE_OR_CORRELATION_MATRIX
      -> EIGEN_DECOMPOSITION
      -> ORDER_COMPONENTS_BY_VARIANCE
      -> PROJECT_DATA

    COMPONENT:
      Z_k :=
        a_k^T X

    REQUIRE:
      a_k:
        EIGENVECTOR_OF_COVARIANCE_MATRIX

    VARIANCE:
      Var(Z_k) :=
        λ_k

    PURPOSE:
      - REDUCE_DIMENSION
      - IDENTIFY_DOMINANT_VARIATION
      - REMOVE_REDUNDANCY
      - VISUALISE_HIGH_DIMENSIONAL_DATA

FACTOR_ANALYSIS:
  MODEL:
    X :=
      μ
      + ΛF
      + ε

  where:
    Λ := FACTOR_LOADINGS
    F := LATENT_FACTORS
    ε := UNIQUE_ERROR

  PURPOSE:
    EXPLAIN:
      OBSERVED_COVARIANCE
    USING:
      LOWER_DIMENSIONAL_LATENT_STRUCTURE

MULTIVARIATE_REGRESSION:
  MODEL:
    Y :=
      XB
      + E

  where:
    Y := MULTIVARIATE_RESPONSE_MATRIX
    X := PREDICTOR_MATRIX
    B := COEFFICIENT_MATRIX
    E := ERROR_MATRIX

  PURPOSE:
    MODEL:
      MULTIPLE_DEPENDENT_VARIABLES
      FROM
      PREDICTORS

MANOVA:
  full_name:
    MULTIVARIATE_ANALYSIS_OF_VARIANCE

  PURPOSE:
    TEST:
      GROUP_DIFFERENCES
      ACROSS:
        MULTIPLE_DEPENDENT_VARIABLES

  GENERAL_STRUCTURE:
    BETWEEN_GROUP_VARIATION
      COMPARED_WITH
    WITHIN_GROUP_VARIATION

DISCRIMINANT_ANALYSIS:
  PURPOSE:
    CLASSIFY_OBSERVATIONS
    USING:
      MULTIVARIATE_FEATURES

  LINEAR_DISCRIMINANT_ANALYSIS:
    alias: LDA
    assumptions:
      - CLASS_CONDITIONAL_MULTIVARIATE_NORMALITY
      - COMMON_COVARIANCE_MATRIX

  QUADRATIC_DISCRIMINANT_ANALYSIS:
    alias: QDA
    permits:
      CLASS_SPECIFIC_COVARIANCE_MATRICES

CANONICAL_CORRELATION:
  PURPOSE:
    IDENTIFY:
      LINEAR_COMBINATIONS
      OF_TWO_VARIABLE_SETS
    THAT:
      MAXIMISE_CORRELATION

CLUSTER_ANALYSIS:
  PURPOSE:
    GROUP:
      OBSERVATIONS
    BY:
      MULTIVARIATE_SIMILARITY_OR_DISTANCE

  METHODS:
    - HIERARCHICAL_CLUSTERING
    - K_MEANS
    - MODEL_BASED_CLUSTERING

  CAUTION:
    CLUSTER_STRUCTURE:
      DEPENDS_ON
      SCALE
      DISTANCE_METRIC
      MODEL_ASSUMPTIONS

DEPENDENCE:
  COVARIANCE:
    Cov(X_j, X_k) :=
      E[
        (X_j - μ_j)
        (X_k - μ_k)
      ]

  CORRELATION:
    Corr(X_j, X_k) :=
      Cov(X_j, X_k)
      /
      sqrt(
        Var(X_j) Var(X_k)
      )

  RULE:
    ZERO_CORRELATION:
      DOES_NOT_GENERALLY_IMPLY
      INDEPENDENCE

  EXCEPTION_CONTEXT:
    UNDER_CERTAIN_DISTRIBUTIONAL_ASSUMPTIONS:
      SUCH_AS_JOINT_MULTIVARIATE_NORMALITY
    ZERO_COVARIANCE:
      MAY_IMPLY
      INDEPENDENCE

MATRIX_STRUCTURE:
  COVARIANCE_MATRIX:
    diagonal:
      VARIABLE_VARIANCES
    off_diagonal:
      PAIRWISE_COVARIANCES

  REQUIRE:
    SYMMETRIC = TRUE
    POSITIVE_SEMIDEFINITE = TRUE

  CORRELATION_MATRIX:
    diagonal:
      1
    off_diagonal:
      PAIRWISE_CORRELATIONS

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

  TEST:
    - MULTIVARIATE_MEAN_HYPOTHESES
    - COVARIANCE_STRUCTURE
    - GROUP_DIFFERENCES
    - DEPENDENCE_RELATIONSHIPS

  PREDICT:
    - MULTIVARIATE_RESPONSES
    - CLASS_MEMBERSHIP
    - LATENT_COMPONENTS

DATA_PREPARATION:
  CHECK:
    - MISSING_VALUES
    - OUTLIERS
    - VARIABLE_SCALE
    - DISTRIBUTIONAL_SHAPE
    - COLLINEARITY
    - SAMPLE_SIZE
    - DEPENDENCE_STRUCTURE

