Measurement Mathematics

# Measurement Mathematics Ontology — Flow Algebra

URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/measurement-mathematics.html

FLOW MeasurementMathematicsOntology {

  DOMAIN:
    Mathematical Ontologies
    -> Measurement & Units Ontologies
    -> Measurement Mathematics Ontology

  PURPOSE:
    Represent the mathematical structures, operations, transformations and inference rules used to express, compare, combine, convert, analyse and propagate measured quantities and their uncertainties. The ontology integrates quantity values, units, dimensions, measurement equations, ratios, products, powers, logarithmic quantities, statistics, probability distributions, covariance, regression, calibration mathematics, uncertainty propagation, numerical methods, dimensional analysis, significant figures, rounding, transformations, time series, sampling and machine-consumable reasoning while preserving the semantic distinction between a physical quantity, its numerical value, its unit, its measurement result and its uncertainty.

  ENTITY MeasurementMathematics {
    TYPE: MathematicalMeasurementDomain

    COMPONENTS:
      quantity_algebra
      unit_algebra
      dimensional_analysis
      measurement_functions
      statistical_inference
      probability
      uncertainty_calculus
      calibration_mathematics
      numerical_methods
      data_transformations
      validation
  }

  ENTITY Quantity {
    TYPE: MeasurableProperty

    COMPONENTS:
      quantity_kind
      magnitude
      unit_or_reference
      system_or_object
      conditions
  }

  ENTITY QuantityValue {
    TYPE: MagnitudeRepresentation

    FORM:
      Q =
        numerical_value * unit

    RULE:
      numerical_value_and_unit MUST remain_semantically_linked
  }

  ENTITY MeasurementResult {
    TYPE: QuantityEstimate

    COMPONENTS:
      measured_quantity_value
      uncertainty
      measurand
      method
      provenance
  }

  ENTITY MeasurementFunction {
    TYPE: MathematicalMapping

    FORM:
      Y =
        f(X_1,X_2,...,X_n)

    RULE:
      input_and_output_quantity_kinds MUST be defined
  }

  QUANTITY_ALGEBRA_MODEL {
    operations:
      addition
      subtraction
      multiplication
      division
      exponentiation
      roots
      functions
      comparison

    RULE:
      operations MUST respect_quantity_kind_and_dimension_semantics
  }

  ADDITION_MODEL {
    A:
      quantity_value

    B:
      quantity_value

    condition:
      dimensions_compatible
      quantity_kinds_additively_meaningful

    result:
      C =
        A+B

    RULE:
      convert_to_compatible_units_before_numeric_addition
  }

  SUBTRACTION_MODEL {
    result:
      C =
        A-B

    condition:
      compatible_dimensions

    RULE:
      affine_quantity_scales_require_special_semantics_for_values_vs_differences
  }

  MULTIPLICATION_MODEL {
    C =
      A*B

    dimension:
      dim(C) =
        dim(A)*dim(B)

    unit:
      unit(C) =
        unit(A)*unit(B)
  }

  DIVISION_MODEL {
    C =
      A/B

    REQUIRE:
      B != 0

    dimension:
      dim(C) =
        dim(A)/dim(B)
  }

  POWER_MODEL {
    C =
      A^n

    dimension:
      dim(C) =
        dim(A)^n

    RULE:
      fractional_power_requires_domain_and_dimension_interpretation
  }

  ROOT_MODEL {
    C =
      A^(1/n)

    RULE:
      resulting_dimension_exponents_may_be_fractional
      unless_quantity_definition_resolves_to_standard_kind
  }

  DIMENSIONLESS_MODEL {
    dimension:
      1

    possible:
      ratio
      angle
      refractive_index
      strain
      probability
      count_ratio

    RULE:
      dimensionless_does_not_mean_semantically_unitless_or_context_free
  }

  RATIO_MODEL {
    R =
      A/B

    same_dimension:
      R_dimensionless

    RULE:
      ratio_requires_denominator_nonzero
  }

  PERCENTAGE_MODEL {
    percent =
      100*ratio

    RULE:
      percentage_is_scaled_dimensionless_quantity
  }

  RATE_MODEL {
    rate =
      delta_quantity/delta_time
      or
      dQ/dt

    RULE:
      average_and_instantaneous_rate MUST be distinguished
  }

  DENSITY_MODEL {
    generic:
      density =
        quantity/measure_of_domain

    examples:
      mass/volume
      current/area
      population/area

    RULE:
      denominator_domain MUST be explicit
  }

  NORMALISATION_MODEL {
    normalized_value =
      value/reference

    RULE:
      reference_value_and_zero_handling MUST be defined
  }

  UNIT_MODEL {
    quantity_value:
      q =
        {q}*[q]

    WHERE:
      {q} = numerical_value
      [q] = unit

    RULE:
      unit_conversion_changes_numerical_value_but_not_underlying_quantity
  }

