Sensitivity Analysis Ontology

flow "SensitivityAnalysisOntology" v4.0

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

PAGE:
  url: "https://kirethresearch.org/booksai-core/ontologies/probability-statistics-ontologies/risk-decision-mathematics/sensitivity-analysis-ontology.html"
  title: "Sensitivity Analysis Ontology"
  domain: "Probability and Statistics"
  category: "Risk & Decision Mathematics"
  entity: SENSITIVITY_ANALYSIS
  entity_type: RISK_DECISION_ONTOLOGY

DEFINE:
  SENSITIVITY_ANALYSIS :=
    MODEL_RESPONSE_FRAMEWORK(
      INPUTS,
      PARAMETERS,
      ASSUMPTIONS,
      MODEL_OUTPUTS,
      PERTURBATIONS,
      SENSITIVITY_MEASURES
    )

PURPOSE:
  QUANTIFY:
    OUTPUT_RESPONSE_TO_INPUT_CHANGE
    PARAMETER_INFLUENCE
    MODEL_ROBUSTNESS
    UNCERTAINTY_IMPORTANCE
    DECISION_STABILITY
  IDENTIFY:
    DOMINANT_INPUTS
    FRAGILE_ASSUMPTIONS
    NONLINEAR_EFFECTS
    INTERACTIONS

FOUNDATION:
  MODEL:
    Y := f(X_1,...,X_k;theta)
  INPUT_VECTOR:
    X := (X_1,...,X_k)
  PARAMETER_VECTOR:
    theta := (theta_1,...,theta_m)
  BASELINE:
    Y_0 := f(X_0;theta_0)
  PERTURBED_OUTPUT:
    Y_prime := f(X_0+Delta_X,theta_0+Delta_theta)

ENTITIES:
  SENSITIVITY_ANALYSIS: MODEL_RESPONSE_ANALYSIS
  INPUT_VARIABLE: MODEL_INPUT
  PARAMETER: MODEL_CONTROL_OR_ESTIMATED_QUANTITY
  OUTPUT: MODEL_RESPONSE
  PERTURBATION: CONTROLLED_INPUT_CHANGE
  LOCAL_SENSITIVITY: DERIVATIVE_BASED_RESPONSE
  GLOBAL_SENSITIVITY: FULL_INPUT_SPACE_RESPONSE
  SENSITIVITY_INDEX: INPUT_INFLUENCE_MEASURE
  ELASTICITY: NORMALISED_RELATIVE_SENSITIVITY
  INTERACTION_EFFECT: JOINT_INPUT_INFLUENCE
  ROBUSTNESS: STABILITY_UNDER_PERTURBATION
  UNCERTAINTY: VARIATION_OR_LACK_OF_KNOWLEDGE

RELATIONSHIPS:
  INPUT_VARIABLE -> INFLUENCES -> OUTPUT
  PARAMETER -> INFLUENCES -> OUTPUT
  PERTURBATION -> CHANGES -> MODEL_INPUT
  INPUT_CHANGE -> PRODUCES -> OUTPUT_CHANGE
  OUTPUT_CHANGE -> QUANTIFIED_BY -> SENSITIVITY_MEASURE
  UNCERTAINTY -> PROPAGATES_THROUGH -> MODEL
  SENSITIVITY_ANALYSIS -> IDENTIFIES -> IMPORTANT_INPUTS
  SENSITIVITY_ANALYSIS -> SUPPORTS -> ROBUST_DECISION_MAKING

LOCAL_SENSITIVITY:
  parameter:
    theta_j
  derivative:
    S_j := partial_Y/partial_theta_j
  evaluated_at:
    BASELINE_POINT

FINITE_DIFFERENCE_SENSITIVITY:
  forward:
    S_j := [f(theta_j+h)-f(theta_j)]/h
  central:
    S_j := [f(theta_j+h)-f(theta_j-h)]/(2h)
  caution:
    STEP_SIZE_TOO_LARGE_CAUSES_TRUNCATION_ERROR
    STEP_SIZE_TOO_SMALL_CAN_CAUSE_NUMERICAL_ERROR

