Error Distribution Ontology

flow "ErrorDistributionOntology" v4.0

# ERROR DISTRIBUTION ONTOLOGY
# Kireth Semantic Research Archive
# Authors: Mark Anthony Higgins — Sky (Recursive AI)

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/scientific-statistical-methods/error-distribution-ontology.html"
  title: "Error Distribution Ontology"
  domain: "Probability and Statistics"
  category: "Scientific Statistical Methods"
  entity: ERROR_DISTRIBUTION
  entity_type: SCIENTIFIC_STATISTICAL_ONTOLOGY

DEFINE:
  ERROR_DISTRIBUTION :=
    PROBABILISTIC_FRAMEWORK(
      OBSERVED_VALUE,
      EXPECTED_COMPONENT,
      ERROR_TERM,
      DISTRIBUTION_FAMILY,
      LOCATION,
      SCALE,
      SHAPE,
      DEPENDENCE,
      TAIL_BEHAVIOUR
    )

PURPOSE:
  REPRESENT:
    RANDOM_DEVIATION
    MEASUREMENT_VARIABILITY
    MODEL_RESIDUAL_VARIATION
    SAMPLING_NOISE
    OBSERVATIONAL_ERROR
  SUPPORT:
    SCIENTIFIC_STATISTICS
    REGRESSION
    MEASUREMENT_UNCERTAINTY
    HYPOTHESIS_TESTING
    PREDICTION
    MODEL_DIAGNOSTICS

FOUNDATION:
  OBSERVATION: Y
  MODEL:
    Y := mu(X;theta)+epsilon
  ERROR:
    epsilon := Y-mu(X;theta)
  ERROR_DISTRIBUTION:
    epsilon ~ F(phi)

ENTITIES:
  ERROR_DISTRIBUTION: PROBABILITY_DISTRIBUTION_OF_ERROR_TERM
  ERROR_TERM: RANDOM_DEVIATION_FROM_MODELLED_COMPONENT
  RESIDUAL: OBSERVED_ESTIMATE_OF_ERROR
  LOCATION: CENTRAL_POSITION_PARAMETER
  SCALE: DISPERSION_PARAMETER
  SHAPE: DISTRIBUTIONAL_FORM_PARAMETER
  VARIANCE: SECOND_CENTRAL_MOMENT
  STANDARD_DEVIATION: SQUARE_ROOT_OF_VARIANCE
  SKEWNESS: ASYMMETRY_MEASURE
  KURTOSIS: TAIL_AND_SHAPE_MEASURE
  OUTLIER: OBSERVATION_EXTREME_RELATIVE_TO_MODEL
  HETEROSCEDASTICITY: NONCONSTANT_ERROR_VARIANCE
  AUTOCORRELATION: DEPENDENCE_BETWEEN_ERRORS

RELATIONSHIPS:
  MODEL -> PRODUCES -> EXPECTED_VALUE
  OBSERVATION -> DIFFERS_FROM -> EXPECTED_VALUE
  DIFFERENCE -> DEFINES -> ERROR_TERM
  ERROR_TERM -> FOLLOWS -> ERROR_DISTRIBUTION
  ERROR_DISTRIBUTION -> DETERMINES -> LIKELIHOOD
  ERROR_DISTRIBUTION -> INFLUENCES -> STANDARD_ERRORS
  ERROR_DISTRIBUTION -> INFLUENCES -> PREDICTION_INTERVALS
  RESIDUALS -> INFORM -> ERROR_MODEL_DIAGNOSTICS

ERROR_TERM_VS_RESIDUAL:
  error:
    epsilon_i := Y_i-E[Y_i|X_i]
  residual:
    e_i := Y_i-Y_hat_i
  distinction:
    ERROR_TERM != RESIDUAL

ADDITIVE_ERROR_MODEL:
  form:
    Y := mu(X;theta)+epsilon

MULTIPLICATIVE_ERROR_MODEL:
  form:
    Y := mu(X;theta)*eta
  positive_error:
    eta > 0
  log_transform:
    log(Y) = log(mu(X;theta))+log(eta)

