Factor Analysis Ontology

flow "FactorAnalysisOntology" v4.0

# ============================================================
# FACTOR ANALYSIS 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/factor-analysis-ontology.html"
  title: "Factor Analysis Ontology"
  domain: "Probability and Statistics"
  category: "Multivariate Statistics"
  entity: FACTOR_ANALYSIS
  entity_type: LATENT_VARIABLE_MODEL

DEFINE FACTOR_ANALYSIS:
  FACTOR_ANALYSIS :=
    MULTIVARIATE_LATENT_VARIABLE_METHOD(
      OBSERVED_VARIABLES,
      LATENT_FACTORS,
      FACTOR_LOADINGS,
      UNIQUE_VARIANCE
    )

  PURPOSE:
    EXPLAIN:
      SHARED_COVARIANCE_AMONG_OBSERVED_VARIABLES
    USING:
      SMALLER_SET_OF_LATENT_FACTORS

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

  LATENT_FACTOR_VECTOR:
    F := [F_1, F_2, ..., F_m]^T

  MODEL:
    X :=
      mu
      + Lambda F
      + epsilon

  where:
    mu := MEAN_VECTOR
    Lambda := FACTOR_LOADING_MATRIX
    F := LATENT_FACTOR_VECTOR
    epsilon := UNIQUE_ERROR_VECTOR

  DIMENSIONS:
    X: p x 1
    mu: p x 1
    Lambda: p x m
    F: m x 1
    epsilon: p x 1

  REQUIRE:
    m < p
    FOR_DIMENSION_REDUCTION_INTERPRETATION

ENTITIES:
  OBSERVED_VARIABLE:
    type: MEASURED_STATISTICAL_VARIABLE

  LATENT_FACTOR:
    type: UNOBSERVED_VARIABLE

  FACTOR_LOADING:
    type: VARIABLE_FACTOR_RELATION

  FACTOR_LOADING_MATRIX:
    type: MATRIX
    symbol: Lambda

  UNIQUE_ERROR:
    type: VARIABLE_SPECIFIC_RANDOM_COMPONENT
    symbol: epsilon

  COMMUNALITY:
    type: SHARED_VARIANCE_MEASURE

  UNIQUENESS:
    type: VARIABLE_SPECIFIC_VARIANCE

  FACTOR_SCORE:
    type: ESTIMATED_LATENT_FACTOR_VALUE

  FACTOR_CORRELATION_MATRIX:
    type: MATRIX
    symbol: Phi

RELATIONSHIPS:
  FACTOR_ANALYSIS
    -> MODELS
    -> OBSERVED_VARIABLES

  OBSERVED_VARIABLES
    -> INFLUENCED_BY
    -> LATENT_FACTORS

  LATENT_FACTOR
    -> CONNECTS_TO
    -> OBSERVED_VARIABLE
    VIA:
      FACTOR_LOADING

  FACTOR_LOADINGS
    -> FORM
    -> FACTOR_LOADING_MATRIX

  OBSERVED_VARIABLE
    -> HAS
    -> COMMUNALITY

  OBSERVED_VARIABLE
    -> HAS
    -> UNIQUENESS

  SHARED_VARIANCE
    -> ATTRIBUTED_TO
    -> COMMON_FACTORS

  RESIDUAL_VARIANCE
    -> ATTRIBUTED_TO
    -> UNIQUE_COMPONENT

COVARIANCE_STRUCTURE:
  GENERAL_MODEL:
    Cov(X) :=
      Lambda Phi Lambda^T
      + Psi

  where:
    Phi :=
      Cov(F)

    Psi :=
      Cov(epsilon)