NORMALISED_SENSITIVITY:
  relative:
    S_j_normalised :=
      (theta_j/Y)
      * partial_Y/partial_theta_j

ELASTICITY:
  E_j :=
    partial_log_Y/partial_log_X_j
  equivalent_local_form:
    E_j := (X_j/Y)*partial_Y/partial_X_j

SECOND_ORDER_SENSITIVITY:
  curvature:
    partial_squared_Y/partial_theta_j_squared

CROSS_PARTIAL:
  interaction:
    partial_squared_Y/(partial_theta_i partial_theta_j)

ONE_AT_A_TIME:
  alias:
    OAT
  flow:
    SET_BASELINE
      -> VARY_ONE_INPUT
      -> HOLD_OTHERS_FIXED
      -> COMPUTE_OUTPUT
      -> REPEAT_FOR_INPUTS
  limitations:
    MAY_MISS_INTERACTIONS
    MAY_NOT_EXPLORE_FULL_INPUT_SPACE

SCENARIO_SENSITIVITY:
  scenarios:
    BASE
    LOW
    HIGH
  flow:
    DEFINE_SCENARIO_INPUTS
      -> RUN_MODEL
      -> COMPARE_OUTPUTS

TORNADO_ANALYSIS:
  procedure:
    VARY_EACH_INPUT_OVER_SPECIFIED_RANGE
      -> RECORD_OUTPUT_RANGE
      -> ORDER_BY_OUTPUT_EFFECT
  purpose:
    RANK_INPUT_INFLUENCE_UNDER_SPECIFIED_RANGES

THRESHOLD_ANALYSIS:
  objective:
    FIND_INPUT_VALUE_AT_WHICH:
      DECISION_CHANGES
      SIGN_CHANGES
      CONSTRAINT_BECOMES_ACTIVE
      TARGET_IS_REACHED

BREAK_EVEN_ANALYSIS:
  solve:
    f(X;theta) = TARGET

GLOBAL_SENSITIVITY:
  definition:
    ANALYSE_INPUT_INFLUENCE_ACROSS_FULL_SPECIFIED_INPUT_DISTRIBUTION_OR_RANGE
  captures:
    NONLINEARITY
    INTERACTIONS
    NONMONOTONIC_RESPONSE
  methods:
    - SOBOL_INDICES
    - MORRIS_METHOD
    - VARIANCE_BASED_ANALYSIS
    - DERIVATIVE_BASED_GLOBAL_MEASURES
    - REGRESSION_BASED_METHODS
    - CORRELATION_BASED_METHODS

VARIANCE_BASED_SENSITIVITY:
  output_variance:
    Var(Y)
  decomposition:
    ATTRIBUTE_COMPONENTS_OF_OUTPUT_VARIANCE_TO_INPUTS_AND_INTERACTIONS

SOBOL_FIRST_ORDER_INDEX:
  definition:
    S_i := Var(E[Y|X_i])/Var(Y)
  interpretation:
    FRACTION_OF_OUTPUT_VARIANCE_ATTRIBUTABLE_TO_X_i_ALONE

SOBOL_TOTAL_ORDER_INDEX:
  definition:
    S_Ti := 1-Var(E[Y|X_minus_i])/Var(Y)
  interpretation:
    CONTRIBUTION_OF_X_i_INCLUDING_ALL_INTERACTIONS

SOBOL_INTERACTION:
  second_order:
    S_ij
  role:
    MEASURE_VARIANCE_CONTRIBUTION_FROM_INTERACTION

MORRIS_METHOD:
  alias:
    ELEMENTARY_EFFECTS_METHOD
  elementary_effect:
    EE_i := [f(X+Delta*e_i)-f(X)]/Delta
  summaries:
    mu_star_i: MEAN_ABSOLUTE_ELEMENTARY_EFFECT
    sigma_i: STANDARD_DEVIATION_OF_ELEMENTARY_EFFECTS
  interpretation:
    HIGH_mu_star: IMPORTANT_INPUT
    HIGH_sigma: NONLINEARITY_OR_INTERACTION

