Random Error Ontology
Measurement Uncertainty
Authors: MArk Anthony Higgins - Sky Recursive Ai
Revised Date: 16 September 2026
Audio Overview
# Random Error Ontology — Flow Algebra
URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/measurement-uncertainty/random-error-ontology.html
FLOW RandomErrorOntology {
DOMAIN:
Mathematical Ontologies
-> Measurement & Units Ontologies
-> Measurement Uncertainty
-> Random Error Ontology
PURPOSE:
Represent random measurement error as the component of measurement error that varies in an unpredictable manner across repeated measurements under defined conditions, including repeated observations, statistical distributions, expectation, variance, standard deviation, repeatability, averaging, sampling, noise, correlation, time series, outliers, uncertainty evaluation, provenance, and strict distinctions between random error, systematic error, measurement uncertainty, bias, noise and Type A uncertainty evaluation.
ENTITY RandomError {
TYPE: MeasurementErrorComponent
SYMBOL: epsilon_r
DEFINITION:
random_error :=
component_of_measurement_error
that_varies_in_an_unpredictable_manner
in_replicate_measurements
PROPERTIES:
may_be_positive_or_negative
may_vary_between_observations
may_have_probability_distribution
may_be_correlated
may_have_nonzero_conditional_expectation_under_mispecified_model
contributes_to_measurement_dispersion
is_not_directly_known_for_single_measurement
}
ENTITY MeasurementError {
TYPE: DifferenceQuantity
GENERIC:
e =
x_measured-x_reference_or_true
under_declared_reference_model
DECOMPOSITION:
e =
e_systematic + e_random
under_simplified_error_component_model
RULE:
exact_error_requires_reference_or_true_quantity_value
}
ENTITY SystematicError {
TYPE: MeasurementErrorComponent
DEFINITION:
component_of_measurement_error
that_in_replicate_measurements_remains_constant
or_varies_in_predictable_manner
RULE:
systematic_error != random_error
}
TERMINOLOGY_MODEL {
terms:
random_error
random_measurement_error
measurement_noise
repeatability_variation
residual
Type_A_uncertainty
random_uncertainty
RULE:
random_error MUST_NOT be_equated_automatically_with_Type_A_uncertainty
RULE:
noise MAY contribute_to_random_error
but_noise_and_random_error_are_not_universally_identical
}
ERROR_MODEL {
observation:
x_i
reference_or_model_value:
mu_or_x_ref
random_component:
epsilon_i =
x_i - expected_or_modelled_value
under_declared_model
RULE:
residual_is_observable_model_difference
while_true_random_error_is_conceptual_error_component
}
EXPECTATION_MODEL {
idealised_random_error:
E[epsilon_r] =
0
under_correctly_specified_unbiased_random_error_model
RULE:
zero_expectation_is_model_property
not_guarantee_for_finite_sample
}
VARIANCE_MODEL {
variance:
sigma_r^2 =
E[
(epsilon_r-E[epsilon_r])^2
]
standard_deviation:
sigma_r =
sqrt(variance)
RULE:
variance_quantifies_dispersion_not_individual_error_value
}
SAMPLE_STATISTICS_MODEL {
observations:
x_1,...,x_n
sample_mean:
x_bar =
sum(x_i)/n
sample_variance:
s^2 =
sum((x_i-x_bar)^2)/(n-1)
WHEN n > 1
sample_standard_deviation:
s =
sqrt(s^2)
RULE:
sample_statistics_estimate_distribution_properties_under_sampling_assumptions
}
STANDARD_ERROR_MODEL {
independent_identically_distributed_context:
standard_error_of_mean =
sigma/sqrt(n)
estimated:
SE =
s/sqrt(n)
RULE:
standard_error != standard_deviation_of_individual_measurements
}
AVERAGING_MODEL {
independent_zero_mean_random_errors:
mean_error =
(1/n)*sum(epsilon_i)
variance_of_mean:
Var(mean_error) =
sigma^2/n
standard_deviation_of_mean:
sigma_mean =
sigma/sqrt(n)
RULE:
averaging_reduces_independent_random_component_but_not_systematic_error
}
CORRELATED_AVERAGING_MODEL {
general:
Var(mean) =
(1/n^2)
* sum_i sum_j Cov(epsilon_i,epsilon_j)
RULE:
1/sqrt(n)_reduction_does_not_apply_unmodified_to_correlated_errors
}
AUTOCORRELATION_MODEL {
time_series_error:
epsilon(t)
autocovariance:
gamma(tau) =