  LINEAR_UNIT_CONVERSION_MODEL {
    y =
      a*x

    uncertainty:
      u_y =
        abs(a)*u_x
      WHEN a_exact

    RULE:
      exact_scale_conversion_preserves_relative_uncertainty
  }

  AFFINE_UNIT_CONVERSION_MODEL {
    y =
      a*x+b

    examples:
      temperature_scales

    uncertainty:
      u_y =
        abs(a)*u_x
      WHEN a_and_b_exact

    RULE:
      quantity_differences_do_not_use_same_offset_semantics_as_absolute_scale_values
  }

  DIMENSION_MODEL {
    base_dimensions:
      L
      M
      T
      I
      Theta
      N
      J

    generic:
      dim(Q) =
        L^a*M^b*T^c*I^d*Theta^e*N^f*J^g
  }

  DIMENSIONAL_CONSISTENCY_MODEL {
    equation:
      left_dimension =
        right_dimension

    RULE:
      every_additive_term MUST have_same_dimension
  }

  DIMENSION_MATRIX_MODEL {
    variables:
      Q_1...Q_n

    matrix:
      rows_or_columns_of_dimension_exponents

    use:
      dependency_analysis
      dimensionless_group_generation

    RULE:
      variable_order MUST be preserved
  }

  BUCKINGHAM_PI_MODEL {
    n_variables:
      n

    rank_of_dimension_matrix:
      r

    dimensionless_groups:
      n-r

    RULE:
      applicability_requires_complete_relevant_variable_set
  }

  SCALAR_MODEL {
    quantity:
      magnitude_without_direction

    examples:
      mass
      temperature
      energy
  }

  VECTOR_MODEL {
    quantity:
      magnitude_and_direction

    representation:
      components_in_coordinate_basis

    RULE:
      coordinate_frame MUST be preserved
  }

  TENSOR_MODEL {
    quantity:
      multilinear_geometric_object

    representation:
      components_depend_on_basis

    RULE:
      tensor_transformation_law MUST be respected
  }

  COORDINATE_TRANSFORMATION_MODEL {
    vector:
      x_prime =
        R*x
      under_linear_rotation

    covariance:
      Sigma_prime =
        R*Sigma*R_transpose

    RULE:
      value_and_uncertainty MUST transform_consistently
  }

  MEASUREMENT_EQUATION_MODEL {
    output:
      Y =
        f(X_1,...,X_n)

    estimate:
      y =
        f(x_1,...,x_n)

    RULE:
      model_parameters_constants_and_corrections MUST be explicit
  }

  DIRECT_MEASUREMENT_MODEL {
    indication:
      maps_directly_to_measurand_through_calibration

    RULE:
      direct_measurement_still_has_measurement_model
  }

  INDIRECT_MEASUREMENT_MODEL {
    measurand:
      computed_from_other_measured_quantities

    RULE:
      uncertainty_requires_propagation_through_function
  }

  DIFFERENTIAL_MODEL {
    differential:
      dY =
        sum_i(
          partial_f/partial_X_i * dX_i
        )

    RULE:
      first_order_differential_is_local_approximation
  }

  JACOBIAN_MODEL {
    vector_function:
      Y =
        f(X)

    Jacobian:
      J_ij =
        partial_f_i/partial_X_j

    RULE:
      variable_order_and_units MUST be preserved
  }

  HESSIAN_MODEL {
    H_ij =
      partial^2_f/
      partial_X_i_partial_X_j

    use:
      curvature
      second_order_uncertainty
      optimisation

    RULE:
      second_order_terms MAY matter_for_nonlinear_measurement_models
  }

  ERROR_MODEL {
    measurement_error:
      e =
        measured_value-reference_or_true_value
      under_declared_reference_context

    RULE:
      error != uncertainty
  }

  ABSOLUTE_ERROR_MODEL {
    abs_error =
      abs(measured-reference)
  }

  RELATIVE_ERROR_MODEL {
    relative_error =
      (measured-reference)/reference
      WHEN reference != 0
  }

  UNCERTAINTY_MODEL {
    standard_uncertainty:
      u(x)

    combined_standard_uncertainty:
      u_c(y)

    expanded_uncertainty:
      U =
        k*u_c

    RULE:
      uncertainty_type MUST be identified
  }

  COVARIANCE_MODEL {
    covariance:
      cov(X_i,X_j)

    covariance_matrix:
      Sigma

    RULE:
      covariance_matrix SHOULD be_symmetric_and_positive_semidefinite
  }

  CORRELATION_MODEL {
    rho_ij =
      cov(X_i,X_j)/
      (sigma_i*sigma_j)
      WHEN sigma_i != 0 AND sigma_j != 0

    bounds:
      -1 <= rho_ij <= 1
  }

  PROPAGATION_MODEL {
    first_order:
      Sigma_Y =
        J*Sigma_X*J_transpose

    scalar:
      u_y^2 =
        sum_i sum_j(
          c_i*c_j*cov(X_i,X_j)
        )