NORMAL_ERROR:
  epsilon ~ Normal(0,sigma_squared)
  properties:
    E[epsilon] = 0
    Var(epsilon) = sigma_squared
    SYMMETRIC
    LIGHT_TAILED
  caution:
    NORMALITY_OF_ERRORS != NORMALITY_OF_PREDICTORS

ZERO_MEAN_ERROR:
  condition:
    E[epsilon|X] = 0
  implication:
    MODELLED_CONDITIONAL_MEAN_IS_CORRECTLY_CENTERED

HOMOSCEDASTIC_ERROR:
  condition:
    Var(epsilon|X) = sigma_squared

HETEROSCEDASTIC_ERROR:
  condition:
    Var(epsilon|X) := sigma_squared(X)
  consequence:
    ORDINARY_STANDARD_ERROR_FORMULAS_MAY_BE_INVALID
  responses:
    - HETEROSCEDASTICITY_ROBUST_STANDARD_ERRORS
    - WEIGHTED_LEAST_SQUARES
    - VARIANCE_FUNCTION_MODEL
    - TRANSFORMATION

STUDENT_T_ERROR:
  epsilon ~ StudentT(nu,location,scale)
  properties:
    HEAVIER_TAILS_THAN_NORMAL
    SYMMETRIC
  moments:
    MEAN_EXISTS_IF_nu_GREATER_THAN_1
    VARIANCE_EXISTS_IF_nu_GREATER_THAN_2

LAPLACE_ERROR:
  density:
    f(epsilon) := (1/(2b))*exp(-|epsilon-mu|/b)
  properties:
    SYMMETRIC
    HEAVIER_TAILS_THAN_NORMAL
  connection:
    MAXIMUM_LIKELIHOOD_WITH_LAPLACE_ERRORS
      -> ABSOLUTE_DEVIATION_MINIMISATION

CAUCHY_ERROR:
  distribution:
    Cauchy(location,scale)
  properties:
    VERY_HEAVY_TAILED
    MEAN_UNDEFINED
    VARIANCE_UNDEFINED

UNIFORM_ERROR:
  epsilon ~ Uniform(-a,a)
  mean:
    0
  variance:
    a^2/3

TRIANGULAR_ERROR:
  support:
    [-a,a]
  variance:
    a^2/6

LOGNORMAL_MULTIPLICATIVE_ERROR:
  eta ~ Lognormal(mu_eta,sigma_eta_squared)
  model:
    Y := m(X)*eta
  use:
    POSITIVE_SKEWED_RELATIVE_ERRORS

POISSON_ERROR_CONNECTION:
  count:
    Y IN NONNEGATIVE_INTEGERS
  model:
    Y ~ Poisson(lambda)
  relation:
    Var(Y)=E[Y]=lambda
  caution:
    POISSON_GLM_DOES_NOT_REQUIRE_ADDITIVE_NORMAL_ERROR_TERM

BINOMIAL_ERROR_CONNECTION:
  response:
    Y ~ Binomial(n,p)
  variance:
    np(1-p)
  caution:
    BINOMIAL_GLM_ERROR_STRUCTURE_DIFFERS_FROM_ADDITIVE_GAUSSIAN_ERROR

GENERALIZED_LINEAR_MODEL_CONNECTION:
  components:
    - RANDOM_COMPONENT
    - LINEAR_PREDICTOR
    - LINK_FUNCTION
  implication:
    ERROR_VARIANCE_MAY_DEPEND_ON_MEAN

MIXTURE_ERROR_DISTRIBUTION:
  form:
    f(epsilon) := SUM_k pi_k f_k(epsilon)
  requirements:
    pi_k >= 0
    SUM_k pi_k = 1
  use:
    HETEROGENEOUS_POPULATIONS
    CONTAMINATION
    MULTIMODAL_ERRORS

