Noise rule
The noise of counts
Convention — Common practice or a guideline in the field, not a rule its sources require.
A count's noise grows with the count: its variance is a straight line in the measured value, a + b * value, measured from how much neighbouring points scatter.
A diffraction or spectroscopy profile counts photons, neutrons or electrons, and a count's variance is the count itself (Poisson). A detector's own noise, or a background that has been taken off, adds a part that does not grow: so the variance is a line in the value, the Poisson-Gaussian model of raw counting data.
It is measured from the data alone. Three neighbouring points y1, y2, y3 give y1 - 2 y2 + y3, which a smooth curve hardly reaches and independent noise of variance s^2 reaches with variance 6 s^2. The points are sorted by value into up to ten groups; in each, the median of those squares over six, over 0.4549 (the median of a chi-square of one degree), is the group's variance, and a line through the groups - its slope never below zero - gives a and b. The median keeps a peak's few steep points from carrying it.
Every curve type of a diffraction or spectroscopy profile names this rule. The verdict on the difference, its shaded stretches, the Difference structure strip, the Residual Spectrum and Fit > Remove Curves By > BIC divide each difference by this noise; the reduced chi-squared, AIC and BIC shown after a fit weigh it by the square root of the value, counting statistics, as they always have.
Limitations
- A peak only a few points wide adds its own curvature to the scatter, so its noise reads somewhat high.
- Data whose noise grows faster than the value - a price, whose scatter is a share of itself - are not counts: the line then falls below zero at small values, where it is held at a quarter of the smallest group's variance, far too small. A curve type for such data names a rule of its own.
- It assumes the noise of neighbouring points is independent.
Sources
- Foi, A., Trimeche, M., Katkovnik, V. and Egiazarian, K., "Practical Poissonian-Gaussian noise modeling and fitting for single-image raw-data", IEEE Transactions on Image Processing 17 (10), 1737-1754 (2008) https://doi.org/10.1109/TIP.2008.2001399