in an elderly population. These are the diagnostic groups for `ADNI-1`. For `ADNI-2`, there are also 3 diagnostic categories[^6]: normal ageing ($0$) (including SMC), mild cognitive impairment (including EMCI and LMCI) ($1$), and AD ($2$).

1ex Specifically, we investigate the following *six* questions:

-   Can the prediction models for regression qualitatively capture similar trends to the regression model obtained by numerical optimization? (Sec. \[sec:regression\_results\])

-   Are atrophy measurements derived from FPSGR biased to overestimate or underestimate volume changes? (Sec. \[sec:bias\])

-   Are FPSGR atrophy measurements consistent with those derived from deformations via numerical optimization (LDDMM) which produced the training dataset? (Sec. \[sec:atrophy\])

-   Are regression results more stable and hence capture trends better than pairwise registrations? (Sec. \[sec:sgr\_justification\])

-   Is the predictive power of the regression models strong enough to forecast deformations for unseen future timepoints (Sec. \[sec:forecast\])

-   Do the prediction results capture expected trends in deformation? (Sec. \[sec:jacobian\])

If these experiments resolve favorably, then the substantially improved computational efficiency of FPSGR justifies its use for large-scale imaging studies. Tables \[tab:dist1\] and \[tab:dist2\] show the distributions of the prediction cases per time-point and the diagnostic groups in `ADNI-1` and `ADNI-2`, respectively.

Regression results {#sec:regression_results}
------------------

![image](image/example_1_update.png){width="\textwidth"}

Table \[tab:1\] indicates 