Uncertainty propagation

tldecpy implements a combined uncertainty budget following the ISO GUM (Guide to the Expression of Uncertainty in Measurement) framework. The combined standard uncertainty \(u_c(T)\) at each temperature channel combines:

1. Model-parameter uncertainty — propagated from the Jacobian covariance.
2. Type-A noise — estimated from detector noise.
3. Type-B contributions — systematic sources (calibration, heating rate, reader drift).

---

Jacobian-based local linearisation

After the least-squares solution \(\hat{\theta}\), the parameter covariance matrix is estimated as:

\[ C_\theta = (J^\top J)^{-1} \, \sigma_\varepsilon^2 \]

where \(J\) is the Jacobian \(\partial \hat{I} / \partial \theta\) at \(\hat{\theta}\) and \(\sigma_\varepsilon^2 = \text{SSR}/(n - k)\) is the residual variance.

The model-parameter contribution to \(u_c(T)\) is:

\[ u_\text{model}(T) = \sqrt{\mathbf{s}(T)^\top C_\theta \, \mathbf{s}(T)} \]

where \(\mathbf{s}(T) = \partial \hat{I}(T) / \partial \theta\) is the sensitivity vector, computed by central differences.

This method is fast (no extra solves) but assumes:

• The model is approximately linear near \(\hat{\theta}\).
• The residuals are approximately normally distributed.

---

Combined uncertainty

Each source \(j\) contributes an absolute standard uncertainty \(u_j(T)\) in detector counts. They are combined in quadrature:

\[ u_c(T) = \sqrt{\sum_j u_j(T)^2 + 2 \sum_{j < k} \rho_{jk}\, u_j(T)\, u_k(T)} \]

where \(\rho_{jk}\) are optional correlation coefficients set via UncertaintyOptions.correlations = {"sourceA:sourceB": rho}.

The relative combined uncertainty (reported in result.uc_curve) is:

\[ u_c^\text{rel}(T) = \frac{u_c(T)}{\hat{I}(T)} \times 100\% \]

---

Global uncertainty criterion

The area-weighted global criterion (analogous to the integral FOM):

\[ u_{c,\text{global}} = \frac{\int u_c(T)\, \hat{I}(T)\, \mathrm{d}T} {\int \hat{I}(T)\, \mathrm{d}T} \times 100\% \]

Available as result.metrics.uc_global.

---

Monte Carlo cross-validation

To verify that the Jacobian linearisation is valid:

uc_opts = tl.UncertaintyOptions(
    enabled=True,
    include_parameter_covariance=True,
    noise_pct=1.0,
    validation_mode="monte_carlo",
    n_validation_samples=200,
    validation_seed=42,
)

MC draws \(N\) parameter vectors from \(\mathcal{N}(\hat{\theta}, C_\theta)\), evaluates the model at each, and estimates \(u_c(T)\) from the spread. result.uncertainty_validation contains the relative L2 difference between MC and linearisation:

val = result.uncertainty_validation
print(f"rel_l2  = {val['rel_l2']:.3f}")   # < 0.10 is good agreement
print(f"rel_max = {val['rel_max']:.3f}")

---

Bootstrap cross-validation

Bootstrap resamples the residuals and re-solves the fit \(N\) times:

uc_opts_boot = tl.UncertaintyOptions(
    enabled=True,
    noise_pct=1.0,
    validation_mode="bootstrap",
    n_validation_samples=100,
    validation_seed=42,
)

Bootstrap is more expensive than MC (requires \(N\) full least-squares solves) but makes fewer distributional assumptions. Use it when the linearisation and MC results disagree.

---

When to trust the Jacobian estimate

Condition	Trust level
jac_cond < 1e8 and converged=True	High
jac_cond 1e8 – 1e10	Moderate — verify with MC
jac_cond > 1e10	Low — parameters are correlated or degenerate
Any hit_bounds is True	Low — re-constrain or fix the offending parameter

---

References

• JCGM 100:2008. Evaluation of measurement data — Guide to the expression of uncertainty in measurement (GUM). BIPM/ISO.
• Peng, J., et al. (2016). Semi-analytical expressions for the one-trap one- recombination centre (OTOR) model. Radiat. Meas. 93, 55.