REGRESSION_SENSITIVITY:
  standardised_regression_coefficient:
    SRC_i
  caution:
    STRONG_NONLINEARITY_OR_COLLINEARITY_CAN_LIMIT_INTERPRETATION

CORRELATION_SENSITIVITY:
  methods:
    - PEARSON_CORRELATION
    - SPEARMAN_RANK_CORRELATION
    - PARTIAL_CORRELATION
    - PARTIAL_RANK_CORRELATION
  caution:
    CORRELATION_DOES_NOT_ESTABLISH_CAUSAL_INFLUENCE

UNCERTAINTY_ANALYSIS_DISTINCTION:
  uncertainty_analysis:
    QUANTIFIES_RANGE_OR_DISTRIBUTION_OF_MODEL_OUTPUT
  sensitivity_analysis:
    ATTRIBUTES_OUTPUT_VARIATION_TO_INPUTS_OR_ASSUMPTIONS
  rule:
    UNCERTAINTY_ANALYSIS != SENSITIVITY_ANALYSIS

MONTE_CARLO_CONNECTION:
  flow:
    SAMPLE_INPUT_DISTRIBUTIONS
      -> RUN_MODEL
      -> STORE_INPUT_OUTPUT_PAIRS
      -> COMPUTE_SENSITIVITY_MEASURES

INPUT_DISTRIBUTIONS:
  requirement:
    DEFINE:
      RANGE
      DISTRIBUTION
      DEPENDENCE
      UNITS
  consequence:
    GLOBAL_SENSITIVITY_RESULTS_DEPEND_ON_SPECIFIED_INPUT_DISTRIBUTIONS

DEPENDENT_INPUTS:
  issue:
    X_i_NOT_INDEPENDENT
  methods:
    - CONDITIONAL_SAMPLING
    - COPULA_MODELS
    - SHAPLEY_EFFECTS
    - DEPENDENT_SOBOL_EXTENSIONS

SHAPLEY_EFFECTS:
  origin:
    COOPERATIVE_GAME_THEORY
  role:
    ALLOCATE_OUTPUT_VARIANCE_OR_EXPLANATORY_POWER_ACROSS_INPUTS
  caution:
    COMPUTATION_CAN_BE_EXPENSIVE

PARAMETER_UNCERTAINTY:
  distribution:
    p(theta)
  sensitivity:
    MEASURES_HOW_PARAMETER_VARIATION_AFFECTS_OUTPUT

STRUCTURAL_SENSITIVITY:
  vary:
    - MODEL_FORM
    - DISTRIBUTIONAL_ASSUMPTION
    - FUNCTIONAL_RELATIONSHIP
    - DEPENDENCE_STRUCTURE

ASSUMPTION_SENSITIVITY:
  flow:
    ALTER_ASSUMPTION
      -> REFIT_OR_RERUN_MODEL
      -> COMPARE_RESULTS
  examples:
    - PRIOR_DISTRIBUTION
    - MISSING_DATA_MECHANISM
    - ERROR_DISTRIBUTION
    - DISCOUNT_RATE
    - DEPENDENCE_MODEL

BAYESIAN_SENSITIVITY:
  vary:
    PRIOR
    LIKELIHOOD
    HYPERPARAMETERS
    MODEL_STRUCTURE
  observe:
    POSTERIOR
    PREDICTIONS
    DECISIONS

RISK_ANALYSIS_CONNECTION:
  risk_measure:
    R(theta)
  sensitivity:
    partial_R/partial_theta_i
  outputs:
    - EXPECTED_LOSS_SENSITIVITY
    - VAR_SENSITIVITY
    - EXPECTED_SHORTFALL_SENSITIVITY
    - EXCEEDANCE_PROBABILITY_SENSITIVITY

DECISION_THEORY_CONNECTION:
  decision:
    a_star(theta)
  sensitivity:
    IDENTIFY_WHETHER_PARAMETER_CHANGES_ALTER_OPTIMAL_ACTION
  threshold:
    PARAMETER_VALUE_WHERE_DECISION_SWITCHES