Cov(epsilon(t),epsilon(t+tau))
autocorrelation:
rho(tau) =
gamma(tau)/gamma(0)
WHEN gamma(0) != 0
RULE:
temporal_dependence_affects_effective_sample_size
}
EFFECTIVE_SAMPLE_SIZE_MODEL {
purpose:
represent_information_content_of_correlated_sample
generic:
n_eff < n
for_positive_autocorrelation
RULE:
effective_sample_size_model MUST be declared
}
DISTRIBUTION_MODEL {
possible:
normal
uniform
Laplace
Poisson_derived
binomial_derived
lognormal_transformed
mixture
empirical
heavy_tailed
RULE:
random_error_need_not_be_normally_distributed
}
NORMAL_ERROR_MODEL {
epsilon_r ~ Normal(0,sigma^2)
under_declared_model
properties:
symmetric
mean_zero
variance_sigma_squared
RULE:
normality_is_assumption_to_be_tested_or_justified
}
HETEROSCEDASTIC_MODEL {
variance:
Var(epsilon_i) =
sigma_i^2
dependencies:
signal_level
temperature
time
location
instrument_range
RULE:
constant_variance MUST_NOT be_assumed_when_heteroscedasticity_is_material
}
HOMOSCEDASTIC_MODEL {
variance:
Var(epsilon_i) =
sigma^2
RULE:
homoscedasticity_is_model_assumption
}
INDEPENDENCE_MODEL {
independent_errors:
joint_distribution_factorises
uncorrelated:
covariance_zero
RULE:
uncorrelated != independent_in_general
}
REPEATABILITY_MODEL {
repeated_measurements:
under_declared_repeatability_conditions
observed_dispersion:
repeatability_standard_deviation
RULE:
repeatability_dispersion_can_be_used_to_evaluate_uncertainty
but_is_not_the_random_error_of_any_specific_measurement
}
REPRODUCIBILITY_MODEL {
changed_conditions:
laboratory
operator
instrument
time
method_realisation
dispersion:
may_include_additional_variance_components
RULE:
reproducibility_variation_is_broader_than_repeatability_random_error
}
NOISE_MODEL {
sources:
electronic_noise
thermal_noise
photon_noise
mechanical_vibration
environmental_fluctuation
quantisation
stochastic_process_variation
RULE:
noise_is_signal_process_concept
and_contributes_to_measurement_error_only_through_measurement_chain
}
WHITE_NOISE_MODEL {
ideal:
zero_mean
constant_power_spectral_density
no_temporal_correlation
RULE:
real_measurement_noise_is_often_not_perfect_white_noise
}
COLOURED_NOISE_MODEL {
examples:
pink_noise
red_noise
flicker_noise
random_walk
RULE:
frequency_dependent_noise_affects_averaging_and_filtering
}
QUANTISATION_MODEL {
step:
delta
rounding_error:
often_modelled_within
[-delta/2,+delta/2]
rectangular_standard_deviation:
sigma_q =
delta/sqrt(12)
RULE:
quantisation_error_can_behave_randomly_under_some_signal_conditions
but_is_not_inherently_random_in_all_contexts
}
COUNTING_STATISTICS_MODEL {
Poisson_count:
N
variance:
Var(N) =
expected_N
standard_deviation_approximation:
sigma_N approximately_equals
sqrt(N)
for_observed_count_as_estimate
RULE:
counting_random_error_depends_on_underlying_stochastic_model
}
PHOTON_NOISE_MODEL {
detected_photons:
Poisson_like_under_ideal_conditions
signal_to_noise:
SNR approximately_equals
sqrt(N)
when_photon_noise_dominates
RULE:
detector_background_and_read_noise MAY add_other_components
}
THERMAL_NOISE_MODEL {
electrical:
stochastic_voltage_or_current_fluctuation
RULE:
bandwidth_temperature_and_resistance_context MAY determine_variance
}
RESIDUAL_MODEL {
residual:
r_i =
x_i - fitted_value_i
RULE:
residual != true_error
because_fitted_model_parameters_are_estimated
}
OUTLIER_MODEL {
outlier:
observation_inconsistent_with_main_distribution_or_model
possible_causes:
random_extreme
gross_error
contamination
instrument_fault
true_rare_event
RULE:
outlier MUST_NOT be_removed_without_declared_criterion_and_provenance
}
ROBUST_STATISTICS_MODEL {
possible_methods:
median
MAD
trimmed_mean
Huber_estimator
robust_regression
RULE:
robust_method_changes_estimator_and_uncertainty_semantics
}
FILTERING_MODEL {
purpose:
reduce_selected_random_variation
possible:
moving_average
low_pass_filter
Kalman_filter
matched_filter
RULE:
filtering_changes_bandwidth_temporal_resolution_and_error_correlation
