Reliability Statistics Ontology

flow "ReliabilityStatisticsOntology" v4.0

# RELIABILITY STATISTICS 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/reliability-statistics-ontology.html"
  title: "Reliability Statistics Ontology"
  domain: "Probability and Statistics"
  category: "Risk & Decision Mathematics"
  entity: RELIABILITY_STATISTICS
  entity_type: RELIABILITY_RISK_ONTOLOGY

DEFINE:
  RELIABILITY_STATISTICS :=
    STATISTICAL_FRAMEWORK(
      TIME_TO_FAILURE,
      SURVIVAL_FUNCTION,
      FAILURE_DISTRIBUTION,
      HAZARD_FUNCTION,
      CENSORING,
      SYSTEM_STRUCTURE,
      RELIABILITY_ESTIMATION
    )

PURPOSE:
  QUANTIFY:
    PROBABILITY_OF_SURVIVAL
    FAILURE_BEHAVIOUR
    TIME_TO_FAILURE
    FAILURE_RATE
    SYSTEM_RELIABILITY
    COMPONENT_RELIABILITY
  SUPPORT:
    RELIABILITY_ENGINEERING
    RISK_ANALYSIS
    MAINTENANCE_PLANNING
    QUALITY_CONTROL
    SURVIVAL_ANALYSIS
    SYSTEM_SAFETY

FOUNDATION:
  FAILURE_TIME: T >= 0
  CDF: F(t) := P(T <= t)
  RELIABILITY_FUNCTION: R(t) := P(T > t)
  relation:
    R(t) := 1-F(t)
  PDF:
    f(t) := dF(t)/dt
  HAZARD_FUNCTION:
    h(t) := f(t)/R(t)
  CUMULATIVE_HAZARD:
    H(t) := INTEGRAL_FROM_0_TO_t h(u)du
  relation_continuous:
    R(t) := exp(-H(t))

ENTITIES:
  RELIABILITY: PROBABILITY_OF_FUNCTIONING_WITHOUT_FAILURE
  FAILURE: LOSS_OF_REQUIRED_FUNCTION
  FAILURE_TIME: RANDOM_TIME_TO_FAILURE
  RELIABILITY_FUNCTION: SURVIVAL_PROBABILITY
  FAILURE_DISTRIBUTION: DISTRIBUTION_OF_TIME_TO_FAILURE
  HAZARD_FUNCTION: INSTANTANEOUS_FAILURE_INTENSITY_CONDITIONAL_ON_SURVIVAL
  CUMULATIVE_HAZARD: ACCUMULATED_HAZARD
  MTTF: MEAN_TIME_TO_FAILURE
  MTBF: MEAN_TIME_BETWEEN_FAILURES
  CENSORING: PARTIAL_FAILURE_TIME_OBSERVATION
  COMPONENT: SYSTEM_ELEMENT
  SYSTEM: COLLECTION_OF_COMPONENTS
  AVAILABILITY: PROBABILITY_SYSTEM_IS_OPERABLE

RELATIONSHIPS:
  FAILURE_TIME -> HAS -> FAILURE_DISTRIBUTION
  FAILURE_DISTRIBUTION -> DETERMINES -> RELIABILITY_FUNCTION
  RELIABILITY_FUNCTION -> CONNECTS_TO -> HAZARD_FUNCTION
  HAZARD_FUNCTION -> INTEGRATES_TO -> CUMULATIVE_HAZARD
  COMPONENT_RELIABILITY -> COMBINES_TO -> SYSTEM_RELIABILITY
  CENSORING -> AFFECTS -> RELIABILITY_ESTIMATION
  MAINTENANCE -> MODIFIES -> FAILURE_PROCESS
  RELIABILITY_STATISTICS -> SUPPORTS -> RISK_ANALYSIS

RELIABILITY_FUNCTION:
  R(t) := P(T > t)
  properties:
    NONINCREASING_IN_t

FAILURE_PROBABILITY:
  F(t) := P(T <= t)
  relation:
    F(t)+R(t)=1

DENSITY_RELATION:
  f(t) := -dR(t)/dt

HAZARD_FUNCTION:
  interpretation:
    INSTANTANEOUS_FAILURE_RATE_CONDITIONAL_ON_SURVIVAL
  caution:
    HAZARD_IS_NOT_A_PROBABILITY

CUMULATIVE_HAZARD:
  H(t) := INTEGRAL_FROM_0_TO_t h(u)du
  relation:
    H(t) := -log(R(t))