  COMMON_ORTHOGONAL_FACTOR_MODEL:
    IF:
      Phi = I

    THEN:
      Sigma :=
        Lambda Lambda^T
        + Psi

  TYPICAL_ASSUMPTION:
    Psi:
      DIAGONAL

  INTERPRETATION:
    Lambda Phi Lambda^T:
      COMMON_COVARIANCE_STRUCTURE

    Psi:
      UNIQUE_OR_ERROR_VARIANCE

COMMUNALITY:
  FOR_VARIABLE_i:
    h_i^2 :=
      VARIANCE_EXPLAINED_BY_COMMON_FACTORS

  ORTHOGONAL_STANDARDISED_CASE:
    h_i^2 :=
      SUM[j=1..m]
        lambda_ij^2

  INTERPRETATION:
    HIGH_COMMUNALITY:
      LARGE_PROPORTION_OF_VARIABLE_VARIANCE
      REPRESENTED_BY:
        COMMON_FACTORS

UNIQUENESS:
  FOR_STANDARDISED_VARIABLE_i:
    psi_i :=
      1 - h_i^2
    UNDER:
      STANDARDISED_ORTHOGONAL_COMMON_FACTOR_MODEL

  GENERAL_ROLE:
    REPRESENT:
      VARIABLE_SPECIFIC_VARIANCE
      + MEASUREMENT_ERROR
      NOT_EXPLAINED_BY_COMMON_FACTORS

FACTOR_LOADINGS:
  loading:
    lambda_ij

  role:
    QUANTIFY:
      RELATIONSHIP_BETWEEN
      OBSERVED_VARIABLE_i
      AND
      LATENT_FACTOR_j

  magnitude:
    LARGER_ABSOLUTE_VALUE
      -> STRONGER_RELATIONSHIP
    SUBJECT_TO:
      MODEL_SCALING_AND_CONVENTION

  sign:
    INDICATES:
      DIRECTION_OF_RELATIONSHIP
    SUBJECT_TO:
      FACTOR_SIGN_INDETERMINACY

EXPLORATORY_FACTOR_ANALYSIS:
  alias: EFA

  purpose:
    DISCOVER:
      POSSIBLE_LATENT_FACTOR_STRUCTURE

  analyst_specifies:
    - NUMBER_OF_FACTORS_OR_SELECTION_RULE
    - EXTRACTION_METHOD
    - ROTATION_METHOD

  structure:
    DATA
      -> ESTIMATE_COMMON_FACTOR_MODEL
      -> DETERMINE_FACTOR_COUNT
      -> ROTATE_LOADINGS
      -> INTERPRET_FACTORS

CONFIRMATORY_FACTOR_ANALYSIS:
  alias: CFA

  purpose:
    TEST:
      PREDEFINED_FACTOR_STRUCTURE

  specifies:
    - WHICH_VARIABLES_LOAD_ON_WHICH_FACTORS
    - FACTOR_CORRELATIONS
    - PARAMETER_CONSTRAINTS

  evaluates:
    MODEL_IMPLIED_COVARIANCE
    AGAINST:
      OBSERVED_COVARIANCE

FACTOR_EXTRACTION:
  methods:
    - PRINCIPAL_AXIS_FACTORING
    - MAXIMUM_LIKELIHOOD_FACTOR_ANALYSIS
    - MINIMUM_RESIDUAL_METHODS

  CAUTION:
    PRINCIPAL_COMPONENT_ANALYSIS:
      IS_NOT:
        COMMON_FACTOR_EXTRACTION_METHOD
      IN_THE_SAME_STATISTICAL_MODEL_SENSE

NUMBER_OF_FACTORS:
  selection_methods:
    - PARALLEL_ANALYSIS
    - SCREE_PLOT
    - INFORMATION_CRITERIA_WHEN_MODEL_BASED
    - LIKELIHOOD_BASED_COMPARISON_WHEN_APPROPRIATE
    - THEORY_AND_INTERPRETABILITY

  CAUTION:
    KAISER_EIGENVALUE_GREATER_THAN_ONE_RULE:
      MAY_BE_USED_AS_HEURISTIC
      BUT:
        SHOULD_NOT_BE_TREATED_AS_UNIVERSAL_DECISION_RULE