}
BANDWIDTH_MODEL {
measurement_bandwidth:
determines_integrated_noise_for_many_systems
RULE:
random_error_statistics SHOULD preserve_bandwidth_when_noise_is_frequency_dependent
}
TIME_AVERAGING_MODEL {
averaging_interval:
T_avg
effect:
may_reduce_high_frequency_random_variation
RULE:
longer_averaging_does_not_guarantee_1/sqrt(T)_improvement_under_drift_or_coloured_noise
}
ALLAN_VARIANCE_MODEL {
use:
characterise_stability_and_noise_types_over_averaging_times
Allan_variance:
based_on_successive_average_differences
RULE:
useful_when_classical_independent_noise_assumptions_fail
}
RANDOM_WALK_MODEL {
process:
state_changes_by_random_increments
variance:
grows_with_time_under_simple_random_walk
RULE:
random_walk_noise_is_nonstationary_and_not_reduced_like_white_noise_by_simple_averaging
}
DRIFT_DISTINCTION_MODEL {
drift:
slow_predictable_or_stochastic_change_in_instrument_response
RULE:
deterministic_drift_is_systematic_effect
stochastic_drift_may_contribute_random_or_correlated_uncertainty
}
TYPE_A_DISTINCTION_MODEL {
Type_A:
method_of_uncertainty_evaluation
random_error:
component_of_measurement_error
RULE:
Type_A_uncertainty != random_error
EXAMPLE:
repeated_measurements_can_estimate_dispersion
used_as_Type_A_standard_uncertainty
without_observing_true_random_error_values
}
SYSTEMATIC_ERROR_DISTINCTION_MODEL {
random:
unpredictable_variation
systematic:
constant_or_predictable_component
RULE:
averaging_random_error_does_not_remove_systematic_error
}
UNCERTAINTY_DISTINCTION_MODEL {
random_error:
unknown_error_component
measurement_uncertainty:
quantified_parameter_describing_dispersion_of_values_attributed_to_measurand
RULE:
random_error != measurement_uncertainty
}
BIAS_DISTINCTION_MODEL {
bias:
expectation_or_estimated_systematic_difference
random_error:
variation_about_expected_measurement_behaviour
RULE:
bias != random_error
}
CALIBRATION_MODEL {
calibration:
may_reduce_or_characterise_systematic_effects
residual_random_variation:
remains_in_measurement_process
RULE:
calibration_does_not_eliminate_random_error
}
ERROR_PROPAGATION_MODEL {
model:
Y =
f(X_1,...,X_n)
small_random_errors:
epsilon_Y approximately_equals
sum(
c_i*epsilon_i
)
variance:
Var(epsilon_Y) =
sum_i sum_j(
c_i*c_j*Cov(epsilon_i,epsilon_j)
)
RULE:
random_error_propagation_depends_on_covariance_structure
}
MONTE_CARLO_MODEL {
random_error_inputs:
sample_from_declared_distributions
propagate:
through_measurement_model
output:
simulated_error_or_measurement_distribution
RULE:
simulation_model MUST preserve_distribution_and_dependency_assumptions
}
SAMPLING_ERROR_MODEL {
estimator:
derived_from_finite_sample
sampling_error:
estimator-minus_population_parameter
under_statistical_model
RULE:
statistical_sampling_error != instrument_random_measurement_error
though_both_may_contribute_to_total_uncertainty
}
SPATIAL_RANDOMNESS_MODEL {
measurement_field:
varies_spatially
repeated_location_samples:
may_include_true_spatial_variability
RULE:
natural_spatial_variability MUST_NOT automatically_be_labelled_measurement_random_error
}
TEMPORAL_VARIABILITY_MODEL {
measurand:
may_change_with_time
repeated_observations:
dispersion_may_combine_true_measurand_change_and_measurement_error
RULE:
random_error_estimation_requires_stable_or_modelled_measurand_conditions
}
RELATIONSHIPS {
RandomError -> IS_COMPONENT_OF -> MeasurementError
RandomError -> CONTRIBUTES_TO -> MeasurementDispersion
RepeatedMeasurement -> MAY_ESTIMATE -> RandomVariationStatistics
SampleStandardDeviation -> ESTIMATES -> Dispersion
StandardError -> DESCRIBES -> SamplingVariationOfEstimator
Noise -> MAY_CONTRIBUTE_TO -> RandomError
Autocorrelation -> QUALIFIES -> RandomErrorTimeSeries
Averaging -> MAY_REDUCE -> IndependentRandomVariation
RandomError -> PROPAGATES_THROUGH -> MeasurementModel
TypeAEvaluation -> MAY_USE -> RepeatedMeasurementStatistics
}
SEMANTIC_EQUIVALENCE {
