Currently, the error that is minimized is a sum of the absolute difference between target and actual region sizes (as a proportion of the overall area of the diagram) for each region:
$$
\sum_{region} \frac{|\text{actual} - \text{target}|}{\text{total diagram area}}
$$
This optimizes for allocating areas proportionally to their targets. If 10% of the represented area is misplaced, that's considered equally bad, no matter where in the diagram the misplaced area is.
However, it's tempting to consider an error of size 0.1 (10% of the overall diagram) to be less bad on a region of size 0.5 (half the diagram, relative error 20%) than on a region of size 0.005 (1/200th of the diagram, relative error 2000%). This is apparent in the MPower example, where:
- overall diagram area is ≈500
- total error is ≈20 (≈4%)
- half the total error comes from a single region ($\text{KRAS} \cap \text{KEAP1} \cap \text{TP53}$) that is supposed to be size 13, but is instead ≈1.
Incorporating a configurable mix of absolute and relative errors lead to more intuitive/aesthetic results, where invariants like "every region is within X% of its desired size" can hold.
As an example, these might all be considered to have equal "cost":
- target 0.5, actual 0.6 (absolute error 0.1, relative error 0.2)
- target 0.1, actual 0.15 (absolute error 0.05, relative error 0.5)
- target 0.02, actual 0.045 (absolute error 0.025, relative error 1.25)
This can be achieved by scaled the absolute error $|A - T|$ by $2^{-\log_5{2T}}$:
$$
\text{error} = |A-T| \left( 2^{-\log_5{2T}} \right)
$$
That exponent, $-\log_5{2T}$, serves to weight absolute errors half as much with each power of 5 that they decrease, and the inner $2T$ aligns the expression to the specific values given above.
TODO: work this into the error computation in step.rs.
Currently, the error that is minimized is a sum of the absolute difference between target and actual region sizes (as a proportion of the overall area of the diagram) for each region:
This optimizes for allocating areas proportionally to their targets. If 10% of the represented area is misplaced, that's considered equally bad, no matter where in the diagram the misplaced area is.
However, it's tempting to consider an error of size 0.1 (10% of the overall diagram) to be less bad on a region of size 0.5 (half the diagram, relative error 20%) than on a region of size 0.005 (1/200th of the diagram, relative error 2000%). This is apparent in the MPower example, where:
Incorporating a configurable mix of absolute and relative errors lead to more intuitive/aesthetic results, where invariants like "every region is within X% of its desired size" can hold.
As an example, these might all be considered to have equal "cost":
This can be achieved by scaled the absolute error$|A - T|$ by $2^{-\log_5{2T}}$ :
That exponent,$-\log_5{2T}$ , serves to weight absolute errors half as much with each power of 5 that they decrease, and the inner $2T$ aligns the expression to the specific values given above.
TODO: work this into the error computation in
step.rs.