ROTATION:
  PURPOSE:
    IMPROVE:
      INTERPRETABILITY_OF_FACTOR_LOADINGS

  ORTHOGONAL_ROTATION:
    factors:
      UNCORRELATED

    examples:
      - VARIMAX
      - QUARTIMAX

  OBLIQUE_ROTATION:
    factors:
      MAY_BE_CORRELATED

    examples:
      - OBLIMIN
      - PROMAX

  RULE:
    ROTATION:
      CHANGES_FACTOR_COORDINATE_REPRESENTATION
      WHILE:
        PRESERVING_MODELLED_COMMON_STRUCTURE
      SUBJECT_TO:
        ROTATION_TYPE_AND_MODEL

FACTOR_SCORES:
  DEFINE:
    ESTIMATED_VALUES_OF:
      LATENT_FACTORS
    FOR:
      INDIVIDUAL_OBSERVATIONS

  methods:
    - REGRESSION_METHOD
    - BARTLETT_METHOD
    - ANDERSON_RUBIN_METHOD

  CAUTION:
    FACTOR_SCORES:
      ARE_ESTIMATES
      NOT:
        DIRECTLY_OBSERVED_VALUES

IDENTIFIABILITY:
  ISSUE:
    FACTOR_MODEL_PARAMETERS:
      MAY_NOT_BE_UNIQUELY_IDENTIFIED
      WITHOUT:
        CONSTRAINTS

  SOURCES:
    - ROTATIONAL_INDETERMINACY
    - FACTOR_SIGN_INDETERMINACY
    - SCALE_INDETERMINACY

  REQUIRE:
    IDENTIFICATION_CONSTRAINTS

ASSUMPTIONS:
  COMMON_FACTOR_MODEL:
    assumes:
      OBSERVED_COVARIATION:
        PARTLY_GENERATED_BY
        LATENT_COMMON_FACTORS

  TYPICAL_CONDITIONS:
    - APPROPRIATE_VARIABLE_RELATIONSHIPS
    - ADEQUATE_SAMPLE_SIZE
    - NO_PATHOLOGICAL_MULTICOLLINEARITY
    - APPROPRIATE_DISTRIBUTIONAL_ASSUMPTIONS_FOR_SELECTED_ESTIMATOR

  MAXIMUM_LIKELIHOOD_FACTOR_ANALYSIS:
    commonly_assumes:
      MULTIVARIATE_NORMALITY
    FOR:
      STANDARD_CLASSICAL_INFERENCE

DATA_SUITABILITY:
  CHECK:
    - CORRELATION_MATRIX
    - SAMPLE_SIZE
    - MISSING_DATA
    - OUTLIERS
    - VARIABLE_DISTRIBUTIONS
    - LINEAR_RELATIONSHIPS
    - EXTREME_MULTICOLLINEARITY
    - NEAR_ZERO_CORRELATIONS

  OPTIONAL_DIAGNOSTICS:
    - KAISER_MEYER_OLKIN_MEASURE
    - BARTLETT_SPHERICITY_TEST

  INTERPRET_WITH:
    METHOD_LIMITATIONS
    SAMPLE_CONTEXT

FACTOR_ANALYSIS_VS_PCA:
  FACTOR_ANALYSIS:
    MODELS:
      COMMON_VARIANCE
      + UNIQUE_VARIANCE

    assumes:
      LATENT_VARIABLE_STRUCTURE

  PCA:
    TRANSFORMS:
      TOTAL_VARIANCE
    INTO:
      ORTHOGONAL_COMPONENTS

  PCA_COMPONENT:
    MATHEMATICAL_LINEAR_COMBINATION

  FACTOR:
    LATENT_MODEL_VARIABLE

  RULE:
    FACTOR_ANALYSIS != PCA

MODEL_FIT:
  IN_CFA_OR_MODEL_BASED_FACTOR_ANALYSIS:
    compare:
      OBSERVED_COVARIANCE_MATRIX
      WITH:
        MODEL_IMPLIED_COVARIANCE_MATRIX

  possible_metrics:
    - CHI_SQUARE_MODEL_TEST
    - RMSEA
    - CFI
    - TLI
    - SRMR