    RULE:
      correlations MUST be included_when_material
  }

  INDEPENDENT_PROPAGATION_MODEL {
    condition:
      cov_ij = 0
      for_i != j

    result:
      u_y =
        sqrt(sum_i(c_i^2*u_i^2))
  }

  SUM_UNCERTAINTY_MODEL {
    Y =
      A+B

    variance:
      u_Y^2 =
        u_A^2+u_B^2+2cov(A,B)
  }

  DIFFERENCE_UNCERTAINTY_MODEL {
    Y =
      A-B

    variance:
      u_Y^2 =
        u_A^2+u_B^2-2cov(A,B)
  }

  PRODUCT_UNCERTAINTY_MODEL {
    Y =
      A*B

    independent_small_relative:
      (u_Y/Y)^2 approximately_equals
        (u_A/A)^2+(u_B/B)^2

    REQUIRE:
      A != 0
      B != 0
  }

  QUOTIENT_UNCERTAINTY_MODEL {
    Y =
      A/B

    independent_small_relative:
      (u_Y/Y)^2 approximately_equals
        (u_A/A)^2+(u_B/B)^2

    REQUIRE:
      A != 0
      B != 0
  }

  POWER_UNCERTAINTY_MODEL {
    Y =
      A^n

    relative:
      u_Y/abs(Y) approximately_equals
        abs(n)*u_A/abs(A)

    REQUIRE:
      A != 0
  }

  PROBABILITY_MODEL {
    random_variable:
      X

    distribution:
      p(X)

    properties:
      expectation
      variance
      quantiles
      moments
  }

  EXPECTATION_MODEL {
    discrete:
      E[X] =
        sum_x x*p(x)

    continuous:
      E[X] =
        integral x*f(x) dx
  }

  VARIANCE_MODEL {
    Var(X) =
      E[(X-E[X])^2]

    identity:
      Var(X) =
        E[X^2]-E[X]^2
      when_moments_exist
  }

  STANDARD_DEVIATION_MODEL {
    sigma =
      sqrt(Var(X))
  }

  NORMAL_DISTRIBUTION_MODEL {
    X ~ Normal(mu,sigma^2)

    RULE:
      normality_is_model_assumption_not_default_truth
  }

  UNIFORM_DISTRIBUTION_MODEL {
    X ~ Uniform(a,b)

    variance:
      (b-a)^2/12
  }

  TRIANGULAR_DISTRIBUTION_MODEL {
    symmetric_half_width:
      a

    standard_deviation:
      a/sqrt(6)
  }

  POISSON_MODEL {
    N ~ Poisson(lambda)

    expectation:
      lambda

    variance:
      lambda
  }

  BINOMIAL_MODEL {
    X ~ Binomial(n,p)

    expectation:
      n*p

    variance:
      n*p*(1-p)
  }

  SAMPLE_MODEL {
    observations:
      x_1,...,x_n

    sample_size:
      n
  }

  SAMPLE_MEAN_MODEL {
    x_bar =
      sum(x_i)/n
  }

  SAMPLE_VARIANCE_MODEL {
    s^2 =
      sum((x_i-x_bar)^2)/(n-1)
    WHEN n > 1
  }

  STANDARD_ERROR_MODEL {
    independent_sample_mean:
      SE =
        s/sqrt(n)

    RULE:
      standard_error != standard_deviation
  }

  WEIGHTED_MEAN_MODEL {
    x_w =
      sum(w_i*x_i)/sum(w_i)

    REQUIRE:
      sum(w_i) != 0

    inverse_variance_weights:
      w_i =
        1/u_i^2
      under_independent_common_measurand_model

    RULE:
      correlated_measurements_require_generalised_weighting
  }

  GENERALIZED_LEAST_SQUARES_MEAN_MODEL {
    covariance:
      Sigma

    ones:
      1_vector

    estimate:
      x_hat =
        (1^T Sigma^-1 x)/
        (1^T Sigma^-1 1)
      when_Sigma_invertible

    RULE:
      covariance_model MUST be valid
  }

  MEDIAN_MODEL {
    median:
      central_order_statistic

    RULE:
      median_and_mean_have_different_estimator_semantics
  }

  QUANTILE_MODEL {
    q_p:
      value_at_cumulative_probability_p

    RULE:
      quantile_definition MAY vary_for_finite_samples
  }

  CONFIDENCE_INTERVAL_MODEL {
    parameter:
      theta

    interval:
      [L,U]

    confidence_level:
      1-alpha

    RULE:
      frequentist_confidence_interval != Bayesian_credible_interval
  }

  STUDENT_T_MODEL {
    mean_interval:
      x_bar +/- t_(1-alpha/2,n-1)*s/sqrt(n)

