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COMBINING THE ERRORS

To get meaningful plots, where error bars faithfully reflect on truth, we ought to better understand our error sources. They appear to be independent, but this observation does not simplify matters.

Let us look at the plots vertically, taking one value of mean pixel displacement at a time. Each value of displacement, denoted by $d$, is produced by warping a set of aligned images, i.e. images where there is no inherent displacement.

Let us define $\sigma_{mi}$ to be the predicted error in the estimate of $m$ for a warp instance $i:(1..N)$. We can then obtain the mean

$$\overline{\sigma_{m}}=\begin{array}{c} \\ \sum\\ i\end{array}\frac{\sigma_{mi}}{N}$$ (1)

and the standard error is thus

$$SE_{\sigma_{m}}=\frac{SD(\sigma_{mi})}{\sqrt{N-1}}.$$ (2)

The above error simply tells us the certainty we have in our values of Specificity, Generalisability, or overlap, averaged over 10 instantiations.

As for the mean of the measurement $m$ for the given displacement value $m$,

$$\overline{m_{d}}=\begin{array}{c} \\ \sum\\ i\end{array}m_{i}/N$$ (3)

and the corresponding standard error

$$SE_{\overline{m}}=\frac{SD(m_{i})}{\sqrt{N-1}}.$$ (4)

This error corresponds to the effect of having 10 different values measured for a given extent of displacement. The values tend to fluctuate slightly, thereby leading to uncertainty. This uncertainty is separate, however, from that which is associated with the computation of the measure itself.