  CAUTION:
    FIT_INDICES:
      MUST_BE_INTERPRETED_TOGETHER
      WITH:
        MODEL_ASSUMPTIONS
        SAMPLE_SIZE
        THEORY
        RESIDUALS

APPLICATIONS:
  FACTOR_ANALYSIS
    -> SUPPORTS
    -> LATENT_CONSTRUCT_DISCOVERY

  FACTOR_ANALYSIS
    -> SUPPORTS
    -> SCALE_DEVELOPMENT

  FACTOR_ANALYSIS
    -> SUPPORTS
    -> PSYCHOMETRICS

  FACTOR_ANALYSIS
    -> SUPPORTS
    -> SOCIAL_SCIENCE_RESEARCH

  FACTOR_ANALYSIS
    -> SUPPORTS
    -> MARKET_RESEARCH

  FACTOR_ANALYSIS
    -> SUPPORTS
    -> DIMENSION_REDUCTION

  FACTOR_ANALYSIS
    -> SUPPORTS
    -> STRUCTURAL_EQUATION_MODELLING

MODEL_CONNECTIONS:
  FACTOR_ANALYSIS
    -> CONNECTS_TO
    -> MULTIVARIATE_STATISTICS

  FACTOR_ANALYSIS
    -> CONNECTS_TO
    -> COVARIANCE_MATRIX

  FACTOR_ANALYSIS
    -> CONNECTS_TO
    -> LATENT_VARIABLE_MODELS

  FACTOR_ANALYSIS
    -> CONNECTS_TO
    -> PRINCIPAL_COMPONENT_ANALYSIS

  FACTOR_ANALYSIS
    -> CONNECTS_TO
    -> STRUCTURAL_EQUATION_MODELLING

  FACTOR_ANALYSIS
    -> CONNECTS_TO
    -> MAXIMUM_LIKELIHOOD

  FACTOR_ANALYSIS
    -> CONNECTS_TO
    -> CORRELATION_MATRIX

SEMANTIC_EQUIVALENCE:
  "factor analysis"
    ~= "common latent factor modelling"

  "factor loading"
    ~= "observed-variable to latent-factor model coefficient"

  "communality"
    ~= "variance represented by common factors"

  "uniqueness"
    ~= "variance not represented by common factors"

  "EFA"
    ~= "exploratory factor analysis"

  "CFA"
    ~= "confirmatory factor analysis"

  EXCLUDE_EQUIVALENCE:
    FACTOR_ANALYSIS != PCA
    LATENT_FACTOR != PRINCIPAL_COMPONENT
    FACTOR_SCORE != DIRECT_OBSERVATION
    COMMUNALITY != TOTAL_VARIANCE
    ROTATION != NEW_RAW_DATA

VALIDATION_RULES:
  RULE_1:
    OBSERVED_VARIABLES:
      MUST_HAVE:
        SUFFICIENT_MEANINGFUL_DEPENDENCE
      FOR:
        COMMON_FACTOR_MODELLING

  RULE_2:
    FACTOR_LOADING_MATRIX:
      DIMENSIONS:
        p x m

  RULE_3:
    MODEL_IMPLIED_COVARIANCE:
      MUST_BE:
        MATHEMATICALLY_VALID

  RULE_4:
    UNIQUENESS_VARIANCES:
      MUST_NOT_BE:
        INVALID_NEGATIVE_VARIANCES
      IN:
        STANDARD_FACTOR_MODEL