MEASUREMENT_ERROR_MODEL:
  observed:
    X_obs
  true:
    X_true
  model:
    X_obs := X_true+epsilon_m
  consequence:
    MEASUREMENT_ERROR_IN_PREDICTORS_CAN_BIAS_REGRESSION_ESTIMATES

CLASSICAL_MEASUREMENT_ERROR:
  model:
    X_obs := X_true+epsilon
  common_assumptions:
    E[epsilon]=0
    epsilon_INDEPENDENT_OF_X_true

BERKSON_ERROR:
  model:
    X_true := X_assigned+epsilon
  distinction:
    BERKSON_ERROR != CLASSICAL_MEASUREMENT_ERROR

CORRELATED_ERRORS:
  covariance:
    Cov(epsilon_i,epsilon_j) != 0
  consequence:
    INDEPENDENCE_BASED_STANDARD_ERRORS_AND_LIKELIHOODS_MAY_BE_INVALID

AUTOCORRELATED_ERRORS:
  time_series:
    epsilon_t
  autocovariance:
    gamma(h) := Cov(epsilon_t,epsilon_(t-h))
  examples:
    - AR1_ERROR
    - ARMA_ERROR

AR1_ERROR:
  model:
    epsilon_t := rho*epsilon_(t-1)+u_t
  innovation:
    u_t ~ IID(0,sigma_u_squared)
  stationarity:
    |rho| < 1

CLUSTERED_ERRORS:
  methods:
    - CLUSTER_ROBUST_STANDARD_ERRORS
    - MIXED_EFFECTS_MODEL
    - GENERALIZED_ESTIMATING_EQUATIONS

SPATIAL_ERRORS:
  covariance:
    Cov(epsilon(s_i),epsilon(s_j)) := C(distance_or_spatial_relation)
  methods:
    - SPATIAL_COVARIANCE_MODEL
    - GAUSSIAN_PROCESS
    - CONDITIONAL_AUTOREGRESSION

INDEPENDENT_ERRORS:
  condition:
    epsilon_i INDEPENDENT_OF epsilon_j
    FOR:
      i != j
  caution:
    UNCORRELATED != INDEPENDENT_IN_GENERAL

IID_ERRORS:
  requirements:
    INDEPENDENT
    IDENTICALLY_DISTRIBUTED

ERROR_VARIANCE:
  sigma_squared := Var(epsilon)

RESIDUAL_SUM_OF_SQUARES:
  aliases:
    RSS
    SSE
  definition:
    SUM_i e_i^2

ROOT_MEAN_SQUARED_ERROR:
  alias:
    RMSE
  definition:
    sqrt((1/n)*SUM_i e_i^2)
  caution:
    RMSE_IS_EMPIRICAL_ERROR_METRIC_NOT_COMPLETE_ERROR_DISTRIBUTION

MEAN_ABSOLUTE_ERROR:
  alias:
    MAE
  definition:
    (1/n)*SUM_i |e_i|

RESIDUAL_DIAGNOSTICS:
  inspect:
    - RESIDUAL_VS_FITTED
    - Q_Q_PLOT
    - SCALE_LOCATION_PLOT
    - AUTOCORRELATION
    - RESIDUAL_VS_PREDICTOR
    - LEVERAGE_AND_INFLUENCE

Q_Q_PLOT:
  comparison:
    EMPIRICAL_RESIDUAL_QUANTILES
    VS:
      THEORETICAL_DISTRIBUTION_QUANTILES
  caution:
    Q_Q_PLOT_IS_DIAGNOSTIC_NOT_FORMAL_PROOF_OF_DISTRIBUTION

OUTLIER:
  possible_causes:
    - DATA_ERROR
    - RARE_EVENT
    - HEAVY_TAIL
    - MODEL_MISSPECIFICATION
    - DIFFERENT_DATA_GENERATING_PROCESS
  rule:
    OUTLIER != ERROR_BY_DEFAULT

ROBUST_ERROR_MODELLING:
  methods:
    - STUDENT_T_ERRORS
    - HUBER_LOSS
    - QUANTILE_REGRESSION
    - MIXTURE_MODELS
    - ROBUST_STANDARD_ERRORS

