Fitted-model covariance propagation

Introduced in: rtd-sensor 0.7.0.

A fitted RTD model does not have perfectly known parameters. When a supported calibration fit retains parameter covariance, rtd-sensor can propagate that covariance into either the resistance predicted by the fitted curve at a selected temperature or the temperature inferred from a fixed resistance.

For a model prediction R(T, θ), covariance propagation is:

u²_fit(R) = J Cov(θ) Jᵀ

J is the vector of resistance sensitivities with respect to the fitted parameters. The full covariance matrix is used, including off-diagonal terms. Those terms matter because fitted parameters are generally correlated. For the IEC-R0 and polynomial fit-space parameterizations, resistance is linear in the retained fitted parameters, so this resistance covariance transformation is exact at fixed temperature under the fit model. Custom CVD covariance is retained in the public R0, A, B, C basis. If R0 and shape coefficients are jointly fitted, those parameters enter the forward equation multiplicatively, so CVD forward propagation is first-order/local in that public basis.

IEC 60751 R0 example

from rtd_sensor import fitting, uncertainty

fit = fitting.fit_iec60751_r0(
    (
        fitting.CalibrationObservation(
            0.0,
            100.037,
            standard_uncertainty_ohms=0.01,
        ),
    ),
    minimum_temperature_c=-50.0,
    maximum_temperature_c=250.0,
)

propagated = uncertainty.propagate_fit_covariance_to_resistance(
    100.0,
    fit_result=fit,
)

print(propagated.resistance_ohms)
print(propagated.resistance_standard_uncertainty_ohms)

For the fixed IEC characteristic, R(T) = R0 × rho(T), so the parameter sensitivity is simply dR/dR0 = rho(T).

Polynomial example

Polynomial covariance is retained in the resistance-space basis

R(T) = a0 + a1*x + a2*x² + ...
x = T - reference_temperature_c

so the sensitivity vector is directly:

J = (1, x, x², ...)

That is why the covariance milestone retained this basis instead of pretending that the normalized deployable polynomial coefficients were independently fitted.

fit = fitting.fit_polynomial(
    (
        fitting.CalibrationObservation(0.0, 100.0, standard_uncertainty_ohms=0.02),
        fitting.CalibrationObservation(50.0, 119.4, standard_uncertainty_ohms=0.02),
        fitting.CalibrationObservation(100.0, 138.5, standard_uncertainty_ohms=0.02),
    ),
    degree=1,
)

propagated = uncertainty.propagate_fit_covariance_to_resistance(
    25.0,
    fit_result=fit,
)

print(propagated.parameter_sensitivity_vector)
print(propagated.resistance_variance_ohms_squared)

Custom CVD example

A fitted CVD result uses the same propagation functions. The sensitivity vector follows exactly the subset recorded by the fit covariance. At a fixed temperature:

dR/dR0 = R/R0
dR/dA  = R0*T
dR/dB  = R0*T²
dR/dC  = R0*(T-100)*T³   for T < 0 °C, otherwise 0

fit = fitting.fit_callendar_van_dusen(
    (
        fitting.CalibrationObservation(-50.0, 80.31, standard_uncertainty_ohms=0.01),
        fitting.CalibrationObservation(0.0, 100.025, standard_uncertainty_ohms=0.01),
        fitting.CalibrationObservation(100.0, 138.56, standard_uncertainty_ohms=0.01),
        fitting.CalibrationObservation(200.0, 175.90, standard_uncertainty_ohms=0.01),
    ),
    fit_parameters=("r0_ohms", "a", "b", "c"),
)

propagated = uncertainty.propagate_fit_covariance_to_resistance(
    -25.0,
    fit_result=fit,
)

Because the exposed CVD covariance is in the physical R0, A, B, C parameter basis, a joint R0 plus shape-coefficient fit is nonlinear in those parameters even though the fitter itself used an exact linearized product basis. That is why CVD forward propagation is described as first-order/local rather than exact in the public covariance basis.

Propagate into inferred temperature

For an inverse conversion, the measured resistance is held fixed while the fitted parameters vary. If the model satisfies

R_model(T, θ) = R_measured

implicit differentiation gives the fitted-parameter temperature sensitivity:

dT/dθ = -(dR/dθ) * (dT/dR)

The full covariance is then propagated with

u²_fit(T) = J_T Cov(θ) J_Tᵀ

resistance = fit.model.celsius_to_resistance(25.0)

temperature_propagated = uncertainty.propagate_fit_covariance_to_temperature(
    resistance,
    fit_result=fit,
)

print(temperature_propagated.temperature_c)
print(temperature_propagated.temperature_standard_uncertainty_c)
print(temperature_propagated.parameter_sensitivity_vector)

Unlike the forward resistance transformation above, this inverse result is a first-order local linearization because inferred temperature is generally nonlinear in the fitted parameters. The result retains both the resistance-side parameter sensitivity vector and the derived temperature-side vector so the calculation can be inspected.

What this uncertainty means

This result describes the uncertainty associated with the fitted parameters under the regression assumptions. It is not a complete uncertainty budget. It does not automatically include:

uncertainty in a later resistance measurement;
uncertainty in the reference temperatures used for calibration;
sensor drift;
self-heating;
lead-wire effects;
IEC tolerance assumptions; or
other acquisition/systematic effects.

Neither propagation function automatically inserts this contribution into temperature_uncertainty_budget(). Whether two contributions are independent is a property of the actual measurement/calibration process, not something the package can infer safely from two numbers.

Covariance must be available

A successful fit does not always have estimable parameter covariance. For example, an unweighted saturated fit has no residual degrees of freedom from which to estimate the unknown residual variance scale. If covariance is unavailable, propagation fails explicitly and reports the reason retained by the fit evidence rather than returning zero uncertainty.

Forward exactness and inverse local linearization

The forward resistance covariance transformation is exact for IEC-R0 and polynomial fit-space representations at fixed temperature. Custom CVD forward propagation is first-order/local when jointly fitted public parameters enter multiplicatively. Inverse temperature propagation is first-order/local because it uses the local implicit sensitivity at the converted temperature. Monte Carlo or other methods may eventually be appropriate when parameter uncertainty is large enough that inverse-model nonlinearity matters.