Applicability

Machine-independent.

Limitations

Says whether the sample can separate the exponents, not whether a fit did; a predictor that barely varies has a large VIF only if it is also correlated with another, so read the spread of each log predictor too. The condition number follows Belsley (columns scaled to unit length, intercept kept); values above about 30 signal harmful collinearity.

Provenance

1. Issue #548 Sec. 3 and 6: B_T stays a fitted variable only if its covariance with I_p and P is quantified.
2. D. A. Belsley, E. Kuh and R. E. Welsch, Regression Diagnostics, Wiley (1980), Ch. 3.

Defaults

n_boot = 2000 and seed = 0: numerical convenience; 2000 resamples put the percentile endpoints within about a percent of their limit.

Applicability

Machine-independent.

Limitations

With a handful of groups the bootstrap distribution is lumpy and its intervals undercover; rejected counts resamples that could not identify every exponent (e.g. a single field value drawn), which is itself identifiability evidence.

Provenance

1. Issue #548 Sec. 6: slices of one shot are not independent, so resampling is by shot.
2. B. Efron and R. J. Tibshirani, An Introduction to the Bootstrap, Chapman & Hall (1993), Ch. 13.

Processing steps

1. Complete the size exponent with the Kadomtsev constraint, $\alpha_R = (\alpha_I + 5\alpha_B + 3\alpha_P + 8\alpha_n + 5)/4$.
2. Substitute into $\mu_\rho(1+\alpha_P) = \alpha_B - \alpha_I - 2\alpha_R + 2\alpha_n
   • 3\alpha_P + 1$, which leaves $\tfrac32\alpha_I + \tfrac32\alpha_B + (\mu_\rho + \tfrac92)\alpha_P
   • 2\alpha_n = -(\mu_\rho + \tfrac32)$.

Applicability

Machine-independent.

Limitations

The closures are alternative hypotheses, not simultaneous constraints; the intermediate $\mu_\rho = -2.5$ is a single index, not an additive Bohm plus gyro-Bohm model (#548 Sec. 8). A density-free fit has $\alpha_n = 0$ by definition, which is part of the hypothesis tested.

Provenance

1. Issue #548 Sec. 8: Bohm, intermediate and gyro-Bohm closures as one parameterized constraint.

Applicability

Machine-independent.

Provenance

1. Issue #548 Sec. 5: the rules and the number each excludes are part of the result.

Processing steps

1. $P_\mathrm{OH} = I_p V_\mathrm{res}$ (vaft.formula.equilibrium.ohmic_heating_power_from_I_p_V_res).
2. $dW/dt$ by smoothed_time_derivative on w_th.
3. $P_\mathrm{net} = P_\mathrm{OH} - dW/dt$ and, where p_rad is finite, $P_\mathrm{transport} = P_\mathrm{net} - P_\mathrm{rad}$ (vaft.formula.equilibrium.loss_power_from_p_heat_dWdt_p_rad).
4. $\tau_E = W/P$ for each loss power, NaN where that power is not positive.

Applicability

Machine-independent. Ohmic plasmas: no auxiliary heating term.

Limitations

v_res carries every assumption of how the inductive voltage was removed; with a loop voltage taken away from the plasma boundary the flux between loop and boundary is still inductive. The transport loss power needs a measured radiated power, which VEST does not map (no bolometer), so on VEST it is NaN.

Provenance

1. Issue #548 Sec. 1: P_net and P_transport as separate, named quantities; W = (3/2) int p dV fixed by #1282/#1410.
2. ITER Physics Expert Groups, Nucl. Fusion 39 (1999) 2175, Ch. 2: the loss power $P_L = P_\mathrm{heat} - dW/dt$ of the confinement database.

Applicability

Machine-independent. The thresholds are the caller’s: none is a database or machine default.

Limitations

Ramp phases are judged by rate against the confinement time, not by a phase label: a short discharge with no flat top (VEST) has no plateau to select, and a rate rule says how far from stationary each slice is.

Provenance

1. Issue #548 Sec. 5: thresholds as arguments, studied by sweep before any is adopted.

Applicability

Machine-independent.