    CONDITIONS:
      independent_normal_population_for_exact_small_sample_form
  }

  REGRESSION_MODEL {
    response:
      y

    predictors:
      X

    linear:
      y =
        X*beta+epsilon

    outputs:
      beta_hat
      covariance
      residuals

    RULE:
      model_assumptions MUST be preserved
  }

  LEAST_SQUARES_MODEL {
    objective:
      minimise_sum_of_squared_residuals

    ordinary:
      beta_hat =
        (X^T X)^-1 X^T y
      when_invertible

    RULE:
      OLS_assumptions MAY be_invalid_for_calibration_with_uncertain_predictors
  }

  WEIGHTED_LEAST_SQUARES_MODEL {
    objective:
      minimise
        sum(w_i*r_i^2)

    RULE:
      weight_definition MUST be documented
  }

  GENERALIZED_LEAST_SQUARES_MODEL {
    covariance:
      Sigma

    objective:
      minimise
        r^T Sigma^-1 r

    RULE:
      covariance_matrix MUST correspond_to_residual_model
  }

  CALIBRATION_MATHEMATICS_MODEL {
    forward:
      indication =
        f(reference_quantity,parameters)

    inverse:
      measurand_estimate =
        g(indication,parameters)

    RULE:
      inverse_calibration_uncertainty MUST include_parameter_covariance
  }

  LINEAR_CALIBRATION_MODEL {
    y =
      a*x+b

    inverse:
      x =
        (y-b)/a
      WHEN a != 0
  }

  POLYNOMIAL_CALIBRATION_MODEL {
    y =
      sum_k(a_k*x^k)

    RULE:
      extrapolation_beyond_calibration_range_requires_caution
  }

  INTERPOLATION_MODEL {
    purpose:
      estimate_between_known_points

    possible:
      linear
      polynomial
      spline
      nearest
      physical_model

    RULE:
      interpolated_value_is_derived_not_directly_measured
  }

  EXTRAPOLATION_MODEL {
    purpose:
      estimate_outside_observed_range

    RULE:
      extrapolation_has_higher_model_risk
  }

  NUMERICAL_DIFFERENTIATION_MODEL {
    central:
      f_prime(x) approximately_equals
        [f(x+h)-f(x-h)]/(2*h)

    RULE:
      h_must_balance_truncation_and_roundoff
  }

  NUMERICAL_INTEGRATION_MODEL {
    possible:
      trapezoidal
      Simpson
      Gaussian_quadrature
      adaptive

    RULE:
      numerical_error SHOULD be estimated_when_material
  }

  TRAPEZOIDAL_MODEL {
    integral:
      approximately
        sum_i(
          (x_(i+1)-x_i)
          *(y_i+y_(i+1))/2
        )

    RULE:
      sample_order_and_spacing MUST be preserved
  }

  DISCRETE_INTEGRATION_MODEL {
    uniform_samples:
      integral approximately_equals
        delta_t*sum(x_n)
      under_rectangle_rule

    RULE:
      integration_rule MUST be identified
  }

  DERIVED_RATE_MODEL {
    finite_difference:
      rate_i approximately_equals
        (x_(i+1)-x_i)/(t_(i+1)-t_i)

    RULE:
      differentiation_amplifies_high_frequency_noise
  }

  SMOOTHING_MODEL {
    possible:
      moving_average
      polynomial
      low_pass
      spline
      robust

    RULE:
      smoothing_changes_data_and_must_preserve_parameters
  }

  FILTERING_MODEL {
    linear:
      y =
        h*x

    frequency:
      Y(f) =
        H(f)*X(f)

    RULE:
      filter_amplitude_phase_and_delay MAY affect_measurement
  }

  FOURIER_MODEL {
    continuous:
      X(f) =
        integral x(t)*exp(-i*2*pi*f*t)dt

    discrete:
      DFT

    RULE:
      transform_normalisation_and_units MUST be defined
  }

  DFT_MODEL {    X_k =
      sum_(n=0)^(N-1)(
        x_n*exp(-i*2*pi*k*n/N)
      )

    frequency_spacing:
      delta_f =
        f_s/N

    RULE:
      windowing_and_record_length_affect_spectral_interpretation
  }

  LOGARITHMIC_MODEL {
    natural:
      y =
        ln(x)
      REQUIRE x > 0

    base10:
      y =
        log10(x)
      REQUIRE x > 0

    RULE:
      logarithm_argument SHOULD be_dimensionless_or_referenced_to_declared_quantity
  }

  DECIBEL_MODEL {
    power_ratio:
      L =
        10*log10(P/P_ref)

    amplitude_ratio:
      L =
        20*log10(A/A_ref)
      under_appropriate_squared_power_relation

    RULE:
      reference_quantity MUST be defined
  }

  EXPONENTIAL_MODEL {
    y =
      exp(x)