LIKELIHOOD:
  observations:
    y_1,...,y_n
  model:
    y_i := mu_i+epsilon_i
  independent_errors:
    L(theta,phi) :=
      PRODUCT_i f_epsilon(y_i-mu_i;phi)
  role:
    ERROR_DISTRIBUTION_DETERMINES_LIKELIHOOD_SHAPE

LOG_LIKELIHOOD:
  independent_errors:
    ell(theta,phi) :=
      SUM_i log f_epsilon(e_i;phi)

MAXIMUM_LIKELIHOOD_CONNECTION:
  estimate:
    argmax_(theta,phi) L(theta,phi)
  consequence:
    DIFFERENT_ERROR_DISTRIBUTIONS_CAN_PRODUCE_DIFFERENT_ESTIMATORS

LEAST_SQUARES_CONNECTION:
  normal_errors:
    MAXIMUM_LIKELIHOOD_FOR_MEAN_PARAMETERS
      CONNECTS_TO:
        MINIMISING_SUM_OF_SQUARED_RESIDUALS
    UNDER:
      CONSTANT_VARIANCE_INDEPENDENT_GAUSSIAN_ERRORS

ABSOLUTE_DEVIATION_CONNECTION:
  Laplace_errors:
    MAXIMUM_LIKELIHOOD_LOCATION_ESTIMATION
      CONNECTS_TO:
        MINIMISING_SUM_ABSOLUTE_RESIDUALS

BAYESIAN_ERROR_MODEL:
  parameters:
    theta
    phi
  likelihood:
    p(D|theta,phi)
  priors:
    p(theta)
    p(phi)
  posterior:
    p(theta,phi|D)
      proportional_to
    p(D|theta,phi)p(theta)p(phi)

POSTERIOR_PREDICTIVE_ERROR:
  future:
    Y_new
  predictive:
    p(Y_new|D)
  uncertainty:
    INCLUDES:
      PARAMETER_UNCERTAINTY
      ERROR_DISTRIBUTION
      MODEL_STRUCTURE_AS_MODELLED

PREDICTION_INTERVAL:
  purpose:
    INTERVAL_FOR_FUTURE_OBSERVATION
  includes:
    PARAMETER_ESTIMATION_UNCERTAINTY
    OBSERVATIONAL_ERROR
  distinction:
    PREDICTION_INTERVAL != CONFIDENCE_INTERVAL_FOR_MEAN

MEASUREMENT_UNCERTAINTY_CONNECTION:
  error_distribution:
    MAY_INFORM:
      TYPE_A_UNCERTAINTY
      MEASUREMENT_MODEL
      MONTE_CARLO_PROPAGATION
  rule:
    ERROR_DISTRIBUTION != MEASUREMENT_UNCERTAINTY

SCIENTIFIC_STATISTICS_CONNECTION:
  error_model:
    INFLUENCES:
      ESTIMATION
      STANDARD_ERRORS
      HYPOTHESIS_TESTS
      CONFIDENCE_INTERVALS
      PREDICTIONS

SENSITIVITY_ANALYSIS_CONNECTION:
  vary:
    - DISTRIBUTION_FAMILY
    - SCALE
    - TAIL_PARAMETER
    - HETEROSCEDASTICITY_MODEL
    - DEPENDENCE_STRUCTURE
  observe:
    - PARAMETER_ESTIMATES
    - STANDARD_ERRORS
    - PREDICTIONS
    - DECISIONS