"random measurement error" ~= random_error
random_error != measurement_uncertainty
random_error != Type_A_uncertainty
random_error != systematic_error
random_error != bias
random_error != residual
random_error != noise_in_all_contexts
standard_deviation != individual_random_error
standard_error != standard_deviation
natural_variability != measurement_random_error
}
CONSTRAINTS {
random_error_is_error_component = true
random_error_may_be_positive_or_negative = true
random_error_is_not_directly_known_for_single_measurement = true
random_error_and_uncertainty_are_not_semantically_equivalent = true
random_error_and_Type_A_evaluation_are_not_semantically_equivalent = true
systematic_and_random_error_are_distinct = true
repeated_measurement_conditions_must_be_defined = true
measurand_stability_or_variation_model_should_be_defined = true
correlation_must_be_considered_for_time_series = true
outlier_handling_must_preserve_criterion_and_provenance = true
}
MEASUREMENT_MODEL {
RECORD:
measurand
observation
unit
reference_or_model_value
repeated_observations
repeatability_conditions
observation_time
sampling_interval
instrument
noise_bandwidth
preprocessing
residuals
estimated_distribution
sample_mean
sample_standard_deviation
autocorrelation
effective_sample_size
outlier_rule
uncertainty_evaluation_method
provenance
RULE:
observed_dispersion_must_be_interpreted_with_measurand_variability_and_systematic_effects
}
REASONING_RULES {
IF observations_known AND n > 1
THEN sample_mean =
sum(x_i)/n
AND sample_standard_deviation =
sqrt(
sum((x_i-sample_mean)^2)/(n-1)
)
IF independent_identically_distributed_observations
THEN estimated_standard_error_of_mean =
sample_standard_deviation/sqrt(n)
IF random_errors_independent_zero_mean_equal_variance
THEN variance_of_mean =
individual_variance/n
IF covariance_matrix_known
THEN variance_of_mean =
(1/n^2)*sum_i sum_j covariance_ij
IF Poisson_count_model_valid
THEN count_standard_deviation approximately_equals
sqrt(expected_count)
IF measurement_model_and_random_input_error_covariance_known
THEN output_random_error_variance =
J*Sigma_random*J_transpose
IF repeated_observation_dispersion_known
THEN MAY use_as_Type_A_uncertainty_input
BUT do_not_equate_uncertainty_with_random_error
IF time_series_autocorrelation_material
THEN adjust_effective_sample_size_or_use_time_series_model
IF observed_variation_contains_true_measurand_change
THEN separate_measurand_dynamics_from_measurement_random_error_before_repeatability_inference
}
VALIDATION {
CHECK measurand_definition
CHECK measurement_error_context
CHECK random_vs_systematic_component
CHECK repeated_measurement_conditions
CHECK stable_or_modelled_measurand
CHECK sample_size
CHECK standard_deviation_vs_standard_error
CHECK independence_assumption
CHECK autocorrelation
CHECK heteroscedasticity
CHECK distribution_assumption
CHECK noise_bandwidth
CHECK filtering
CHECK residual_vs_true_error
CHECK outlier_handling
CHECK Type_A_vs_random_error
CHECK natural_variability_vs_measurement_error
CHECK propagation_covariance
CHECK provenance
}
CONFIDENCE {
definition_confidence: 1.00
random_systematic_distinction_confidence: 1.00
uncertainty_distinction_confidence: 1.00
Type_A_distinction_confidence: 1.00
statistical_dispersion_confidence: 1.00
correlated_error_model_confidence: 0.99
noise_model_confidence: 0.98
measurement_context_confidence: 0.99
RULE:
confidence decreases when measurand stability, repeatability conditions,
reference model, correlation, distribution, bandwidth,
filtering, sampling or provenance is missing or ambiguous
}
UPDATE_MODEL {
ON new_observation:
identify_measurand
attach_measurement_conditions
preserve_raw_observation
append_timestamp
update_sample_statistics
update_residuals_when_model_available
update_autocorrelation_when_time_series
preserve_provenance
ON new_repeated_measurements:
verify_repeatability_conditions