    RULE:
      exponent_argument MUST be_dimensionless
  }

  ANGLE_MODEL {
    plane_angle:
      dimensionless_with_named_unit_radian

    RULE:
      trigonometric_calculus_uses_radian_measure
  }

  PERIODIC_MODEL {
    frequency:
      f =
        1/T
      WHEN T > 0

    angular_frequency:
      omega =
        2*pi*f
  }

  TIME_SERIES_MODEL {
    observations:
      x(t_i)

    components:
      trend
      periodicity
      noise
      drift
      events
  }

  AUTOCOVARIANCE_MODEL {
    gamma(k) =
      Cov(X_t,X_(t+k))
  }

  AUTOCORRELATION_MODEL {
    rho(k) =
      gamma(k)/gamma(0)
      WHEN gamma(0) != 0
  }

  SAMPLING_MODEL {
    uniform:
      t_n =
        t_0+n*T_s

    sample_rate:
      f_s =
        1/T_s
      WHEN T_s > 0
  }

  NYQUIST_MODEL {
    ideal_bandlimited:
      f_s > 2*f_max

    RULE:
      practical_anti_alias_margin_required
  }

  ALIASING_MODEL {
    sampled_frequencies:
      may_be_indistinguishable_modulo_sampling_frequency

    RULE:
      aliasing_is_not_corrected_by_simple_interpolation
  }

  MONTE_CARLO_MODEL {
    inputs:
      joint_distribution

    process:
      sample_inputs
      evaluate_measurement_function
      derive_output_distribution

    outputs:
      expectation
      standard_deviation
      quantiles
      covariance

    RULE:
      dependencies MUST be preserved
  }

  RANDOM_NUMBER_MODEL {
    simulation:
      pseudorandom_or_quasirandom_samples

    metadata:
      generator
      seed
      sequence
      algorithm

    RULE:
      reproducible_simulation_requires_configuration_provenance
  }

  OPTIMISATION_MODEL {
    objective:
      minimise_or_maximise
        F(theta)

    constraints:
      equality
      inequality
      bounds

    RULE:
      optimisation_solution_is_model_dependent
  }

  ROOT_FINDING_MODEL {
    solve:
      f(x)=0

    possible:
      bisection
      Newton
      secant

    RULE:
      convergence_and_domain MUST be checked
  }

  NEWTON_MODEL {
    x_(n+1) =
      x_n-f(x_n)/f_prime(x_n)

    REQUIRE:
      f_prime(x_n) != 0

    RULE:
      convergence_not_guaranteed_for_arbitrary_initial_value
  }

  MATRIX_MODEL {
    quantities:
      vector_or_multichannel_measurements

    operations:
      multiplication
      inversion
      decomposition
      eigenanalysis

    RULE:
      matrix_shape_variable_order_and_units MUST be preserved
  }

  LINEAR_SYSTEM_MODEL {
    A*x =
      b

    solution:
      x =
        A^-1*b
      only_when_A_invertible

    preferred:
      solve_without_explicit_inverse

    RULE:
      condition_number SHOULD be assessed
  }

  CONDITION_NUMBER_MODEL {
    kappa(A):
      sensitivity_of_solution_to_input_perturbations

    RULE:
      large_condition_number_indicates_numerical_or_model_sensitivity
  }

  LEAST_SQUARES_SYSTEM_MODEL {
    overdetermined:
      minimise
        ||A*x-b||^2

    RULE:
      QR_or_SVD_MAY be_preferred_to_normal_equations_for_numerical_stability
  }

  SVD_MODEL {
    A =
      U*Sigma*V_transpose

    uses:
      rank
      pseudoinverse
      conditioning
      dimensional_reduction
  }

  PSEUDOINVERSE_MODEL {
    x =
      A_plus*b

    RULE:
      pseudoinverse_solution_semantics_depend_on_rank_and_norm_criterion
  }

  ROUNDING_MODEL {
    value:
      rounded_to_declared_increment_or_digits

    possible:
      nearest
      floor
      ceiling
      truncate
      ties_to_even

    RULE:
      rounding_mode MUST be known_for_exact_reconstruction
  }

  SIGNIFICANT_FIGURES_MODEL {
    reporting:
      digits_consistent_with_measurement_resolution_and_uncertainty

    RULE:
      significant_figures_are_reporting_convention_not_substitute_for_uncertainty
  }

  NUMERICAL_PRECISION_MODEL {
    floating_point:
      finite_representation

    sources:
      roundoff
      cancellation
      overflow
      underflow

    RULE:
      computational_precision != measurement_precision
  }

  CANCELLATION_MODEL {
    subtraction:
      nearly_equal_numbers

    risk:
      loss_of_significant_numeric_digits

    RULE:
      stable_algebraic_reformulation MAY be_required
  }

  ERROR_BOUND_MODEL {
    deterministic:
      abs(error) <= B

    RULE:
      deterministic_bound != standard_uncertainty
  }

  INTERVAL_ARITHMETIC_MODEL {
    input:
      X in [L,U]

    output:
      enclosure_under_interval_operations

    RULE:
      dependency_problem_can_widen_intervals
  }

  BAYESIAN_MODEL {
    posterior:
      p(theta|data)
      proportional_to
      p(data|theta)*p(theta)

    outputs:
      posterior_mean
      credible_interval
      predictive_distribution