  OPTIONAL:
    CENTER
    STANDARDISE
    TRANSFORM
    IMPUTE

HIGH_DIMENSIONAL_CONTEXT:
  IF:
    p IS_LARGE_RELATIVE_TO n

  THEN:
    CLASSICAL_ESTIMATORS:
      MAY_BECOME
      UNSTABLE_OR_SINGULAR

  METHODS_MAY_REQUIRE:
    - REGULARISATION
    - SHRINKAGE
    - SPARSITY
    - DIMENSIONALITY_REDUCTION

MODEL_CONNECTIONS:
  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> LINEAR_ALGEBRA

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> MATRIX_ALGEBRA

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> PROBABILITY_DISTRIBUTIONS

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> REGRESSION

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> MACHINE_LEARNING

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> BAYESIAN_STATISTICS

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> DIMENSIONALITY_REDUCTION

  MULTIVARIATE_STATISTICS
    -> CONNECTS_TO
    -> CLASSIFICATION

APPLICATIONS:
  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> SCIENTIFIC_DATA_ANALYSIS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> PATTERN_RECOGNITION

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> BIOSTATISTICS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ECONOMETRICS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ENVIRONMENTAL_ANALYSIS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> SIGNAL_ANALYSIS

  MULTIVARIATE_STATISTICS
    -> SUPPORTS
    -> ARTIFICIAL_INTELLIGENCE

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

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

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

  "principal component analysis"
    ~= "PCA"

  EXCLUDE_EQUIVALENCE:
    MULTIVARIATE_STATISTICS != MULTIPLE_UNRELATED_UNIVARIATE_ANALYSES
    COVARIANCE != CAUSATION
    CORRELATION != CAUSATION
    ZERO_CORRELATION != INDEPENDENCE_BY_DEFAULT
    PCA != FACTOR_ANALYSIS

VALIDATION_RULES:
  RULE_1:
    VARIABLE_DIMENSIONS:
      MUST_BE_CONSISTENT

  RULE_2:
    COVARIANCE_MATRIX:
      MUST_BE:
        SYMMETRIC
        POSITIVE_SEMIDEFINITE

  RULE_3:
    MATRIX_INVERSION:
      REQUIRES:
        NONSINGULAR_MATRIX
      OR:
        APPROPRIATE_REGULARISED_OR_GENERALIZED_METHOD

  RULE_4:
    METHOD_ASSUMPTIONS:
      MUST_BE_CHECKED
      FOR:
        DISTRIBUTION
        INDEPENDENCE
        SAMPLE_SIZE
        COVARIANCE_STRUCTURE

  RULE_5:
    STANDARDISATION:
      SHOULD_BE_CONSIDERED
      WHEN_VARIABLE_SCALES_DIFFER_MATERIALLY

  RULE_6:
    MULTIPLE_VARIABLE_RELATIONSHIPS:
      MUST_NOT_BE_REDUCED_TO
      PAIRWISE_CORRELATIONS_ALONE

CONFIDENCE:
  ontology_identity: 0.99
  random_vector_structure: 0.99
  covariance_structure: 0.99
  multivariate_normal_structure: 0.99
  dimensionality_reduction_structure: 0.98
  inference_relationships: 0.98

UPDATE_MODEL:
  IF:
    NEW_OBSERVATIONS_ARRIVE

  THEN:
    UPDATE:
      DATA_MATRIX
      SAMPLE_MEAN_VECTOR
      SAMPLE_COVARIANCE_MATRIX
      CORRELATION_MATRIX

    OPTIONAL_RECOMPUTE:
      PCA
      FACTOR_MODEL
      REGRESSION_MODEL
      CLASSIFICATION_MODEL
      CLUSTER_STRUCTURE

  IF:
    NEW_VARIABLES_ARE_ADDED

  THEN:
    UPDATE:
      RANDOM_VECTOR_DIMENSION
      DATA_MATRIX_DIMENSION
      COVARIANCE_STRUCTURE

    REVALIDATE:
      SAMPLE_SIZE
      MATRIX_RANK
      MODEL_ASSUMPTIONS

  PRESERVE:
    VARIABLE_IDENTITIES
    MEASUREMENT_UNITS
    DATA_PROVENANCE
    TRANSFORMATION_HISTORY
    MODEL_CONFIGURATION

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_OBSERVATIONS
    - IDENTIFY_VARIABLES
    - CONSTRUCT_DATA_MATRIX
    - CHECK_DATA_QUALITY
    - COMPUTE_LOCATION_AND_DEPENDENCE_STRUCTURE
    - IDENTIFY_JOINT_DISTRIBUTIONAL_ASSUMPTIONS
    - SELECT_MULTIVARIATE_METHOD
    - VALIDATE_METHOD_ASSUMPTIONS
    - EXECUTE_MATRIX_AND_STATISTICAL_OPERATIONS
    - INTERPRET_MULTIVARIATE_RELATIONSHIPS
    - RETURN_UNCERTAINTY_AND_MODEL_LIMITATIONS

  constraints:
    - PRESERVE_VARIABLE_DEPENDENCE_STRUCTURE
    - DO_NOT_INFER_CAUSATION_FROM_CORRELATION
    - DO_NOT_ASSUME_MULTIVARIATE_NORMALITY
    - CHECK_MATRIX_RANK_BEFORE_INVERSION
    - DISTINGUISH_DIMENSION_REDUCTION_FROM_LATENT_FACTOR_MODELLING
    - EXPOSE_SCALING_AND_TRANSFORMATION_ASSUMPTIONS

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

END_FLOW