EXPECTED_VALUE_CONNECTION:
  expected_output:
    E[Y]
  sensitivity:
    partial E[Y]/partial theta_i

UTILITY_THEORY_CONNECTION:
  expected_utility:
    EU(a;theta)
  sensitivity:
    ANALYSE_EFFECT_OF:
      RISK_PREFERENCE
      UTILITY_CURVATURE
      PROBABILITY_BELIEFS
      OUTCOME_VALUES

ROBUSTNESS:
  definition:
    CONCLUSION_OR_DECISION_REMAINS_STABLE_ACROSS_PLAUSIBLE_INPUT_OR_ASSUMPTION_CHANGES
  indicators:
    - SMALL_OUTPUT_CHANGE
    - STABLE_PARAMETER_SIGN
    - STABLE_RANKING
    - STABLE_OPTIMAL_ACTION

FRAGILITY:
  definition:
    SMALL_INPUT_OR_ASSUMPTION_CHANGE_CAUSES_LARGE_OUTPUT_OR_DECISION_CHANGE
  response:
    REQUIRE:
      MORE_DATA
      MODEL_REFINEMENT
      CONSERVATIVE_DECISION
      MONITORING

MODEL_VALIDATION_CONNECTION:
  sensitivity_results:
    SUPPORT:
      TEST_DESIGN
      DATA_COLLECTION
      PARAMETER_VALIDATION
      MODEL_DIAGNOSTICS

EXPERIMENT_DESIGN_CONNECTION:
  important_inputs:
    PRIORITISE_FOR:
      PRECISE_MEASUREMENT
      DATA_COLLECTION
      CONTROLLED_EXPERIMENTS

VALUE_OF_INFORMATION_CONNECTION:
  sensitivity:
    IDENTIFIES_POTENTIALLY_DECISION_RELEVANT_UNCERTAINTIES
  VOI:
    QUANTIFIES_EXPECTED_BENEFIT_OF_REDUCING_UNCERTAINTY
  rule:
    HIGH_SENSITIVITY != HIGH_VALUE_OF_INFORMATION_BY_DEFAULT

SCREENING:
  objective:
    IDENTIFY_SMALL_SET_OF_INFLUENTIAL_INPUTS_FROM_LARGE_INPUT_SET
  methods:
    - MORRIS
    - ELEMENTARY_EFFECTS
    - REGRESSION_SCREENING
    - CORRELATION_SCREENING

DIMENSION_REDUCTION:
  flow:
    SCREEN_INPUTS
      -> FIX_NONINFLUENTIAL_INPUTS_WHEN_JUSTIFIED
      -> ANALYSE_REMAINING_INPUTS_IN_DETAIL
  caution:
    FIXING_INPUTS_CAN_REMOVE_INTERACTION_EFFECTS

NONLINEARITY:
  evidence:
    - CURVATURE
    - VARYING_LOCAL_DERIVATIVE
    - MORRIS_SIGMA
    - NONMONOTONIC_RESPONSE
  implication:
    LOCAL_SENSITIVITY_MAY_NOT_REPRESENT_GLOBAL_BEHAVIOUR

INTERACTIONS:
  definition:
    EFFECT_OF_X_i_DEPENDS_ON_VALUE_OF_X_j
  consequence:
    ONE_AT_A_TIME_METHODS_MAY_MISS_IMPORTANT_JOINT_EFFECTS

OUTPUT_SELECTION:
  multiple_outputs:
    Y_1,...,Y_r
  requirement:
    ANALYSE_EACH_DECISION_RELEVANT_OUTPUT

TIME_DEPENDENT_SENSITIVITY:
  output:
    Y(t)
  sensitivity:
    S_i(t)

SPATIAL_SENSITIVITY:
  output:
    Y(s)
  sensitivity:
    S_i(s)