    RULE:
      prior_and_likelihood MUST be preserved
  }

  MAXIMUM_LIKELIHOOD_MODEL {
    estimate:
      theta_hat =
        argmax_theta L(theta|data)

    RULE:
      likelihood_model MUST be defined
  }

  INFORMATION_MODEL {
    Fisher_information:
      expected_curvature_of_log_likelihood

    RULE:
      regularity_conditions_required_for_standard_asymptotic_results
  }

  RESIDUAL_MODEL {
    r_i =
      observed_i-predicted_i

    diagnostics:
      bias
      variance
      autocorrelation
      outliers
      model_structure
  }

  GOODNESS_OF_FIT_MODEL {
    possible:
      residual_analysis
      chi_square
      R_squared
      likelihood
      information_criteria

    RULE:
      no_single_metric_establishes_model_validity
  }

  CHI_SQUARE_MODEL {
    weighted_residual:
      chi2 =
        sum((r_i/u_i)^2)
      under_independent_standardised_normal_context

    RULE:
      covariance_requires_generalised_quadratic_form
  }

  COVARIANCE_CHI_SQUARE_MODEL {
    chi2 =
      r_transpose*Sigma^-1*r
      when_Sigma_invertible

    RULE:
      degrees_of_freedom_depend_on_model_and_fitted_parameters
  }

  OUTLIER_MODEL {
    observation:
      unusually_inconsistent_with_model_or_distribution

    RULE:
      exclusion_requires_declared_criterion_and_provenance
  }

  ROBUST_STATISTICS_MODEL {
    possible:
      median
      MAD
      Huber
      Theil_Sen
      robust_regression

    RULE:
      robust_method_changes_estimator_semantics
  }

  UNCERTAINTY_BUDGET_MODEL {
    fields:
      input_quantity
      estimate
      unit
      distribution
      standard_uncertainty
      sensitivity
      covariance
      contribution
      provenance

    output:
      combined_standard_uncertainty
  }

  COVERAGE_MODEL {
    expanded:
      U =
        k*u_c

    RULE:
      coverage_factor_and_probability_basis MUST be reported
  }

  DEGREES_OF_FREEDOM_MODEL {
    sample_variance:
      nu =
        n-1

    combined:
      may_use_effective_degrees_of_freedom

    RULE:
      degrees_of_freedom_are_model_specific
  }

  WELCH_SATTERTHWAITE_MODEL {
    nu_eff =
      u_c^4/
      sum_i(
        c_i^4*u_i^4/nu_i
      )
      under_applicable_independent_component_context
  }

  INFORMATION_PRESERVATION_MODEL {
    measurement_record:
      retain
        raw_values
        units
        transformations
        uncertainty
        model
        versions
        provenance

    RULE:
      mathematical_transformation MUST be reversible_or_documented_when_information_is_lost
  }

  SEMANTIC_EQUIVALENCE {
    "measurement mathematics" ~= mathematical_framework_for_quantity_measurement_and_uncertainty
    "measurement equation" ~= measurement_function WHEN declared_metrological_context

    quantity != numerical_value
    quantity != unit
    measurement_result != raw_indication
    error != uncertainty
    standard_deviation != standard_error
    confidence_interval != credible_interval
    interpolation != measurement
    computational_precision != measurement_precision
    deterministic_error_bound != standard_uncertainty
  }

  RELATIONSHIPS {
    QuantityValue -> HAS -> NumericalValue
    QuantityValue -> HAS -> Unit
    Quantity -> HAS -> Dimension
    MeasurementResult -> ESTIMATES -> Measurand
    MeasurementFunction -> MAPS -> InputQuantitiesToOutputQuantity
    Jacobian -> MAPS -> InputPerturbationsToOutputPerturbations
    CovarianceMatrix -> PROPAGATES_THROUGH -> Jacobian
    CalibrationFunction -> MAPS -> IndicationToQuantityEstimate
    ProbabilityDistribution -> REPRESENTS -> QuantityValueDispersion
    UncertaintyBudget -> COMBINES -> UncertaintyComponents
    Sampling -> PRODUCES -> Observations
    NumericalMethod -> COMPUTES -> DerivedMeasurementQuantity
  }

  CONSTRAINTS {
    numerical_value_and_unit_must_remain_linked = true
    additive_operations_require_dimension_compatibility = true
    measurement_function_inputs_and_outputs_must_be_defined = true
    denominator_must_be_nonzero_for_division = true
    logarithm_arguments_must_satisfy_domain = true
    trigonometric_calculus_requires_radian_semantics = true
    covariance_dependencies_must_be_preserved = true
    error_and_uncertainty_are_not_semantically_equivalent = true
    standard_deviation_and_standard_error_are_not_semantically_equivalent = true
    computational_and_measurement_precision_are_not_semantically_equivalent = true
    numerical_method_and_approximation_error_should_be_preserved = true
    provenance_must_be_preserved = true
  }

