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fitting:r2-vs-qfactor
Goodness-of-Fit: R² vs Q-Factor
R² — Coefficient of Determination
$$R^2 = 1 - \frac{\sum(y_i - \hat{y}_i)^2}{\sum(y_i - \bar{y})^2}$$
| Parameter | Description |
|---|---|
| $y_i$ | measured data points |
| $\hat{y}_i$ | model predicted values |
| $\bar{y}$ | mean of measured data |
- Range: $0 \leq R^2 \leq 1$
- Does not require knowledge of measurement uncertainties
- Measures fraction of variance explained by the model
- $R^2 \to 1$ always interpreted as a good fit
Q-Factor — Chi-Squared Probability
$$Q = 1 - P\!\left(\frac{\nu}{2},\, \frac{\chi^2}{2}\right) \qquad \chi^2 = \sum_{i=1}^{N}\left(\frac{y_i - \hat{y}_i}{\sigma_i}\right)^2$$
| Parameter | Description |
|---|---|
| $y_i$ | measured data points |
| $\hat{y}_i$ | model predicted values |
| $\sigma_i$ | measurement uncertainty of point $i$ |
| $\nu$ | degrees of freedom $= N - p$ |
| $N$ | number of data points |
| $p$ | number of free parameters in model |
| $P(a,x)$ | lower regularized incomplete gamma function |
- Requires calibrated uncertainties $\sigma_i$
- Good fit: $Q \approx 0.1 \ldots 0.9$
- $Q \to 0$ : errors underestimated or model is wrong
- $Q \to 1$ : errors overestimated or model over-constrained
Comparison
| Property | R² | Q-Factor |
|---|---|---|
| Requires $\sigma_i$ | No | Yes |
| Scale-aware | No | Yes |
| Overfitting detection | Poor | Yes ($Q\to1$) |
| “Too good” warning | No | Yes |
| Error calibration check | No | Yes |
| Distribution assumption | None | Gaussian errors |
| Typical good range | $\to 1$ | $0.1 \ldots 0.9$ |
Rule of Thumb
<note important>
If measurement uncertainties $\sigma_i$ are well calibrated, Q-factor is far more informative than R².
R² is the fallback when uncertainties are unknown.
</note>
fitting/r2-vs-qfactor.txt · Last modified: 2026/05/20 00:57 by Vitaliy Pipich
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