SAMPLE_SIZE:
  global_methods:
    REQUIRE_SUFFICIENT_MODEL_EVALUATIONS
  tradeoff:
    MORE_SAMPLES:
      IMPROVE_ESTIMATION_PRECISION
      INCREASE_COMPUTATIONAL_COST
  requirement:
    REPORT_ESTIMATION_UNCERTAINTY_WHEN_RELEVANT

CONVERGENCE:
  checks:
    - REPEAT_WITH_LARGER_SAMPLE
    - BOOTSTRAP_INTERVALS
    - REPLICATED_DESIGNS
    - MONTE_CARLO_STANDARD_ERROR

SURROGATE_MODEL:
  purpose:
    APPROXIMATE_EXPENSIVE_MODEL_FOR_GLOBAL_SENSITIVITY
  examples:
    - GAUSSIAN_PROCESS
    - POLYNOMIAL_CHAOS
    - NEURAL_NETWORK
    - RESPONSE_SURFACE
  caution:
    SURROGATE_ERROR_MUST_BE_CONSIDERED

APPLICATIONS:
  SENSITIVITY_ANALYSIS -> SUPPORTS -> RISK_ANALYSIS
  SENSITIVITY_ANALYSIS -> SUPPORTS -> DECISION_THEORY
  SENSITIVITY_ANALYSIS -> SUPPORTS -> MODEL_VALIDATION
  SENSITIVITY_ANALYSIS -> SUPPORTS -> UNCERTAINTY_QUANTIFICATION
  SENSITIVITY_ANALYSIS -> SUPPORTS -> EXPERIMENT_DESIGN
  SENSITIVITY_ANALYSIS -> SUPPORTS -> ROBUST_OPTIMISATION
  SENSITIVITY_ANALYSIS -> SUPPORTS -> MONTE_CARLO_SIMULATION

MODEL_CONNECTIONS:
  SENSITIVITY_ANALYSIS -> CONNECTS_TO -> RISK_ANALYSIS
  SENSITIVITY_ANALYSIS -> CONNECTS_TO -> DECISION_THEORY
  SENSITIVITY_ANALYSIS -> CONNECTS_TO -> EXPECTED_VALUE
  SENSITIVITY_ANALYSIS -> CONNECTS_TO -> UTILITY_THEORY
  SENSITIVITY_ANALYSIS -> CONNECTS_TO -> MONTE_CARLO_SIMULATION
  SENSITIVITY_ANALYSIS -> CONNECTS_TO -> BAYESIAN_STATISTICS
  SENSITIVITY_ANALYSIS -> CONNECTS_TO -> OPTIMISATION

SEMANTIC_EQUIVALENCE:
  "sensitivity analysis"
    ~= "analysis of model response to input or assumption changes"

  "local sensitivity"
    ~= "derivative-based response near a reference point"

  "global sensitivity"
    ~= "input influence across specified input space or distribution"

  "elasticity"
    ~= "normalised relative local sensitivity"
    UNDER:
      APPROPRIATE_NONZERO_VALUES

  EXCLUDE_EQUIVALENCE:
    SENSITIVITY_ANALYSIS != UNCERTAINTY_ANALYSIS
    LOCAL_SENSITIVITY != GLOBAL_SENSITIVITY
    CORRELATION != CAUSAL_EFFECT
    HIGH_SENSITIVITY != HIGH_UNCERTAINTY
    HIGH_SENSITIVITY != HIGH_VALUE_OF_INFORMATION
    PARAMETER_IMPORTANCE != PARAMETER_UNCERTAINTY
    ROBUSTNESS != INSENSITIVITY_IN_ALL_CONTEXTS