MEAN_TIME_TO_FAILURE:
  alias: MTTF
  definition:
    E[T]
  identity:
    E[T] := INTEGRAL_FROM_0_TO_INFINITY R(t)dt
    WHEN:
      EXPECTATION_FINITE

B_QUANTILE_LIFE:
  examples:
    B10_LIFE:
      P(T <= t_B10) := 0.10
    B50_LIFE:
      P(T <= t_B50) := 0.50

EXPONENTIAL_RELIABILITY:
  T ~ Exponential(lambda)
  hazard:
    h(t) := lambda
  reliability:
    R(t) := exp(-lambda*t)
  MTTF:
    1/lambda
  property:
    MEMORYLESS

WEIBULL_RELIABILITY:
  parameters:
    shape: beta > 0
    scale: eta > 0
  reliability:
    R(t) := exp(-(t/eta)^beta)
  hazard:
    h(t) := (beta/eta)*(t/eta)^(beta-1)
  shape_interpretation:
    beta < 1: DECREASING_HAZARD
    beta = 1: CONSTANT_HAZARD
    beta > 1: INCREASING_HAZARD

LOGNORMAL_RELIABILITY:
  IF:
    log(T) ~ Normal(mu,sigma_squared)
  THEN:
    T ~ Lognormal(mu,sigma_squared)
  use:
    POSITIVE_SKEWED_FAILURE_TIMES

BATHTUB_CURVE:
  phases:
    EARLY_FAILURE
    USEFUL_LIFE
    WEAR_OUT
  hazard_pattern:
    DECREASING
      -> APPROXIMATELY_CONSTANT
      -> INCREASING
  caution:
    NOT_ALL_SYSTEMS_FOLLOW_BATHTUB_HAZARD

CENSORING:
  types:
    - RIGHT_CENSORING
    - LEFT_CENSORING
    - INTERVAL_CENSORING

RIGHT_CENSORING:
  observation:
    T > C
  recorded:
    observed_time := C
    event_indicator := 0

COMPLETE_FAILURE:
  recorded:
    observed_time := T
    event_indicator := 1

KAPLAN_MEIER_ESTIMATOR:
  alias:
    PRODUCT_LIMIT_ESTIMATOR
  estimate:
    R_hat(t) :=
      PRODUCT_over_failure_times_t_i_less_equal_t (1-d_i/n_i)
  where:
    d_i := failures_at_t_i
    n_i := at_risk_before_t_i

NELSON_AALEN_ESTIMATOR:
  cumulative_hazard:
    H_hat(t) :=
      SUM_over_t_i_less_equal_t d_i/n_i

PARAMETRIC_RELIABILITY_ESTIMATION:
  distributions:
    - EXPONENTIAL
    - WEIBULL
    - LOGNORMAL
    - GAMMA
  methods:
    - MAXIMUM_LIKELIHOOD
    - BAYESIAN_ESTIMATION
  requirement:
    CENSORING_CONTRIBUTION_MUST_BE_INCLUDED_IN_LIKELIHOOD

LIKELIHOOD_WITH_RIGHT_CENSORING:
  contribution:
    f(t_i)^delta_i * R(t_i)^(1-delta_i)
  full_likelihood:
    PRODUCT_i [f(t_i)^delta_i R(t_i)^(1-delta_i)]

ACCELERATED_LIFE_TESTING:
  alias:
    ALT
  purpose:
    OBSERVE_FAILURES_MORE_QUICKLY_UNDER_ELEVATED_STRESS
  stresses:
    - TEMPERATURE
    - VOLTAGE
    - LOAD
    - HUMIDITY
    - PRESSURE

SERIES_SYSTEM:
  requirement:
    SYSTEM_FUNCTIONS_ONLY_IF_ALL_REQUIRED_COMPONENTS_FUNCTION
  independent_components:
    R_system(t) := PRODUCT_i R_i(t)

PARALLEL_SYSTEM:
  requirement:
    SYSTEM_FUNCTIONS_IF_AT_LEAST_ONE_PARALLEL_COMPONENT_FUNCTIONS
  independent_components:
    R_system(t) := 1-PRODUCT_i(1-R_i(t))

K_OUT_OF_N_SYSTEM:
  system_success:
    AT_LEAST_k_OF_n_COMPONENTS_FUNCTION
  identical_independent_components:
    R_system :=
      SUM[j=k..n] COMBINATION(n,j)R^j(1-R)^(n-j)

