PHALIN THAKKAR
A good fit does not name the cause
What does a residual pattern suggest, and what additional check would test its cause?
Note 3 · Explanatory note
Analytical result · Hypothetical examples are not physical measurements.
Define a residual as ri = yi − Ipred(ai). For unequal known reading uncertainties, also examine standardized residuals ri/σi.
A constant positive residual may suggest an omitted additive background. A wrong intensity scale tends to leave a pattern following cos²θ. A small angular zero offset δ can produce:
This expression uses small δ in radians.
These patterns help form hypotheses. They do not uniquely identify a faulty component. Different instrumental effects can produce similar patterns, while fitting an extra parameter can absorb a bias.
In the existing example, every point is 8 counts too high. An added background term removes the constant residual. A separate dark reading would help test whether detector background explains it. If the source drifts during a sweep, randomizing or reversing the measurement order provides a different check.
- Synthetic reading minus prediction
- Constant residual
Calculated data: Synthetic residual illustration: an added 8 counts
| Series | Analyzer angle (degrees) | Residual (counts) |
|---|---|---|
| Synthetic reading minus prediction | 0 | 8 |
| Synthetic reading minus prediction | 15 | 8 |
| Synthetic reading minus prediction | 30 | 8 |
| Synthetic reading minus prediction | 45 | 8 |
| Synthetic reading minus prediction | 60 | 8 |
| Synthetic reading minus prediction | 75 | 8 |
| Synthetic reading minus prediction | 90 | 8 |
| Synthetic reading minus prediction | 105 | 8 |
| Synthetic reading minus prediction | 120 | 8 |
| Synthetic reading minus prediction | 135 | 8 |
| Synthetic reading minus prediction | 150 | 8 |
| Synthetic reading minus prediction | 165 | 8 |
| Constant residual | 0 | 8 |
| Constant residual | 180 | 8 |