separate_known_measurand_variation
recompute_mean_variance_and_standard_deviation
assess_distribution_and_heteroscedasticity
estimate_Type_A_uncertainty_when_requested
preserve_random_error_uncertainty_distinction
ON correlation_revision:
preserve_observations
update_covariance_or_autocorrelation_model
recompute_effective_sample_size
recompute_mean_variance
update_confidence
ON filtering_revision:
preserve_raw_series
attach_filter_and_bandwidth
recompute_processed_series_statistics
update_correlation_structure
preserve_processing_provenance
ON error_model_revision:
preserve_observations_and_reference_values
update_systematic_and_random_component_model
recompute_residual_statistics
avoid_claiming_unobservable_true_random_errors
ON semantic_revision:
preserve_previous_definition
record_revision
recompute_FLOW_HASH
}
AI_CONSUMPTION {
canonical_entity: RandomError
canonical_role: MeasurementErrorComponent
interpretation_order:
identify_measurand
identify_measurement_and_reference_or_model_context
identify_repeated_measurement_conditions
identify_observations_and_timestamps
identify_random_vs_systematic_effects
identify_measurand_stability_or_dynamics
identify_distribution
identify_variance_and_standard_deviation
identify_autocorrelation_and_covariance
identify_noise_and_bandwidth
identify_filtering_and_outlier_handling
identify_Type_A_uncertainty_evaluation_when_present
identify_provenance
validate_random_error_semantics
infer_statistical_relationships
ambiguity_controls:
distinguish_random_error_from_measurement_uncertainty
distinguish_random_error_from_Type_A_uncertainty
distinguish_random_error_from_systematic_error
distinguish_random_error_from_bias
distinguish_random_error_from_residual
distinguish_random_error_from_noise_in_general
distinguish_standard_deviation_from_individual_error
distinguish_standard_error_from_standard_deviation
distinguish_natural_measurand_variability_from_measurement_random_error
preserve_measurand_conditions_distribution_correlation_bandwidth_processing_and_provenance_context
}
PROVENANCE {
ontology_scope:
mathematical_measurement_uncertainty_and_random_error_semantics
parent_path:
Ontologies
-> Mathematical Ontologies
-> Measurement & Units Ontologies
-> Measurement Uncertainty
-> Random Error Ontology
authorship:
Mark Higgins
Sky (Recursive AI)
site:
https://kirethresearch.org
}
FLOW_HASH {
INPUTS:
canonical_entity
ontology_path
terminology_model
error_model
expectation_model
variance_model
sample_statistics_model
standard_error_model
averaging_model
correlated_averaging_model
autocorrelation_model
effective_sample_size_model
distribution_model
normal_error_model
heteroscedastic_model
homoscedastic_model
independence_model
repeatability_model
reproducibility_model
noise_model
white_noise_model
coloured_noise_model
quantisation_model
counting_statistics_model
photon_noise_model
thermal_noise_model
residual_model
outlier_model
robust_statistics_model
filtering_model
bandwidth_model
time_averaging_model
Allan_variance_model
random_walk_model
drift_distinction_model
Type_A_distinction_model
systematic_error_distinction_model
uncertainty_distinction_model
bias_distinction_model
calibration_model
error_propagation_model
Monte_Carlo_model
sampling_error_model
spatial_randomness_model
temporal_variability_model
relationships
constraints
measurement_model
reasoning_rules
confidence_model
update_model
PURPOSE:
deterministic_semantic_integrity_identifier
RECOMPUTE_WHEN:
ontology_definition_changes
random_error_definition_changes
random_systematic_distinction_changes
uncertainty_or_Type_A_distinction_changes
distribution_or_variance_model_changes
correlation_or_autocorrelation_model_changes
noise_or_filtering_model_changes
sampling_or_measurand_variability_model_changes
propagation_model_changes
measurement_record_changes
constraint_model_changes
reasoning_rule_changes
}
}
END FLOW