  MEASUREMENT_MODEL {
    RECORD:
      measurand
      quantity_kind
      numerical_value
      unit
      dimension
      measurement_function
      input_quantities
      input_values
      input_units
      constants
      corrections
      covariance_matrix
      uncertainty_components
      standard_uncertainty
      combined_uncertainty
      expanded_uncertainty
      coverage_factor
      statistical_model
      sample_size
      degrees_of_freedom
      calibration_model
      numerical_method
      transformation_history
      rounding_mode
      significant_figures
      validation_results
      provenance

    RULE:
      derived_result MUST remain_linked_to_inputs_model_units_uncertainty_and_computation
  }

  REASONING_RULES {
    IF compatible_quantity_values_added
      THEN convert_to_common_unit
      AND add_numerical_values

    IF quantities_multiplied
      THEN multiply_numerical_values
      AND multiply_dimensions_and_units

    IF quantities_divided
      THEN verify_denominator_nonzero
      AND divide_values_dimensions_and_units

    IF measurement_function_and_input_values_known
      THEN compute_output_estimate

    IF Jacobian_and_input_covariance_known
      THEN output_covariance =
        Jacobian*input_covariance*transpose(Jacobian)

    IF inputs_uncorrelated
      THEN combined_variance =
        sum(sensitivity_i^2*input_variance_i)

    IF exact_linear_unit_conversion_applied
      THEN scale_absolute_uncertainty_by_abs(conversion_factor)
      AND preserve_relative_uncertainty

    IF repeated_independent_measurements_known
      THEN compute_mean_sample_variance_and_standard_error

    IF calibration_model_known
      THEN map_indication_to_quantity_estimate
      AND propagate_parameter_covariance

    IF nonlinear_model_or_nonGaussian_input_material
      THEN MAY use_Monte_Carlo_distribution_propagation

    IF sample_rate_known
      THEN derive_sampling_interval_and_Nyquist_frequency

    IF numerical_algorithm_used
      THEN preserve_algorithm_version_tolerance_and_approximation_error

    IF rounding_required
      THEN round_after_primary_computation_when_possible
      and_preserve_unrounded_internal_value
  }

  VALIDATION {
    CHECK quantity_kind
    CHECK numerical_value_and_unit
    CHECK dimensional_consistency
    CHECK measurement_function
    CHECK input_domains
    CHECK division_by_zero
    CHECK logarithm_and_power_domains
    CHECK coordinate_frames
    CHECK covariance_matrix
    CHECK uncertainty_propagation
    CHECK statistical_assumptions
    CHECK standard_deviation_vs_standard_error
    CHECK calibration_model
    CHECK interpolation_vs_extrapolation
    CHECK sampling_and_aliasing
    CHECK numerical_conditioning
    CHECK numerical_error
    CHECK rounding_and_significant_figures
    CHECK confidence_and_coverage_semantics
    CHECK transformation_history
    CHECK provenance
  }

  CONFIDENCE {
    quantity_algebra_confidence: 1.00
    unit_dimension_confidence: 1.00
    measurement_function_confidence: 1.00
    uncertainty_propagation_confidence: 1.00
    statistical_model_confidence: 0.99
    calibration_mathematics_confidence: 0.99
    numerical_method_confidence: 0.99
    semantic_distinction_confidence: 1.00

    RULE:
      confidence decreases when units, dimensions, model, covariance,
      statistical assumptions, numerical method, transformation history
      or provenance is missing or ambiguous
  }

  UPDATE_MODEL {
    ON new_measurement:
      identify_measurand
      attach_value_unit_and_dimension
      identify_measurement_function
      attach_input_quantities
      attach_uncertainties_and_covariance
      compute_derived_result
      validate_dimensions
      preserve_provenance

    ON model_revision:
      preserve_previous_measurement_function
      update_equation_parameters_and_dependencies
      recompute_Jacobian_and_output
      recompute_uncertainty
      record_revision

    ON unit_revision:
      preserve_underlying_quantity
      apply_exact_or_uncertain_conversion
      update_numerical_value_and_uncertainty
      preserve_original_representation

    ON statistical_revision:
      preserve_raw_data
      update_distribution_estimator_or_covariance_model
      recompute_statistics_and_intervals
      preserve_method_version

    ON numerical_method_revision:
      preserve_inputs_and_model
      update_algorithm_tolerance_or_discretisation
      recompute_result
      compare_numerical_error
      preserve_previous_result

    ON semantic_revision:
      preserve_previous_definition
      record_revision
      recompute_FLOW_HASH
  }