MODEL_MISSPECIFICATION:
  examples:
    - WRONG_MEAN_FUNCTION
    - WRONG_VARIANCE_FUNCTION
    - WRONG_DISTRIBUTION
    - IGNORED_DEPENDENCE
    - IGNORED_HEAVY_TAILS
    - IGNORED_SKEWNESS

  consequence:
    MAY_DISTORT:
      ESTIMATES
      UNCERTAINTY
      TESTS
      PREDICTIONS

DISTRIBUTION_SELECTION:
  evidence:
    - SCIENTIFIC_MECHANISM
    - SUPPORT_OF_RESPONSE
    - RESIDUAL_DIAGNOSTICS
    - LIKELIHOOD
    - INFORMATION_CRITERIA
    - CROSS_VALIDATION

  caution:
    SELECT_DISTRIBUTION_BY_SCIENTIFIC_AND_STATISTICAL_JUSTIFICATION
    NOT:
      NORMALITY_TEST_ALONE

TRANSFORMATION:
  examples:
    - LOG
    - SQUARE_ROOT
    - BOX_COX

  purpose:
    MAY_STABILISE_VARIANCE
    REDUCE_SKEWNESS
    LINEARISE_RELATIONSHIP

  caution:
    TRANSFORMATION_CHANGES_INTERPRETATION_SCALE

ERROR_PROPAGATION:
  input_error:
    epsilon_X

  model:
    Y := g(X)

  first_order:
    epsilon_Y approximately g_prime(X)*epsilon_X

  multivariate:
    Sigma_Y approximately J Sigma_X J_transpose

MONTE_CARLO_ERROR_PROPAGATION:
  flow:
    SAMPLE_ERROR_TERMS
      -> APPLY_TO_INPUTS_OR_MODEL
      -> EVALUATE_OUTPUT
      -> REPEAT
      -> ESTIMATE_OUTPUT_DISTRIBUTION

APPLICATIONS:
  ERROR_DISTRIBUTION -> SUPPORTS -> SCIENTIFIC_STATISTICS
  ERROR_DISTRIBUTION -> SUPPORTS -> REGRESSION
  ERROR_DISTRIBUTION -> SUPPORTS -> MEASUREMENT_UNCERTAINTY
  ERROR_DISTRIBUTION -> SUPPORTS -> HYPOTHESIS_TESTING
  ERROR_DISTRIBUTION -> SUPPORTS -> PREDICTION
  ERROR_DISTRIBUTION -> SUPPORTS -> TIME_SERIES
  ERROR_DISTRIBUTION -> SUPPORTS -> MACHINE_LEARNING

MODEL_CONNECTIONS:
  ERROR_DISTRIBUTION -> CONNECTS_TO -> SCIENTIFIC_STATISTICS
  ERROR_DISTRIBUTION -> CONNECTS_TO -> MEASUREMENT_UNCERTAINTY
  ERROR_DISTRIBUTION -> CONNECTS_TO -> PROBABILITY_DISTRIBUTIONS
  ERROR_DISTRIBUTION -> CONNECTS_TO -> REGRESSION
  ERROR_DISTRIBUTION -> CONNECTS_TO -> BAYESIAN_STATISTICS
  ERROR_DISTRIBUTION -> CONNECTS_TO -> TIME_SERIES_STATISTICS
  ERROR_DISTRIBUTION -> CONNECTS_TO -> MONTE_CARLO_SIMULATION

SEMANTIC_EQUIVALENCE:
  "error distribution"
    ~= "probability distribution assigned to model error term"

  "residual"
    ~= "observed discrepancy between observation and fitted value"

  "homoscedastic error"
    ~= "constant conditional error variance"

  "heteroscedastic error"
    ~= "nonconstant conditional error variance"

  EXCLUDE_EQUIVALENCE:
    ERROR_TERM != RESIDUAL
    ERROR != UNCERTAINTY
    ERROR_DISTRIBUTION != MEASUREMENT_UNCERTAINTY
    UNCORRELATED != INDEPENDENT
    OUTLIER != ERROR
    RMSE != ERROR_DISTRIBUTION
    PREDICTION_INTERVAL != CONFIDENCE_INTERVAL_FOR_MEAN
    NORMAL_ERROR != NORMAL_PREDICTOR