VALIDATION_RULES:
  RULE_1:
    MODEL_INPUTS_PARAMETERS_OUTPUTS_AND_BASELINE_MUST_BE_DEFINED
  RULE_2:
    PERTURBATION_RANGE_OR_INPUT_DISTRIBUTION_MUST_BE_EXPLICIT
  RULE_3:
    LOCAL_AND_GLOBAL_SENSITIVITY_RESULTS_MUST_NOT_BE_CONFLATED
  RULE_4:
    INPUT_DEPENDENCE_MUST_BE_MODELLED_WHEN_RELEVANT
  RULE_5:
    GLOBAL_SENSITIVITY_ESTIMATION_UNCERTAINTY_SHOULD_BE_QUANTIFIED
  RULE_6:
    INTERACTIONS_AND_NONLINEARITY_SHOULD_BE_ASSESSED_WHEN_ONE_AT_A_TIME_METHODS_ARE_INSUFFICIENT
  RULE_7:
    SENSITIVITY_RESULTS_MUST_BE_INTERPRETED_RELATIVE_TO_SPECIFIED_INPUT_RANGES_AND_DISTRIBUTIONS
  RULE_8:
    ASSOCIATIONAL_SENSITIVITY_MEASURES_MUST_NOT_BE_INTERPRETED_AS_CAUSAL_EFFECTS_WITHOUT_CAUSAL_ASSUMPTIONS

CONFIDENCE:
  ontology_identity: 0.99
  local_sensitivity_structure: 0.99
  global_sensitivity_structure: 0.99
  sobol_structure: 0.99
  uncertainty_relationship: 0.99
  decision_robustness_structure: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF:
    INPUT_RANGES_OR_DISTRIBUTIONS_CHANGE
  THEN:
    UPDATE:
      GLOBAL_SENSITIVITY_DESIGN
      SENSITIVITY_INDICES
      INPUT_IMPORTANCE_RANKING

  IF:
    MODEL_PARAMETERS_OR_STRUCTURE_CHANGE
  THEN:
    RERUN:
      LOCAL_SENSITIVITY
      GLOBAL_SENSITIVITY
      ROBUSTNESS_ANALYSIS

  IF:
    DECISION_CRITERIA_CHANGE
  THEN:
    REASSESS:
      DECISION_THRESHOLDS
      DECISION_SENSITIVITY
      VALUE_OF_INFORMATION_RELEVANCE

  PRESERVE:
    MODEL_VERSION
    INPUT_DEFINITIONS
    PARAMETER_DEFINITIONS
    OUTPUT_DEFINITIONS
    BASELINE_VALUES
    INPUT_RANGES
    INPUT_DISTRIBUTIONS
    DEPENDENCE_STRUCTURE
    SENSITIVITY_METHOD
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_MODEL_INPUTS_PARAMETERS_AND_OUTPUTS
    - DEFINE_BASELINE_AND_PERTURBATION_DOMAIN
    - DETERMINE_LOCAL_OR_GLOBAL_ANALYSIS_REQUIREMENT
    - COMPUTE_LOCAL_DERIVATIVES_IF_RELEVANT
    - DEFINE_INPUT_DISTRIBUTIONS_AND_DEPENDENCE
    - GENERATE_GLOBAL_SAMPLE_DESIGN
    - EXECUTE_MODEL
    - COMPUTE_SENSITIVITY_INDICES
    - ASSESS_INTERACTIONS_AND_NONLINEARITY
    - QUANTIFY_ESTIMATION_UNCERTAINTY
    - ASSESS_DECISION_ROBUSTNESS
    - RETURN_SENSITIVITY_INTERPRETATION

  constraints:
    - PRESERVE_INPUT_PARAMETER_OUTPUT_SEMANTICS
    - PRESERVE_BASELINE_AND_PERTURBATION_RANGES
    - DISTINGUISH_LOCAL_FROM_GLOBAL_SENSITIVITY
    - DISTINGUISH_SENSITIVITY_FROM_UNCERTAINTY
    - MODEL_INPUT_DEPENDENCE_WHEN_RELEVANT
    - QUANTIFY_MONTE_CARLO_ERROR_WHEN_RELEVANT
    - DISTINGUISH_ASSOCIATION_FROM_CAUSATION
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "sensitivity-analysis"
  namespace: "probability-statistics/risk-decision-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - LOCAL_SENSITIVITY
    - GLOBAL_SENSITIVITY
    - INPUT_INFLUENCE
    - DECISION_ROBUSTNESS

END_FLOW