Uncertainty analysis

Bias

Precision

Statistical uncertainty

Bootstrap

Independence issues

Typical discussions of uncertainty analysis assume independence between samples. Our measurements are time and space correlated, so we can not make this assumption.

If we are extracting Eulerian data from the tracks database, e. g. mean$( v_x|y )$ we need to be careful. Consider a single track. We find v(t), v(t+1), ... v(t+n), but for spans of time less than the integral time-scale of velocity we cannot assume independence.

To be very strict with independence we should:

  • Take a measurement point

  • Ignore the rest of the points on the track until t > correlation time

  • It is not a practical concern in low-particle-density images, but we should also consider the space correlations, i.e. ignore all other points in the spacial correlation length of the initial measurement point. This plays a role because we assume the flow in statistically 1D, so we spatially average in x and z.

However, for two-time statistics independence is more easily handled. For example, in general we measure v(t)v(t+tau) once for each track, and if we consider the tracks to be independent than these statistics are also independent.