  AI_CONSUMPTION {
    canonical_domain: MeasurementMathematics

    interpretation_order:
      identify_quantity_and_measurand
      identify_numerical_value_unit_and_dimension
      identify_measurement_function
      identify_input_quantities_and_dependencies
      identify_algebraic_operations
      identify_unit_transformations
      identify_statistical_and_probability_model
      identify_covariance_and_uncertainty
      identify_calibration_mathematics
      identify_sampling_and_time_series_context
      identify_numerical_method
      identify_rounding_and_reporting
      identify_validation
      identify_provenance
      validate_dimensional_and_semantic_consistency
      compute_or_transform_measurement_result

    ambiguity_controls:
      distinguish_quantity_from_numerical_value
      distinguish_quantity_from_unit
      distinguish_measurement_result_from_raw_indication
      distinguish_error_from_uncertainty
      distinguish_standard_deviation_from_standard_error
      distinguish_confidence_interval_from_credible_interval
      distinguish_interpolation_from_measurement
      distinguish_computational_precision_from_measurement_precision
      distinguish_deterministic_error_bound_from_standard_uncertainty
      preserve_quantity_unit_dimension_model_covariance_method_transformation_and_provenance_context
  }

  PROVENANCE {
    ontology_scope:
      mathematical_measurement_units_and_measurement_mathematics_semantics

    parent_path:
      Ontologies
      -> Mathematical Ontologies
      -> Measurement & Units Ontologies
      -> Measurement Mathematics Ontology

    authorship:
      Mark Higgins
      Sky (Recursive AI)

    site:
      https://kirethresearch.org
  }

  FLOW_HASH {
    INPUTS:
      canonical_domain
      ontology_path
      quantity_algebra_model
      addition_model
      subtraction_model
      multiplication_model
      division_model
      power_model
      root_model
      dimensionless_model
      ratio_model
      percentage_model
      rate_model
      density_model
      normalisation_model
      unit_model
      linear_unit_conversion_model
      affine_unit_conversion_model
      dimension_model
      dimensional_consistency_model
      dimension_matrix_model
      Buckingham_Pi_model
      scalar_model
      vector_model
      tensor_model
      coordinate_transformation_model
      measurement_equation_model
      direct_measurement_model
      indirect_measurement_model
      differential_model
      Jacobian_model
      Hessian_model
      error_model
      absolute_error_model
      relative_error_model
      uncertainty_model
      covariance_model
      correlation_model
      propagation_model
      independent_propagation_model
      sum_uncertainty_model
      difference_uncertainty_model
      product_uncertainty_model
      quotient_uncertainty_model
      power_uncertainty_model
      probability_model
      expectation_model
      variance_model
      standard_deviation_model
      normal_distribution_model
      uniform_distribution_model
      triangular_distribution_model
      Poisson_model
      binomial_model
      sample_model
      sample_mean_model
      sample_variance_model
      standard_error_model
      weighted_mean_model
      generalized_least_squares_mean_model
      median_model
      quantile_model
      confidence_interval_model      Student_t_model
      regression_model
      least_squares_model
      weighted_least_squares_model
      generalized_least_squares_model
      calibration_mathematics_model
      linear_calibration_model
      polynomial_calibration_model
      interpolation_model
      extrapolation_model
      numerical_differentiation_model
      numerical_integration_model
      trapezoidal_model
      discrete_integration_model
      derived_rate_model
      smoothing_model
      filtering_model
      Fourier_model
      DFT_model
      logarithmic_model
      decibel_model
      exponential_model
      angle_model
      periodic_model
      time_series_model
      autocovariance_model
      autocorrelation_model
      sampling_model
      Nyquist_model
      aliasing_model
      Monte_Carlo_model
      random_number_model
      optimisation_model
      root_finding_model
      Newton_model
      matrix_model
      linear_system_model
      condition_number_model
      least_squares_system_model
      SVD_model
      pseudoinverse_model
      rounding_model
      significant_figures_model
      numerical_precision_model
      cancellation_model
      error_bound_model
      interval_arithmetic_model
      Bayesian_model
      maximum_likelihood_model
      information_model
      residual_model
      goodness_of_fit_model
      chi_square_model
      covariance_chi_square_model
      outlier_model
      robust_statistics_model
      uncertainty_budget_model
      coverage_model
      degrees_of_freedom_model
      Welch_Satterthwaite_model
      information_preservation_model
      semantic_equivalence
      relationships
      constraints
      measurement_model
      reasoning_rules
      confidence_model
      update_model

    PURPOSE:
      deterministic_semantic_integrity_identifier

    RECOMPUTE_WHEN:
      ontology_definition_changes
      quantity_unit_or_dimension_semantics_change
      measurement_function_or_dependency_changes
      algebraic_or_transformation_model_changes
      probability_statistical_or_covariance_model_changes
      uncertainty_propagation_changes
      calibration_mathematics_changes
      sampling_time_series_or_signal_model_changes
      numerical_method_or_precision_changes
      rounding_or_reporting_semantics_change
      measurement_record_changes
      constraint_model_changes
      reasoning_rule_changes
  }

}

END FLOW