Calibration, Identifiability and Uncertainty
What is calibrated and what remains frozen
Calibration means estimating parameters from a defined set of observations, an objective function and declared assumptions. It does not mean selecting constants until a favourable case is reproduced. Validation requires evaluating predictions against information that did not influence those decisions. In this edition, the proprietary commercial T1 2.3 engine remains frozen: the alternatives in this chapter are research proposals, not executed adjustments or publicly verified calibration of its constants.
The first step would not be opening an optimizer but building a correspondence table linking each parameter, proposed mechanism and identifying observation. A parameter may be computable yet unidentifiable from the available observations. When different combinations produce responses indistinguishable within experimental error, publishing more decimal places does not resolve ambiguity. The useful conclusion is to design the next comparison or reduce the number of free parameters.
Ceiling and rate: Structural confounding
To examine identifiability, consider the proposed family r(c)=theta*(1-exp(-beta*c)). theta represents operational amplitude and beta controls the approach with exposure. Here beta is not the recirculation ratio used in other hydraulic balances. In T1, the approach term corresponds algebraically to beta=ln(2) while the other rules remain constant. Allowing a free beta would be a calibration alternative, not a description of an implemented modification.
If beta*c is small, a first-order expansion gives r(c)≈theta*beta*c. Early points mainly inform the product of theta and beta. A high ceiling with slow approach can resemble a low ceiling with fast approach. Distinguishing them requires a broader exposure range while maintaining comparable conditions affecting theta. An apparent plateau helps only when it exceeds resolution limitations and cannot be explained by dose or matrix changes.
The sensitivities are dr/dtheta=1-exp(-beta*c) and dr/dbeta=theta*c*exp(-beta*c). Near the origin both contain c and provide similar information; at high exposure, sensitivity to beta declines. Concentrating all samples at the beginning or all at the end can therefore waste identification capacity. Point placement should address the question, not just sampling convenience.
Example 1: One datum, several curves
In a synthetic exercise, the algebraic response r(2)=0.30 is specified without attribution to any experiment. With theta=0.60, the result is beta=-ln(1-0.30/0.60)/2=0.3466; with theta=0.80, it is beta=-ln(1-0.30/0.80)/2=0.2350. Both curves pass through the same point. This equality shows that one observation cannot identify amplitude and approach simultaneously, even when the fitted residual is exactly zero.
The resulting mini-protocol adds different exposure conditions and independent replicates, preserving dose and matrix or recording their variations. Before measurement, it defines which divergence between curves would be resolvable. If the feasible operating range cannot distinguish them, a set of compatible parameters is reported or one is fixed through an explicit hypothesis. A parameter selected by numerical convenience is not declared identified.
Removed fraction, concentration or load
Fitting r=1-Cout/Cin facilitates fraction comparison but places Cin in the denominator and shares its error with the response. At low inlet concentrations, the ratio can amplify uncertainty. Fitting Cout in physical units changes weighting: the same absolute deviation has different relative meanings across matrices. Fitting loads incorporates flow and volume but also their errors. No transformation automatically eliminates all these problems.
The objective function must correspond to intended use. If final quality is the prediction target, concentration error deserves prominence. If removed mass and specific consumption matter, comparable loads are required. Both outcomes should be reported when adequate data exist, without changing metrics after discovering which favours the model. Uncertainty-based weights require justified uncertainties; they must not be invented to improve the appearance of a fit.
Example 2: Same percentage, different mass significance
Assume two synthetic scenarios with the same net flow of 10 m3/day and inlet concentrations of 100 and 1000 mg/L. A ten-percentage-point removal error corresponds respectively to 10 and 100 mg/L outlet error, or 0.1 and 1 kg/day load error. An unweighted fraction regression would assign them the same squared error although their mass differences are tenfold apart.
This exercise does not prescribe always favouring the larger load: a low-concentration objective could make the first scenario more important. It requires the purpose of the statistical loss to be declared. Cross-matrix comparisons should retain error tables by concentration, load and group instead of summarizing the entire campaign with a single coefficient of determination.
Separating training from evaluation
Rows from the same facility, batch or campaign share characteristics. Randomly distributing aliquots from the same batch between training and validation causes leakage: during fitting, the model sees part of what it was supposedly expected to predict. Splitting should use groups consistent with future use. For transfer between facilities, facilities are held out; for new batches at a known facility, complete batches or periods are reserved.
Variable selection, normalization, imputation and hyperparameter choice take place within the training set. If validation results repeatedly inform decisions, that set ceases to be an independent final assessment. A final reserved group can protect evaluation, provided its size and diversity address the question. With few groups, transfer variability may remain highly uncertain.
A proposed grouped validation could leave each facility out in turn to explore transferability, keeping its campaigns together. This is a possible design, not an analysis performed here. The report should explain whether the objective is interpolation within known conditions or extrapolation to new matrices and scales; good interpolation does not automatically establish the latter capability.
Conditional uncertainty, not invented intervals
Uncertainty arises from analysis, sampling, operating inputs, parameters and structure. An interval propagating only flow and concentration errors does not cover the possibility that the equation is inappropriate. It must be labelled as conditional on the version, assumptions and distributions used. Deliberately varying inputs over a grid produces sensitivity scenarios, not probabilities or confidence intervals.
Censored results must retain their limits in proposed fitting, through a censoring-compatible objective or justified interval analysis. Replacing all such values with zero artificially changes the residual. Grouped resampling could evaluate variability between independent units, but it cannot create nonexistent groups or correct missing coverage. Without sufficient data, the honest outcome is unquantified uncertainty, accompanied by the measurement that could reduce it.
Sensitivity and discontinuous code rules
A local derivative can be calculated within a smooth branch. T1 2.3 contains rule changes requiring a different interpretation: the exposure multiplier changes at c=12, stage groupings may change discretely and outputs apply floors. A finite difference crossing these boundaries combines local sensitivity with a programmed jump. It must not be interpreted as a kinetic constant or an observed physical transition.
The floors are 3 for BOD, 15 for COD, 5 for TSS, 2 for N and 0.2 for P, using each parameter's corresponding units. They are not analytical limits. When an inlet lies below its floor, the calculated outlet can exceed it; this behaviour is retained as a code limitation and excluded from physical interpretation, without modifying the engine for this chapter.
Example 3: Audit boundaries without recalibration
The third mini-protocol runs the same scenario at c=11.999, 12 and 12.001, keeping all other inputs fixed. It records output, active branch and differences without fitting a smooth curve through the three points. Another series places concentrations on either side of each floor and checks whether the calculation represents interpretable removal. These cases are software tests, not water experiments.
Next, one input at a time is varied within interior regions, supplemented by selected combinations because interactions may exist. The effective configuration is preserved and outputs compared with the reproducibility bank. Any future alternative, such as a smooth approach or explicit residual model, would require a new version, prior criteria and held-out evaluation. This chapter proposes how to learn from data; it does not attribute a calibration to T1 2.3 that is not yet publicly documented.