  RULE_5:
    FACTOR_COUNT:
      MUST_NOT_BE_SELECTED
      SOLELY_FROM:
        ARBITRARY_UNIVERSAL_THRESHOLD

  RULE_6:
    ROTATION_METHOD:
      MUST_BE_REPORTED

  RULE_7:
    FACTOR_INTERPRETATION:
      MUST_BE_SUPPORTED_BY:
        LOADING_PATTERN
        DOMAIN_MEANING
        MODEL_CONTEXT

  RULE_8:
    FACTOR_SIGN:
      MAY_BE_REVERSED
      WITHOUT:
        CHANGING_MODEL_MEANING
      WHEN:
        CORRESPONDING_LOADINGS_AND_SCORES_ARE_REVERSED

CONFIDENCE:
  ontology_identity: 0.99
  factor_model_definition: 0.99
  covariance_structure: 0.99
  loading_structure: 0.99
  efa_cfa_distinction: 0.99
  pca_distinction: 0.99
  rotation_structure: 0.98

UPDATE_MODEL:
  IF:
    NEW_OBSERVATIONS_ARRIVE

  THEN:
    UPDATE:
      SAMPLE_COVARIANCE_MATRIX
      SAMPLE_CORRELATION_MATRIX

    OPTIONAL_REFIT:
      FACTOR_LOADINGS
      UNIQUENESS
      FACTOR_CORRELATIONS
      FACTOR_SCORES

  IF:
    NEW_VARIABLES_ARRIVE

  THEN:
    EXPAND:
      OBSERVED_VECTOR
      LOADING_MATRIX
      COVARIANCE_STRUCTURE

    REASSESS:
      NUMBER_OF_FACTORS
      MODEL_IDENTIFICATION
      MODEL_FIT

  IF:
    FACTOR_STRUCTURE_CHANGES

  THEN:
    REESTIMATE:
      MODEL_PARAMETERS

    REVALIDATE:
      IDENTIFIABILITY
      FIT
      INTERPRETABILITY

  PRESERVE:
    VARIABLE_IDENTITIES
    FACTOR_IDENTITIES
    EXTRACTION_METHOD
    ROTATION_METHOD
    ESTIMATION_METHOD
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_OBSERVED_VARIABLES
    - ASSESS_DATA_SUITABILITY
    - COMPUTE_COVARIANCE_OR_CORRELATION_MATRIX
    - DETERMINE_EFA_OR_CFA_CONTEXT
    - SELECT_FACTOR_COUNT_OR_SPECIFIED_STRUCTURE
    - SELECT_EXTRACTION_OR_ESTIMATION_METHOD
    - ESTIMATE_FACTOR_MODEL
    - APPLY_ROTATION_IF_REQUIRED
    - VALIDATE_IDENTIFIABILITY
    - ASSESS_MODEL_FIT
    - INTERPRET_LOADINGS_AND_COMMUNALITIES
    - ESTIMATE_FACTOR_SCORES_IF_REQUIRED
    - RETURN_LATENT_STRUCTURE_AND_UNCERTAINTY

  constraints:
    - DISTINGUISH_FACTOR_ANALYSIS_FROM_PCA
    - DO_NOT_TREAT_LATENT_FACTORS_AS_DIRECTLY_OBSERVED
    - EXPOSE_FACTOR_COUNT_DECISION
    - EXPOSE_EXTRACTION_METHOD
    - EXPOSE_ROTATION_METHOD
    - ACCOUNT_FOR_ROTATIONAL_AND_SIGN_INDETERMINACY
    - PRESERVE_COMMON_AND_UNIQUE_VARIANCE_DISTINCTION

FLOW_HASH:
  canonical_entity: "factor-analysis"
  namespace: "probability-statistics/multivariate-statistics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - LATENT_FACTOR_STRUCTURE
    - FACTOR_LOADING_MATRIX
    - COMMON_AND_UNIQUE_VARIANCE

END_FLOW