SYSTEM_RELIABILITY:
  requirement:
    DEPENDENCE_BETWEEN_COMPONENT_FAILURES_MUST_BE_MODELLED_WHEN_RELEVANT

RELIABILITY_BLOCK_DIAGRAM:
  alias: RBD
  structures:
    - SERIES
    - PARALLEL
    - SERIES_PARALLEL
    - K_OUT_OF_N

FAULT_TREE_CONNECTION:
  top_event:
    SYSTEM_FAILURE
  gates:
    - AND
    - OR

COMMON_CAUSE_FAILURE:
  definition:
    SINGLE_CAUSE_INDUCES_MULTIPLE_COMPONENT_FAILURES
  consequence:
    INDEPENDENCE_ASSUMPTION_MAY_OVERSTATE_REDUNDANCY_BENEFIT

DEPENDENT_FAILURES:
  methods:
    - CONDITIONAL_PROBABILITY
    - COPULA
    - SHARED_FRAILTY
    - COMMON_CAUSE_MODEL
    - MULTISTATE_MODEL

REPAIRABLE_SYSTEM:
  cycles:
    FAILURE
      -> REPAIR
      -> RETURN_TO_SERVICE
  metrics:
    - MTBF
    - MTTR
    - AVAILABILITY
    - ROCOF

MEAN_TIME_BETWEEN_FAILURES:
  alias: MTBF
  context:
    REPAIRABLE_SYSTEM
  caution:
    MTBF != MTTF_BY_DEFAULT

MEAN_TIME_TO_REPAIR:
  alias: MTTR
  definition:
    EXPECTED_REPAIR_DURATION

AVAILABILITY:
  point_availability:
    A(t) := P(SYSTEM_OPERABLE_AT_TIME_t)
  steady_state_simple_two_state:
    A := MTTF/(MTTF+MTTR)
    UNDER:
      APPROPRIATE_ALTERNATING_RENEWAL_ASSUMPTIONS

UNAVAILABILITY:
  Q(t) := 1-A(t)

RENEWAL_PROCESS_CONNECTION:
  renewal_function:
    m(t) := E[N(t)]
  use:
    REPAIRABLE_OR_REPLACED_COMPONENT_MODELLING

ROCOF:
  full_name:
    RATE_OF_OCCURRENCE_OF_FAILURES
  caution:
    ROCOF != NONREPAIRABLE_COMPONENT_HAZARD_BY_DEFAULT

MAINTENANCE:
  types:
    - CORRECTIVE
    - PREVENTIVE
    - CONDITION_BASED
    - PREDICTIVE
  effect:
    MAY_CHANGE:
      FAILURE_PROCESS
      AVAILABILITY
      COST
      RESIDUAL_LIFE

PREVENTIVE_MAINTENANCE:
  objective:
    REDUCE_FAILURE_PROBABILITY_OR_CONSEQUENCE_BEFORE_FAILURE
  decision:
    BALANCE:
      MAINTENANCE_COST
      FAILURE_COST
      DOWNTIME
      RELIABILITY

REMAINING_USEFUL_LIFE:
  alias: RUL
  at_age:
    t_0
  survival:
    P(T-t_0 > u | T > t_0)
      =
    R(t_0+u)/R(t_0)

CONDITIONAL_RELIABILITY:
  given_survival_to_t:
    P(T > t+s | T > t)
      =
    R(t+s)/R(t)
  exponential_case:
    equals:
      R(s)

RELIABILITY_GROWTH:
  objective:
    IMPROVE_RELIABILITY_THROUGH_TEST_FIX_RETEST_CYCLES
  models:
    - DUANE_MODEL
    - CROW_AMSAA

RELIABILITY_DEMONSTRATION_TEST:
  objective:
    COLLECT_TEST_EVIDENCE_THAT_RELIABILITY_REQUIREMENT_IS_MET
  design_depends_on:
    - TARGET_RELIABILITY
    - CONFIDENCE_LEVEL
    - TEST_DURATION
    - FAILURE_COUNT
    - DISTRIBUTIONAL_ASSUMPTIONS

ZERO_FAILURE_TEST:
  observation:
    ZERO_FAILURES_OVER_TEST_EXPOSURE
  caution:
    ZERO_OBSERVED_FAILURES != ZERO_FAILURE_PROBABILITY

CONFIDENCE_INTERVAL:
  parameter:
    RELIABILITY
    FAILURE_RATE
    MTTF
    QUANTILE_LIFE
  methods:
    - EXACT
    - ASYMPTOTIC
    - PROFILE_LIKELIHOOD
    - BOOTSTRAP
    - BAYESIAN_CREDIBLE_INTERVAL