Provenance

1. Issue #548 Sec. 5: the slice-quality evidence, kept apart from the acceptance decision as #253 asks.

Processing steps

1. Complete the size exponent with infer_kadomtsev_size_exponent.
2. Map the completed engineering exponents to dimensionless indices with vaft.formula.equilibrium.dimensionless_scaling_coeffs_from_engineering_scaling_coeffs (exact here: a completed scaling satisfies the constraint), and $\mu_q = -\alpha_I/(1+\alpha_P)$.
3. Propagate cov through the map’s Jacobian (central differences, the size exponent completed at every evaluation).

Applicability

Machine-independent.

Limitations

Inherits the assumed size exponent: the indices are those of the Kadomtsev-completed scaling, not measured dimensionless dependences. The linear propagation is poor when $1 + \alpha_P$ is within a few standard errors of zero, where every index diverges.

Provenance

1. Issue #548 Sec. 7 (layer E): dimensionless conversion with uncertainties and covariances.
2. T. C. Luce, C. C. Petty and J. G. Cordey, Plasma Phys. Control. Fusion 50 (2008) 043001.

Processing steps

1. Keep rows where the response and every predictor are finite and positive; take logs and prepend an intercept column.
2. Fit by ordinary least squares (or Huber IRLS with robust).
3. Covariance clustered by groups (statsmodels cov_type="cluster", CR1 small-sample correction); 95 % intervals from Student’s t with $G - 1$ degrees of freedom.
4. Leverage and Cook’s distance from the least-squares influence of the same design (also for a robust fit, as a diagnostic).

With robust the cluster covariance is that of a weighted fit with the converged Huber weights held fixed, which slightly understates the error when rows are down-weighted; stderr_iid is then the Huber fit’s own.

Applicability

Machine-independent.

Limitations

With few groups the cluster-robust error is itself noisy (it has $G - 1$ degrees of freedom); read it together with the group bootstrap and leave-one-group-out results. Errors in the predictors are ignored, so exponents are attenuated where a predictor is noisy; with $\tau_E = W/P$ and $P$ a predictor, the $P$ exponent is biased towards $-1$ by any error in $P$.

Provenance

1. Issue #548 Sec. 6: covariance, intervals, influence and robust regression with grouped samples.
2. A. C. Cameron and D. L. Miller, J. Human Resources 50 (2015) 317: cluster-robust inference and the few-clusters problem.

Processing steps

1. Keep rows where everything is finite and positive; take logs.
2. Partial the exact predictors (zero error) and the intercept out of the response and the noisy predictors by least squares.
3. Scale each residual column by its error and take the right singular vector $v$ of the smallest singular value (total least squares); the noisy slopes are $b_j = -(v_j/\sigma_j)/(v_y/\sigma_y)$.
4. The exact predictors’ slopes and the intercept by least squares of $y - \sum_{j\in\mathrm{noisy}} b_j x_j$ on them.
5. With per-row response errors steps 2-4 have no closed form: the same linear model is solved as weighted orthogonal distance regression: the maximum-likelihood criterion of a linear model with independent Gaussian errors, $\sum_i (y_i - \beta\cdot x_i)^2 / (\sigma_{y,i}^2 + \sum_j \beta_j^2\sigma_{x,j}^2)$, minimised from the least-squares start (exact predictors carry $\sigma = 0$). For a constant per-row error it reproduces steps 2-4.

Applicability

Machine-independent.

Limitations

The answer depends only on the assumed error ratios, so report it as a function of them. Understating the equation error of the response over-corrects the attenuation and inflates the noisy slopes, the classic pitfall of errors-in-variables confinement fits. Errors are assumed independent between variables. Fitting $\tau_E = W/P$ against $P$ violates that, which is why the response should be $W$ (its $P$ exponent is $1 + \alpha_P$). No standard error is returned: rows of one shot are not independent, so use a shot bootstrap around this function.