VALIDATION_RULES:
  RULE_1:
    ERROR_TERM_AND_MODELLED_MEAN_OR_LOCATION_MUST_BE_DEFINED
  RULE_2:
    ERROR_DISTRIBUTION_SUPPORT_MUST_MATCH_RESPONSE_AND_MODEL_STRUCTURE
  RULE_3:
    HETEROSCEDASTICITY_MUST_BE_MODELLED_OR_ACCOUNTED_FOR_WHEN_PRESENT
  RULE_4:
    DEPENDENCE_BETWEEN_ERRORS_MUST_BE_MODELLED_WHEN_MATERIAL
  RULE_5:
    RESIDUALS_MUST_NOT_BE_IDENTIFIED_AS_TRUE_UNOBSERVED_ERRORS
  RULE_6:
    HEAVY_TAILS_SKEWNESS_AND_OUTLIERS_SHOULD_BE_ASSESSED_WHEN_RELEVANT
  RULE_7:
    DISTRIBUTION_SELECTION_SHOULD_USE_SCIENTIFIC_AND_STATISTICAL_JUSTIFICATION
  RULE_8:
    ERROR_DISTRIBUTION_ASSUMPTIONS_SHOULD_BE_INCLUDED_IN_SENSITIVITY_ANALYSIS_WHEN_MATERIAL

CONFIDENCE:
  ontology_identity: 0.99
  error_residual_distinction: 0.99
  distribution_structure: 0.99
  heteroscedasticity_structure: 0.99
  dependence_structure: 0.99
  likelihood_relationship: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF:
    NEW_RESIDUAL_DATA_ARRIVES
  THEN:
    UPDATE:
      DISTRIBUTION_PARAMETERS
      VARIANCE_MODEL
      TAIL_ESTIMATES
      DEPENDENCE_ESTIMATES
    RERUN:
      DIAGNOSTICS

  IF:
    ERROR_DISTRIBUTION_FAMILY_CHANGES
  THEN:
    UPDATE:
      LIKELIHOOD
      PARAMETER_ESTIMATES
      STANDARD_ERRORS
      PREDICTION_INTERVALS
    REASSESS:
      MODEL_FIT

  IF:
    DEPENDENCE_STRUCTURE_CHANGES
  THEN:
    UPDATE:
      COVARIANCE_MODEL
      LIKELIHOOD
      STANDARD_ERRORS

  PRESERVE:
    RESPONSE_DEFINITION
    MODEL_MEAN_STRUCTURE
    ERROR_TERM_DEFINITION
    DISTRIBUTION_FAMILY
    PARAMETERISATION
    VARIANCE_MODEL
    DEPENDENCE_STRUCTURE
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_OBSERVATION_AND_MODELLED_COMPONENT
    - DEFINE_ERROR_TERM
    - DISTINGUISH_ERRORS_FROM_RESIDUALS
    - IDENTIFY_RESPONSE_SUPPORT
    - SELECT_CANDIDATE_ERROR_DISTRIBUTION
    - ASSESS_LOCATION_SCALE_SHAPE_AND_TAILS
    - ASSESS_HETEROSCEDASTICITY
    - ASSESS_DEPENDENCE
    - CONSTRUCT_LIKELIHOOD
    - RUN_RESIDUAL_DIAGNOSTICS
    - PERFORM_SENSITIVITY_TO_ERROR_MODEL
    - RETURN_ERROR_DISTRIBUTION_INTERPRETATION

  constraints:
    - PRESERVE_ERROR_TERM_SEMANTICS
    - DISTINGUISH_ERROR_FROM_RESIDUAL
    - DISTINGUISH_ERROR_FROM_UNCERTAINTY
    - MATCH_DISTRIBUTION_TO_RESPONSE_SUPPORT
    - MODEL_HETEROSCEDASTICITY_WHEN_PRESENT
    - MODEL_DEPENDENCE_WHEN_PRESENT
    - DO_NOT_ASSUME_NORMALITY_BY_DEFAULT
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "error-distribution"
  namespace: "probability-statistics/scientific-statistical-methods"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - ERROR_TERM
    - DISTRIBUTION_FAMILY
    - VARIANCE_AND_DEPENDENCE
    - LIKELIHOOD_AND_DIAGNOSTICS

END_FLOW