BAYESIAN_RELIABILITY:
  prior:
    p(theta)
  data:
    D
  posterior:
    p(theta|D)
  posterior_reliability:
    DISTRIBUTION_OF_R(t;theta)
  posterior_predictive:
    P(T_new > t | D)

RELIABILITY_TESTING:
  data_sources:
    - LAB_TEST
    - FIELD_FAILURES
    - WARRANTY_DATA
    - SENSOR_DATA
    - ACCELERATED_TESTS
  caution:
    OPERATING_CONDITIONS_MAY_DIFFER_BETWEEN_TEST_AND_FIELD

FIELD_RELIABILITY:
  requirement:
    ACCOUNT_FOR:
      USAGE
      ENVIRONMENT
      LOAD
      MAINTENANCE
      POPULATION_HETEROGENEITY

RISK_ANALYSIS_CONNECTION:
  failure_probability:
    F(t)
  consequence:
    C
  risk:
    MAY_USE:
      F(t)*C
  caution:
    RELIABILITY_PROBABILITY_ALONE_DOES_NOT_DEFINE_CONSEQUENCE_RISK

DECISION_THEORY_CONNECTION:
  decisions:
    - REPLACE
    - REPAIR
    - INSPECT
    - REDESIGN
    - CONTINUE_OPERATION
  objective:
    BALANCE_EXPECTED_COST_RISK_AND_AVAILABILITY

SENSITIVITY_ANALYSIS_CONNECTION:
  vary:
    - FAILURE_RATE
    - WEIBULL_SHAPE
    - SCALE_PARAMETER
    - REPAIR_RATE
    - COMPONENT_DEPENDENCE
  observe:
    - SYSTEM_RELIABILITY
    - AVAILABILITY
    - MTTF
    - RISK

MONTE_CARLO_CONNECTION:
  flow:
    SAMPLE_FAILURE_TIMES
      -> SIMULATE_SYSTEM_LOGIC
      -> RECORD_SYSTEM_FAILURE
      -> REPEAT
      -> ESTIMATE_RELIABILITY_AND_RISK

APPLICATIONS:
  RELIABILITY_STATISTICS -> SUPPORTS -> RELIABILITY_ENGINEERING
  RELIABILITY_STATISTICS -> SUPPORTS -> RISK_ANALYSIS
  RELIABILITY_STATISTICS -> SUPPORTS -> MAINTENANCE_PLANNING
  RELIABILITY_STATISTICS -> SUPPORTS -> QUALITY_CONTROL
  RELIABILITY_STATISTICS -> SUPPORTS -> SURVIVAL_ANALYSIS
  RELIABILITY_STATISTICS -> SUPPORTS -> SYSTEM_SAFETY
  RELIABILITY_STATISTICS -> SUPPORTS -> WARRANTY_ANALYSIS

MODEL_CONNECTIONS:
  RELIABILITY_STATISTICS -> CONNECTS_TO -> RISK_ANALYSIS
  RELIABILITY_STATISTICS -> CONNECTS_TO -> SURVIVAL_ANALYSIS
  RELIABILITY_STATISTICS -> CONNECTS_TO -> STOCHASTIC_PROCESSES
  RELIABILITY_STATISTICS -> CONNECTS_TO -> RENEWAL_PROCESSES
  RELIABILITY_STATISTICS -> CONNECTS_TO -> MONTE_CARLO_SIMULATION
  RELIABILITY_STATISTICS -> CONNECTS_TO -> BAYESIAN_STATISTICS
  RELIABILITY_STATISTICS -> CONNECTS_TO -> DECISION_THEORY

SEMANTIC_EQUIVALENCE:
  "reliability function"
    ~= "survival function"
    IN:
      TIME_TO_FAILURE_CONTEXT

  "failure rate"
    ~= "hazard rate"
    IN:
      COMMON_NONREPAIRABLE_CONTINUOUS_CONTEXT

  "mean time to failure"
    ~= "expected failure time"

  EXCLUDE_EQUIVALENCE:
    RELIABILITY != AVAILABILITY
    MTTF != MTBF_BY_DEFAULT
    HAZARD != PROBABILITY
    ROCOF != HAZARD_BY_DEFAULT
    ZERO_FAILURES != ZERO_FAILURE_PROBABILITY
    COMPONENT_RELIABILITY != SYSTEM_RELIABILITY
    RELIABILITY != RISK