Provenance

1. Issue #548 Sec. 6: errors-in-variables regression, and the coupling from $\tau_E = W/P$ with $P$ a predictor.
2. S. Van Huffel and J. Vandewalle, The Total Least Squares Problem, SIAM (1991), Ch. 3: mixed exact/noisy columns (partial TLS); for a linear model with known error ratios it is the orthogonal distance (maximum-likelihood) solution.

Processing steps

1. The unconstrained least-squares fit and its clustered covariance $V$ (fit_confinement_scaling).
2. Restricted least squares, $\beta_c = \beta - (X^\top X)^{-1}c^\top(c(X^\top X)^{-1}c^\top)^{-1}(c\beta - r)$.
3. Covariance $MVM^\top$ with $M = I - (X^\top X)^{-1}c^\top(c(X^\top X)^{-1}c^\top)^{-1}c$.
4. Residuals, $R^2$, the restricted leverage $H - XAc^\top(cAc^\top)^{-1}cAX^\top$ ($A = (X^\top X)^{-1}$) and Cook’s distance with $k - 1$ free coefficients; stderr_iid from $s^2 MAM^\top$ with the constrained residual variance.

Applicability

Machine-independent.

Limitations

Least squares only (no robust option).

Provenance

1. Issue #548 Sec. 8: closures fitted as constraints and compared.
2. W. H. Greene, Econometric Analysis, 7th ed., Pearson (2012), Sec. 5.5: restricted least squares.

Applicability

Machine-independent.

Limitations

An assumed size exponent, not a measured one: one machine’s shot-to-shot radius changes are not a size scan, and a Kadomtsev-completed scaling satisfies the constraint by construction, so a zero residual is no evidence for it. Label every use accordingly (#548 Sec. 7).

Provenance

1. Issue #548 Sec. 7: the Connor-Taylor/Kadomtsev completion and its covariance propagation.
2. B. B. Kadomtsev, Sov. J. Plasma Phys. 1 (1975) 295.

Applicability

Machine-independent.

Limitations

A held-out shot whose predictor values lie outside the others’ range tests extrapolation, not interpolation; the prediction error of such a group dominates rmse_log.

Provenance

1. Issue #548 Sec. 6 and 8: leave-one-shot-out error as a model comparison criterion.

Processing steps

1. $dl_{i,3}/dt$ by smoothed_time_derivative over rate_window_s (NaN with fewer than two samples).
2. resistive_loop_voltage: $V_\mathrm{ind} = L_i\,dI_p/dt + \tfrac14\mu_0 R_0 I_p\,dl_{i,3}/dt$, $V_\mathrm{res} = V_\mathrm{loop} - V_\mathrm{ind}$, $R_p = V_\mathrm{res}/I_p$.
3. confinement_power_balance: $P_\mathrm{OH} = I_p V_\mathrm{res}$, $dW/dt$ over the same window, $P_\mathrm{net} = P_\mathrm{OH} - dW/dt$, $P_\mathrm{transport} = P_\mathrm{net} - P_\mathrm{rad}$, $\tau_E = W/P$ where that power is positive.
4. With a single sample no rate exists: $P_\mathrm{OH}$ is kept (inductance held fixed), $dW/dt$ and everything built on it are NaN. A missing energy rate is never replaced by zero.

Defaults

rate_polyorder = 1 and hold_li3_where_missing = True are the VEST Tier A reference settings of #548 (3 ms holds about three EFIT slices), numerical choices recorded in settings, not physical constants.

Applicability

Machine-independent. Ohmic plasmas: no auxiliary heating and no non-inductive current (with one, $R_p$ needs $I_p - I_\mathrm{ni}$).

Limitations

v_res is inferred, and exact only for a loop on the plasma boundary (recorded in assumptions['v_res']). Holding $l_{i,3}$ fixed is a numerical fallback, not evidence of a stationary current profile. Smoothing windows bias rates that change within them.

Provenance

1. J. A. Romero et al., Nucl. Fusion 50 (2010) 115002, eq. 24.
2. Issue #548 Sec. 1 and workflow/confinement_scaling: the VEST Tier A reference chain this function was moved out of.
3. Issue #1905 Sec. 3: the shared confinement contract and its required result fields.

Processing steps

1. Take logs, centre and divide each column by its standard deviation.
2. Singular-value decompose; a direction with a small singular value is a combination of predictors the sample barely varies, so the matching combination of exponents is poorly determined (PCR diagnostic). With fewer rows than predictors the missing directions are returned with singular value zero: they are the unconstrained ones.
3. With groups, split each direction’s spread into the part carried by the group means (each row taking its group’s mean) and the part carried by the within-group deviations, both with the pooled centring and scale.

Applicability

Machine-independent.

Limitations

Directions are in standardised log coordinates; a direction mixes exponents in units of each predictor’s spread, so convert with scale before reading it as a physical scan. It diagnoses the design, not a fit: it says which scan would add information, not which exponent is right.

Provenance

1. Issue #1621 Secs. 8 and 12: PCR-style identifiability and the weak singular directions that point to the next scan.
2. D. A. Belsley, E. Kuh and R. E. Welsch, Regression Diagnostics, Wiley (1980), Ch. 3.

Processing steps

1. $L_i = \mu_0 R_0 l_{i,3}/2$, so $W_\mathrm{int} = L_i I_p^2/2$.
2. $V_\mathrm{ind} = (1/I_p)\,dW_\mathrm{int}/dt = L_i\,dI_p/dt + \tfrac14\mu_0 R_0 I_p\,dl_{i,3}/dt$, the second term only where dli3_dt is given and finite (or, with hold_li3_where_missing, zero where it is not).
3. $V_\mathrm{res} = V_\mathrm{loop} - V_\mathrm{ind}$ and $R_p = V_\mathrm{res}/I_p$.

Applicability

Machine-independent.

Limitations

Exact (Romero’s eq. 24) only for the loop voltage at the plasma boundary. A flux loop away from the boundary also sees the external inductive flux between loop and boundary, which this does not remove.

Provenance

1. J. A. Romero et al., Nucl. Fusion 50 (2010) 115002, eq. 24: the inductive voltage of the internal poloidal-field energy.
2. Issue #548 Sec. 1: P_OH from the loop voltage with the inductive part removed, the Spitzer path only for comparison (#1188).

Processing steps

1. A thermal scaling takes tau_thermal; nothing else stands in for it.
2. A global scaling takes tau_global where it is finite. Elsewhere it is NaN (strict), or the thermal time with THERMAL_AS_GLOBAL recorded when thermal_as_global is set.
3. An unaudited scaling has no known basis: NaN unless thermal_as_global, when the thermal time is used and recorded.

Applicability

Machine-independent. The thermal-for-global substitution is defensible for ohmic plasmas without fast ions (VEST Tier A) and wrong for beam- or RF-heated plasmas with a fast-ion population.

Limitations

Resolves which time to use; it does not construct $W_{global}$ or $W_{th}$ from a reconstruction. w_mhd_J from a magnetics EFIT is total kinetic pressure and is not by itself a thermal energy.

Provenance

1. Issue #1713: thermal and global energy confinement must not be silently interchanged in scaling comparisons.

Processing steps

1. Without window_s: the weighted central difference at the samples (one-sided at the ends).
2. With window_s: at each evaluation time $t_0$, the finite samples with $|t - t_0| \le$ window_s/2 are fitted by a degree- polyorder polynomial in $t - t_0$ (least squares, unweighted); the derivative is its linear coefficient. On a uniform grid away from the ends this is the Savitzky-Golay first derivative; at the ends and across gaps the window is simply one-sided or sparser.

Defaults

polyorder = 2: numerical convenience, the lowest degree whose derivative is unbiased for a quantity with constant curvature.

Applicability

Machine-independent.

Limitations

Smoothing biases a derivative that changes within the window; the window width belongs in the provenance of every quantity built on it. Near the ends the window is one-sided and the derivative noisier.

Provenance

1. A. Savitzky and M. J. E. Golay, Anal. Chem. 36 (1964) 1627: the local least-squares polynomial derivative.