VALIDATION_RULES:
  RULE_1:
    FAILURE_DEFINITION_AND_TIME_OR_USAGE_SCALE_MUST_BE_EXPLICIT
  RULE_2:
    CENSORING_AND_TRUNCATION_MUST_BE_MODELLED_WHEN_PRESENT
  RULE_3:
    DISTRIBUTION_PARAMETERISATION_MUST_BE_EXPLICIT
  RULE_4:
    COMPONENT_DEPENDENCE_MUST_BE_MODELLED_WHEN_MATERIAL
  RULE_5:
    REPAIRABLE_AND_NONREPAIRABLE_SYSTEM_METRICS_MUST_NOT_BE_CONFLATED
  RULE_6:
    ACCELERATED_TEST_EXTRAPOLATION_REQUIRES_JUSTIFIED_STRESS_LIFE_MODEL
  RULE_7:
    UNCERTAINTY_INTERVALS_SHOULD_ACCOMPANY_RELIABILITY_ESTIMATES_WHEN_INFERENTIAL_CLAIMS_ARE_MADE
  RULE_8:
    TEST_AND_FIELD_OPERATING_CONDITIONS_MUST_BE_DISTINGUISHED

CONFIDENCE:
  ontology_identity: 0.99
  reliability_function_structure: 0.99
  hazard_structure: 0.99
  censoring_structure: 0.99
  system_reliability_structure: 0.99
  repairable_system_structure: 0.99
  semantic_distinctions: 0.99

UPDATE_MODEL:
  IF:
    NEW_FAILURE_OR_CENSORING_DATA_ARRIVES
  THEN:
    UPDATE:
      FAILURE_DISTRIBUTION
      RELIABILITY_FUNCTION
      HAZARD_FUNCTION
      MTTF
      PARAMETER_UNCERTAINTY

  IF:
    COMPONENT_STRUCTURE_CHANGES
  THEN:
    UPDATE:
      SYSTEM_LOGIC
      SYSTEM_RELIABILITY
      REDUNDANCY_MODEL

  IF:
    MAINTENANCE_POLICY_CHANGES
  THEN:
    REASSESS:
      FAILURE_PROCESS
      AVAILABILITY
      MTBF
      RESIDUAL_LIFE
      EXPECTED_COST

  PRESERVE:
    FAILURE_DEFINITION
    TIME_OR_USAGE_SCALE
    DISTRIBUTION_FAMILY
    PARAMETERISATION
    CENSORING_STATUS
    COMPONENT_STRUCTURE
    DEPENDENCE_ASSUMPTIONS
    OPERATING_CONDITIONS
    DATA_PROVENANCE

AI_CONSUMPTION:
  parse_as: FORMAL_SEMANTIC_SPECIFICATION
  execution_mode: RECURSIVE_REASONING

  preferred_reasoning_order:
    - IDENTIFY_FAILURE_DEFINITION_AND_TIME_SCALE
    - IDENTIFY_FAILURE_AND_CENSORING_DATA
    - SELECT_RELIABILITY_DISTRIBUTION_OR_NONPARAMETRIC_METHOD
    - ESTIMATE_RELIABILITY_FUNCTION
    - ESTIMATE_HAZARD_AND_CUMULATIVE_HAZARD
    - COMPUTE_MTTF_AND_LIFE_QUANTILES
    - MODEL_COMPONENT_DEPENDENCE_AND_SYSTEM_STRUCTURE
    - COMPUTE_SYSTEM_RELIABILITY_OR_AVAILABILITY
    - QUANTIFY_PARAMETER_UNCERTAINTY
    - ASSESS_MAINTENANCE_AND_RISK_IMPLICATIONS
    - RETURN_RELIABILITY_INTERPRETATION

  constraints:
    - PRESERVE_FAILURE_SEMANTICS
    - PRESERVE_TIME_OR_USAGE_SCALE
    - DISTINGUISH_RELIABILITY_FROM_AVAILABILITY
    - DISTINGUISH_MTTF_FROM_MTBF
    - DISTINGUISH_HAZARD_FROM_PROBABILITY
    - MODEL_CENSORING_CORRECTLY
    - MODEL_COMPONENT_DEPENDENCE_WHEN_RELEVANT
    - PRESERVE_DATA_PROVENANCE

FLOW_HASH:
  canonical_entity: "reliability-statistics"
  namespace: "probability-statistics/risk-decision-mathematics"
  version: "4.0"
  identity_basis:
    - PAGE_URL
    - CANONICAL_ENTITY
    - ENTITY_TYPE
    - RELIABILITY_FUNCTION
    - HAZARD_FUNCTION
    - FAILURE_TIME
    - SYSTEM_RELIABILITY

END_FLOW